Analysis of a diffusive Leslie–Gower predator–prey model with nonmonotonic functional response

In this paper, a diffusive Leslie–Gower predator–prey system with nonmonotonic functional respond is studied. We obtain the persistence of this model and show the local asymptotic stability of positive constant equilibrium by linearized analysis and the global stability by constructing Liapunov function. Besides, Turing instability of this equilibrium is obtained. The existence and nonexistence of positive nonconstant steady states of this model are established. Furthermore, by numerical simulations we illustrate the patterns of prey and predator.     

A bioeconomic model of a ratio-dependent predator-prey system and optimal harvesting

This paper deals with the problem of a ratio-dependent prey-predator model with combined harvesting. The existence of steady states and their stability are studied using eigenvalue analysis. Boundedness of the exploited system is examined. We derive conditions for persistence and global stability of the system. The possibility of existence of bionomic equilibria has been considered. The problem of optimal harvest policy is then solved by using Pontryagin’s maximal principle.     

A competition un-stirred chemostat model with virus in an aquatic system

A reaction–diffusion system of two bacteria species competing a single limiting nutrient with the consideration of virus infection is derived and analysed. Firstly, the well-posedness of the system, the existence of the trivial and semi-trivial steady states, and some prior estimations of the steady states are given. Secondly, a single species subsystem with virus is studied. The stability of the trivial and semi-trivial steady states and the uniform persistence of the subsystem are obtained. Further, taking the infective ability of virus as a bifurcation parameter, the global structure of the positive steady states and the effect of virus on the positive steady states are established via bifurcation theory and limiting arguments. It shows that the backward bifurcation may occur. Some sufficient conditions for the existence, uniqueness and stability of the positive steady state are also obtained. Finally, some sufficient conditions on the existence of the positive steady states for the full system are derived by using the fixed point index theory. Some results on persistence or extinction for the full system are also obtained.     

Dynamics of a spatiotemporal model on populations in a polluted river

In this paper, we investigate the dynamics for a reaction–diffusion–advection system which models populations in a polluted river. More precisely, we study the stability of steady states, which yields sufficient conditions that lead to population persistence or extinction. Furthermore, some dependence of the stability of the toxicant-only steady state and the population-toxicant coexistence steady state on the model parameters are given.     

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.     

Global stability and travelling waves of a predator-prey model with diffusion and nonlocal maturation delay

In this paper, a diffusive predator-prey system with nonlocal maturation delay is investigated. By analyzing the corresponding characteristic equations, the local stability of each of uniform steady states of the system is discussed. Sufficient conditions are derived for the global stability of the positive steady state and the semi-trivial steady state of the system by using the method of upper–lower solutions and its associated monotone iteration scheme, respectively. The existence of travelling wave solution of the system is established by using the geometric singular perturbation theory. Numerical simulations are carried out to illustrate the theoretical results.     

A non-standard finite difference scheme for a diffusive HBV infection model with capsids and time delay

In this paper, a non-standard finite difference (NSFD) scheme for a delayed diffusive hepatitis B virus (HBV) infection model with intracellular HBV DNA-containing capsids is proposed. Dynamic consistency of this NSFD scheme is achieved by showing that the scheme preserves the non-negativity and boundedness of the solutions and the global stability of the homogeneous steady states of the corresponding continuous model without any restriction on spatial and temporal grid sizes. We prove the global stability of the steady states by constructing suitable discrete Lyapunov functions.     

Stability and traveling waves of an epidemic model with relapse and spatial diffusion

An epidemic model with relapse and spatial diffusion is studied. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. By using the linearized method, the local stability of each of feasible steady states to this model is investigated. It is proven that if the basic reproduction number is less than unity, the disease-free steady state is locally asymptotically stable; and if the basic reproduction number is greater than unity, the endemic steady state is locally asymptotically stable. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder's fixed point theorem, the existence of a traveling wave solution which connects the two steady states is established. Furthermore, numerical simulations are carried out to complement the main results.     

Global boundedness and stability of solutions for prey-taxis model with handling and searching predators

In this paper, we show the global boundedness and stability of solutions for prey-taxis model with handling and searching predators in a two-dimensional bounded domain with smooth boundary. First, entropy-like equations and boundedness criteria are derived, and it is proved that the system has a unique uniformly bounded global classical solution. In addition, we show that prey-only steady state is globally asymptotically stable if the predator is weak. The convergence rate of solutions to the steady states is derived in the paper.     

Dynamics and steady-state analysis of an unstirred chemostat model with internal storage and toxin mortality

This study proposes and analyzes a reaction–diffusion system describing the competition of two species for a single limiting nutrient that is stored internally in an unstirred chemostat, in which each species also produces a toxin that increases the mortality of its competitors. The possibility of coexistence and bistability for the model system is studied by the theory of uniform persistence and topological degree theory in cones, respectively. More precisely, the sharp a priori estimates for nonnegative solutions of the system are first established, which assure that all of nonnegative solutions belong to a special cone. Then it turns out that coexistence and bistability can be determined by the sign of the principal eigenvalues associated with specific nonlinear eigenvalue problems in the special positive cones. The local stability of two semi-trivial steady states cannot be studied via the technique of linearization since a singularity arises from the linearization around those steady states. Instead, we introduce a 1-homogeneous operator to rigorously investigate their local stability.     

一类具有阶段结构的捕食-被捕食模型的全局稳定性（英文）

A Holling type III predator-prey model with stage structure for prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. By using the uniformly persistence theory, the system is proven to be permanent if the coexistence equilibrium exists. By using Lyapunov functionals and La Salle's invariance principle, it is shown that the two boundary equilibria is globally asymptotically stable when the coexistence equilibrium is not feasible. By using compound matrix theory, the sufficient conditions are obtained for the global stability of the coexistence equilibrium. At last, numerical simulations are carried out to illustrate the main results.     

A note on a diffusive predator-prey model and its steady-state system

In this note, a diffusive predator-prey model subject to the homogeneous Neumann bound- ary condition is investigated and some qualitative analysis of solutions to this reaction-diffusion system and its corresponding steady-state problem is presented. In particular, by use of a Lyapunov function, the global stability of the constant positive steady state is discussed. For the associated steady state problem, a priori estimates for positive steady states are derived and some non-existence results for non-constant positive steady states are also established when one of the diffusion rates is large enough. Consequently, our results extend and complement the existing ones on this model.     

Global bifurcation and stability of steady states for a bacterial colony model with density-suppressed motility

We investigate the structure and stability of the steady states for a bacterial colony model with density-suppressed motility. We treat the growth rate of bacteria as a bifurcation parameter to explore the local and global structure of the steady states. Relying on asymptotic analysis and the theory of Fredholm solvability, we derive the second-order approximate expression of the steady states. We analytically establish the stability criterion of the bifurcation solutions, and show that sufficiently large growth rate of bacteria leads to a stable uniform steady state. While the growth rate of bacteria is less than some certain value, there is pattern formation with the admissible wave mode. All the analytical results are corroborated by numerical simulations from different stages.     

Two-prey one-predator model

In this paper we propose a new multi-team prey–predator model, in which the prey teams help each other. We study its local stability. In the absence of predator, there is no help between the prey teams. So, we study the global stability and persistence of the model without help.     

Global stability and Hopf bifurcation of a plankton model with time delay

A differential delay equation model with a discrete time delay and a distributed time delay is introduced to simulate zooplankton–nutrient interaction. The differential inequalities’ methods and standard Hopf bifurcation analysis are applied. Some sufficient conditions are obtained for persistence and for the global stability of the unique positive steady state, respectively. It was shown that there is a Hopf bifurcation in the model by using the discrete time delay as a bifurcation parameter.     

Stability of Non-constant Equilibrium Solutions for Bipolar Full Compressible Navier–Stokes–Maxwell Systems

We study the stability of smooth solutions near non-constant equilibrium states for a bipolar full compressible Navier–Stokes–Maxwell system in a three-dimensional torus \(\mathbb {T}= (\mathbb {R}/\mathbb {Z})^3\). This system is quasilinear hyperbolic-parabolic. In the first part, by using the maximum principle, we find a non-constant steady state solution with small amplitude for this system. In the second part, with the help of suitable choices of symmetrizers and classic energy estimates, we prove that global smooth solutions exist and converge to the non-constant steady states as the time goes to infinity. As a byproduct, we obtain the global stability for the bipolar full compressible Navier–Stokes–Poisson system.     

基于食饵保护的食饵-捕食捕鱼模型分析(英文)

In this paper,we consider a prey-predator fishery model with prey dispersal in a two-patch environment,one is assumed to be a free fishing zone and the other is a reserved zone where fishing and other extractive activities are prohibited.The existence of possible steady states of the system is discussed.The local and global stability analysis has been carried out.An optimal harvesting policy is given using Pontryagin s maximum principle.     

A stage-structured predator-prey model with Beddington-DeAngelis functional response

In this paper, a stage-structured predator-prey model with Beddington-DeAngelis functional response is proposed and analyzed. It is assumed in the model that the individuals in each specie may belong to one of two classes: the immature and the mature, the age of maturity is represented by a time delay. By using the persistence theory for infinite dimensional systems, necessary and sufficient conditions for the permanence of the system are obtained. By constructing suitable Lyapunov functions and using an iterative technique, a set of easily verifiable sufficient conditions is also obtained for the local asymptotic stability and the global attractiveness of the positive equilibrium of the model.     

具有避难所的非自治竞争系统的持续生存和全局稳定性

考虑一个四缀块模型，其中一缀块里有三个竞争种群，另外三个分别是它们的避难所，并且种群能在竞争缀块和各自的避难所间相互扩散．在一定的条件下，我们给出了此模型的持续生存，周期性和全局稳定性．     

Stability analysis of diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.