Dynamics in a diffusive predator–prey system with ratio-dependent predator influence

Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment

“…Many researchers have paid more attention to the effect of interspecies interaction on the dynamic behaviors of reaction-diffusion population models in which it is assumed that the environment is spatially homogeneous, that is, all the coefficients are constant [12,13,14,15,23,28,30,37,38]. It has been observed in many scientific experiments that spatial heterogeneity has a profound effect on ecosystems.…”

Bifurcation and stability of a two-species diffusive Lotka-Volterra model

“…For this purpose, define a nonlinear mapping F : Clearly, the Crandall-Rabinowitz bifurcation theorem [3,4] does not work here. Hence, we shall resort to Lyapunov-Schmidt reduction method [12,13,26,14,28,44,50,51]. Firstly, we define the operator P by P U = u, ϕ 1 Φ + v, ψ 1 Ψ for U = (u, v) T ∈ X and decompose X as X = X 1 ⊕X 2 with X 1 = P X and X 2 = (I−P )X.…”

Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps

“…Throughout this paper, we always assume that lim S→0 g(S, I)/S exists for all I ≥ 0, g (S, 0) lim I→0 g(S, I)/I > 0 for all S > 0, and g(S, I)/I is monotone nonincreasing with respect to I ∈ (0, ∞) for S ∈ (0, ∞). Obviously, the function g includes some special incidence rates [10,11,15,17,25,26,40,41], such as bilinear incidence rate g(S, I) = SI, saturation incidence rate g(S, I) = SI/(1 + mI) with a positive constant m denoting the half-saturation constant, Holling type II incidence rate g(S, I) = SI/(1 + mS), Beddington-DeAngelis incidence rate g(S, I) = SI/(aS + bI + c) with positive constants a, b, and c, and some other kinds of incidence rates like g(S, I) = e −mI SI and g(S, I) = SI/(1 + mI θ ) with positive constants m and θ (see, for example [8,30,36]). In [8], model (1) with bilinear incidence rate always has a globally asymptotically stable disease-free equilibrium E 0 and so the disease disappears if the basic reproduction number R 0 = Λβ µ(µ+α+λ) ≤ 1.…”