1
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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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2
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Barraquand F. No sensitivity to functional forms in the Rosenzweig-MacArthur model with strong environmental stochasticity. J Theor Biol 2023; 572:111566. [PMID: 37422068 DOI: 10.1016/j.jtbi.2023.111566] [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: 11/03/2022] [Revised: 06/04/2023] [Accepted: 06/26/2023] [Indexed: 07/10/2023]
Abstract
The classic Rosenzweig-MacArthur predator-prey model has been shown to exhibit, like other coupled nonlinear ordinary differential equations (ODEs) from ecology, worrying sensitivity to model structure. This sensitivity manifests as markedly different community dynamics arising from saturating functional responses with nearly identical shapes but different mathematical expressions. Using a stochastic differential equation (SDE) version of the Rosenzweig-MacArthur model with the three functional responses considered by Fussmann & Blasius (2005), I show that such sensitivity seems to be solely a property of ODEs or stochastic systems with weak noise. SDEs with strong environmental noise have by contrast very similar fluctuations patterns, irrespective of the mathematical formula used. Although eigenvalues of linearized predator-prey models have been used as an argument for structural sensitivity, they can also be an argument against structural sensitivity. While the sign of the eigenvalues' real part is sensitive to model structure, its magnitude and the presence of imaginary parts are not, which suggests noise-driven oscillations for a broad range of carrying capacities. I then discuss multiple other ways to evaluate structural sensitivity in a stochastic setting, for predator-prey or other ecological systems.
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Affiliation(s)
- Frédéric Barraquand
- Institute of Mathematics of Bordeaux, CNRS & University of Bordeaux, Talence, France.
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3
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Medvinsky AB, Nurieva NI, Adamovich BV, Radchikova NP, Rusakov AV. Direct input of monitoring data into a mechanistic ecological model as a way to identify the phytoplankton growth-rate response to temperature variations. Sci Rep 2023; 13:10124. [PMID: 37349488 PMCID: PMC10287759 DOI: 10.1038/s41598-023-36950-3] [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: 01/23/2023] [Accepted: 06/13/2023] [Indexed: 06/24/2023] Open
Abstract
We present an approach (knowledge-and-data-driven, KDD, modeling) that allows us to get closer to understanding the processes that affect the dynamics of plankton communities. This approach, based on the use of time series obtained as a result of ecosystem monitoring, combines the key features of both the knowledge-driven modeling (mechanistic models) and data-driven (DD) modeling. Using a KDD model, we reveal the phytoplankton growth-rate fluctuations in the ecosystem of the Naroch Lakes and determine the degree of phase synchronization between fluctuations in the phytoplankton growth rate and temperature variations. More specifically, we estimate a numerical value of the phase locking index (PLI), which allows us to assess how temperature fluctuations affect the dynamics of phytoplankton growth rates. Since, within the framework of KDD modeling, we directly include the time series obtained as a result of field measurements in the model equations, the dynamics of the phytoplankton growth rate obtained from the KDD model reflect the behavior of the lake ecosystem as a whole, and PLI can be considered as a holistic parameter.
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Affiliation(s)
- Alexander B Medvinsky
- Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Russia, 142290.
| | - Nailya I Nurieva
- Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Russia, 142290
| | - Boris V Adamovich
- Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Russia, 142290
- Belarusian State University, 220010, Minsk, Belarus
| | - Nataly P Radchikova
- Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Russia, 142290
- Moscow State University of Psychology and Education, Moscow, Russia, 127051
| | - Alexey V Rusakov
- Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Russia, 142290
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4
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Kuiper JJ, Kooi BW, Peterson GD, Mooij WM. Bridging Theories for Ecosystem Stability Through Structural Sensitivity Analysis of Ecological Models in Equilibrium. Acta Biotheor 2022; 70:18. [PMID: 35737146 PMCID: PMC9225980 DOI: 10.1007/s10441-022-09441-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2021] [Accepted: 05/27/2022] [Indexed: 11/24/2022]
Abstract
Ecologists are challenged by the need to bridge and synthesize different approaches and theories to obtain a coherent understanding of ecosystems in a changing world. Both food web theory and regime shift theory shine light on mechanisms that confer stability to ecosystems, but from different angles. Empirical food web models are developed to analyze how equilibria in real multi-trophic ecosystems are shaped by species interactions, and often include linear functional response terms for simple estimation of interaction strengths from observations. Models of regime shifts focus on qualitative changes of equilibrium points in a slowly changing environment, and typically include non-linear functional response terms. Currently, it is unclear how the stability of an empirical food web model, expressed as the rate of system recovery after a small perturbation, relates to the vulnerability of the ecosystem to collapse. Here, we conduct structural sensitivity analyses of classical consumer-resource models in equilibrium along an environmental gradient. Specifically, we change non-proportional interaction terms into proportional ones, while maintaining the equilibrium biomass densities and material flux rates, to analyze how alternative model formulations shape the stability properties of the equilibria. The results reveal no consistent relationship between the stability of the original models and the proportionalized versions, even though they describe the same biomass values and material flows. We use these findings to critically discuss whether stability analysis of observed equilibria by empirical food web models can provide insight into regime shift dynamics, and highlight the challenge of bridging alternative modelling approaches in ecology and beyond.
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Affiliation(s)
- Jan J Kuiper
- Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, SE 10691, Stockholm, Sweden.
- Department of Aquatic Ecology, Netherlands Institute of Ecology, P.O. Box 50, 6700 AB, Wageningen, The Netherlands.
| | - Bob W Kooi
- Faculty of Science, VU University, de Boelelaan 1085, 1081 HV, Amsterdam, The Netherlands
| | - Garry D Peterson
- Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, SE 10691, Stockholm, Sweden
| | - Wolf M Mooij
- Department of Aquatic Ecology, Netherlands Institute of Ecology, P.O. Box 50, 6700 AB, Wageningen, The Netherlands
- Aquatic Ecology and Water Quality Management Group, Department of Environmental Sciences, Wageningen University, P.O. Box 47, 6700 AA, Wageningen, The Netherlands
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5
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Golas BD, Goodell B, Webb CT. Host adaptation to novel pathogen introduction: Predicting conditions that promote evolutionary rescue. Ecol Lett 2021; 24:2238-2255. [PMID: 34310798 DOI: 10.1111/ele.13845] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2020] [Revised: 01/04/2021] [Accepted: 06/09/2021] [Indexed: 02/02/2023]
Abstract
Novel pathogen introduction can have drastic consequences for naive host populations, and outcomes can be difficult to predict. Evolutionary rescue (ER) provides a foundation for understanding whether hosts are driven to extinction or survive via adaptation. Currently, patterns of host population dynamics alongside evidence of adaptation are used to infer ER. However, the gap between established ER theory and complexity inherent in natural systems makes interpreting empirical patterns difficult because they can be confounded with ecological drivers of survival under current theory. To bridge this gap, we expand ER theory to include biological selective agents, such as pathogens. We find birth processes to be more important than previously theorised in determining ER potential. We employ a novel framework evaluating ER potential within natural systems and gain ability to identify system characteristics that make ER possible. Identifying these characteristics allows a shift from retrospective observation to a predictive mindset, and our findings suggest that ER occurrence may be more limited than previously thought. We use the plague system of Yersinia pestis infecting Cynomys ludovicianus (black-tailed prairie dogs) and Spermophilus beecheyi (California ground squirrels) as a case study.
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6
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Stouffer DB, Novak M. Hidden layers of density dependence in consumer feeding rates. Ecol Lett 2021; 24:520-532. [PMID: 33404158 DOI: 10.1111/ele.13670] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/26/2020] [Revised: 11/26/2020] [Accepted: 12/07/2020] [Indexed: 01/16/2023]
Abstract
Functional responses relate a consumer's feeding rates to variation in its abiotic and biotic environment, providing insight into consumer behaviour and fitness, and underpinning population and food-web dynamics. Despite their broad relevance and long-standing history, we show here that the types of density dependence found in classic resource- and consumer-dependent functional-response models equate to strong and often untenable assumptions about the independence of processes underlying feeding rates. We first demonstrate mathematically how to quantify non-independence between feeding and consumer interference and between feeding on multiple resources. We then analyse two large collections of functional-response data sets to show that non-independence is pervasive and borne out in previously hidden forms of density dependence. Our results provide a new lens through which to view variation in consumer feeding rates and disentangle the biological underpinnings of species interactions in multi-species contexts.
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Affiliation(s)
- Daniel B Stouffer
- Centre for Integrative Ecology, School of Biological Sciences, University of Canterbury, Christchurch, 8041, New Zealand
| | - Mark Novak
- Department of Integrative Biology, Oregon State University, Corvallis, OR, 97331, USA
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7
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Adamson MW, Morozov AY. Identifying the sources of structural sensitivity in partially specified biological models. Sci Rep 2020; 10:16926. [PMID: 33037267 PMCID: PMC7547730 DOI: 10.1038/s41598-020-73710-z] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2020] [Accepted: 09/14/2020] [Indexed: 12/02/2022] Open
Abstract
Biological systems are characterised by a high degree of uncertainty and complexity, which implies that exact mathematical equations to describe biological processes cannot generally be justified. Moreover, models can exhibit sensitivity to the precise formulations of their component functions—a property known as structural sensitivity. Structural sensitivity can be revealed and quantified by considering partially specified models with uncertain functions, but this goes beyond well-established, parameter-based sensitivity analysis, and currently presents a mathematical challenge. Here we build upon previous work in this direction by addressing the crucial question of identifying the processes which act as the major sources of model uncertainty and those which are less influential. To achieve this goal, we introduce two related concepts: (1) the gradient of structural sensitivity, accounting for errors made in specifying unknown functions, and (2) the partial degree of sensitivity with respect to each function, a global measure of the uncertainty due to possible variation of the given function while the others are kept fixed. We propose an iterative framework of experiments and analysis to inform a heuristic reduction of structural sensitivity in a model. To demonstrate the framework introduced, we investigate the sources of structural sensitivity in a tritrophic food chain model.
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Affiliation(s)
- Matthew W Adamson
- Institute of Mathematics, Institute of Environmental Systems Research, University of Osnabrück, Osnabrück, 49076, Germany.
| | - Andrew Yu Morozov
- Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK.,Institute of Ecology and Evolution, Russian Academy of Sciences, 33 Leninskii pr., Moscow, Russia, 119071.,N.I. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
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8
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Sandhu SK, Morozov A, Juan L. Exploring the role of spatial and stoichiometric heterogeneity in the top-down control in eutrophic planktonic ecosystems. J Theor Biol 2020; 499:110311. [PMID: 32437709 DOI: 10.1016/j.jtbi.2020.110311] [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] [Received: 11/13/2019] [Revised: 04/27/2020] [Accepted: 04/28/2020] [Indexed: 10/24/2022]
Abstract
Understanding the impact of eutrophication on the dynamics of aquatic food webs, remains a long-term challenge in ecology. Mathematical models generally predict the destabilisation of such webs, under increasing eutrophication levels, with large oscillations of species densities that can result in their extinction. This is at odds with a number of ecological observations that show stable dynamics even for high nutrient loads. The apparent discrepancy between theory and observations is known as the Rosenzweig's 'paradox of enrichment' and various solutions to the problem have been proposed over the years. In this study, we explore the stabilisation of dynamics of a tri-trophic plankton model in a eutrophic environment which occurs as a result of interplay of space heterogeneity, ecological stoichiometry, and food taxis of predators. We build a variety of models of increasing complexity, to explore various scenarios of phytoplankton growth, zooplankton food-dependent vertical movement, and different stoichiometric limitations of zooplankton. We show that the synergy among the vertical gradient in phytoplankton growth, phytoplankton structuring in terms of their stoichiometric ratio, and food-dependent vertical movement of zooplankton, would result in a postponing of destabilisation of eutrophic systems as compared to a well-mixed system. Our approach reveals a high complexity of the bifurcation structure of the system when key model parameters, such as the degree of eutrophication and light shading, are varied. We find coexistence of limit cycles and stable equilibria and that the possibility of multiple attractors in the system can result in hysteresis phenomena when the nutrient load is manipulated. These results are relevant and should be taken into account in lake restoration programs.
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Affiliation(s)
| | - Andrew Morozov
- Department of Mathematics, University of Leicester, UK; Institute of Ecology and Evolution, Russian Academy of Sciences, Moscow, Russia.
| | - Lourdes Juan
- Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA
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9
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Medvinsky AB, Adamovich BV, Rusakov AV, Tikhonov DA, Nurieva NI, Tereshko VM. Population Dynamics: Mathematical Modeling and Reality. Biophysics (Nagoya-shi) 2019. [DOI: 10.1134/s0006350919060150] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022] Open
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10
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Morozov A, Abbott K, Cuddington K, Francis T, Gellner G, Hastings A, Lai YC, Petrovskii S, Scranton K, Zeeman ML. Long transients in ecology: Theory and applications. Phys Life Rev 2019; 32:1-40. [PMID: 31982327 DOI: 10.1016/j.plrev.2019.09.004] [Citation(s) in RCA: 61] [Impact Index Per Article: 12.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2019] [Accepted: 09/09/2019] [Indexed: 11/15/2022]
Abstract
This paper discusses the recent progress in understanding the properties of transient dynamics in complex ecological systems. Predicting long-term trends as well as sudden changes and regime shifts in ecosystems dynamics is a major issue for ecology as such changes often result in population collapse and extinctions. Analysis of population dynamics has traditionally been focused on their long-term, asymptotic behavior whilst largely disregarding the effect of transients. However, there is a growing understanding that in ecosystems the asymptotic behavior is rarely seen. A big new challenge for theoretical and empirical ecology is to understand the implications of long transients. It is believed that the identification of the corresponding mechanisms along with the knowledge of scaling laws of the transient's lifetime should substantially improve the quality of long-term forecasting and crisis anticipation. Although transient dynamics have received considerable attention in physical literature, research into ecological transients is in its infancy and systematic studies are lacking. This text aims to partially bridge this gap and facilitate further progress in quantitative analysis of long transients in ecology. By revisiting and critically examining a broad variety of mathematical models used in ecological applications as well as empirical facts, we reveal several main mechanisms leading to the emergence of long transients and hence lays the basis for a unifying theory.
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Affiliation(s)
- Andrew Morozov
- Mathematics, University of Leicester, UK; Shirshov Institute of Oceanology, Moscow, Russia
| | | | | | - Tessa Francis
- Tacoma Puget Sound Institute, University of Washington, USA
| | | | - Alan Hastings
- Environmental Science and Policy, University of California, Davis, USA; Santa Fe Institute, Santa Fe, New Mexico, USA
| | - Ying-Cheng Lai
- Electrical, Computer and Energy Engineering, Arizona State University, Tempe, USA
| | - Sergei Petrovskii
- Mathematics, University of Leicester, UK; Peoples Friendship University of Russia (RUDN University), Moscow, Russia.
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11
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Letten AD, Stouffer DB. The mechanistic basis for higher-order interactions and non-additivity in competitive communities. Ecol Lett 2019; 22:423-436. [PMID: 30675983 DOI: 10.1111/ele.13211] [Citation(s) in RCA: 67] [Impact Index Per Article: 13.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2018] [Revised: 09/18/2018] [Accepted: 11/27/2018] [Indexed: 11/27/2022]
Abstract
Motivated by both analytical tractability and empirical practicality, community ecologists have long treated the species pair as the fundamental unit of study. This notwithstanding, the challenge of understanding more complex systems has repeatedly generated interest in the role of so-called higher-order interactions (HOIs) imposed by species beyond the focal pair. Here we argue that HOIs - defined as non-additive effects of density on per capita growth - are best interpreted as emergent properties of phenomenological models (e.g. Lotka-Volterra competition) rather than as distinct 'ecological processes' in their own right. Using simulations of consumer-resource models, we explore the mechanisms and system properties that give rise to HOIs in observational data. We demonstrate that HOIs emerge under all but the most restrictive of assumptions, and that incorporating non-additivity into phenomenological models improves the quantitative and qualitative accuracy of model predictions. Notably, we also observe that HOIs derive primarily from mechanisms and system properties that apply equally to single-species or pairwise systems as they do to more diverse communities. Consequently, there exists a strong mandate for further recognition of non-additive effects in both theoretical and empirical research.
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Affiliation(s)
- Andrew D Letten
- Centre for Integrative Ecology, University of Canterbury, Christchurch, 8140, New Zealand.,Institute of Integrative Biology, Department of Environmental Systems Science, ETH Zürich, 8092, Zürich, Switzerland
| | - Daniel B Stouffer
- Centre for Integrative Ecology, University of Canterbury, Christchurch, 8140, New Zealand
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12
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Aldebert C, Stouffer DB. Community dynamics and sensitivity to model structure: towards a probabilistic view of process-based model predictions. J R Soc Interface 2018; 15:rsif.2018.0741. [PMID: 30518566 DOI: 10.1098/rsif.2018.0741] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/05/2018] [Accepted: 11/05/2018] [Indexed: 11/12/2022] Open
Abstract
Statistical inference and mechanistic, process-based modelling represent two philosophically different streams of research whose primary goal is to make predictions. Here, we merge elements from both approaches to keep the theoretical power of process-based models while also considering their predictive uncertainty using Bayesian statistics. In environmental and biological sciences, the predictive uncertainty of process-based models is usually reduced to parametric uncertainty. Here, we propose a practical approach to tackle the added issue of structural sensitivity, the sensitivity of predictions to the choice between quantitatively close and biologically plausible models. In contrast to earlier studies that presented alternative predictions based on alternative models, we propose a probabilistic view of these predictions that include the uncertainty in model construction and the parametric uncertainty of each model. As a proof of concept, we apply this approach to a predator-prey system described by the classical Rosenzweig-MacArthur model, and we observe that parametric sensitivity is regularly overcome by structural sensitivity. In addition to tackling theoretical questions about model sensitivity, the proposed approach can also be extended to make probabilistic predictions based on more complex models in an operational context. Both perspectives represent important steps towards providing better model predictions in biology, and beyond.
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Affiliation(s)
- Clement Aldebert
- Mediterranean Institute of Oceanography, Aix-Marseille University, Toulon University, CNRS/INSU, IRD, MIO, UM 110, 13288 Cedex 09, Marseille, France
| | - Daniel B Stouffer
- School of Biological Sciences, University of Canterbury, Christchurch 8140, New Zealand
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13
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Egilmez HI, Morozov AY, Clokie MRJ, Shan J, Letarov A, Galyov EE. Temperature-dependent virus lifecycle choices may reveal and predict facets of the biology of opportunistic pathogenic bacteria. Sci Rep 2018; 8:9642. [PMID: 29941954 PMCID: PMC6018541 DOI: 10.1038/s41598-018-27716-3] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2017] [Accepted: 06/08/2018] [Indexed: 12/21/2022] Open
Abstract
Melioidosis, a serious illness caused by Burkholderia pseudomallei, results in up to 40% fatality in infected patients. The pathogen is found in tropical water and soil. Recent findings demonstrated that bacterial numbers can be regulated by a novel clade of phages that are abundant in soil and water. These phages differentially infect their bacterial hosts causing lysis at high temperatures and lysogeny at lower temperatures. Thus seasonal and daily temperature variations would cause switches in phage-bacteria interactions. We developed mathematical models using realistic parameters to explore the impact of phages on B. pseudomallei populations in the surface water of rice fields over time and under seasonally changing environmental conditions. Historical records were used to provide UV radiation levels and temperature for two Thailand provinces. The models predict seasonal variation of phage-free bacterial numbers correlates with the higher risk of melioidosis acquisition during the “warm and wet” season. We find that enrichment of the environment may lead to irregular large amplitude pulses of bacterial numbers that could significantly increase the probability of disease acquisition. Our results suggest that the phages may regulate B. pseudomallei populations throughout the seasons, and these data can potentially help improve the melioidosis prevention efforts in Southeast Asia.
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Affiliation(s)
- Halil I Egilmez
- Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
| | - Andrew Yu Morozov
- Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK.
| | - Martha R J Clokie
- Department of Infection, Immunity and Inflammation, University of Leicester, Leicester, LE1 7RH, UK
| | - Jinyu Shan
- Department of Infection, Immunity and Inflammation, University of Leicester, Leicester, LE1 7RH, UK
| | - Andrey Letarov
- Winogradsky Institute of Microbiology, RC Biotechnology RAS, Moscow, Russia.,Faculty of Biology, Lomonosov Moscow State University, Leninskie Gory, 1, Moscow, 119991, Russia.,Moscow Institute of Physics and Technology, Institutskiy per., 9, Dolgoprudny, Moscow Region, 141701, Russia
| | - Edouard E Galyov
- Department of Infection, Immunity and Inflammation, University of Leicester, Leicester, LE1 7RH, UK
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14
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Aldebert C, Kooi BW, Nerini D, Poggiale JC. Is structural sensitivity a problem of oversimplified biological models? Insights from nested Dynamic Energy Budget models. J Theor Biol 2018; 448:1-8. [PMID: 29550453 DOI: 10.1016/j.jtbi.2018.03.019] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2017] [Revised: 03/01/2018] [Accepted: 03/13/2018] [Indexed: 10/17/2022]
Abstract
Many current issues in ecology require predictions made by mathematical models, which are built on somewhat arbitrary choices. Their consequences are quantified by sensitivity analysis to quantify how changes in model parameters propagate into an uncertainty in model predictions. An extension called structural sensitivity analysis deals with changes in the mathematical description of complex processes like predation. Such processes are described at the population scale by a specific mathematical function taken among similar ones, a choice that can strongly drive model predictions. However, it has only been studied in simple theoretical models. Here, we ask whether structural sensitivity is a problem of oversimplified models. We found in predator-prey models describing chemostat experiments that these models are less structurally sensitive to the choice of a specific functional response if they include mass balance resource dynamics and individual maintenance. Neglecting these processes in an ecological model (for instance by using the well-known logistic growth equation) is not only an inappropriate description of the ecological system, but also a source of more uncertain predictions.
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Affiliation(s)
- Clement Aldebert
- Mediterranean Institute of Oceanography, Aix-Marseille University, Toulon University, CNRS/INSU,IRD, MIO, UM 110, Marseille, Cedex 09 13288, France; University of Zurich, Institute of Evolutionary Biology and Environmental Studies, Winterthurerstrasse 190, Zurich 8057, Switzerland.
| | - Bob W Kooi
- Faculty of Science, VU University, de Boelelaan 1085,HV Amsterdam 1081, The Netherlands
| | - David Nerini
- Mediterranean Institute of Oceanography, Aix-Marseille University, Toulon University, CNRS/INSU,IRD, MIO, UM 110, Marseille, Cedex 09 13288, France.
| | - Jean-Christophe Poggiale
- Mediterranean Institute of Oceanography, Aix-Marseille University, Toulon University, CNRS/INSU,IRD, MIO, UM 110, Marseille, Cedex 09 13288, France.
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15
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Introduction to the themed issue: Uncertainty, sensitivity and predictability in ecology: Mathematical challenges and ecological applications. ECOLOGICAL COMPLEXITY 2017. [DOI: 10.1016/j.ecocom.2017.11.004] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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16
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Ben-Haim Y. Does a better model yield a better argument? An info-gap analysis. Proc Math Phys Eng Sci 2017. [DOI: 10.1098/rspa.2016.0890] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Theories, models and computations underlie reasoned argumentation in many areas. The possibility of error in these arguments, though of low probability, may be highly significant when the argument is used in predicting the probability of rare high-consequence events. This implies that the choice of a theory, model or computational method for predicting rare high-consequence events must account for the probability of error in these components. However, error may result from lack of knowledge or surprises of various sorts, and predicting the probability of error is highly uncertain. We show that the putatively best, most innovative and sophisticated argument may not actually have the lowest probability of error. Innovative arguments may entail greater uncertainty than more standard but less sophisticated methods, creating an innovation dilemma in formulating the argument. We employ info-gap decision theory to characterize and support the resolution of this problem and present several examples.
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17
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18
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Aldebert C, Nerini D, Gauduchon M, Poggiale J. Structural sensitivity and resilience in a predator–prey model with density-dependent mortality. ECOLOGICAL COMPLEXITY 2016. [DOI: 10.1016/j.ecocom.2016.05.004] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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19
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Morozov AY, Kuzenkov OA. Towards developing a general framework for modelling vertical migration in zooplankton. J Theor Biol 2016; 405:17-28. [PMID: 26804642 DOI: 10.1016/j.jtbi.2016.01.011] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2015] [Revised: 12/22/2015] [Accepted: 01/09/2016] [Indexed: 11/19/2022]
Abstract
Diel vertical migration (DVM) of zooplankton is a widespread phenomenon in both oceans and lakes, and is generally considered to be the largest synchronized movement of biomass on Earth. Most existing mathematical models of DVM are based on the assumption that animals maximize a certain criterion such as the expected reproductive value, the venturous revenue, the ratio of energy gain/mortality or some predator avoidance function when choosing their instantaneous depth. The major shortcoming of this general point of view is that the predicted DVM may be strongly affected by a subjective choice of a particular optimization criterion. Here we argue that the optimal strategy of DVM can be unambiguously obtained as an outcome of selection in the underlying equations of genotype/traits frequency dynamics. Using this general paradigm, we explore the optimal strategy for the migration across different depths by zooplankton grazers throughout the day. To illustrate our ideas we consider four generic DVM models, each making different assumptions on the population dynamics of zooplankton, and demonstrate that in each model we need to maximize a particular functional to find the optimal strategy. Surprisingly, patterns of DVM obtained for different models greatly differ in terms of their parameters dependence. We then show that the infinite dimensional trait space of different zooplankton trajectories can be projected onto a low dimensional space of generalized parameters and the genotype evolution dynamics can be easily followed using this low-dimensional space. Using this space of generalized parameters we explore the influence of mutagenesis on evolution of DVM, and we show that strong mutagenesis allows the coexistence of an infinitely large number of strategies whereas for weak mutagenesis the selection results in the extinction of most strategies, with the surviving strategies all staying close to the optimal strategy in the corresponding mutagenesis-free system.
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Affiliation(s)
- Andrew Yu Morozov
- Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK.
| | - Oleg A Kuzenkov
- Lobachevsky State University of Nizhniy Novgorod, Nizhniy Novgorod, Russia; Shirshov Institute of Oceanology, Moscow, Russia.
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Adamson MW, Morozov AY, Kuzenkov OA. Quantifying uncertainty in partially specified biological models: how can optimal control theory help us? Proc Math Phys Eng Sci 2016; 472:20150627. [PMID: 27713655 PMCID: PMC5046979 DOI: 10.1098/rspa.2015.0627] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2015] [Accepted: 08/18/2016] [Indexed: 11/12/2022] Open
Abstract
Mathematical models in biology are highly simplified representations of a complex underlying reality and there is always a high degree of uncertainty with regards to model function specification. This uncertainty becomes critical for models in which the use of different functions fitting the same dataset can yield substantially different predictions-a property known as structural sensitivity. Thus, even if the model is purely deterministic, then the uncertainty in the model functions carries through into uncertainty in model predictions, and new frameworks are required to tackle this fundamental problem. Here, we consider a framework that uses partially specified models in which some functions are not represented by a specific form. The main idea is to project infinite dimensional function space into a low-dimensional space taking into account biological constraints. The key question of how to carry out this projection has so far remained a serious mathematical challenge and hindered the use of partially specified models. Here, we propose and demonstrate a potentially powerful technique to perform such a projection by using optimal control theory to construct functions with the specified global properties. This approach opens up the prospect of a flexible and easy to use method to fulfil uncertainty analysis of biological models.
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Affiliation(s)
- M. W. Adamson
- Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
| | - A. Y. Morozov
- Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
- Shirshov Institute of Oceanology, Moscow, 117997, Russia
| | - O. A. Kuzenkov
- Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod, Russia
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Morozov AY, Banerjee M, Petrovskii SV. Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect. J Theor Biol 2016; 396:116-24. [PMID: 26921467 DOI: 10.1016/j.jtbi.2016.02.016] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2015] [Revised: 01/27/2016] [Accepted: 02/12/2016] [Indexed: 10/22/2022]
Abstract
Traditionally, mathematical modeling in population ecology is mainly focused on asymptotic behavior of the model, i.e. as given by the system attractors. Recently, however, transient regimes and especially long-term transients have been recognized as playing a crucial role in the dynamics of ecosystems. In particular, long-term transients are a potential explanation of ecological regime shifts, when an apparently healthy population suddenly collapses and goes extinct. In this paper, we show that the interplay between delay in maturation and a strong Allee effect can result in long-term transients in a single species system. We first derive a simple 'conceptual' model of the population dynamics that incorporates both a strong Allee effect and maturation delay. Unlike much of the previous work, our approach is not empirical since our model is derived from basic principles. We show that the model exhibits a high complexity in its asymptotic dynamics including multi-periodic and chaotic attractors. We then show the existence of long-term transient dynamics in the system, when the population size oscillates for a long time between locally stable stationary states before it eventually settles either at the persistence equilibrium or goes extinct. The parametric space of the model is found to have a complex structure with the basins of attraction corresponding to the persistence and extinction states being of a complicated shape. This impedes the prediction of the eventual fate of the population, as a small variation in the maturation delay or the initial population size can either bring the population to extinction or ensure its persistence.
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Affiliation(s)
- A Yu Morozov
- Department of Mathematics, University of Leicester, LE1 7RH, UK
| | - M Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India.
| | - S V Petrovskii
- Department of Mathematics, University of Leicester, LE1 7RH, UK
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Medvinsky AB, Adamovich BV, Chakraborty A, Lukyanova EV, Mikheyeva TM, Nurieva NI, Radchikova NP, Rusakov AV, Zhukova TV. Chaos far away from the edge of chaos: A recurrence quantification analysis of plankton time series. ECOLOGICAL COMPLEXITY 2015. [DOI: 10.1016/j.ecocom.2015.07.001] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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23
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Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect. THEOR ECOL-NETH 2014. [DOI: 10.1007/s12080-014-0222-z] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Bifurcation analysis of models with uncertain function specification: how should we proceed? Bull Math Biol 2014; 76:1218-40. [PMID: 24789567 DOI: 10.1007/s11538-014-9951-9] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2013] [Accepted: 03/13/2014] [Indexed: 10/25/2022]
Abstract
When we investigate the bifurcation structure of models of natural phenomena, we usually assume that all model functions are mathematically specified and that the only existing uncertainty is with respect to the parameters of these functions. In this case, we can split the parameter space into domains corresponding to qualitatively similar dynamics, separated by bifurcation hypersurfaces. On the other hand, in the biological sciences, the exact shape of the model functions is often unknown, and only some qualitative properties of the functions can be specified: mathematically, we can consider that the unknown functions belong to a specific class of functions. However, the use of two different functions belonging to the same class can result in qualitatively different dynamical behaviour in the model and different types of bifurcation. In the literature, the conventional way to avoid such ambiguity is to narrow the class of unknown functions, which allows us to keep patterns of dynamical behaviour consistent for varying functions. The main shortcoming of this approach is that the restrictions on the model functions are often given by cumbersome expressions and are strictly model-dependent: biologically, they are meaningless. In this paper, we suggest a new framework (based on the ODE paradigm) which allows us to investigate deterministic biological models in which the mathematical formulation of some functions is unspecified except for some generic qualitative properties. We demonstrate that in such models, the conventional idea of revealing a concrete bifurcation structure becomes irrelevant: we can only describe bifurcations with a certain probability. We then propose a method to define the probability of a bifurcation taking place when there is uncertainty in the parameterisation in our model. As an illustrative example, we consider a generic predator-prey model where the use of different parameterisations of the logistic-type prey growth function can result in different dynamics in terms of the type of the Hopf bifurcation through which the coexistence equilibrium loses stability. Using this system, we demonstrate a framework for evaluating the probability of having a supercritical or subcritical Hopf bifurcation.
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Adamson MW, Morozov AY. Defining and detecting structural sensitivity in biological models: developing a new framework. J Math Biol 2014; 69:1815-48. [PMID: 24448657 DOI: 10.1007/s00285-014-0753-3] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2013] [Revised: 12/21/2013] [Indexed: 11/25/2022]
Abstract
When we construct mathematical models to represent biological systems, there is always uncertainty with regards to the model specification--whether with respect to the parameters or to the formulation of model functions. Sometimes choosing two different functions with close shapes in a model can result in substantially different model predictions: a phenomenon known in the literature as structural sensitivity, which is a significant obstacle to improving the predictive power of biological models. In this paper, we revisit the general definition of structural sensitivity, compare several more specific definitions and discuss their usefulness for the construction and analysis of biological models. Then we propose a general approach to reveal structural sensitivity with regards to certain system properties, which considers infinite-dimensional neighbourhoods of the model functions: a far more powerful technique than the conventional approach of varying parameters for a fixed functional form. In particular, we suggest a rigorous method to unearth sensitivity with respect to the local stability of systems' equilibrium points. We present a method for specifying the neighbourhood of a general unknown function with [Formula: see text] inflection points in terms of a finite number of local function properties, and provide a rigorous proof of its completeness. Using this powerful result, we implement our method to explore sensitivity in several well-known multicomponent ecological models and demonstrate the existence of structural sensitivity in these models. Finally, we argue that structural sensitivity is an important intrinsic property of biological models, and a direct consequence of the complexity of the underlying real systems.
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Affiliation(s)
- M W Adamson
- Department of Mathematics, University of Leicester, University Road, Leicester , LE1 7RH, UK,
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Morozov A, Petrovskii S. Feeding on multiple sources: towards a universal parameterization of the functional response of a generalist predator allowing for switching. PLoS One 2013; 8:e74586. [PMID: 24086356 PMCID: PMC3783441 DOI: 10.1371/journal.pone.0074586] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/14/2013] [Accepted: 08/06/2013] [Indexed: 11/18/2022] Open
Abstract
Understanding of complex trophic interactions in ecosystems requires correct descriptions of the rate at which predators consume a variety of different prey species. Field and laboratory data on multispecies communities are rarely sufficient and usually cannot provide an unambiguous test for the theory. As a result, the conventional way of constructing a multi-prey functional response is speculative, and often based on assumptions that are difficult to verify. Predator responses allowing for prey selectivity and active switching are thought to be more biologically relevant compared to the standard proportion-based consumption. However, here we argue that the functional responses with switching may not be applicable to communities with a broad spectrum of resource types. We formulate a set of general rules that a biologically sound parameterization of a predator functional response should satisfy, and show that all existing formulations for the multispecies response with prey selectivity and switching fail to do so. Finally, we propose a universal framework for parameterization of a multi-prey functional response by combining patterns of food selectivity and proportion-based feeding.
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Affiliation(s)
- Andrew Morozov
- Department of Mathematics, University of Leicester, Leicester, United Kingdom
| | - Sergei Petrovskii
- Department of Mathematics, University of Leicester, Leicester, United Kingdom
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Barabás G, Ostling A. Community robustness in discrete-time periodic environments. ECOLOGICAL COMPLEXITY 2013. [DOI: 10.1016/j.ecocom.2013.07.001] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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