Existence Of Non-constant Positive Solutions Research Articles

Spatiotemporal dynamic analysis of an extended water-plant model with power exponent plant growth and nonlocal plant loss

In this paper, we offer a detailed mathematical analysis of an extended water-plant model to describe vegetation pattern formation in arid and semi-arid grazing ecosystems. We present some fundamental analytic properties of the ordinary differential equations and the nonlocal reaction-diffusion equations. The results indicate that when infiltration parameter is equal to one, the unique positive equilibrium is locally stable and where diffusion cannot lead to Turing instability. On the contrary, when infiltration parameter is greater than one, the existence of both positive equilibriums and Turing instability can be observed under further parametric constraints. In addition, we report some characterizations for the non-constant positive steady state solutions, including a priori estimate of the positive solutions and the non-existence and existence of non-constant positive solutions. Some notes on numerical simulation are given, based on different diffusion coefficients, infiltration parameter and rain fall parameter.

Non-constant positive solutions of a general Gause-type predator-prey system with self- and cross-diffusions

In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to the homogeneous Neumann boundary condition. In the system, the cross-diffusions are introduced in such a way that the prey runs away from the predator, while the predator moves away from a large group of preys. Firstly, we establisha prioriestimate for the positive solutions. Secondly, we study the non-existence results of non-constant positive solutions. Finally, we consider the existence of non-constant positive solutions and discuss the Turing instability of the positive constant solution.

Qualitative analysis of a Belousov–Zhabotinskii reaction model

This paper deals with one kind of Belousov–Zhabotinskii reaction model. Linear stability is discussed for the spatially homogeneous problem firstly. Then we focus on the stationary problem with diffusion. Non-existence and existence of non-constant positive solutions are obtained by using implicit function theorem and Leray–Schauder degree theory, respectively.

Effect of predator cannibalism and prey growth on the dynamic behavior for a predator-stage structured population model with diffusion

In this paper, a predator-prey model with predator-stage structured and diffusion is concerned. We deal with the system by endowing it with the homogeneous Neumann boundary conditions. We first give the a priori estimates of positive solutions for the reduced reaction diffusion system. Secondly, we discuss the effects of predator cannibalism and prey growth on the stability of nonnegative constant steady states of the model in detail. Thirdly, we investigate the nonexistence and existence of nonconstant positive solutions. Finally, we discuss the Hopf bifurcation created by diffusion.

On the steady state of a shadow system to the SKT competition model

We study a boundary value problem with an integral constraint that arises from the modelings of species competition proposed by Lou and Ni in [10]. Through local and global bifurcation theories, we obtain the existence of non-constant positive solutions to this problem, which are small perturbations from its positive constant solution, over a one-dimensional domain. Moreover, we investigate the stability of these bifurcating solutions. Finally, for the diffusion rate being sufficiently small, we construct infinitely many positive solutions with single transition layer, which is represented as an approximation of a step function. The transition-layer solution can be used to model the segregation phenomenon through interspecific competition.

The estimates on the energy functional of an elliptic system with Neumann boundary conditions

We consider an elliptic system of the form − ε 2 Δu+u=f(v), − ε 2 Δv+v=g(u) in Ω with Neumann boundary conditions, where Ω is a C 2 domain in R N , f and g are nonlinearities having superlinear and subcritical growth at infinity. We prove the existence of nonconstant positive solutions of the system, and estimate the energy functional on a configuration space H ¯ by a different technique, which is an important step in the proof of the solution’s concentrative property. We conclude that the least energy solutions of the system concentrate at the point of boundary, which maximizes the mean curvature of ∂ Ω.

Stationary pattern of a diffusive prey–predator model with trophic intersections of three levels

The paper is concerned with a diffusive prey–predator model subject to the homogeneous Neumann boundary condition, which models the trophic intersections of three levels. We will prove that under certain assumptions, even though the unique positive constant steady state is globally asymptotically stable for the dynamics with diffusion, the non-constant positive steady state can exist due to the emergence of cross-diffusion. We demonstrate that the cross-diffusion can create stationary pattern. Moreover, we treat the cross-diffusion parameter as a bifurcation parameter and discuss the existence of non-constant positive solutions to the system with cross-diffusion.

Spike-layered solutions for an elliptic system with Neumann boundary conditions

We prove the existence of nonconstant positive solutions for a system of the form−ε2Δu+u=g(v)-\varepsilon ^2\Delta u + u = g(v),−ε2Δv+v=f(u)-\varepsilon ^2\Delta v + v = f(u)inΩ\Omega, with Neumann boundary conditions on∂Ω\partial \Omega, whereΩ\Omegais a smooth bounded domain andff,ggare power-type nonlinearities having superlinear and subcritical growth at infinity. For small values ofε\varepsilon, the corresponding solutionsuεu_{\varepsilon }andvεv_{\varepsilon }admit a unique maximum point which is located at the boundary ofΩ\Omega.