Non-constant Positive Steady-state Solutions Research Articles

Positive steady states of a density-dependent predator-prey model with diffusion

In this paper, we investigate the rich dynamics of a diffusive Holling type-Ⅱ predator-prey model with density-dependent death rate for the predator under homogeneous Neumann boundary condition. The value of this study lies in two-aspects. Mathematically, we show the stability of the constant positive steady state solution, the existence and nonexistence, the local and global structure of nonconstant positive steady state solutions. And biologically, we find that Turing instability is induced by the density-dependent death rate, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion.

Dynamics and pattern formation in a modified Leslie–Gower model with Allee effect and Bazykin functional response

In this paper, we study the dynamics of a diffusive modified Leslie–Gower model with the multiplicative Allee effect and Bazykin functional response. We give detailed study on the stability of equilibria. Non-existence of non-constant positive steady state solutions are shown to identify the rage of parameters of spatial pattern formation. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns.

Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics

In this paper, a reaction-diffusion system known as an autocatalytic reaction model is considered. The model is characterized by a system of two differential equations which describe a type of complex biochemical reaction. Firstly, some basic characterizations of steady-state solutions of the model are presented. And then, the stability of positive constant steady-state solution and the non-existence, existence of non-constant positive steady-state solutions are discussed. Meanwhile, the bifurcation solution which emanates from positive constant steady-state is investigated, and the global analysis to the system is given in one dimensional case. Finally, a few numerical examples are provided to illustrate some corresponding analytic results.

Spatiotemporal Patterns of a Predator–Prey System with an Allee Effect and Holling Type III Functional Response

A diffusive Gause type predator–prey system with Allee effect in prey growth and Holling type III response subject to Neumann boundary conditions is investigated. Existence of nonconstant positive steady state solutions is proved by Leray–Schauder degree theory and bifurcation theory. Global stability of the positive equilibrium of the system is also investigated. Moreover, bifurcations of spatially homogeneous and nonhomogeneous periodic solutions are analyzed. Our rigorous results justify some recent ecological observations.

Analysis of a nutrient-phytoplankton model in the presence of viral infection

In this paper, a system of reaction-diffusion equations arising in a nutrient-phytoplankton populations is investigated. The equations model a situation in which phytoplankton population is divided into two groups, namely susceptible phytoplankton and infected phytoplankton. A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given. If the diffusion coefficient of the infected phytoplankton is treated as bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.

The global stability and pattern formations of a predator–prey system with consuming resource

The paper deals with a model that describes a predator–prey system with a common consuming resource. We use Lyapunov functions to prove the global stability of the kinetic system and the diffusive system. The existence of non-constant positive steady state solutions is shown to identify the range of parameters of spatial pattern formation for the cross-diffusion system.

Global bifurcation analysis and pattern formation in homogeneous diffusive predator–prey systems

The dynamics of a general diffusive predator–prey system is considered. Existence and nonexistence of non-constant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation. Bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as non-constant steady state solutions are studied.

Numerical simulation and qualitative analysis for a predator–prey model with B–D functional response

We consider a predator–prey model with Beddington–DeAngelis functional response subject to the homogeneous Neumann boundary condition. We study the local and global stability of the unique positive constant solution, the nonexistence of nonconstant positive solution. By bifurcation theorem we obtain one sufficient condition of the existence of nonconstant positive steady state solution in one dimensional case. Furthermore, in numerical simulation, we design a path to analyze and complement our theoretical results.

Global asymptotic behavior in a Lotka–Volterra competition system with spatio-temporal delays

This paper is concerned with a Lotka–Volterra competition system with spatio-temporal delays. By using the linearization method, we show the local asymptotic behavior of the nonnegative steady-state solutions. Especially, the global asymptotic stability of the positive steady-state solution is investigated by the method of upper and lower solutions. The result of global asymptotic stability implies that the system has no nonconstant positive steady-state solution.

Spatiotemporal pattern formation of a diffusive bimolecular model with autocatalysis and saturation law

In this paper, a reaction–diffusion bimolecular model with autocatalysis and saturation law is investigated. Firstly, we provide some conditions for the stability/Turing instability of the constant positive solution. Then we mainly consider Hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive solution of the system. Our results suggest the existence of spatially non-homogeneous periodic orbit and non-constant positive steady state solutions, which implies the possibility of rich spatiotemporal patterns in this diffusive bimolecular model. Numerical examples are presented to support our theoretical analysis. Furthermore, non-existence of non-constant steady state is investigated in terms of parameters.

Turing pattern formation in a three species model with generalist predator and cross-diffusion

In a natural ecosystem, specialist predators feed almost exclusively on one species of prey. But generalist predators feed on many types of species. Consequently, their dynamics is not coupled to the dynamics of a specific prey population. However, the defense of prey formed by congregating made the predator tend to move in the direction of lower concentration of prey species. This is described by cross-diffusion in a generalist predator–prey model. First, the positive equilibrium solution is globally asymptotically stable for the ODE system and for the reaction–diffusion system without cross-diffusion, respectively, hence it does not belong to the classical Turing instability scheme. But it becomes linearly unstable only when cross-diffusion also plays a role. This implies that cross–diffusion can lead to the occurrence and disappearance of the instability. Our results exhibit some interesting combining effects of cross-diffusion, predations and intra-species interactions. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions.

Qualitative analysis of a Hölling–Tanner predator–prey model with mutual interference

A reaction-diffusion system arising from the Hölling–Tanner predator–prey model is proposed and mutual interference among predators is considered. Conditions for the local and global asymptotical stability of equilibrium solutions are given by means of analyzing eigenvalue spectrum as well as Lyapunov functionals. Some non-existence results of non-constant positive steady-state solutions are obtained. Also, the numerical simulation is given to verify the theoretical results.

Global Stability of a Reaction-diffusion System of a Competitor-Competitor-Mutualist Model

In this paper, we study a reaction-diffusion system of a competitorcompetitor-mutualist model with Neumann boundary condition. Using iteration method, we investigate the global asymptotic stability of the unique positive constant steady-state solution under some assumptions. We also give some sufficient conditions under which there are no nonconstant positive steady-state solution exist.

Stationary patterns and stability in a tumor-immune interaction model with immunotherapy

This paper examines a diffusive tumor-immune system with immunotherapy under homogeneous Neumann boundary conditions. We first investigate the large-time behavior of nonnegative equilibria and then explore the persistence of solutions to the time-dependent system. In particular, we present the sufficient conditions for tumor-free states. We also determine whether nonconstant positive steady-state solutions (i.e., a stationary pattern) exist for this coupled reaction–diffusion system when the parameter of the immunotherapy effect is small. The results indicate that this stationary pattern is driven by diffusion effects. For this study, we employ the comparison principle for parabolic systems and the Leray–Schauder degree.

Dynamics and pattern formation in a diffusive predator–prey system with strong Allee effect in prey

The dynamics of a reaction–diffusion predator–prey system with strong Allee effect in the prey population is considered. Nonexistence of nonconstant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation. Bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied. These results show that the impact of the Allee effect essentially increases the system spatiotemporal complexity.

Analysis of a prey–predator model with disease in prey

In this paper, a system of reaction–diffusion equations arising in eco-epidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the predator is treated as bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.

Turing patterns of a strongly coupled predator–prey system with diffusion effects

The main purpose of this work is to investigate the effects of cross-diffusion in a strongly coupled predator–prey system. By a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, it is shown that Turing instability of the reaction–diffusion system can disappear due to the presence of the cross-diffusion, which implies that the cross-diffusion induced stability can be regarded as the cross-stability of the corresponding reaction–diffusion system. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions. These results exhibit interesting and very different roles of the cross-diffusion in the formation and the disappearance of the Turing instability.

MULTIPLE BIFURCATION ANALYSIS AND SPATIOTEMPORAL PATTERNS IN A 1-D GIERER–MEINHARDT MODEL OF MORPHOGENESIS

A reaction–diffusion Gierer–Meinhardt model of morphogenesis subject to Dirichlet fixed boundary condition in the one-dimensional spatial domain is considered. We perform a detailed Hopf bifurcation analysis and steady state bifurcation analysis to the system. Our results suggest the existence of spatially nonhomogenous periodic orbits and nonconstant positive steady state solutions, which imply the possibility of complex spatiotemporal patterns of the system. Numerical simulations are carried out to support our theoretical analysis.

Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model

In this paper, we have investigated a homogeneous reaction–diffusion bimolecular model with autocatalysis and saturation law subject to Neumann boundary conditions. We mainly consider Hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive equilibrium solution of the system. Our results suggest the existence of spatially non-homogeneous periodic orbits and non-constant positive steady state solutions, which implies the possibility of rich spatiotemporal patterns in this diffusive biomolecular system. Numerical examples are presented to support our theoretical analysis.

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.