Non-constant Positive Steady-state Solutions Research Articles

Asymptotic behavior of solutions for a Lotka–Volterra mutualism reaction–diffusion system with time delays

This paper is to investigate the asymptotic behavior of solutions for a time-delayed Lotka–Volterra N -species mutualism reaction–diffusion system with homogeneous Neumann boundary condition. It is shown, under a simple condition on the reaction rates, that the system has a unique bounded time-dependent solution and a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the constant positive steady-state solution as time tends to infinity. This convergence result implies that the trivial steady-state solution and all forms of semitrivial steady-state solutions are unstable, and moreover, the system has no nonconstant positive steady-state solution. A condition ensuring the convergence of the time-dependent solution to one of nonnegative semitrivial steady-state solutions is also given.

Non-existence of non-constant positive steady states of two Holling type-II predator–prey systems: Strong interaction case

We prove the non-existence of non-constant positive steady state solutions of two reaction–diffusion predator–prey models with Holling type-II functional response when the interaction between the predator and the prey is strong. The result implies that the global bifurcating branches of steady state solutions are bounded loops.

Nonconstant positive steady states for a ratio-dependent predator–prey system with cross-diffusion

We have investigated a ratio-dependent predator–prey system with diffusion in [X.Z. Zeng, A ratio-dependent predator–prey system with diffusion, Nonlinear Anal. RWA 8 (6) (2007) 1062–1078] and obtained that the system with diffusion can admit nonconstant positive steady-state solutions when a 0 ( b ) < a < m 1 , whereas for a > m 1 , the system with diffusion has no nonconstant positive steady-state solution. In the present paper, we continue to investigate a ratio-dependent predator–prey system with cross-diffusion for a > m 1 , where the cross-diffusion represents that the predator moves away from a large group of prey. We obtain that there exist positive constants D 1 0 and D 3 0 such that for max { m 1 − m 2 2 , 0 } < b < 2 m 1 , m 1 < a < a 2 ( b ) , d 1 < D 1 0 and d 3 > D 3 0 , the system with cross-diffusion admits nonconstant positive steady-state solutions for some ( d 1 , d 2 , d 3 ) ; whereas, for b ≥ 2 m 1 or a ≥ a 2 ( b ) or d 1 ≥ D 1 0 or d 3 ≤ D 3 0 , the system with cross-diffusion still has no nonconstant positive steady-state solution. Our results show that this kind of cross-diffusion is helpful to create nonconstant positive steady-state solutions for the predator–prey system.

A qualitative study on general Gause-type predator–prey models with non-monotonic functional response

In this paper, we study a diffusive predator–prey model with general growth rates and non-monotonic functional response under homogeneous Neumann boundary condition. A local existence of periodic solutions and the asymptotic behavior of spatially inhomogeneous solutions are investigated. Moreover, we show the existence and non-existence of non-constant positive steady-state solutions. Especially, to show the existence of non-constant positive steady-states, the fixed point index theory is used without estimating the lower bounds of positive solutions. More precisely, calculating the indexes at the trivial, semi-trivial and positive constant solutions, some sufficient conditions for the existence of non-constant positive steady-state solutions are studied. This is in contrast to the works in previous papers. Furthermore, on obtaining these results, we can observe that the monotonicity of a prey isocline at the positive constant solution plays an important role.

A qualitative study on general Gause-type predator–prey models with constant diffusion rates

In this paper, we study the qualitative behavior of non-constant positive solutions on a general Gause-type predator–prey model with constant diffusion rates under homogeneous Neumann boundary condition. We show the existence and non-existence of non-constant positive steady-state solutions by the effects of the induced diffusion rates. In addition, we investigate the asymptotic behavior of spatially inhomogeneous solutions, local existence of periodic solutions, and diffusion-driven instability in some eigenmode.

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. (An erratum to this article has been appended at the end of the pdf file.)

Qualitative analysis of a predator–prey model with Holling type II functional response incorporating a prey refuge

We study a predator–prey model with Holling type II functional response incorporating a prey refuge under homogeneous Neumann boundary condition. We show the existence and non-existence of non-constant positive steady-state solutions depending on the constant m ∈ ( 0 , 1 ] , which provides a condition for protecting ( 1 − m ) u of prey u from predation. Moreover, we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions.

A ratio-dependent predator–prey model with diffusion

This paper deals with the existence and nonexistence of nonconstant positive steady-state solutions to a ratio-dependent predator–prey model with diffusion and with the homogeneous Neumann boundary condition. We demonstrate that there exists a 0 ( b ) satisfying 0 < a 0 ( b ) < m 1 for 0 < b < m 1 , such that if 0 < b < m 1 and a 0 ( b ) < a < m 1 , then the diffusion can create nonconstant positive steady-state solutions; whereas the diffusion cannot do provided a > m 1 .

Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion

This paper deals with non-constant positive steady-state solutions of a predator-prey system with non-monotonic functional response, also called Holling type-IV interaction terms, and diffusion under the homogeneous Neumann boundary condition. We first establish positive upper and lower bounds for such solutions, and then study their non-existence, global existence and bifurcation.

Positive steady-state solutions of the Noyes–Field model for Belousov–Zhabotinskii reaction

In the paper, we investigate the Noyes–Field model for Belousov–Zhabotinskii reaction and study positive steady-state solutions of this model with the homogeneous Neumann boundary condition. We obtain the existence and non-existence of non-constant positive steady-state solutions.