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Lehnertz K. Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems. CHAOS (WOODBURY, N.Y.) 2024; 34:072102. [PMID: 38985967 DOI: 10.1063/5.0214733] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2024] [Accepted: 06/21/2024] [Indexed: 07/12/2024]
Abstract
Real-world non-autonomous systems are open, out-of-equilibrium systems that evolve in and are driven by temporally varying environments. Such systems can show multiple timescale and transient dynamics together with transitions to very different and, at times, even disastrous dynamical regimes. Since such critical transitions disrupt the systems' intended or desired functionality, it is crucial to understand the underlying mechanisms, to identify precursors of such transitions, and to reliably detect them in time series of suitable system observables to enable forecasts. This review critically assesses the various steps of investigation involved in time-series-analysis-based detection of critical transitions in real-world non-autonomous systems: from the data recording to evaluating the reliability of offline and online detections. It will highlight pros and cons to stimulate further developments, which would be necessary to advance understanding and forecasting nonlinear behavior such as critical transitions in complex systems.
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Ngonghala CN, Enright H, Prosper O, Zhao R. Modeling the synergistic interplay between malaria dynamics and economic growth. Math Biosci 2024; 372:109189. [PMID: 38580079 DOI: 10.1016/j.mbs.2024.109189] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2023] [Revised: 03/28/2024] [Accepted: 04/01/2024] [Indexed: 04/07/2024]
Abstract
The mosquito-borne disease (malaria) imposes significant challenges on human health, healthcare systems, and economic growth/productivity in many countries. This study develops and analyzes a model to understand the interplay between malaria dynamics, economic growth, and transient events. It uncovers varied effects of malaria and economic parameters on model outcomes, highlighting the interdependence of the reproduction number (R0) on both malaria and economic factors, and a reciprocal relationship where malaria diminishes economic productivity, while higher economic output is associated with reduced malaria prevalence. This emphasizes the intricate interplay between malaria dynamics and socio-economic factors. The study offers insights into malaria control and underscores the significance of optimizing external aid allocation, especially favoring an even distribution strategy, with the most significant reduction observed in an equal monthly distribution strategy compared to longer distribution intervals. Furthermore, the study shows that controlling malaria in high mosquito biting areas with limited aid, low technology, inadequate treatment, or low economic investment is challenging. The model exhibits a backward bifurcation implying that sustainability of control and mitigation measures is essential even when R0 is slightly less than one. Additionally, there is a parameter regime for which long transients are feasible. Long transients are critical for predicting the behavior of dynamic systems and identifying factors influencing transitions; they reveal reservoirs of infection, vital for disease control. Policy recommendations for effective malaria control from the study include prioritizing sustained control measures, optimizing external aid allocation, and reducing mosquito biting.
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Affiliation(s)
- Calistus N Ngonghala
- Department of Mathematics, University of Florida, Gainesville, FL 32611, USA; Emerging Pathogens Institute, University of Florida, Gainesville, FL 32610, USA.
| | - Hope Enright
- Department of Mathematics and Statistics, Minnesota State University, Mankato, MN 56001, USA
| | - Olivia Prosper
- Department of Mathematics, University of Tennessee, Knoxville, TN 37916, USA
| | - Ruijun Zhao
- Department of Mathematics and Statistics, Minnesota State University, Mankato, MN 56001, USA
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3
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Sadhu S, Chakraborty Thakur S. Analysis of long transients and detection of early warning signals of extinction in a class of predator-prey models exhibiting bistable behavior. J Math Biol 2024; 88:70. [PMID: 38668899 DOI: 10.1007/s00285-024-02095-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2023] [Revised: 04/01/2024] [Accepted: 04/09/2024] [Indexed: 05/12/2024]
Abstract
In this paper, we develop a method of analyzing long transient dynamics in a class of predator-prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a singular zero-Hopf bifurcation of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator-prey model considered in Sadhu (Discrete Contin Dyn Syst B 26:5251-5279, 2021) and we find that our theory is in good agreement with the numerical simulations carried out for this model.
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Affiliation(s)
- S Sadhu
- Department of Mathematics, Georgia College & State University, Milledgeville, GA, 31061, USA.
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4
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Lin Q, Zhang K, Giguet-Covex C, Arnaud F, McGowan S, Gielly L, Capo E, Huang S, Ficetola GF, Shen J, Dearing JA, Meadows ME. Transient social-ecological dynamics reveal signals of decoupling in a highly disturbed Anthropocene landscape. Proc Natl Acad Sci U S A 2024; 121:e2321303121. [PMID: 38640342 PMCID: PMC11046650 DOI: 10.1073/pnas.2321303121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2023] [Accepted: 03/19/2024] [Indexed: 04/21/2024] Open
Abstract
Understanding the transient dynamics of interlinked social-ecological systems (SES) is imperative for assessing sustainability in the Anthropocene. However, how to identify critical transitions in real-world SES remains a formidable challenge. In this study, we present an evolutionary framework to characterize these dynamics over an extended historical timeline. Our approach leverages multidecadal rates of change in socioeconomic data, paleoenvironmental, and cutting-edge sedimentary ancient DNA records from China's Yangtze River Delta, one of the most densely populated and intensively modified landscapes on Earth. Our analysis reveals two significant social-ecological transitions characterized by contrasting interactions and feedback spanning several centuries. Initially, the regional SES exhibited a loosely connected and ecologically sustainable regime. Nevertheless, starting in the 1950s, an increasingly interconnected regime emerged, ultimately resulting in the crossing of tipping points and an unprecedented acceleration in soil erosion, water eutrophication, and ecosystem degradation. Remarkably, the second transition occurring around the 2000s, featured a notable decoupling of socioeconomic development from ecoenvironmental degradation. This decoupling phenomenon signifies a more desirable reconfiguration of the regional SES, furnishing essential insights not only for the Yangtze River Basin but also for regions worldwide grappling with similar sustainability challenges. Our extensive multidecadal empirical investigation underscores the value of coevolutionary approaches in understanding and addressing social-ecological system dynamics.
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Affiliation(s)
- Qi Lin
- Key Laboratory of Lake and Watershed Science for Water Security, Nanjing Institute of Geography and Limnology, Chinese Academy of Sciences, Nanjing210008, People’s Republic of China
| | - Ke Zhang
- Key Laboratory of Lake and Watershed Science for Water Security, Nanjing Institute of Geography and Limnology, Chinese Academy of Sciences, Nanjing210008, People’s Republic of China
| | - Charline Giguet-Covex
- Laboratoire Environnements, Dyamiques et Teritoires de la Montagne, Université Savoie Mont Blanc, CNRS, Chambéry73000, France
| | - Fabien Arnaud
- Laboratoire Environnements, Dyamiques et Teritoires de la Montagne, Université Savoie Mont Blanc, CNRS, Chambéry73000, France
| | - Suzanne McGowan
- Department of Aquatic Ecology, Netherlands Institute of Ecology, Wageningen6708PB, Netherlands
| | - Ludovic Gielly
- Laboratoire d’Écologie Alpine, CNRS, Université Grenoble Alpes, GrenobleF-38000, France
| | - Eric Capo
- Department of Ecology and Environmental Sciences, Umeå University, UmeåSE-90187, Sweden
| | - Shixin Huang
- Key Laboratory of Lake and Watershed Science for Water Security, Nanjing Institute of Geography and Limnology, Chinese Academy of Sciences, Nanjing210008, People’s Republic of China
| | - Gentile Francesco Ficetola
- Laboratoire d’Écologie Alpine, CNRS, Université Grenoble Alpes, GrenobleF-38000, France
- Department of Environmental Science and Policy, University of Milan, Milan20133, Italy
| | - Ji Shen
- School of Geography and Ocean Science, Nanjing University, Nanjing210023, People’s Republic of China
| | - John A. Dearing
- School of Geography and Environmental Science, University of Southampton, SouthamptonSO17 1BJ, United Kingdom
| | - Michael E. Meadows
- School of Geography and Ocean Science, Nanjing University, Nanjing210023, People’s Republic of China
- Department of Environmental & Geographical Science, University of Cape Town, Rondebosch7701, South Africa
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Karádi-Kovács K, Szivák I, Bozóki T, Kovács K, Móra A, Padisák J, Selmeczy GB, Schmera D, Boda P. Long-term recovery dynamics determined by the degree of the disturbance - Ten years tracking of aquatic macroinvertebrate recolonisation after an industrial disaster (Red Sludge Disaster, Hungary). THE SCIENCE OF THE TOTAL ENVIRONMENT 2024; 921:171071. [PMID: 38378064 DOI: 10.1016/j.scitotenv.2024.171071] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2023] [Revised: 01/29/2024] [Accepted: 02/16/2024] [Indexed: 02/22/2024]
Abstract
A ten-year-long examination of macroinvertebrate community recovery was conducted following a catastrophic spill of highly alkaline red sludge (pH >13) into lowland streams. Our primary objective was to compare recovery patterns after coarse- and fine-grain disturbances, focusing on two aspects: i) trend analysis to reveal long-term changes of six community parameters, and ii) variation analyses to assess parameter changes over time. We conducted statistical analysis on long-term data series of macroinvertebrates obtained from quantitative samples collected at four sections with varying degrees of disturbance along the impacted stream sections. We developed a comprehensive theoretical framework comprising a series of sequential phases: Ramp-up, Overshoot, and Oscillation Phases. i) A trend analysis revealed that disturbances show a gradual recovery pattern, while variance analyses showed an asymptotic convergence to an equilibrium. ii) Evaluating these trends across phases unveiled that the initial recovery phase exhibited a steep trajectory, lasting 4-9 months, irrespective of disturbance severity. Coarse-grain disturbances induced a remarkable Overshoot phenomenon across all community metrics. The more severe the disturbance, the greater the height and duration of the Overshoot. Our results suggest that the presence or absence of Overshoot can serve as an indicator for coarse-grain disturbances in the context of large and infrequent disturbances (LID). The entire recovery process lasts for 2.5-3 years irrespective of the severity of the LID. In conclusion, a minimum survey duration of two and half years is deemed imperative to capture the phases of recovery, and changes associated with LID are not expected to extend beyond the three-year threshold. The theoretical framework, including Overshoot parameters, may assist future studies in comparing recovery patterns of different LID types. Furthermore, our theoretical framework is likely to be applicable to other groups of organisms given a sufficiently long monitoring of recovery, influenced also by the length of reproductive cycles.
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Affiliation(s)
- Kata Karádi-Kovács
- HUN-REN Balaton Limnological Research Institute, Klebelsberg Kuno utca 3, H-8237 Tihany, Hungary.
| | - Ildikó Szivák
- HUN-REN Balaton Limnological Research Institute, Klebelsberg Kuno utca 3, H-8237 Tihany, Hungary
| | - Tamás Bozóki
- HUN-REN Centre for Ecological Research, Institute of Aquatic Ecology, Bem tér 18/c, H-4026 Debrecen, Hungary
| | - Krisztián Kovács
- Laboratory for Environmental Protection, Government Office of Győr-Moson-Sopron County, Tatai út 3, H-9028 Győr, Hungary
| | - Arnold Móra
- University of Pécs, Department of Hydrobiology, Ifjúság útja 6, H-7624 Pécs, Hungary
| | - Judit Padisák
- University of Pannonia, Center for Natural Science, Research Group of Limnology, Egyetem u. 10, H-8200 Veszprém, Hungary; HUN-REN-PE Limnoecology Research Group, Egyetem u. 10, H-8200 Veszprém, Hungary
| | - Géza Balázs Selmeczy
- University of Pannonia, Center for Natural Science, Research Group of Limnology, Egyetem u. 10, H-8200 Veszprém, Hungary; HUN-REN-PE Limnoecology Research Group, Egyetem u. 10, H-8200 Veszprém, Hungary
| | - Dénes Schmera
- HUN-REN Balaton Limnological Research Institute, Klebelsberg Kuno utca 3, H-8237 Tihany, Hungary; HUN-REN Balaton Limnological Research Institute, National Laboratory for Water Science and Security, Klebelsberg Kuno utca 3, H-8237 Tihany, Hungary
| | - Pál Boda
- HUN-REN Centre for Ecological Research, Institute of Aquatic Ecology, Bem tér 18/c, H-4026 Debrecen, Hungary
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Barabás G. Parameter Sensitivity of Transient Community Dynamics. Am Nat 2024; 203:473-489. [PMID: 38489777 DOI: 10.1086/728764] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/17/2024]
Abstract
AbstractTransient dynamics have always intrigued ecologists, but current rapid environmental change (inducing transients even in previously undisturbed systems) has highlighted their importance more than ever. Here, I introduce a method for analyzing the sensitivity of transient ecological dynamics to parameter perturbations. The question the method answers is: how would the community dynamics have unfolded for some time horizon had the parameters been slightly different? I apply the method to three empirically parameterized models: competition between native forbs and exotic grasses in California, a host-parasitoid system, and an experimental chemostat predator-prey model. These applications showcase the ecological insights one can gain from models using transient sensitivity analysis. First, one can find parameters and their combinations whose perturbations disproportionately affect a system. Second, one can identify particular windows of time during which the predicted deviation from the unperturbed trajectories is especially large and utilize this information for management purposes. Third, there is an inverse relationship between transient and long-term sensitivities whenever the interacting populations are ecologically similar; paradoxically, the smaller the immediate response of the system, the more extreme its long-term response will be.
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Johnson CR, Dudgeon S. Understanding change in benthic marine systems. ANNALS OF BOTANY 2024; 133:131-144. [PMID: 38079203 PMCID: PMC10921837 DOI: 10.1093/aob/mcad187] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/2023] [Accepted: 12/10/2023] [Indexed: 03/09/2024]
Abstract
BACKGROUND The unprecedented influence of human activities on natural ecosystems in the 21st century has resulted in increasingly frequent large-scale changes in ecological communities. This has heightened interest in understanding such changes and effective means to manage them. Accurate interpretation of state changes is challenging because of difficulties translating theory to empirical study, and most theory emphasizes systems near equilibrium, which may not be relevant in rapidly changing environments. SCOPE We review concepts of long-transient stages and phase shifts between stable community states, both smooth, continuous and discontinuous shifts, and the relationships among them. Three principal challenges emerge when applying these concepts. The first is how to interpret observed change in communities - distinguishing multiple stable states from long transients, or reversible shifts in the phase portrait of single attractor systems. The second is how to quantify the magnitudes of three sources of variability that cause switches between community states: (1) 'noise' in species' abundances, (2) 'wiggle' in system parameters and (3) trends in parameters that affect the topography of the basin of attraction. The third challenge is how variability of the system shapes evidence used to interpret community changes. We outline a novel approach using critical length scales to potentially address these challenges. These concepts are highlighted by a review of recent examples involving macroalgae as key players in marine benthic ecosystems. CONCLUSIONS Real-world examples show three or more stable configurations of ecological communities may exist for a given set of parameters, and transient stages may persist for long periods necessitating their respective consideration. The characteristic length scale (CLS) is a useful metric that uniquely identifies a community 'basin of attraction', enabling phase shifts to be distinguished from long transients. Variabilities of CLSs and time series data may likewise provide proactive management measures to mitigate phase shifts and loss of ecosystem services. Continued challenges remain in distinguishing continuous from discontinuous phase shifts because their respective dynamics lack unique signatures.
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Affiliation(s)
- Craig R Johnson
- Institute for Marine & Antarctic Studies, University of Tasmania, Private Bag 129, Hobart, Tasmania, Australia 7001, and
| | - Steve Dudgeon
- Department of Biology, California State University, Northridge, CA 91330-8303, USA
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Poulsen GR, Plunkett CE, Reimer JR. First Passage Times of Long Transient Dynamics in Ecology. Bull Math Biol 2024; 86:34. [PMID: 38396166 DOI: 10.1007/s11538-024-01259-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2023] [Accepted: 01/10/2024] [Indexed: 02/25/2024]
Abstract
Long transient dynamics in ecological models are characterized by extended periods in one state or regime before an eventual, and often abrupt, transition. One mechanism leading to long transient dynamics is the presence of ghost attractors, states where system dynamics slow down and the system lingers before eventually transitioning to the true attractor. This transition results solely from system dynamics rather than external factors. This paper investigates the dynamics of a classical herbivore-grazer model with the potential for ghost attractors or alternative stable states. We propose an intuitive threshold for first passage time analysis applicable to both bistable and ghost attractor regimes. By formulating the first passage time problem as a backward Kolmogorov equation, we examine how the mean first passage time changes as parameters are varied from the ghost attractor regime to the bistable one, through a saddle-node bifurcation. Our results reveal that the mean and variance of first passage times vary smoothly across the bifurcation threshold, eliminating the deterministic distinction between ghost attractors and bistable regimes. This work suggests that first passage time analysis can be an informative way to classify the length of a long transient. A better understanding of the duration of long transients may contribute to greater ecological understanding and more effective environmental management.
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Affiliation(s)
- Grant R Poulsen
- Department of Mathematics, University of Utah, Salt Lake City, UT, USA
| | - Claire E Plunkett
- Department of Mathematics, University of Utah, Salt Lake City, UT, USA
| | - Jody R Reimer
- Department of Mathematics, University of Utah, Salt Lake City, UT, USA.
- School Of Biological Sciences, University of Utah, Salt Lake City, UT, USA.
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Potts JR, Painter KJ. Distinguishing Between Long-Transient and Asymptotic States in a Biological Aggregation Model. Bull Math Biol 2024; 86:28. [PMID: 38341397 PMCID: PMC10858835 DOI: 10.1007/s11538-023-01254-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2023] [Accepted: 12/30/2023] [Indexed: 02/12/2024]
Abstract
Aggregations are emergent features common to many biological systems. Mathematical models to understand their emergence are consequently widespread, with the aggregation-diffusion equation being a prime example. Here we study the aggregation-diffusion equation with linear diffusion in one spatial dimension. This equation is known to support solutions that involve both single and multiple aggregations. However, numerical evidence suggests that the latter, which we term 'multi-peaked solutions' may often be long-transient solutions rather than asymptotic steady states. We develop a novel technique for distinguishing between long transients and asymptotic steady states via an energy minimisation approach. The technique involves first approximating our study equation using a limiting process and a moment closure procedure. We then analyse local minimum energy states of this approximate system, hypothesising that these will correspond to asymptotic patterns in the aggregation-diffusion equation. Finally, we verify our hypotheses through numerical investigation, showing that our approximate analytic technique gives good predictions as to whether a state is asymptotic or transient. Overall, we find that almost all twin-peaked, and by extension multi-peaked, solutions are transient, except for some very special cases. We demonstrate numerically that these transients can be arbitrarily long-lived, depending on the parameters of the system.
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Affiliation(s)
- Jonathan R Potts
- School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, Sheffield, S3 7RH, UK.
| | - Kevin J Painter
- Dipartimento Interateneo di Scienze, Progetto e Politiche del Territorio (DIST), Politecnico di Torino, Turin, Italy
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Storm L, Linander H, Bec J, Gustavsson K, Mehlig B. Finite-Time Lyapunov Exponents of Deep Neural Networks. PHYSICAL REVIEW LETTERS 2024; 132:057301. [PMID: 38364126 DOI: 10.1103/physrevlett.132.057301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2023] [Revised: 11/05/2023] [Accepted: 01/03/2024] [Indexed: 02/18/2024]
Abstract
We compute how small input perturbations affect the output of deep neural networks, exploring an analogy between deep feed-forward networks and dynamical systems, where the growth or decay of local perturbations is characterized by finite-time Lyapunov exponents. We show that the maximal exponent forms geometrical structures in input space, akin to coherent structures in dynamical systems. Ridges of large positive exponents divide input space into different regions that the network associates with different classes. These ridges visualize the geometry that deep networks construct in input space, shedding light on the fundamental mechanisms underlying their learning capabilities.
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Affiliation(s)
- L Storm
- Department of Physics, University of Gothenburg, 41296 Gothenburg, Sweden
| | - H Linander
- Department of Mathematical Sciences, Chalmers Technical University and University of Gothenburg, Gothenburg, Sweden
| | - J Bec
- MINES Paris, PSL Research University, CNRS, Cemef, Valbonne, F-06900, France
- Université Côte d'Azur, Inria, CNRS, Cemef, Valbonne, F-06900, France
| | - K Gustavsson
- Department of Physics, University of Gothenburg, 41296 Gothenburg, Sweden
| | - B Mehlig
- Department of Physics, University of Gothenburg, 41296 Gothenburg, Sweden
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11
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Panahi S, Do Y, Hastings A, Lai YC. Rate-induced tipping in complex high-dimensional ecological networks. Proc Natl Acad Sci U S A 2023; 120:e2308820120. [PMID: 38091288 PMCID: PMC10743502 DOI: 10.1073/pnas.2308820120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2023] [Accepted: 11/15/2023] [Indexed: 12/24/2023] Open
Abstract
In an ecosystem, environmental changes as a result of natural and human processes can cause some key parameters of the system to change with time. Depending on how fast such a parameter changes, a tipping point can occur. Existing works on rate-induced tipping, or R-tipping, offered a theoretical way to study this phenomenon but from a local dynamical point of view, revealing, e.g., the existence of a critical rate for some specific initial condition above which a tipping point will occur. As ecosystems are subject to constant disturbances and can drift away from their equilibrium point, it is necessary to study R-tipping from a global perspective in terms of the initial conditions in the entire relevant phase space region. In particular, we introduce the notion of the probability of R-tipping defined for initial conditions taken from the whole relevant phase space. Using a number of real-world, complex mutualistic networks as a paradigm, we find a scaling law between this probability and the rate of parameter change and provide a geometric theory to explain the law. The real-world implication is that even a slow parameter change can lead to a system collapse with catastrophic consequences. In fact, to mitigate the environmental changes by merely slowing down the parameter drift may not always be effective: Only when the rate of parameter change is reduced to practically zero would the tipping be avoided. Our global dynamics approach offers a more complete and physically meaningful way to understand the important phenomenon of R-tipping.
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Affiliation(s)
- Shirin Panahi
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ85287
| | - Younghae Do
- Department of Mathematics, Nonlinear Dynamics Mathematical Application Center, Kyungpook National University, Daegu41566, Republic of Korea
| | - Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA95616
- Santa Fe Institute, Santa Fe, NM87501
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ85287
- Department of Physics, Arizona State University, Tempe, AZ85287
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Milles A, Banitz T, Bielcik M, Frank K, Gallagher CA, Jeltsch F, Jepsen JU, Oro D, Radchuk V, Grimm V. Local buffer mechanisms for population persistence. Trends Ecol Evol 2023; 38:1051-1059. [PMID: 37558537 DOI: 10.1016/j.tree.2023.06.006] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2023] [Revised: 06/22/2023] [Accepted: 06/29/2023] [Indexed: 08/11/2023]
Abstract
Assessing and predicting the persistence of populations is essential for the conservation and control of species. Here, we argue that local mechanisms require a better conceptual synthesis to facilitate a more holistic consideration along with regional mechanisms known from metapopulation theory. We summarise the evidence for local buffer mechanisms along with their capacities and emphasise the need to include multiple buffer mechanisms in studies of population persistence. We propose an accessible framework for local buffer mechanisms that distinguishes between damping (reducing fluctuations in population size) and repelling (reducing population declines) mechanisms. We highlight opportunities for empirical and modelling studies to investigate the interactions and capacities of buffer mechanisms to facilitate better ecological understanding in times of ecological upheaval.
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Affiliation(s)
- Alexander Milles
- Department of Plant Ecology and Nature Conservation, University of Potsdam, Am Muhlenberg 3, 14476, Potsdam-Golm, Germany; Department of Ecological Modelling, Helmholtz Centre for Environmental Research - UFZ, Permoserstr. 15, 04318 Leipzig, Germany; Nationalparkamt Hunsrück-Hochwald, Research, Biotope- and Wildlife Management, Brückener Straße 24, 55765 Birkenfeld, Germany.
| | - Thomas Banitz
- Department of Ecological Modelling, Helmholtz Centre for Environmental Research - UFZ, Permoserstr. 15, 04318 Leipzig, Germany
| | - Milos Bielcik
- Freie Universität Berlin, Institute of Biology, Altensteinstr. 6, 14195 Berlin, Germany; Berlin-Brandenburg Institute of Advanced Biodiversity Research (BBIB), 14195 Berlin, Germany
| | - Karin Frank
- Department of Ecological Modelling, Helmholtz Centre for Environmental Research - UFZ, Permoserstr. 15, 04318 Leipzig, Germany; University of Osnabrück, Institute for Environmental Systems Research, Barbarastr. 12, 49076 Osnabrück, Germany; German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Puschstr. 4, 04103 Leipzig, Germany
| | - Cara A Gallagher
- Department of Plant Ecology and Nature Conservation, University of Potsdam, Am Muhlenberg 3, 14476, Potsdam-Golm, Germany
| | - Florian Jeltsch
- Department of Plant Ecology and Nature Conservation, University of Potsdam, Am Muhlenberg 3, 14476, Potsdam-Golm, Germany; Berlin-Brandenburg Institute of Advanced Biodiversity Research (BBIB), 14195 Berlin, Germany
| | - Jane Uhd Jepsen
- Department of Arctic Ecology, Norwegian Institute for Nature Research, Fram Centre, Hjalmar Johansens gt.14, 9007 Tromsø, Norway
| | - Daniel Oro
- Centre d'Estudis Avançats de Blanes (CEAB - CSIC), Acces Cala Sant Francesc 14, 17300 Blanes, Girona, Spain.
| | - Viktoriia Radchuk
- Ecological Dynamics Department, Leibniz Institute for Zoo and Wildlife Research, 10315 Berlin, Germany
| | - Volker Grimm
- Department of Plant Ecology and Nature Conservation, University of Potsdam, Am Muhlenberg 3, 14476, Potsdam-Golm, Germany; Department of Ecological Modelling, Helmholtz Centre for Environmental Research - UFZ, Permoserstr. 15, 04318 Leipzig, Germany; German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Puschstr. 4, 04103 Leipzig, Germany
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Glimm T, Kaźmierczak B, Newman SA, Bhat R. A two-galectin network establishes mesenchymal condensation phenotype in limb development. Math Biosci 2023; 365:109054. [PMID: 37544500 DOI: 10.1016/j.mbs.2023.109054] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2023] [Revised: 06/09/2023] [Accepted: 07/24/2023] [Indexed: 08/08/2023]
Abstract
Previous work showed that Gal-1A and Gal-8, two proteins belonging to the galactoside-binding galectin family, are the earliest determinants of the patterning of the skeletal elements of embryonic chicken limbs, and further, that their experimentally determined interactions in the embryonic limb bud can be interpreted via a reaction-diffusion-adhesion (2GL: two galectin plus ligands) model. Here, we use an ordinary differential equation-based approach to analyze the intrinsic switching modality of the 2GL network and characterize the network behavior independent of the diffusive and adhesive arms of the patterning mechanism. We identify two states: where the concentrations of both the galectins are respectively, negligible, and very high. This bistable switch-like system arises via a saddle-node bifurcation from a monostable state. For the case of mass-action production terms, we provide an explicit Lyapunov function for the system, which shows that it has no periodic solutions. Our model therefore predicts that the galectin network may exist in low expression and high expression states separated in space or time, without any intermediate states. We test these predictions in experiments performed with high density cultures of chick limb mesenchymal cells and observe that cells inside precartilage protocondensations express Gal-1A at a much higher rate than those outside, for which it was negligible. The Gal-1A and -8-based patterning network is therefore sufficient to partition the mesenchymal cell population into two discrete cell states with different developmental (chondrogenic vs. non-chondrogenic) fates. When incorporated into an adhesion and diffusion-enabled framework this system can generate a spatially patterned limb skeleton.
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Affiliation(s)
- T Glimm
- Department of Mathematics, Western Washington University, Bellingham, WA, 98229, USA
| | - B Kaźmierczak
- Institute of Fundamental Technological Research Polish Academy of Sciences, 02-106, Warsaw, Poland
| | - S A Newman
- Department of Cell Biology and Anatomy, New York Medical College, Valhalla, New York, NY, 10595, USA
| | - R Bhat
- Department of Developmental Biology and Genetics, Indian Institute of Science, Bangalore 560012, India; Department of Bioengineering, Indian Institute of Science, Bangalore 560012, India.
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14
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Sudakow I, Myers C, Petrovskii SV, Sumrall CD, Witts J. Mathematical modeling is an efficient research tool to address challenges in mass extinction research: Reply to comments on "Knowledge gaps and missing links in understanding mass extinctions: Can mathematical modeling help?". Phys Life Rev 2023; 46:5-7. [PMID: 37244153 DOI: 10.1016/j.plrev.2023.05.005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2023] [Accepted: 05/07/2023] [Indexed: 05/29/2023]
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15
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Dey S, Ghorai S, Banerjee M. Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth. J Math Biol 2023; 87:21. [PMID: 37395822 DOI: 10.1007/s00285-023-01957-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2023] [Revised: 06/06/2023] [Accepted: 06/20/2023] [Indexed: 07/04/2023]
Abstract
Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigenmodes act as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions. Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response.
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Affiliation(s)
- Subrata Dey
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India
| | - S Ghorai
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India
| | - Malay Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India.
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16
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Liu A, Magpantay FMG, Abdella K. A framework for long-lasting, slowly varying transient dynamics. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:12130-12153. [PMID: 37501436 DOI: 10.3934/mbe.2023540] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/29/2023]
Abstract
Much of the focus of applied dynamical systems is on asymptotic dynamics such as equilibria and periodic solutions. However, in many systems there are transient phenomena, such as temporary population collapses and the honeymoon period after the start of mass vaccination, that can last for a very long time and play an important role in ecological and epidemiological applications. In previous work we defined transient centers which are points in state space that give rise to arbitrarily long and arbitrarily slow transient dynamics. Here we present the mathematical properties of transient centers and provide further insight into these special points. We show that under certain conditions, the entire forward and backward trajectory of a transient center, as well as all its limit points must also be transient centers. We also derive conditions that can be used to verify which points are transient centers and whether those are reachable transient centers. Finally we present examples to demonstrate the utility of the theory, including applications to predatory-prey systems and disease transmission models, and show that the long transience noted in these models are generated by transient centers.
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Affiliation(s)
- Ankai Liu
- Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada
| | | | - Kenzu Abdella
- Department of Mathematics, Trent University, Peterborough, ON, K9L 0G2, Canada
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17
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Pattanayak D, Mishra A, Bairagi N, Dana SK. Multimodal distribution of transient time of predator extinction in a three-species food chain. CHAOS (WOODBURY, N.Y.) 2023; 33:043122. [PMID: 37097935 DOI: 10.1063/5.0136372] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2022] [Accepted: 03/20/2023] [Indexed: 06/19/2023]
Abstract
The transient dynamics capture the time history in the behavior of a system before reaching an attractor. This paper deals with the statistics of transient dynamics in a classic tri-trophic food chain with bistability. The species of the food chain model either coexist or undergo a partial extinction with predator death after a transient time depending upon the initial population density. The distribution of transient time to predator extinction shows interesting patterns of inhomogeneity and anisotropy in the basin of the predator-free state. More precisely, the distribution shows a multimodal character when the initial points are located near a basin boundary and a unimodal character when chosen from a location far away from the boundary. The distribution is also anisotropic because the number of modes depends on the direction of the local of initial points. We define two new metrics, viz., homogeneity index and local isotropic index, to characterize the distinctive features of the distribution. We explain the origin of such multimodal distributions and try to present their ecological implications.
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Affiliation(s)
- Debarghya Pattanayak
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - Arindam Mishra
- Department of Physics, National University of Singapore, Singapore 117551
| | - Nandadulal Bairagi
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - Syamal K Dana
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
- Department of Mathematics, National Institute of Technology, Durgapur 713209, India
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18
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Sadhu S. Viewing mass extinctions through the lens of mathematical modeling: Comment on "Knowledge gaps and missing links in understanding mass extinctions: Can mathematical modeling help?" by I. Sudakow, C. Myers, S. Petrovskii, C.D. Sumrall and J. Witts. Phys Life Rev 2023; 44:204-206. [PMID: 36791570 DOI: 10.1016/j.plrev.2023.01.022] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2023] [Accepted: 01/30/2023] [Indexed: 02/05/2023]
Affiliation(s)
- Susmita Sadhu
- Department of Mathematics, Georgia College & State University, Milledgeville, GA, USA.
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19
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Valdovinos FS, Dritz S, Marsland R. Transient dynamics in plant–pollinator networks: fewer but higher quality of pollinator visits determines plant invasion success. OIKOS 2023. [DOI: 10.1111/oik.09634] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/23/2023]
Affiliation(s)
| | - Sabine Dritz
- Dept of Environmental Science and Policy, Univ. of California, Davis Davis CA USA
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20
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Huo Y, Li C, Li Y, Li X, Xu P, Bao Z, Liu W. Detecting early-warning signals for influenza by dysregulated dynamic network biomarkers. Brief Funct Genomics 2023:7036269. [PMID: 36787234 DOI: 10.1093/bfgp/elad006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Revised: 01/15/2023] [Accepted: 01/25/2023] [Indexed: 02/15/2023] Open
Abstract
As a dynamical system, complex disease always has a sudden state transition at the tipping point, which is the result of the long-term accumulation of abnormal regulations. This paper proposes a novel approach to detect the early-warning signals of influenza A (H3N2 and H1N1) outbreaks by dysregulated dynamic network biomarkers (dysregulated DNBs) for individuals. The results of cross-validation show that our approach can detect early-warning signals before the symptom appears successfully. Unlike the traditional DNBs, our dysregulated DNBs are anchored and very few, which is essential for disease early diagnosis in clinical practice. Moreover, the genes of dysregulated DNBs are significantly enriched in the influenza-related pathways. The source code of this paper can be freely downloaded from https://github.com/YanhaoHuo/dysregulated-DNBs.git.
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Affiliation(s)
- Yanhao Huo
- Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
| | - Chuchu Li
- Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
| | - Yujie Li
- Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
| | - Xianbin Li
- Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China.,School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Duyun, China
| | - Peng Xu
- Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China.,School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Duyun, China
| | - Zhenshen Bao
- Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China.,School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Duyun, China
| | - Wenbin Liu
- Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
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21
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Heterogeneity of interaction strengths and its consequences on ecological systems. Sci Rep 2023; 13:1905. [PMID: 36732566 PMCID: PMC9895049 DOI: 10.1038/s41598-023-28473-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2022] [Accepted: 01/19/2023] [Indexed: 02/04/2023] Open
Abstract
Ecosystems are formed by networks of species and their interactions. Traditional models of such interactions assume a constant interaction strength between a given pair of species. However, there is often significant trait variation among individual organisms even within the same species, causing heterogeneity in their interaction strengths with other species. The consequences of such heterogeneous interactions for the ecosystem have not been studied systematically. As a theoretical exploration, we analyze a simple ecosystem with trophic interactions between two predators and a shared prey, which would exhibit competitive exclusion in models with homogeneous interactions. We consider several scenarios where individuals of the prey species differentiate into subpopulations with different interaction strengths. We show that in all these cases, whether the heterogeneity is inherent, reversible, or adaptive, the ecosystem can stabilize at a new equilibrium where all three species coexist. Moreover, the prey population that has heterogeneous interactions with its predators reaches a higher density than it would without heterogeneity, and can even reach a higher density in the presence of two predators than with just one. Our results suggest that heterogeneity may be a naturally selected feature of ecological interactions that have important consequences for the stability and diversity of ecosystems.
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22
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Dynamical modelling of street protests using the Yellow Vest Movement and Khabarovsk as case studies. Sci Rep 2022; 12:20447. [PMID: 36443352 PMCID: PMC9705368 DOI: 10.1038/s41598-022-23917-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2022] [Accepted: 11/07/2022] [Indexed: 11/29/2022] Open
Abstract
Social protests, in particular in the form of street protests, are a frequent phenomenon of modern world often making a significant disruptive effect on the society. Understanding the factors that can affect their duration and intensity is therefore an important problem. In this paper, we consider a mathematical model of protests dynamics describing how the number of protesters change with time. We apply the model to two events such as the Yellow Vest Movement 2018-2019 in France and Khabarovsk protests 2019-2020 in Russia. We show that in both cases our model provides a good description of the protests dynamics. We consider how the model parameters can be estimated by solving the inverse problem based on the available data on protesters number at different time. The analysis of parameter sensitivity then allows for determining which factor(s) may have the strongest effect on the protests dynamics.
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23
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Blanchard G, Munoz F. Revisiting extinction debt through the lens of multitrophic networks and meta‐ecosystems. OIKOS 2022. [DOI: 10.1111/oik.09435] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
Affiliation(s)
- Grégoire Blanchard
- AMAP, Univ. Montpellier, CIRAD, CNRS, INRAE, IRD Montpellier France
- AMAP, IRD, Herbier de Nouvelle Calédonie Nouméa Nouvelle Calédonie
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24
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Rubin JE, Earn DJD, Greenwood PE, Parsons TL, Abbott KC. Irregular population cycles driven by environmental stochasticity and saddle crawlbys. OIKOS 2022. [DOI: 10.1111/oik.09290] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
| | - David J. D. Earn
- Dept of Mathematics & Statistics, McMaster Univ. Hamilton ON Canada
| | | | - Todd L. Parsons
- Laboratoire de Probabilités, Statistique et Modélisation (UMR 8001), CNRS&Sorbonne Univ. Paris France
| | - Karen C. Abbott
- Dept of Biology, Case Western Reserve Univ. Cleveland OH USA
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25
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Arroyo-Esquivel J, Hastings A, Baskett ML. Characterizing Long Transients in Consumer–Resource Systems With Group Defense and Discrete Reproductive Pulses. Bull Math Biol 2022; 84:102. [PMID: 35964274 PMCID: PMC9376152 DOI: 10.1007/s11538-022-01059-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2022] [Accepted: 07/25/2022] [Indexed: 11/24/2022]
Abstract
AbstractDuring recent years, the study of long transients has been expanded in ecological theory to account for shifts in long-term behavior of ecological systems. These long transients may lead to regime shifts between alternative states that resemble the dynamics of alternative stable states for a prolonged period of time. One dynamic that potentially leads to long transients is the group defense of a resource in a consumer–resource interaction. Furthermore, time lags in the population caused by discrete reproductive pulses have the potential to produce long transients, either independently or in conjunction to the transients caused by the group defense. In this work, we analyze the potential for long transients in a model for a consumer–resource system in which the resource exhibits group defense and reproduces in discrete reproductive pulses. This system exhibits crawl-by transients near the extinction and carrying capacity states of resource, and a transcritical bifurcation, under which a ghost limit cycle appears. We estimate the transient time of our system from these transients using perturbation theory. This work advances an understanding of how systems shift between alternate states and their duration of staying in a given regime and what ecological dynamics may lead to long transients.
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Affiliation(s)
| | - Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA, 95616, USA
| | - Marissa L Baskett
- Department of Environmental Science and Policy, University of California, Davis, CA, 95616, USA
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26
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Ji J, Lin G, Wang L, Mai A. Spatiotemporal dynamics induced by intraguild predator diffusion in an intraguild predation model. J Math Biol 2022; 85:1. [PMID: 35767083 DOI: 10.1007/s00285-022-01772-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2021] [Revised: 06/06/2022] [Accepted: 06/10/2022] [Indexed: 11/26/2022]
Abstract
An intraguild predation model with intraguild predator diffusion is proposed and studied in this work. It is shown that the local system can have four boundary equilibria and at most two interior equilibria. The interior equilibria may exist even when the system is not uniformly persistent. When only intraguild predator diffusion is incorporated into our three-species model, the resulting model is a partially degenerate reaction-diffusion system. For this partially degenerate system, we show that the solution semiflow is bounded dissipative and the positive orbits of bounded sets are bounded. We also demonstrate that intraguild predator diffusion can lead to the occurrence of spatially nonhomogeneous oscillations and spatiotemporal chaos. Further, we show that intraguild predator diffusion can induce transitions between spatially homogeneous oscillations, spatially nonhomogeneous oscillations and chaos.
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Affiliation(s)
- Juping Ji
- Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada.
| | - Genghong Lin
- School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China
- Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, China
| | - Lin Wang
- Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada
| | - Ali Mai
- School of Mathematics and Information Technology, Yuncheng University, Yuncheng, 044000, China
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27
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Inderjit, Callaway RM, Meron E. Vegetation patterning and biodiversity of plant communities. Phys Life Rev 2022; 42:29-32. [DOI: 10.1016/j.plrev.2022.05.008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2022] [Accepted: 05/23/2022] [Indexed: 11/30/2022]
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28
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Khalighi M, Sommeria-Klein G, Gonze D, Faust K, Lahti L. Quantifying the impact of ecological memory on the dynamics of interacting communities. PLoS Comput Biol 2022; 18:e1009396. [PMID: 35658019 PMCID: PMC9200327 DOI: 10.1371/journal.pcbi.1009396] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2021] [Revised: 06/15/2022] [Accepted: 05/12/2022] [Indexed: 12/21/2022] Open
Abstract
Ecological memory refers to the influence of past events on the response of an ecosystem to exogenous or endogenous changes. Memory has been widely recognized as a key contributor to the dynamics of ecosystems and other complex systems, yet quantitative community models often ignore memory and its implications. Recent modeling studies have shown how interactions between community members can lead to the emergence of resilience and multistability under environmental perturbations. We demonstrate how memory can be introduced in such models using the framework of fractional calculus. We study how the dynamics of a well-characterized interaction model is affected by gradual increases in ecological memory under varying initial conditions, perturbations, and stochasticity. Our results highlight the implications of memory on several key aspects of community dynamics. In general, memory introduces inertia into the dynamics. This favors species coexistence under perturbation, enhances system resistance to state shifts, mitigates hysteresis, and can affect system resilience both ways depending on the time scale considered. Memory also promotes long transient dynamics, such as long-standing oscillations and delayed regime shifts, and contributes to the emergence and persistence of alternative stable states. Our study highlights the fundamental role of memory in communities, and provides quantitative tools to introduce it in ecological models and analyse its impact under varying conditions. An ecosystem is said to exhibit ecological memory when its future states do not only depend on its current state but also on its initial state and trajectory. Memory may arise through various mechanisms as organisms adapt to their environment, modify it, and accumulate biotic and abiotic material. It may also emerge from phenotypic heterogeneity at the population level. Despite its commonness in nature, ecological memory and its potential influence on ecosystem dynamics have been so far overlooked in many applied contexts. Here, we use modeling to investigate how memory can influence the dynamics, composition, and stability landscape of communities. We incorporate long-term memory effects into a multi-species model recently introduced to investigate alternative stable states in microbial communities. We assess the impact of memory on key aspects of model behavior and further examine our findings using a model parameterized by empirical data from the human gut microbiota. Our approach for modeling long-term memory and studying its implications has the potential to improve our understanding of microbial community dynamics and ultimately our ability to predict, manipulate, and experimentally design microbial ecosystems. It could also be applied more broadly in the study of systems composed of interacting components.
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Affiliation(s)
- Moein Khalighi
- Department of Computing, Faculty of Technology, University of Turku, Turku, Finland
- * E-mail: (MK); (LL)
| | | | - Didier Gonze
- Unité de Chronobiologie Théorique, Faculté des Sciences CP 231, Université Libre de Bruxelles, Brussels, Belgium
| | - Karoline Faust
- Laboratory of Molecular Bacteriology (Rega Institute), Department of Microbiology, Immunology and Transplantation, KU Leuven, Leuven, Belgium
| | - Leo Lahti
- Department of Computing, Faculty of Technology, University of Turku, Turku, Finland
- * E-mail: (MK); (LL)
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29
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Potts JR, Börger L, Strickland BK, Street GM. Assessing the predictive power of step selection functions: how social and environmental interactions affect animal space use. Methods Ecol Evol 2022. [DOI: 10.1111/2041-210x.13904] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Jonathan R. Potts
- School of Mathematics and Statistics University of Sheffield, Hicks Building, Hounsfield Road Sheffield UK
| | - Luca Börger
- Department of Biosciences College of Science Swansea University, Singleton Park Swansea Wales UK
- Centre for Biomathematics College of Science Swansea University, Singleton Park Swansea Wales UK
| | - Bronson K. Strickland
- Department of Wildlife, Fisheries, and Aquaculture Mississippi State University Mississippi State MS USA
| | - Garrett M. Street
- Department of Wildlife, Fisheries, and Aquaculture Mississippi State University Mississippi State MS USA
- Quantitative Ecology and Spatial Technologies Laboratory Mississippi State University Mississippi State MS USA
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30
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Chen R, Tu C, Liu QX. Transient perturbations reveal distinct strategies for reserve benefits in life history-dependent ecosystems. Ecol Modell 2022. [DOI: 10.1016/j.ecolmodel.2022.109895] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
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31
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Knowledge Gaps and Missing Links in Understanding Mass Extinctions: Can Mathematical Modeling Help? Phys Life Rev 2022; 41:22-57. [DOI: 10.1016/j.plrev.2022.04.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2022] [Accepted: 04/11/2022] [Indexed: 11/20/2022]
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32
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Budzinski RC, Nguyen TT, Đoàn J, Mináč J, Sejnowski TJ, Muller LE. Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks. CHAOS (WOODBURY, N.Y.) 2022; 32:031104. [PMID: 35364855 PMCID: PMC8947818 DOI: 10.1063/5.0078791] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2021] [Accepted: 01/31/2022] [Indexed: 05/26/2023]
Abstract
One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original, real-valued nonlinear Kuramoto model and a corresponding complex-valued system that permits describing the system in terms of a linear operator and iterative update rule. We now use this description to investigate three major synchronization phenomena in Kuramoto networks (phase synchronization, chimera states, and traveling waves), not only in terms of steady state solutions but also in terms of transient dynamics and individual simulations. These results provide new mathematical insight into how sophisticated behaviors arise from connection patterns in nonlinear networked systems.
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Affiliation(s)
| | | | | | - Ján Mináč
- Department of Mathematics, Western University, London, Ontario N6A 3K7, Canada
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33
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Effect of Slow–Fast Time Scale on Transient Dynamics in a Realistic Prey-Predator System. MATHEMATICS 2022. [DOI: 10.3390/math10050699] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/27/2023]
Abstract
Systems with multiple time scales, often referred to as `slow–fast systems’, have been a focus of research for about three decades. Such systems show a variety of interesting, sometimes counter-intuitive dynamical behaviors and are believed to, in many cases, provide a more realistic description of ecological dynamics. In particular, the presence of slow–fast time scales is known to be one of the main mechanisms resulting in long transients—dynamical behavior that mimics a system’s asymptotic regime but only lasts for a finite (albeit very long) time. A prey–predator system where the prey growth rate is much larger than that of the predator is a paradigmatic example of slow–fast systems. In this paper, we provide detailed investigation of a more advanced variant of prey–predator system that has been overlooked in previous studies, that is, where the predator response is ratio-dependent and the predator mortality is nonlinear. We perform a comprehensive analytical study of this system to reveal a sequence of bifurcations that are responsible for the change in the system dynamics from a simple steady state and/or a limit cycle to canards and relaxation oscillations. We then consider how those changes in the system dynamics affect the properties of long transient dynamics. We conclude with a discussion of the ecological implications of our findings, in particular to argue that the changes in the system dynamics in response to an increase of the time scale ratio are counter-intuitive or even paradoxical.
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Abbott KC, Cuddington K, Hastings A. Transients in ecology: stochasticity, management, and understanding. THEOR ECOL-NETH 2021; 14:623-624. [PMID: 34904021 PMCID: PMC8653802 DOI: 10.1007/s12080-021-00524-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/13/2022]
Affiliation(s)
- Karen C Abbott
- Department of Biology, Case Western Reserve University, Cleveland, OH 44106 USA
| | - Kim Cuddington
- Department of Biology, University of Waterloo, Waterloo, ON N2L 3G1 Canada
| | - Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA 95616 USA.,Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 USA
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35
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Morozov A. Towards creating a mechanistic predictive theory of self-organized vegetation patterns: Comment on "Belowground feedbacks as drivers of spatial self-organization and community assembly" by Inderjit, Callaway and Meron. Phys Life Rev 2021; 40:54-56. [PMID: 34838506 DOI: 10.1016/j.plrev.2021.11.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2021] [Accepted: 11/09/2021] [Indexed: 11/29/2022]
Affiliation(s)
- Andrew Morozov
- Institute of Ecology and Evolution, Russian Academy of Sciences, Moscow, Russia; School of Computing and Mathematical Sciences, University of Leicester, UK.
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36
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Wojcik LA, Ceulemans R, Gaedke U. Functional diversity buffers the effects of a pulse perturbation on the dynamics of tritrophic food webs. Ecol Evol 2021; 11:15639-15663. [PMID: 34824780 PMCID: PMC8601937 DOI: 10.1002/ece3.8214] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2021] [Revised: 09/08/2021] [Accepted: 09/21/2021] [Indexed: 11/28/2022] Open
Abstract
Biodiversity decline causes a loss of functional diversity, which threatens ecosystems through a dangerous feedback loop: This loss may hamper ecosystems' ability to buffer environmental changes, leading to further biodiversity losses. In this context, the increasing frequency of human-induced excessive loading of nutrients causes major problems in aquatic systems. Previous studies investigating how functional diversity influences the response of food webs to disturbances have mainly considered systems with at most two functionally diverse trophic levels. We investigated the effects of functional diversity on the robustness, that is, resistance, resilience, and elasticity, using a tritrophic-and thus more realistic-plankton food web model. We compared a non-adaptive food chain with no diversity within the individual trophic levels to a more diverse food web with three adaptive trophic levels. The species fitness differences were balanced through trade-offs between defense/growth rate for prey and selectivity/half-saturation constant for predators. We showed that the resistance, resilience, and elasticity of tritrophic food webs decreased with larger perturbation sizes and depended on the state of the system when the perturbation occurred. Importantly, we found that a more diverse food web was generally more resistant and resilient but its elasticity was context-dependent. Particularly, functional diversity reduced the probability of a regime shift toward a non-desirable alternative state. The basal-intermediate interaction consistently determined the robustness against a nutrient pulse despite the complex influence of the shape and type of the dynamical attractors. This relationship was strongly influenced by the diversity present and the third trophic level. Overall, using a food web model of realistic complexity, this study confirms the destructive potential of the positive feedback loop between biodiversity loss and robustness, by uncovering mechanisms leading to a decrease in resistance, resilience, and potentially elasticity as functional diversity declines.
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Affiliation(s)
- Laurie Anne Wojcik
- Ecology and Ecosystem Modelling GroupUniversity of PotsdamPotsdamGermany
| | - Ruben Ceulemans
- Ecology and Ecosystem Modelling GroupUniversity of PotsdamPotsdamGermany
| | - Ursula Gaedke
- Ecology and Ecosystem Modelling GroupUniversity of PotsdamPotsdamGermany
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37
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Hoyer-Leitzel A, Iams S. Impulsive Fire Disturbance in a Savanna Model: Tree-Grass Coexistence States, Multiple Stable System States, and Resilience. Bull Math Biol 2021; 83:113. [PMID: 34591211 PMCID: PMC8484106 DOI: 10.1007/s11538-021-00944-x] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2021] [Accepted: 09/10/2021] [Indexed: 11/30/2022]
Abstract
Savanna ecosystems are shaped by the frequency and intensity of regular fires. We model savannas via an ordinary differential equation (ODE) encoding a one-sided inhibitory Lotka-Volterra interaction between trees and grass. By applying fire as a discrete disturbance, we create an impulsive dynamical system that allows us to identify the impact of variation in fire frequency and intensity. The model exhibits three different bistability regimes: between savanna and grassland; two savanna states; and savanna and woodland. The impulsive model reveals rich bifurcation structures in response to changes in fire intensity and frequency-structures that are largely invisible to analogous ODE models with continuous fire. In addition, by using the amount of grass as an example of a socially valued function of the system state, we examine the resilience of the social value to different disturbance regimes. We find that large transitions ("tipping") in the valued quantity can be triggered by small changes in disturbance regime.
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Affiliation(s)
- Alanna Hoyer-Leitzel
- Department of Mathematics and Statistics, Mount Holyoke College, 50 College Street, South Hadley, MA, 01075, USA.
| | - Sarah Iams
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA
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38
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Hastings A, Abbott KC, Cuddington K, Francis TB, Lai YC, Morozov A, Petrovskii S, Zeeman ML. Effects of stochasticity on the length and behaviour of ecological transients. J R Soc Interface 2021; 18:20210257. [PMID: 34229460 DOI: 10.1098/rsif.2021.0257] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Abstract
There is a growing recognition that ecological systems can spend extended periods of time far away from an asymptotic state, and that ecological understanding will therefore require a deeper appreciation for how long ecological transients arise. Recent work has defined classes of deterministic mechanisms that can lead to long transients. Given the ubiquity of stochasticity in ecological systems, a similar systematic treatment of transients that includes the influence of stochasticity is important. Stochasticity can of course promote the appearance of transient dynamics by preventing systems from settling permanently near their asymptotic state, but stochasticity also interacts with deterministic features to create qualitatively new dynamics. As such, stochasticity may shorten, extend or fundamentally change a system's transient dynamics. Here, we describe a general framework that is developing for understanding the range of possible outcomes when random processes impact the dynamics of ecological systems over realistic time scales. We emphasize that we can understand the ways in which stochasticity can either extend or reduce the lifetime of transients by studying the interactions between the stochastic and deterministic processes present, and we summarize both the current state of knowledge and avenues for future advances.
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Affiliation(s)
- Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA 95616, USA.,Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
| | - Karen C Abbott
- Department of Biology, Case Western Reserve University, Cleveland, OH 44106, USA
| | - Kim Cuddington
- Department of Biology, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
| | - Tessa B Francis
- Puget Sound Institute, University of Washington Tacoma, Tacoma, WA 98421, USA
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA
| | - Andrew Morozov
- School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK.,Institute of Ecology and Evolution, Russian Academy of Sciences, Leninsky pr. 33, Moscow 117071, Russia
| | - Sergei Petrovskii
- School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK.,Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Str., Moscow 117198, Russia
| | - Mary Lou Zeeman
- Department of Mathematics, Bowdoin College, Brunswick, ME 04011, USA
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39
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Vidiella B, Fontich E, Valverde S, Sardanyés J. Habitat loss causes long extinction transients in small trophic chains. THEOR ECOL-NETH 2021. [DOI: 10.1007/s12080-021-00509-7] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
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40
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Tao Y, Hite JL, Lafferty KD, Earn DJD, Bharti N. Transient disease dynamics across ecological scales. THEOR ECOL-NETH 2021; 14:625-640. [PMID: 34075317 PMCID: PMC8156581 DOI: 10.1007/s12080-021-00514-w] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2020] [Accepted: 05/04/2021] [Indexed: 11/25/2022]
Abstract
Analyses of transient dynamics are critical to understanding infectious disease transmission and persistence. Identifying and predicting transients across scales, from within-host to community-level patterns, plays an important role in combating ongoing epidemics and mitigating the risk of future outbreaks. Moreover, greater emphases on non-asymptotic processes will enable timely evaluations of wildlife and human diseases and lead to improved surveillance efforts, preventive responses, and intervention strategies. Here, we explore the contributions of transient analyses in recent models spanning the fields of epidemiology, movement ecology, and parasitology. In addition to their roles in predicting epidemic patterns and endemic outbreaks, we explore transients in the contexts of pathogen transmission, resistance, and avoidance at various scales of the ecological hierarchy. Examples illustrate how (i) transient movement dynamics at the individual host level can modify opportunities for transmission events over time; (ii) within-host energetic processes often lead to transient dynamics in immunity, pathogen load, and transmission potential; (iii) transient connectivity between discrete populations in response to environmental factors and outbreak dynamics can affect disease spread across spatial networks; and (iv) increasing species richness in a community can provide transient protection to individuals against infection. Ultimately, we suggest that transient analyses offer deeper insights and raise new, interdisciplinary questions for disease research, consequently broadening the applications of dynamical models for outbreak preparedness and management. SUPPLEMENTARY INFORMATION The online version contains supplementary material available at 10.1007/s12080-021-00514-w.
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Affiliation(s)
- Yun Tao
- Intelligence Community Postdoctoral Research Fellowship Program, Department of Ecology, Evolution and Marine Biology, University of California, Santa Barbara, CA 93106 USA
| | - Jessica L. Hite
- School of Veterinary Medicine, Department of Pathobiological Sciences, University of Wisconsin, Madison, WI 53706 USA
| | - Kevin D. Lafferty
- Western Ecological Research Center at UCSB Marine Science Institute, U.S. Geological Survey, CA 93106 Santa Barbara, USA
| | - David J. D. Earn
- Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1 Canada
| | - Nita Bharti
- Department of Biology Center for Infectious Disease Dynamics, Penn State University, University Park, PA 16802 USA
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41
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Zincenko A, Petrovskii S, Volpert V, Banerjee M. Turing instability in an economic-demographic dynamical system may lead to pattern formation on a geographical scale. J R Soc Interface 2021; 18:20210034. [PMID: 33906386 DOI: 10.1098/rsif.2021.0034] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
Spatial distribution of the human population is distinctly heterogeneous, e.g. showing significant difference in the population density between urban and rural areas. In the historical perspective, i.e. on the timescale of centuries, the emergence of densely populated areas at their present locations is widely believed to be linked to more favourable environmental and climatic conditions. In this paper, we challenge this point of view. We first identify a few areas at different parts of the world where the environmental conditions (quantified by the temperature, precipitation and elevation) show a relatively small variation in space on the scale of thousands of kilometres. We then examine the population distribution across those areas to show that, in spite of the approximate homogeneity of the environment, it exhibits a significant variation revealing a nearly periodic spatial pattern. Based on this apparent disagreement, we hypothesize that there may exist an inherent mechanism that may lead to pattern formation even in a uniform environment. We consider a mathematical model of the coupled demographic-economic dynamics and show that its spatially uniform, locally stable steady state can give rise to a periodic spatial pattern due to the Turing instability, the spatial scale of the emerging pattern being consistent with observations. Using numerical simulations, we show that, interestingly, the emergence of the Turing patterns may eventually lead to the system collapse.
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Affiliation(s)
- Anna Zincenko
- School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK.,School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
| | - Sergei Petrovskii
- School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK
| | - Vitaly Volpert
- Institut Camille Jordan, UMR, 5208, CNRS, University Lyon 1, 69622 Villeurbanne, France.,INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France.,Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russian Federation
| | - Malay Banerjee
- Department of Mathematics and Statistics, IIT Kanpur, Kanpur 208016, India
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42
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Francis TB, Abbott KC, Cuddington K, Gellner G, Hastings A, Lai YC, Morozov A, Petrovskii S, Zeeman ML. Management implications of long transients in ecological systems. Nat Ecol Evol 2021; 5:285-294. [PMID: 33462492 DOI: 10.1038/s41559-020-01365-0] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2019] [Accepted: 11/16/2020] [Indexed: 01/29/2023]
Abstract
The underlying biological processes that govern many ecological systems can create very long periods of transient dynamics. It is often difficult or impossible to distinguish this transient behaviour from similar dynamics that would persist indefinitely. In some cases, a shift from the transient to the long-term, stable dynamics may occur in the absence of any exogenous forces. Recognizing the possibility that the state of an ecosystem may be less stable than it appears is crucial to the long-term success of management strategies in systems with long transient periods. Here we demonstrate the importance of considering the potential of transient system behaviour for management actions across a range of ecosystem organizational scales and natural system types. Developing mechanistic models that capture essential system dynamics will be crucial for promoting system resilience and avoiding system collapses.
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Affiliation(s)
- Tessa B Francis
- Puget Sound Institute, University of Washington, Tacoma, WA, USA.
| | - Karen C Abbott
- Department of Biology, Case Western Reserve University, Cleveland, OH, USA
| | - Kim Cuddington
- Department of Biology, University of Waterloo, Waterloo, Ontario, Canada
| | - Gabriel Gellner
- Department of Integrative Biology, University of Guelph, Guelph, Ontario, Canada
| | - Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA, USA.,Santa Fe Institute, Santa Fe, NM, USA
| | - Ying-Cheng Lai
- School of Electrical Computer and Energy Engineering, Arizona State University, Tempe, AZ, USA
| | - Andrew Morozov
- School of Mathematics and Actuarial Science, University of Leicester, Leicester, UK.,Institute of Ecology and Evolution, Russian Academy of Sciences, Moscow, Russia
| | - Sergei Petrovskii
- School of Mathematics and Actuarial Science, University of Leicester, Leicester, UK
| | - Mary Lou Zeeman
- Department of Mathematics, Bowdoin College, Brunswick, ME, USA
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43
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Meng Y, Grebogi C. Control of tipping points in stochastic mutualistic complex networks. CHAOS (WOODBURY, N.Y.) 2021; 31:023118. [PMID: 33653048 DOI: 10.1063/5.0036051] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2020] [Accepted: 01/26/2021] [Indexed: 06/12/2023]
Abstract
Nonlinear stochastic complex networks in ecological systems can exhibit tipping points. They can signify extinction from a survival state and, conversely, a recovery transition from extinction to survival. We investigate a control method that delays the extinction and advances the recovery by controlling the decay rate of pollinators of diverse rankings in a pollinators-plants stochastic mutualistic complex network. Our investigation is grounded on empirical networks occurring in natural habitats. We also address how the control method is affected by both environmental and demographic noises. By comparing the empirical network with the random and scale-free networks, we also study the influence of the topological structure on the control effect. Finally, we carry out a theoretical analysis using a reduced dimensional model. A remarkable result of this work is that the introduction of pollinator species in the habitat, which is immune to environmental deterioration and that is in mutualistic relationship with the collapsed ones, definitely helps in promoting the recovery. This has implications for managing ecological systems.
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Affiliation(s)
- Yu Meng
- Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
| | - Celso Grebogi
- Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
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44
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Zhao LX, Zhang K, Siteur K, Li XZ, Liu QX, van de Koppel J. Fairy circles reveal the resilience of self-organized salt marshes. SCIENCE ADVANCES 2021; 7:7/6/eabe1100. [PMID: 33547078 PMCID: PMC7864568 DOI: 10.1126/sciadv.abe1100] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2020] [Accepted: 12/17/2020] [Indexed: 05/21/2023]
Abstract
Spatial patterning is a fascinating theme in both theoretical and experimental ecology. It reveals resilience and stability to withstand external disturbances and environmental stresses. However, existing studies mainly focus on well-developed persistent patterns rather than transient patterns in self-organizing ecosystems. Here, combining models and experimental evidence, we show that transient fairy circle patterns in intertidal salt marshes can both infer the underlying ecological mechanisms and provide a measure of resilience. The models based on sulfide accumulation and nutrient depletion mechanisms reproduced the field-observed fairy circles, providing a generalized perspective on the emergence of transient patterns in salt marsh ecosystems. Field experiments showed that nitrogen fertilization mitigates depletion stress and shifts plant growth from negative to positive in the center of patches. Hence, nutrient depletion plays an overriding role, as only this process can explain the concentric rings. Our findings imply that the emergence of transient patterns can identify the ecological processes underlying pattern formation and the factors determining the ecological resilience of salt marsh ecosystems.
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Affiliation(s)
- Li-Xia Zhao
- State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai 200241, China
- Shanghai Key Laboratory for Urban Ecological Processes and Eco-Restoration and Center for Global Change and Ecological Forecasting, School of Ecological and Environmental Sciences, East China Normal University, Shanghai 200241, China
- Yangtze Delta Estuarine Wetland Ecosystem Observation and Research Station, Ministry of Education and Shanghai Science and Technology Committee, Shanghai, China
| | - Kang Zhang
- Shanghai Key Laboratory for Urban Ecological Processes and Eco-Restoration and Center for Global Change and Ecological Forecasting, School of Ecological and Environmental Sciences, East China Normal University, Shanghai 200241, China
| | - Koen Siteur
- Shanghai Key Laboratory for Urban Ecological Processes and Eco-Restoration and Center for Global Change and Ecological Forecasting, School of Ecological and Environmental Sciences, East China Normal University, Shanghai 200241, China
- Department of Estuarine and Delta Systems, Royal Netherlands Institute of Sea Research, Yerseke 4401 NT, Netherlands
| | - Xiu-Zhen Li
- State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai 200241, China
- Yangtze Delta Estuarine Wetland Ecosystem Observation and Research Station, Ministry of Education and Shanghai Science and Technology Committee, Shanghai, China
| | - Quan-Xing Liu
- State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai 200241, China.
- Shanghai Key Laboratory for Urban Ecological Processes and Eco-Restoration and Center for Global Change and Ecological Forecasting, School of Ecological and Environmental Sciences, East China Normal University, Shanghai 200241, China
- Yangtze Delta Estuarine Wetland Ecosystem Observation and Research Station, Ministry of Education and Shanghai Science and Technology Committee, Shanghai, China
| | - Johan van de Koppel
- Department of Estuarine and Delta Systems, Royal Netherlands Institute of Sea Research, Yerseke 4401 NT, Netherlands.
- Groningen Institute for Evolutionary Life Sciences, University of Groningen, Groningen 9700 CC, Netherlands
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45
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Viewing communities as coupled oscillators: elementary forms from Lotka and Volterra to Kuramoto. THEOR ECOL-NETH 2021. [DOI: 10.1007/s12080-020-00493-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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46
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Borics G, Abonyi A, Salmaso N, Ptacnik R. Freshwater phytoplankton diversity: models, drivers and implications for ecosystem properties. HYDROBIOLOGIA 2021; 848:53-75. [PMID: 32836348 PMCID: PMC7334633 DOI: 10.1007/s10750-020-04332-9] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2020] [Revised: 06/09/2020] [Accepted: 06/13/2020] [Indexed: 05/20/2023]
Abstract
Our understanding on phytoplankton diversity has largely been progressing since the publication of Hutchinson on the paradox of the plankton. In this paper, we summarise some major steps in phytoplankton ecology in the context of mechanisms underlying phytoplankton diversity. Here, we provide a framework for phytoplankton community assembly and an overview of measures on taxonomic and functional diversity. We show how ecological theories on species competition together with modelling approaches and laboratory experiments helped understand species coexistence and maintenance of diversity in phytoplankton. The non-equilibrium nature of phytoplankton and the role of disturbances in shaping diversity are also discussed. Furthermore, we discuss the role of water body size, productivity of habitats and temperature on phytoplankton species richness, and how diversity may affect the functioning of lake ecosystems. At last, we give an insight into molecular tools that have emerged in the last decades and argue how it has broadened our perspective on microbial diversity. Besides historical backgrounds, some critical comments have also been made.
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Affiliation(s)
- Gábor Borics
- Department of Tisza Research, Centre for Ecological Research, Danube Research Institute, Bem tér 18/c, 4026 Debrecen, Hungary
- GINOP Sustainable Ecosystems Group, Centre for Ecological Research, Klebelsberg Kuno u. 3, 8237 Tihany, Hungary
| | - András Abonyi
- Centre for Ecological Research, Institute of Ecology and Botany, Alkotmány u. 2-4, 2163 Vácrátót, Hungary
- WasserCluster Lunz – Biologische Station GmbH, Dr. Carl Kupelwieser-Promenade 5, 3293 Lunz am See, Austria
| | - Nico Salmaso
- Research and Innovation Centre, Fondazione Edmund Mach, Via E. Mach 1, 38010 San Michele all’Adige, Italy
| | - Robert Ptacnik
- WasserCluster Lunz – Biologische Station GmbH, Dr. Carl Kupelwieser-Promenade 5, 3293 Lunz am See, Austria
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47
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Ray A, Pal A, Ghosh D, Dana SK, Hens C. Mitigating long transient time in deterministic systems by resetting. CHAOS (WOODBURY, N.Y.) 2021; 31:011103. [PMID: 33754784 DOI: 10.1063/5.0038374] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/23/2020] [Accepted: 12/24/2020] [Indexed: 06/12/2023]
Abstract
How long does a trajectory take to reach a stable equilibrium point in the basin of attraction of a dynamical system? This is a question of quite general interest and has stimulated a lot of activities in dynamical and stochastic systems where the metric of this estimation is often known as the transient or first passage time. In nonlinear systems, one often experiences long transients due to their underlying dynamics. We apply resetting or restart, an emerging concept in statistical physics and stochastic process, to mitigate the detrimental effects of prolonged transients in deterministic dynamical systems. We show that resetting the intrinsic dynamics intermittently to a spatial control line that passes through the equilibrium point can dramatically expedite its completion, resulting in a huge reduction in mean transient time and fluctuations around it. Moreover, our study reveals the emergence of an optimal restart time that globally minimizes the mean transient time. We corroborate the results with detailed numerical studies on two canonical setups in deterministic dynamical systems, namely, the Stuart-Landau oscillator and the Lorenz system. The key features-expedition of transient time-are found to be very generic under different resetting strategies. Our analysis opens up a door to control the mean and fluctuations in transient time by unifying the original dynamics with an external stochastic or periodic timer and poses open questions on the optimal way to harness transients in dynamical systems.
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Affiliation(s)
- Arnob Ray
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Arnab Pal
- School of Chemistry, Faculty of Exact Sciences and The Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Syamal K Dana
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - Chittaranjan Hens
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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48
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Kovács T. How can contemporary climate research help understand epidemic dynamics? Ensemble approach and snapshot attractors. J R Soc Interface 2020; 17:20200648. [PMID: 33292097 DOI: 10.1098/rsif.2020.0648] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023] Open
Abstract
Standard epidemic models based on compartmental differential equations are investigated under continuous parameter change as external forcing. We show that seasonal modulation of the contact parameter superimposed upon a monotonic decay needs a different description from that of the standard chaotic dynamics. The concept of snapshot attractors and their natural distribution has been adopted from the field of the latest climate change research. This shows the importance of the finite-time chaotic effect and ensemble interpretation while investigating the spread of a disease. By defining statistical measures over the ensemble, we can interpret the internal variability of the epidemic as the onset of complex dynamics-even for those values of contact parameters where originally regular behaviour is expected. We argue that anomalous outbreaks of the infectious class cannot die out until transient chaos is presented in the system. Nevertheless, this fact becomes apparent by using an ensemble approach rather than a single trajectory representation. These findings are applicable generally in explicitly time-dependent epidemic systems regardless of parameter values and time scales.
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Affiliation(s)
- T Kovács
- Institute for Theoretical Physics, Eötvös University, Pázmány P. s. 1A, H-1117 Budapest, Hungary
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49
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Spatial Dynamics and Spread of Ecosystem Engineers: Two Patch Analysis. Bull Math Biol 2020; 82:149. [DOI: 10.1007/s11538-020-00833-9] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2020] [Accepted: 11/02/2020] [Indexed: 10/22/2022]
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50
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Chen R. Transient inconsistency between population density and fisheries yields without bycatch species extinction. Ecol Evol 2020; 10:12372-12384. [PMID: 33209295 PMCID: PMC7663084 DOI: 10.1002/ece3.6868] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/27/2020] [Revised: 09/10/2020] [Accepted: 09/11/2020] [Indexed: 11/06/2022] Open
Abstract
Recent studies have demonstrated the great advantages of marine reserves in solving bycatch problems by maintaining the persistence (i.e., avoid extinction) of endangered species without sacrificing the fisheries yields of target species. However, transient phenomena rather than equilibrium states of population dynamics still require further research. Here, with a simple and general model, the transient dynamics of the target fish species are investigated under management which minimizes extinction risk of the bycatch species. An interesting finding is that fisheries yields can strongly fluctuate even if population density both inside and outside marine reserve only slightly varies (or vice versa), leading to transient inconsistency between the population densities and fisheries yields. This finding suggests that population density dynamics of the target fish species cannot be used to predict the transient phenomena of fisheries yields (or vice versa) in fisheries management. However, the unpredictability can be receded as the sensitivity analyses show that a large marine reserve size and low escapement rate can shorten the transient duration.
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Affiliation(s)
- Renfei Chen
- School of Life ScienceShanxi Normal UniversityLinfenChina
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