Abstract
This paper investigates a stochastic two-patch predator-prey model with ratio-dependent functional responses. First, the existence of a unique global positive solution is proved via the stochastic comparison theorem. Then, two different methods are used to discuss the long-time properties of the solutions pathwise. Next, sufficient conditions for extinction and persistence in mean are obtained. Moreover, stochastic persistence of the model is discussed. Furthermore, sufficient conditions for the existence of an ergodic stationary distribution are derived by a suitable Lyapunov function. Next, we apply the main results in some special models. Finally, some numerical simulations are introduced to support the main results obtained. The results in this paper generalize and improve the previous related results.
Highlights
Xi denotes the density of prey in patch i (i 1, 2), and y represents the density of predators. v (0 ≤ v ≤ 1) is the proportion of time that predators stay in patch 1 on average; ri (i 1, 2) is the intrinsic growth rate of prey in patch i; ai is the intraspecific competition coefficient of the prey in patch i; si is the attacking rate of the predators in patch i; ei is the expected biomass of the prey converted to predators in patch i; mi is the per capita mortality rate of predators in patch i; and hi is the handling time of the predation in patch i, respectively
We first apply the main results to two stochastic two-species predator-prey models. en, we present the application of the main results to stochastic two-patch predator-prey model (4)
By using the comparison theorem of stochastic differential equations, we show that the model has a unique global positive solution. en, the long-time properties of the solutions are discussed pathwise
Summary
We first show that model (5) has a unique positive global solution by the stochastic comparison theorem. en, we discuss the long-time properties of the solutions pathwise. We first show that model (5) has a unique positive global solution by the stochastic comparison theorem. ∞ a.s. We consider the following two stochastic differential systems:. Us, from the stochastic comparison theorem (see eorem 3.1 in [18]), it follows that 0 < φi(t) ≤ xi(t) ≤ Φi(t) (i 1, 2) and 0 < ψ(t) ≤ y(t) ≤ Ψ(t) almost surely for t ∈ [ 0, τe ). For any (x10, x20, y0) ∈ R3+, let (x1(t), x2(t), y(t)) be the solution of model (5) with initial value (x10, x20, y0). For i 1, 2, if λi − (σ2i /2) > 0, lim In xi(t) 0, t⟶∞ t
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