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

Journal of Mathematical Biology - Tập 86 - Trang 1-28 - 2023
Nicholas W. Barendregt1, Peter J. Thomas2
1Department of Applied Mathematics, University of Colorado Boulder, Boulder, USA
2Department of Mathematics, Applied Mathematics, and Statistics; Department of Biology; Department of Cognative Science; Department of Data and Computer Science; Department of Electrical, Control and Systems Engineering, Case Western Reserve University, Cleveland, USA

Tóm tắt

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.

Tài liệu tham khảo

Anderson DF, Kurtz TG (2015) Stochastic analysis of biochemical systems. Springer

Anderson DF, Ermentrout B, Thomas PJ (2015) Stochastic representations of ion channel kinetics and exact stochastic simulation of neuronal dynamics. J Comput Neurosci 38(1):67–82

Bressloff PC (2014) Stochastic processes in cell biology. Springer

De Candia A, Sarracino A, Apicella I, et al (2021) Critical behaviour of the stochastic Wilson–Cowan model. bioRxiv

Gardiner C (2009) Stochastic methods. Springer, Berlin

Gill JP (2020) Neural correlates of adaptive responses to changing load in feeding Aplysia. PhD thesis, Case Western Reserve University

Hofbauer J, Sigmund K et al (1998) Evolutionary games and population dynamics. Cambridge University Press

Rabinovich M, Huerta R, Laurent G (2008) Trans Dyn Neural Process. Science pp 48–50

Van Kampen NG (1992) Stochastic processes in physics and chemistry. Elsevier

Volterra V (1926) Variazioni e fluttuazioni del numero d’individui in specie animali conviventi