Global asymptotic stability for predator-prey systems whose prey receives time-variation of the environment

Abstract

Abstract.A predator-prey model with prey receiving time-variation of the environment is considered. Such a system is shown to have a unique interior equilibrium that is globally asymptotically stable if the time-variation is bounded and weakly integrally positive. In particular, the result tells us that the equilibrium point can be stabilized even by nonnegative functions that make the limiting system structurally unstable. The method that is used to obtain the result is an analysis of asymptotic behavior of the solutions of an equivalent system to the predator-prey model.

A necessary and sufficient condition for global asymptotic stability of time-varying Lotka–Volterra predator–prey systems

“…For example, 1/(1 + t) ∈ F [WIP] and sin 2 t/(1 + t) ∈ F [WIP] (for the proof, see [15, Proposition 2.1]). Sugie et al [19] obtained the following result (see also [6,10,18]). …”

“…To show that the interior equilibrium (c/d, a/b) of (E) is stable (to be precise, uniformly stable), it is enough only to assume that the function h is nonnegative (for the proof, see [19,Proposition 2]). To show that the interior equilibrium (c/d, a/b) of (E) is stable (to be precise, uniformly stable), it is enough only to assume that the function h is nonnegative (for the proof, see [19,Proposition 2]).…”

An eco-epidemiological model with general functional response of predator to prey

“…We now consider an example where the uninfected model is nonautonomous. For this model we will be able to obtain explicit thresholds based on the study of the underlying susceptible prey/predator subsystem in [16]. Assuming that g(t, S) = (p + qh(t) − dh(t)S)S with h(t) continuous and satisfying h ℓ < h(t) < h u for some constants h ℓ , h u > 0, f (S, P ) = S, a(t) = b, h(t, P ) = −q and γ(t) = d/b in (1), we obtain the following particular model:…”

“…To illustrate our findings, in section 2, several predator-prey models available in the literature, satisfying our assumptions, are considered and thresholds conditions for the corresponding eco-epidemiological model automatically obtained from our results: in our Example 1, we consider the situation where f ≡ 0 in system (1), corresponding to a generalized version of the situation studied in [12]; in Example 2, we obtain a particular form for the threshold conditions in the context of periodic models and particularize our result for a model constructed from the predator-prey model in [5]; in Example 3, we start with an uninfected subsystem with Gausetype interaction (a predator-prey model with Holling type II functional response of predator to prey, logistic growth of prey in the absence of predators and exponential extinction of predator in the absence of prey) and, using [10], obtain the corresponding results for the eco-epidemiological model; in Example 4, we consider the eco-epidemiological model obtained from an uninfected subsystem with ratiodependent functional response of predator to prey, a type interaction considered as an attempt to overcome some know biological paradoxes observed in models with Gause-type interaction and again obtain the corresponding results for the ecoepidemiological model, based on the discussion of ratio-dependent predator-prey systems in [7]; finally, in Examples 5 and 6, we consider eco-epidemiological models, based on the discussion of the corresponding predator-prey models in [14,16] where the uninfected subsystem has some specific type of non-autonomy in the prey equation (Example 5) or the predator equation (Example 6). For all these examples we present some simulation that corroborate our conclusions.…”

Razumikhin method to global exponential stability for coupled neutral stochastic delayed systems on networks

“…On the basis of the Lyapunov functional method, lots of results on the stability have been reported in the literature. Here, we only refer the reader to . However, another method commonly used to investigate stability is the Razumikhin–Lyapunov function method that allows one to utilize simple functions instead of functionals.…”