Abstract
In this paper, we consider the dynamics of a stochastic Gilpin–Ayala model with regime switching and impulsive perturbations. The Gilpin–Ayala parameter is also allowed to switch. Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are provided. The critical number among the extinction, nonpersistence in the mean, and weak persistence is obtained. Our results demonstrate that the dynamics of the model have close relations with the impulses and the Markov switching.
Highlights
Population ecology is an important branch of ecology; people often use some basic models in mathematical ecology and the mathematical analysis methods to study the living environment conditions of organisms
In [22], they presented a stochastic Gilpin–Ayala model under regime switching on patches, studied the effect of environmental noises in the asymptotic properties of the model on patches under regime switching, and established some sufficient conditions for extinction and persistence of species
5 Conclusions and further remarks This paper is concerned with the dynamics of a stochastic Gilpin–Ayala model with regime switching and impulsive perturbations
Summary
Population ecology is an important branch of ecology; people often use some basic models in mathematical ecology and the mathematical analysis methods to study the living environment conditions of organisms. Liu et al [17, 20] considered the following stochastic Gilpin–Ayala model under regime switching: dx(t) = x(t) r γ (t) – a γ (t) xλ(t) dt + σ1 γ (t) x(t) dB1(t) + σ2 γ (t) x1+θ (t) dB2(t). They studied the existence of global positive solution, extinction, nonpersistence, and weak persistence of the species, and obtained the stochastic permanence of the species under the conditions that 0 < θ ≤ 1 and 0 < λ ≤ 1+θ .
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