Effect of prey‐taxis and diffusion on positive steady states for a predator‐prey system

Abstract

In this paper, we consider a generalized predator-prey system with prey-taxis under the Neumann boundary condition. We investigate the local and global asymptotical stability of constant steady states (including trivial, semitrivial, and interior constant steady states). On the basis of a priori estimate and the fixed-point index theory, several sufficient conditions for the nonexistence/existence of nonconstant positive solutions are given.

Multi-stable and spatiotemporal staggered patterns in a predator-prey model with predator-taxis and delay

“…When 𝑟 0 < 𝑟 0 , then 𝑎 ū − 𝜛 < 0. According to (9), 𝑇 𝑅 𝑘 < 0 for 𝑘 ∈ ℕ 0 also holds. Whereas, there exists some 𝑘 ∈ ℕ such that 𝐷𝐸𝑇 𝑘 < 0, meaning that Turing bifurcation can occur for system (2).…”

“…20 showed that large prey-taxis tends to stabilize the coexistence equilibrium when 𝑓 1 , 𝑓 2 and 𝑝 correspond to logistic growth rate, constant mortality, and ratio-dependent forms. Moreover, Gao and Guo 9 demonstrated that the local stability of the constant steady state is enhanced by the presence of prey-taxis. Subsequently, Qiu et al 27 showed that prey-taxis can suppress the globally asymptotical stability of the coexistence steady state, and pointed out that due to the effect of prey-taxis, periodic solutions bifurcating from the coexistence steady state via Hopf bifurcation can be spatially inhomogeneous.…”

Multi-stable spatial patterns and spatiotemporal staggered periodic patterns in a predator-taxis model with delay

“…[23] showed that large prey-taxis tends to stabilize the coexistence equilibrium when f 1 , f 2 and p correspond to logistic growth rate, constant mortality, and ratio-dependent forms. Moreover, Gao and Guo [12] demonstrated that the local stability of the constant steady state is enhanced by the presence of prey-taxis. Subsequently, Qiu et al [26] showed that prey-taxis can suppress the globally asymptotical stability of the coexistence steady state, and pointed out that due to the effect of prey-taxis, periodic solutions bifurcating from the coexistence steady state via Hopf bifurcation can be spatially inhomogeneous.…”

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.…”

“…For any initial values x > 0, equation 10is asymptotically stable in distribution. Moreover, there is a unique invariant measure π * for the process (ϕ x (t), r(t)) in equation (10).…”