Positive Constant Equilibrium Research Articles

Bifurcation analysis of a diffusive ratio-dependent predator–prey model

In this paper, a ratio-dependent predator–prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction.

Cross-diffusion induced stationary patterns in a prey–predator system with parental care for predators

This paper is concerned about a strongly-coupled nonlinear reaction–diffusion system, which describes a prey–predator model with parental care for predators. The long time behaviors of the solution are discussed, stability and instability of the positive constant equilibrium are studied. Our results show that even though the unique positive constant steady-state is stable for the kinetic system and for the self-diffusion reaction system, cross-diffusion can generate the stationary patterns (nonconstant positive steady states). The numerical simulation is also given to illustrate the theoretical result.

Dynamics of a diffusive predator–prey model with modified Leslie–Gower schemes and additive Allee effect

In this paper, we consider a diffusive predator–prey model with modified Leslie–Gower schemes and additive Allee effect on prey under homogeneous Neumann boundary condition. Firstly, we investigate the qualitative properties of the system including the persistent property, and local and global asymptotical stability of the unique positive constant equilibrium point of the system. Next, we also show the existence and nonexistence of nonconstant positive steady states of the reaction–diffusion system, which reflect the effect of large diffusivity.

Effects of Delay and Diffusion on the Dynamics of a Leslie–Gower Type Predator–Prey Model

In this paper, we consider the effects of time delay and space diffusion on the dynamics of a Leslie–Gower type predator–prey system. It is shown that under homogeneous Neumann boundary condition the occurrence of space diffusion does not affect the stability of the positive constant equilibrium of the system. However, we find that the incorporation of a discrete delay representing the gestation of prey species can not only destabilize the positive constant equilibrium of the system but can also cause a Hopf bifurcation at the positive constant equilibrium as it crosses some critical values. In particular, we prove that these Hopf bifurcations' periodic solutions are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as periodic solutions of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system will generate spatially nonhomogeneous periodic solutions. The results in this work demonstrate that diffusion plays an important role in deriving complex spatiotemporal dynamics.

Stability and bifurcation in a ratio-dependent Holling-III system with diffusion and delay

A diffusive ratio-dependent predator-prey system with Holling-III functional response and delay effects is considered. Global stability of the boundary equilibrium and the stability of the unique positive steady state and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated in detail, by the maximum principle and the characteristic equations. Ratio-dependent functional response exhibits rich spatiotemporal patterns. It is found that, the system without delay is dissipative and uniformly permanent under certain conditions, the delay can destabilize the positive constant equilibrium and spatial Hopf bifurcations occur as the delay crosses through some critical values. Then, the direction and the stability of Hopf bifurcations are determined by applying the center manifold reduction and the normal form theory for partial functional differential equations. Some numerical simulations are carried out to illustrate the theoretical results.

Stationary patterns in a two-cell coupled isothermal chemical system with arbitrary powers of autocatalysis

Stationary patterns of a two-cell coupled isothermal chemical system with arbitrary powers of autocatalysis are considered. Firstly, we obtain the stability of the unique positive constant equilibrium solution for the system. Then, based on a priori estimates, non-existence and existence of nontrivial steady state solutions are shown by using implicit function theorem and topological degree theory, respectively. The effects of autocatalysis order and diffusion coefficients to the pattern formation are discussed.

Stability and traveling waves of a stage-structured predator–prey model with Holling type-II functional response and harvesting

In this paper, we consider a reaction–diffusion predator–prey model with stage-structure, Holling type-II functional response, nonlocal spatial impact and harvesting. The stability of the equilibria is investigated. Furthermore, by the cross-iteration scheme companied with a pair of admissible upper and lower solutions and Schauder fixed point theorem, we deduce the existence of traveling wave solution which connects the zero solution and the positive constant equilibrium.

Global stability of the positive equilibrium for a non-cooperative model of nuclear reactors

In this paper, we investigate the non-cooperative reaction-diffusion model of nuclear reactors subject to the homogeneous Neumann boundary condition. By establishing appropriate Lyapunov functions, we prove the global stability of the unique positive constant equilibrium

Coexistence of a diffusive predator–prey model with Holling type-II functional response and density dependent mortality

In this paper, we consider a two competitor–one prey diffusive model in which both competitors exhibit Holling type-II functional response and one of the competitors exhibits density dependent mortality rate. First, we study the local and global existence of strong solution by using the C 0 analytic semigroup. Then, we consider the local and global stability of the positive constant equilibrium by using the linearization method and Laypunov functional method, respectively. Furthermore, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain species is small or large. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system.

Multiple bifurcations and spatiotemporal patterns for a coupled two-cell Brusselator model

A coupled two-cell Brusselator model with diffusion effect subject to Neumann boundary condition is considered. Hopf bifurcations and global steady state bifurcations which bifurcate from the unique positive constant equilibrium point are investigated in detail. Meanwhile, Turing instability occurs when diffusion is present. Particularly, we show the existence of spatially inhomogeneous periodic solutions and non-constant steady state solutions, which exhibit rich spatiotemporal patterns in this coupled Brusselator system. Some numerical simulations are presented to illustrate the theoretical results obtained.

Positive steady states of a diffusive predator–prey system with modified Holling–Tanner functional response

In this paper, we study a diffusive predator–prey system with modified Holling–Tanner functional response under homogeneous Neumann boundary condition. The qualitative properties, including the global attractor, persistence property, local and global asymptotic stability of the unique positive constant equilibrium are obtained. We also establish the existence and nonexistence of nonconstant positive steady states of this reaction–diffusion system, which indicates the effect of large diffusivity.

Traveling wavefronts for a two-species ratio-dependent predator–prey system with diffusion terms and stage structure

In this paper, we considered an important nonlinear reaction-diffusion equations describing a two-species ratio-dependent predator–prey system with diffusion terms and stage structure. By using the linearized method, we investigated the locally asymptotical stability of the nonnegative equilibria of the above mentioned system and obtained the locally stable conditions. And by combining the approach introduced by J. Canosa (see [J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Deve. 17 (1973) 307–313]) with the method of upper and lower solutions, we proved that the traveling wavefronts which connect the zero solution with the positive constant equilibrium of the system exist and appear to be monotone. Finally, we gave a conclusion to summarize the achievements of the work.

Qualitative analysis for a ratio-dependent predator–prey model with stage structure and diffusion

This paper is purported to study a reaction diffusion system arising from a ratio-dependent predator–prey model with stage structure. The global asymptotic stability of the unique positive constant equilibrium is obtained under suitable conditions; existence and non-existence results of non-trivial solution are derived.

Global asymptotic stability of a ratio-dependent predator–prey system with diffusion

This paper is concerned with a predator–prey system with diffusion. The global asymptotic stability of the unique positive constant equilibrium is obtained under certain conditions. The method used here is the upper and lower solutions combined with the monotone iteration and constructing suitable Lyapunov functions.