A method to deal with the critical case in stochastic population dynamics

Dang Nguyen (1) , Edouard Strickler (2, 3)

1 UA - University of Alabama [Tuscaloosa]
2 IECL - Institut Élie Cartan de Lorraine
3 TOSCA - TO Simulate and CAlibrate stochastic models

In numerous papers, the behaviour of stochastic population models is investigated through the sign of a real quantity which is the growth rate of the population near the extinction set. In many cases, it is proven that when this growth rate is positive, the process is persistent in the long run, while if it is negative, the process converges to extinction. However, the critical case when the growth rate is null is rarely treated. The aim of this paper is to provide a method that can be applied in many situations to prove that in the critical case, the process con-gerves in temporal average to extinction. A number of applications are given, for Stochastic Dierential Equations and Piecewise Deterministic Markov Processes modelling prey-predator, epidemilogical or structured population dynamics.

Mots clés: Stochastic Environment; SEIR; SIRS; SIS; Epidemiology; Stochastic Differential Equation; Piecewise deterministic Markov processes; Rosenszweig- McArthur; Lyapunov Exponents; Stochastic Persistence

Domaines: Probabilités [math.PR]

Dang Nguyen, Edouard Strickler. A method to deal with the critical case in stochastic population dynamics. SIAM Journal on Applied Mathematics, 2020, 80 (3), pp.1567-1589. ⟨10.1137/20M131134X⟩. ⟨hal-02427636v2⟩