Stationary distributions of persistent ecological systems

Hening, Alexandru; Li, Yao

doi:10.1007/s00285-021-01613-2

Cited by 11 publications

(5 citation statements)

References 58 publications

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“…Discontinuous dynamical systems find pervasive applications across various domains, from electrical circuits and mechanical systems to biological processes, see [1][2][3]. These systems frequently exhibit abrupt changes, discontinuities, or switching phenomena, leading to sudden state transitions or impacts.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line

Casimiro,

Llibre

2024

Axioms

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In this article, we study the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems separated by a straight line. We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously degree one, two or three. We obtain that in these three cases, this maximum number is zero, one and three, respectively. Moreover, we prove that there are discontinuous piecewise differential systems realizing these maximum number of limit cycles. Note that we have solved the extension of the 16th Hilbert problem about the maximum number of limit cycles that these three classes of discontinuous piecewise differential systems separated by one straight line and formed by two Hamiltonian systems with a degree either one, two, or three, which such systems can exhibit.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line

Casimiro,

Llibre

2024

Axioms

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“…Such hybrid systems occur in neuroscience (Anderson et al. 2015b ), cell biology (Bressloff 2014 ), and ecology (Hening and Li 2021 ; Hening et al. 2021 ).…”

Section: Discussionmentioning

confidence: 99%

“…Recent work by Hening et al has further extended this work to stochastic switching systems that can exhibit discontinuous changes in conditionally deterministic dynamics. Such hybrid systems occur in neuroscience (Anderson et al 2015b), cell biology (Bressloff 2014), and ecology (Hening and Li 2021;. In an ecological context, Hening et al (2020) classified the resulting systems by analyzing their stationary distributions.…”

Section: Discussionmentioning

confidence: 99%

“…Kerr et al ( 2002 ) observed a similar behavior in colicinogenic E. coli : a toxin-producing strain kills a susceptible population, a toxin-resistant population grows faster than the toxin-producing population, and the susceptible population grows faster than the resistant population. Rock-paper-scissors models have also been considered subject to environmental stochasticity (Hening and Li 2021 ). In computational neuroscience, heteroclinic cycling has been proposed as an alternative to classic “winner-take-all” models for neural networks.…”

Section: Introductionmentioning

confidence: 99%

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Heteroclinic cycling and extinction in May–Leonard models with demographic stochasticity

Barendregt

Thomas

2023

J. Math. Biol.

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May and Leonard (SIAM J Appl Math 29:243–253, 1975) introduced a three-species Lotka–Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete each “cycle”, passing closer and closer to unstable fixed points in which one population dominates and the others approach zero. Aperiodic heteroclinic dynamics have subsequently been studied in ecological systems (side-blotched lizards; colicinogenic Escherichia coli), in the immune system, in neural information processing models (“winnerless competition”), and in models of neural central pattern generators. Yet as May and Leonard observed “Biologically, the behavior (produced by the model) is nonsense. Once it is conceded that the variables represent animals, and therefore cannot fall below unity, it is clear that the system will, after a few cycles, converge on some single population, extinguishing the other two.” Here, we explore different ways of introducing discrete stochastic dynamics based on May and Leonard’s ODE model, with application to ecological population dynamics, and to a neuromotor central pattern generator system. We study examples of several quantitatively distinct asymptotic behaviors, including total extinction of all species, extinction to a single species, and persistent cyclic dominance with finite mean cycle length.

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“…These results are generalized in [10] to allow parameters to change according to a piecewise deterministic Markov chain, and a fast algorithm for computing the stationary EJP 27 (2022), paper 85. distributions is also given. I expect that for density-dependent Markov chains and SDEs of the form (2.1), since N → 0 as N → ∞ a similar result holds if we use the invasion rates defined for the corresponding deterministic system x = F (x), and define permanence and impermanence according to whether the time to hit the boundary is long or short as a function of N .…”

Section: Related Workmentioning

confidence: 99%

Fixation time of the rock-paper-scissors model: rigorous results in the well-mixed setting

Foxall

Madani

Roemer

2022

Electron. J. Probab.

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The rock-paper-scissors model simulates the effect of cyclic dominance in a finite population of size N and has received considerable attention in applied literature. In the well-mixed version of the model, population densities fluctuate around periodic orbits of a deterministic ODE approximation, and for large N the time to fixation (complete dominance by one species) has been observed by simulation to be approximately τ N where τ is a positive, finite random variable. We give a rigorous proof of this observation by establishing a slow diffusion limit for a conserved quantity of the deterministic approximation, together with a careful analysis of the behaviour at the boundary, both in the limit as N → ∞, and for large but finite N .

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Stationary distributions of persistent ecological systems

Cited by 11 publications

References 58 publications

Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line

Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line

Heteroclinic cycling and extinction in May–Leonard models with demographic stochasticity

Fixation time of the rock-paper-scissors model: rigorous results in the well-mixed setting

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