Successional stability of vector fields in dimension three

Schreiber, Sebastian J.

doi:10.1090/s0002-9939-99-05169-2

Cited by 6 publications

(1 citation statement)

References 34 publications

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“…There has been a significant amount of work dedicated to the classification of the long term behavior of deterministic Lotka-Volterra systems (Bomze 1983, 1995, Zeeman 1993, Hofbauer & So 1994, Takeuchi 1996, Hofbauer & Sigmund 1998. While there is a full classification in dimension two (Bomze 1983(Bomze , 1995, the classification is still incomplete for three dimensions even in the special case of competitive systems (Zeeman 1993, Zeeman & van den Driessche 1998, Hofbauer & So 1994, Schreiber 1999, Xiao & Li 2000, Gyllenberg et al 2006, Gyllenberg & Yan 2009.…”

Section: Introductionmentioning

confidence: 99%

A classification of the dynamics of three-dimensional stochastic ecological systems

Hening¹,

Nguyen²,

Schreiber³

2020

Preprint

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The classification of the long-term behavior of dynamical systems is a fundamental problem in mathematics. For both deterministic and stochastic dynamics specific classes of models verify Palis' conjecture: the long-term behavior is determined by a finite number of stationary distributions. In this paper we consider the classification problem for stochastic models of interacting species. For a large class of three-species, stochastic differential equation models, we prove a variant of Palis' conjecture: the long-term statistical behavior is determined by a finite number of stationary distributions and, generically, three general types of behavior are possible: 1) convergence to a unique stationary distribution that supports all species, 2) convergence to one of a finite number of stationary distributions supporting two or fewer species, 3) convergence to convex combinations of single species, stationary distributions due to a rock-paper-scissors type of dynamic. Moreover, we prove that the classification reduces to computing Lyapunov exponents (external Lyapunov exponents) that correspond to the average per-capita growth rate of species when rare. Our results stand in contrast to the deterministic setting where the classification is incomplete even for three-dimensional, competitive Lotka-Volterra systems. For these SDE models, our results also provide a rigorous foundation for ecology's modern coexistence theory (MCT) which assumes the external Lyapunov exponents determine long-term ecological outcomes.

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Section: Introductionmentioning

confidence: 99%

A classification of the dynamics of three-dimensional stochastic ecological systems

Hening¹,

Nguyen²,

Schreiber³

2020

Preprint

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Criteria for Cr Robust Permanence

Schreiber

2000

Journal of Differential Equations

135

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Let x* i =x i f i (x) (i=1, ..., n) be a C r vector field that generates a dissipative flow , on the positive cone of R n . , is called permanent if the boundary of the positive cone is repelling. , is called C r robustly permanent if , remains permanent for sufficiently small C r perturbations of the vector field. A necessary condition and a sufficient condition for C r robust permanence involving the average per-capita growth rates f i d+ with respect to invariant measures + are derived. The necessary condition requires that inf + max i f i d+>0, where the infimum is taken over ergodic measures with compact support in the boundary of the positive cone. The sufficient condition requires that the boundary flow admit a Morse decomposition [M 1 , ..., M k ] such that every M j satisfies min + max i f i d+>0 where the minimum is taken over invariant measures with support in M j . As applications, we provide a sufficient condition for C r robust permanence of Lotka Volterra models and a topological characterization of C r robust permanence for food chain models. Academic Press

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Kolmogorov Vector Fields with Robustly Permanent Subsystems

Mierczyński¹,

Schreiber²

2002

Journal of Mathematical Analysis and Applications

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Successional stability of vector fields in dimension three

Cited by 6 publications

References 34 publications

A classification of the dynamics of three-dimensional stochastic ecological systems

A classification of the dynamics of three-dimensional stochastic ecological systems

Criteria for Cr Robust Permanence

Kolmogorov Vector Fields with Robustly Permanent Subsystems

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