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Narayan Chattopadhyay S, Kumar Gupta A. Tipping points, multistability, and stochasticity in a two-dimensional traffic network dynamics. CHAOS (WOODBURY, N.Y.) 2024; 34:073107. [PMID: 38949532 DOI: 10.1063/5.0202785] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2024] [Accepted: 06/08/2024] [Indexed: 07/02/2024]
Abstract
Mitigating traffic jams is a critical step for the betterment of the urban transportation system, which comprises a large number of interconnected routes to form an intricate network. To understand distinct features of vehicular traffic flow on a network, a macroscopic two-dimensional traffic network model is proposed incorporating intra-nodal and inter-nodal vehicular interaction. Utilizing the popular techniques of nonlinear dynamics, we investigate the impact of different parameters like occupancy, entry rates, and exit rates of vehicles. The existence of saddle-node, Hopf, homoclinic, Bogdanov-Takens, and cusp bifurcations have been shown using single or biparametric bifurcation diagrams. The occurrences of different multistability (bistability/tristability) phenomena, stochastic switching, and critical transitions are explored in detail. Further, we calculate the possibility of achieving each alternative state using the basin stability metric to characterize multistability. In addition, critical transitions from free flow to congestion are identified at different magnitudes of stochastic fluctuations. The applicability of critical slowing down based generic indicators, e.g., variance, lag-1 autocorrelation, skewness, kurtosis, and conditional heteroskedasticity are investigated to forewarn the critical transition from free flow to traffic congestion. It is demonstrated through the use of simulated data that not all of the measures exhibit sensitivity to rapid phase transitions in traffic flow. Our study reveals that traffic congestion emerges because of either bifurcation or stochasticity. The result provided in this study may serve as a paradigm to understand the qualitative behavior of traffic jams and to explore the tipping mechanisms occurring in transport phenomena.
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Bieg C, Gellner G, McCann KS. Stability of consumer-resource interactions in periodic environments. Proc Biol Sci 2023; 290:20231636. [PMID: 37752846 PMCID: PMC10523078 DOI: 10.1098/rspb.2023.1636] [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: 07/21/2023] [Accepted: 08/31/2023] [Indexed: 09/28/2023] Open
Abstract
Periodic fluctuations in abiotic conditions are ubiquitous across a range of temporal scales and regulate the structure and function of ecosystems through dynamic biotic responses that are adapted to these external forces. Research has suggested that certain environmental signatures may play a crucial role in the maintenance of biodiversity and the stability of food webs, while others argue that coupled oscillators ought to promote chaos. As such, numerous uncertainties remain regarding the intersection of temporal environmental patterns and biological responses, and we lack a general understanding of the implications for food web stability. Alarmingly, global change is altering the nature of both environmental rhythms and biological rates. Here, we develop a general theory for how continuous periodic variation in productivity, across temporal scales, influences the stability of consumer-resource interactions: a fundamental building block of food webs. Our results suggest that consumer-resource dynamics under environmental forcing are highly complex and depend on asymmetries in both the speed of forcing relative to underlying dynamics and in local stability properties. These asymmetries allow for environmentally driven stabilization under fast forcing, relative to underlying dynamics, as well as extremely complex and unstable dynamics at slower periodicities. Our results also suggest that changes in naturally occurring periodicities from climate change may lead to precipitous shifts in dynamics and stability.
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Affiliation(s)
- Carling Bieg
- Department of Integrative Biology, University of Guelph, Guelph, Ontario, Canada N1G 2W1
- Ecology and Evolutionary Biology, Yale University, New Haven, CT, USA
| | - Gabriel Gellner
- Department of Integrative Biology, University of Guelph, Guelph, Ontario, Canada N1G 2W1
| | - Kevin S. McCann
- Department of Integrative Biology, University of Guelph, Guelph, Ontario, Canada N1G 2W1
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Alkhayuon H, Marley J, Wieczorek S, Tyson RC. Stochastic resonance in climate reddening increases the risk of cyclic ecosystem extinction via phase-tipping. GLOBAL CHANGE BIOLOGY 2023; 29:3347-3363. [PMID: 37021593 DOI: 10.1111/gcb.16679] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2022] [Revised: 02/24/2023] [Accepted: 03/02/2023] [Indexed: 05/16/2023]
Abstract
Human activity is leading to changes in the mean and variability of climatic parameters in most locations around the world. The changing mean has received considerable attention from scientists and climate policy makers. However, recent work indicates that the changing variability, that is, the amplitude and the temporal autocorrelation of deviations from the mean, may have greater and more imminent impact on ecosystems. In this paper, we demonstrate that changes in climate variability alone could drive cyclic predator-prey ecosystems to extinction via so-called phase-tipping (P-tipping), a new type of instability that occurs only from certain phases of the predator-prey cycle. We construct a mathematical model of a variable climate and couple it to two self-oscillating paradigmatic predator-prey models. Most importantly, we combine realistic parameter values for the Canada lynx and snowshoe hare with actual climate data from the boreal forest. In this way, we demonstrate that critically important species in the boreal forest have increased likelihood of P-tipping to extinction under predicted changes in climate variability, and are most vulnerable during stages of the cycle when the predator population is near its maximum. Furthermore, our analysis reveals that stochastic resonance is the underlying mechanism for the increased likelihood of P-tipping to extinction.
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Affiliation(s)
- Hassan Alkhayuon
- School of Mathematical Sciences, University College Cork, Western Road, Cork, T12 XF62, Ireland
| | - Jessa Marley
- CMPS Department (Mathematics), University of British Columbia Okanagan, Kelowna, British Columbia, Canada
| | - Sebastian Wieczorek
- School of Mathematical Sciences, University College Cork, Western Road, Cork, T12 XF62, Ireland
| | - Rebecca C Tyson
- CMPS Department (Mathematics), University of British Columbia Okanagan, Kelowna, British Columbia, Canada
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Kaur T, Sharathi Dutta P. Critical rates of climate warming and abrupt collapse of ecosystems. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2022.0086] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In the age of climate warming, comprehension of ecosystems’ future is one of the pressing challenges to humanity. While most studies on climate warming focus on the ‘magnitude of change’ of the Earth’s temperature, the ‘rate’ at which it is increasing cannot be ruled out. Rapid warming has already caused sudden ecosystem transitions at numerous biodiversity hot spots; a mechanistic understanding of such transitions is crucial. Here, we study a slow–fast consumer–resource ecosystem interacting in rapid warming scenarios. Employing geometric singular perturbation theory, we find that while a gradual change in mean temperature may accord population persistence, a critical warming rate can drive the resource’s sudden collapse, termed a warming-induced abrupt transition. This further triggers the bottom-up effect, resulting in the extinction of the consumer. The difference between the optimum temperature of the resource’s growth rate and the habitat temperature is crucial in deciding the critical rate of warming. Consequently, species inhabiting extreme temperature regions are more susceptible to warming-induced collapse than those within intermediate temperature ranges. We find that stochastic fluctuations in the system can advance warming-induced transitions, and the efficacy of generic early warning signals to anticipate sudden transitions is challenged.
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Affiliation(s)
- Taranjot Kaur
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140 001, India
| | - Partha Sharathi Dutta
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140 001, India
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Li Y, Buenzli PR, Simpson MJ. Interpreting how nonlinear diffusion affects the fate of bistable populations using a discrete modelling framework. Proc Math Phys Eng Sci 2022; 478:20220013. [PMID: 35702596 PMCID: PMC9185834 DOI: 10.1098/rspa.2022.0013] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2022] [Accepted: 04/28/2022] [Indexed: 12/11/2022] Open
Abstract
Understanding whether a population will survive or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction–diffusion equations, where migration is usually represented as a linear diffusion term, and birth–death is represented with a nonlinear source term. While linear diffusion is most commonly employed to study migration, there are several limitations of this approach, such as the inability of linear diffusion-based models to predict a well-defined population front. One way to overcome this is to generalize the constant diffusivity, D, to a nonlinear diffusivity function D(C), where C>0 is the population density. While the choice of D(C) affects long-term survival or extinction of a bistable population, working solely in a continuum framework makes it difficult to understand how the choice of D(C) affects survival or extinction. We address this question by working with a discrete simulation model that is easy to interpret. This approach provides clear insight into how the choice of D(C) either encourages or suppresses population extinction relative to the classical linear diffusion model.
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Affiliation(s)
- Yifei Li
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD 4001, Australia
| | - Pascal R Buenzli
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD 4001, Australia
| | - Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD 4001, Australia
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van Belzen J, Fivash GS, Hu Z, Bouma TJ, Herman PMJ. A probabilistic framework for windows of opportunity: the role of temporal variability in critical transitions. J R Soc Interface 2022; 19:20220041. [PMID: 35506213 PMCID: PMC9065964 DOI: 10.1098/rsif.2022.0041] [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: 01/16/2022] [Accepted: 04/04/2022] [Indexed: 11/13/2022] Open
Abstract
The establishment of young organisms in harsh environments often requires a window of opportunity (WoO). That is, a short time window in which environmental conditions drop long enough below the hostile average level, giving the organism time to develop tolerance and transition into stable existence. It has been suggested that this kind of establishment dynamics is a noise-induced transition between two alternate states. Understanding how temporal variability (i.e. noise) in environmental conditions affects establishment of organisms is therefore key, yet not well understood or included explicitly in the WoO framework. In this paper, we develop a coherent theoretical framework for understanding when the WoO open or close based on simple dichotomous environmental variation. We reveal that understanding of the intrinsic timescales of both the developing organism and the environment is fundamental to predict if organisms can or cannot establish. These insights have allowed us to develop statistical laws for predicting establishment probabilities based on the period and variance of the fluctuations in naturally variable environments. Based on this framework, we now get a clear understanding of how changes in the timing and magnitude of climate variability or management can mediate establishment chances.
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Affiliation(s)
- Jim van Belzen
- Department of Estuarine and Delta Systems, Royal Netherlands Institute for Sea Research (NIOZ), 4401 NT Yerseke, The Netherlands
| | - Gregory S. Fivash
- Department of Estuarine and Delta Systems, Royal Netherlands Institute for Sea Research (NIOZ), 4401 NT Yerseke, The Netherlands
| | - Zhan Hu
- School of Marine Sciences, Sun Yat-Sen University, and Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, People's Republic of China
- Guangdong Provincial Key Laboratory of Marine Resources and Coastal Engineering, Guangzhou, People's Republic of China
- Pearl River Estuary Marine Ecosystem Research Station, Ministry of Education, Zhuhai, People's Republic of China
| | - Tjeerd J. Bouma
- Department of Estuarine and Delta Systems, Royal Netherlands Institute for Sea Research (NIOZ), 4401 NT Yerseke, The Netherlands
- Faculty of Geosciences, Department of Physical Geography, Utrecht University, 3508 TC Utrecht, The Netherlands
| | - Peter M. J. Herman
- Department of Hydraulic Engineering, Delft University of Technology, 2628 CN, Delft, The Netherlands
- Unit of Marine and Coastal Systems, Deltares, 2600 MH, Delft, The Netherlands
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Alkhayuon H, Tyson RC, Wieczorek S. Phase tipping: how cyclic ecosystems respond to contemporary climate. Proc Math Phys Eng Sci 2021; 477:20210059. [PMID: 35153584 PMCID: PMC8511773 DOI: 10.1098/rspa.2021.0059] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2021] [Accepted: 09/06/2021] [Indexed: 11/30/2022] Open
Abstract
We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.
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Affiliation(s)
- Hassan Alkhayuon
- University College Cork, School of Mathematical Sciences, Western Road, Cork T12 XF62, Ireland
| | - Rebecca C Tyson
- CMPS Department (Mathematics), University of British Columbia Okanagan, Kelowna, British Columbia, Canada
| | - Sebastian Wieczorek
- University College Cork, School of Mathematical Sciences, Western Road, Cork T12 XF62, Ireland
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