Rui Peng

Positive steady states of the Holling–Tanner prey–predator model with diffusion

This paper is concerned with the Holling–Tanner prey–predator model with diffusion subject to the homogeneous Neumann boundary condition. We obtain the existence and non-existence of positive non-constant steady states.

Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model

Abstract This work is concerned with a Holling–Tanner prey–predator model with diffusion subject to the homogeneous Neumann boundary condition. We will obtain some results for the global stability of the unique positive equilibrium of this model, and thus improve some previous results.

On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law

Understanding of spatial and temporal behaviour of interacting species or reactants in ecological or chemical systems has become a central issue, and rigorously determining the formation of patterns in models from various mechanisms is of particular interest to applied mathematicians. In this paper, we study a bimolecular autocatalytic reaction–diffusion model with saturation law and are mainly concerned with the corresponding steady-state problem subject to the homogeneous Neumann boundary condition. In particular, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain reactant is large or small. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system. Our theoretical analysis shows that the diffusion rate of this reactant and the size of the reactor play decisive roles in leading to the formation of stationary patterns.

STATIONARY PATTERN OF A RATIO-DEPENDENT FOOD CHAIN MODEL WITH DIFFUSION ∗

In the paper, we investigate a three-species food chain model with diffusion and ratio-dependent predation functional response. We mainly focus on the coexistence of the three species. For this coupled reaction-diffusion system, we study the persistent property of the solution, the stability of the constant positive steady state solution, and the existence and nonexistence of nonconstant positive steady state solutions. Both the general stationary pattern and Turing pattern are observed as a result of diffusion. Our results also exhibit some interesting effects of diffusion and functional responses on pattern formation.

Stationary patterns of the Holling-Tanner prey-predator model with diffusion and cross-diffusion

This paper is concerned with the Holling–Tanner prey–predator model subject to the homogeneous Neumann boundary condition. We will show that under certain hypotheses, even though the unique positive constant steady state is uniformly asymptotically stable for the dynamics with diffusion, non-constant positive steady states can exist due to the emergence of cross-diffusion. The biological implication of cross-diffusion means that the prey species exercise a self-defense mechanism to protect themselves from the attack of the predator. Our result also demonstrates that cross-diffusion can create Stationary Patterns.

The periodic logistic equation with spatial and temporal degeneracies

In this article, we study the degenerate periodic logistic equation with homogeneous Neumann boundary conditions: uf8f1 uf8f4uf8f2 uf8f4uf8f3 ∂tu−∆u = au− b(x, t)u in Ω× (0,∞), ∂νu = 0 on ∂Ω× (0,∞), u(x, 0) = u0(x) ≥, 6≡ 0 in Ω, where Ω ⊂ R (N ≥ 2) is a bounded domain with smooth boundary ∂Ω, a and p > 1 are constants. The function b ∈ C(Ω × R) (0 < θ < 1) is T-periodic in t, nonnegative, and vanishes (i.e., has a degeneracy) in some subdomain of Ω×R. We examine the effects of various natural spatial and temporal degeneracies of b(x, t) on the long-time dynamical behavior of the positive solutions. Our analysis leads to a new eigenvalue problem for periodic-parabolic operators over a varying cylinder and certain parabolic boundary blow-up problems not known before. The investigation in this paper shows that the temporal degeneracy causes a fundamental change of the dynamical behavior of the equation only when spatial degeneracy also exists; but in sharp contrast, whether or not temporal degeneracy appears in the equation, the spatial degeneracy always induces fundamental changes of the behavior of the equation, though such changes differ significantly according to whether or not there is temporal degeneracy.

Some nonexistence results for nonconstant stationary solutions to the Gray-Scott model in a bounded domain

Abstract In the present paper, we are concerned with a reaction–diffusion system well-known as the Gray–Scott model in a bounded domain and study the corresponding steady-state problem. We establish some results for the nonexistence of nonconstant positive stationary solutions.

Uniqueness and stability of steady states for a predator-prey model in heterogeneous environment

In this paper, we deal with a predator-prey model with diffusion in a heterogeneous environment, and we study the uniqueness and stability of positive steady states as the diffusion coefficient of the predator is small enough.

Positive steady-state solutions of the Sel'kov model

In the paper, we investigate a reaction-diffusion system known as the Selkov model under the homogeneous Dirichlet boundary condition. First, we give a priori estimates of non-negative steady-state solutions. Then, we obtain some results of the non-existence and existence of positive steady-state solutions.

Positive solutions of a diffusive prey-predator model in a heterogeneous environment

In this paper, we investigate a diffusive prey-predator model in a spatially degenerate heterogeneous environment. We are concerned with the positive solutions of the model, and obtain some results for the existence and non-existence of positive solutions. Moreover, the multiplicity, stability and asymptotical behaviors of positive solutions with respect to the parameters are also studied.