一类具时滞和阶段结构的捕食模型的稳定性与Hopf分支

研究一类具有时滞和阶段结构的捕食模型的稳定性和Hopf分支的存在性问题.通过分析特征方程,得到了正平衡点局部稳定的条件.同时,应用中心流形定理和规范型理论,得到了确定Hopf分支方向和分支周期解的稳定性的计算公式.最后对所得理论结果进行了数值模拟.     

Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay

In this paper, a modified Holling-Tanner predator-prey model with time delay is considered. By regarding the delay as the bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.     

Effect of time delay on a harvested predator-prey model

The predator-prey systems with harvesting have received a great deal of attentions for last few decades. Incorporating discrete time delays into predator-prey models could induce instability and bifurcation. In this paper we are interested in studying the combined effects of harvesting and discrete time delay on the dynamics of a predator-prey model. A comparative analysis is provided for stability behaviour in absence as well as in presence of time delay. The length of discrete time delay to preserve stability of the model system is obtained. Existence of Hopf-bifurcating small amplitude periodic solutions is derived by taking discrete time delay as a bifurcation parameter.     

Effect of herd shape in a diffusive predator-prey model with time delay

In this paper, we deal with the effect of the shape of herd behavior on the interaction between predator and prey. The model analysis was studied in three parts. The first, The analysis of the system in the absence of spatial diffusion and the time delay, where the local stability of the equilibrium states, the existence of Hopf bifurcation have been investigated. For the second part, the spatiotemporal dynamics introduce by self diffusion was determined, where the existence of Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been proved. Further, the order of Hopf bifurcation points and regions of the stability of the non trivial equilibrium state was given. In the last part of the paper, we studied the delay effect on the stability of the non trivial equilibrium, where we proved that the delay can lead to the instability of interior equilibrium state, and also the existence of Hopf bifurcation. A numerical simulation was carried out to insure the theoretical results.     

Traveling waves of a diffusive predator-prey model with nonlocal delay and stage structure

In this paper, a diffusive predator-prey model with nonlocal delay and stage structure is investigated. By using the cross iteration method and Schauder's fixed point theorem, we reduce the existence of traveling wave solutions to the existence of a pair of upper-lower solutions. Numerical simulations are carried out to illustrate the theoretical results.     

Stability and Hopf bifurcation of a modified delay predator-prey model with stage structure

In this paper, a modified delay predator-prey model with stage structure is established, which involves the economic factor and internal competition of all the prey and predator populations. By the methods of normal form and characteristic equation, we obtain the stability of the positive equilibrium point and the sufficient condition of the existence of Hopf bifurcation. We analyze the influence of the time delay on the equation and show the occurrence of Hopf bifurcation periodic solution. The simulation gives a visual understanding for the existence and direction of Hopf bifurcation of the model.     

Stable periodic traveling waves for a predator-prey model with non-constant death rate and delay

In this paper we will consider a predator-prey model with a non-constant death rate and distributed delay, described by a partial integro-differential system. The main goal of this work is to prove that the partial integro-differential system has periodic orbitally asymptotically stable solutions in the form of periodic traveling waves; i.e. N(x, t) = N(σt − μ · x), P(x, t) = P(σt − μ · x), where σ > 0 is the angular frequency and μ is the vector number of the plane wave, which propagates in the direction of the vector μ with speed c = σ/∥μ∥; and N(x, t) and P(x, t) are the spatial population densities of the prey and the predator species, respectively. In order to achieve our goal we will use singular perturbation’s techniques.     

一类基于比率的且具有收获率和时滞的捕食系统的周期解

研究一类基于比率且具有收获和时滞的捕食系统．证明了系统正周期解的存在性,并通过构造适当的Lyapunov泛函,给出了正周期解全局稳定的充分条件．     

Persistence and extinction of a stochastic delay predator-prey model under regime switching

The paper is concerned with a stochastic delay predator-prey model under regime switching. Sufficient conditions for extinction and non-persistence in the mean of the system are established. The threshold between persistence and extinction is also obtained for each population. Some numerical simulations are introduced to support our main results.     

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.     

Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator-prey model

The interactions of diffusion-driven Turing instability and delay-induced Hopf bifurcation always give rise to rich spatiotemporal dynamics. In this paper, we first derive the algorithm for the normal forms associated with the Turing-Hopf bifurcation in the reaction-diffusion system with delay, which can be used to investigate the spatiotemporal dynamical classification near the Turing-Hopf bifurcation point in the parameter plane. Then, we consider a diffusive predator-prey model with weak Allee effect and delay. Through investigating the dynamics of the corresponding normal form of Turing-Hopf bifurcation induced by diffusion and delay, the spatiotemporal dynamics near this bifurcation point can be divided into six categories. Especially, stable spatially homogeneous/inhomogeneous periodic solutions and steady states, coexistence of two stable spatially inhomogeneous periodic solutions, coexistence of two stable spaially inhomogeneous steady states and the transition from one kind of spatiotemporal patterns to another are found.     

Periodic solution for a three-stage-structured predator-prey system with time delay

By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for a three-stage-structured predator-prey system with time delay is established.     

A new predator-prey model with a profitless delay of digestion and impulsive perturbation on the prey

In this paper, we formulate a robust prey-dependent consumption predator-prey model with a delay of digestion and impulsive perturbation on the prey. Using the discrete dynamical system determined by the stroboscopic map, we obtain a ‘predator-eradication’ periodic solution and show that the ‘predator-eradication’ periodic solution is globally attractive when harvesting for the prey is over certain value. Using a new qualitative analysis method for impulsive and delay differential equations, we prove the system is uniformly persistent when harvesting for the prey is under certain value. Further, we show the delay of digestion is a “profitless” time delay. Moreover, we show our theoretical results by numerical simulation. In this paper, the main feature is that we introduce a delay of digestion and impulsive effects into the predator-prey model and exhibit a new mathematical method which is applied to investigate the system which is governed by both impulsive and delay differential equations.     

A stage-structured predator-prey system with time delay

A stage-structured predator-prey system with time delay is considered. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated. The existence of Hopf bifurcations is established. Formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results. Based on the global Hopf bifurcation theorem for general functional differential equations, the global existence of periodic solutions is established.     

Periodic solutions of a periodic delay predator-prey system

The existence of a positive periodic solution for

is established, where , , , , are positive periodic continuous functions with period , and , are periodic continuous functions with period .

     

Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation

In this paper, a mathematical model for tumor growth with time delay in proliferation under indirect effect of inhibitor is studied. The delay represents the time taken for cells to undergo mitosis. Nonnegativity of solutions is investigated. The steady-state analysis is presented with respect to the magnitude of the delay. Existence of Hopf bifurcation is proved for some parameter values. Local and global stability of the stationary solutions are proved for other ones. The analysis of the effect of inhibitor's parameters on tumor's growth is presented. The results show that dynamical behavior of solutions of this model is similar to that of solutions for corresponding non-retarded problems for some parameter values.     

一类带时滞的Watt型功能性反应的捕食系统的Hopf分支

研究一类具有时滞的Watt型功能性反应的捕食模型,通过分析正平衡点处的特征方程,讨论了该系统正平衡点的稳定性.应用Hopf分支理论,得到了该系统产生Hopf分支的条件.     

The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay

In this paper, we consider an autonomous predator-prey Lotka-Volterra system in which individuals in the population may belong to one of two classes: the immatures and the matures, the age to maturity is represented by a time delay. By using eigenvalue analysis, principal term analyze method, reduction to absurdity, and iterative method, we obtain some simple conditions for global asymptotic stability of the unique positive equilibrium point. Moreover, a condition that the prey population in the system get extinction and the predator population in the system get permanence will be obtained. Meanwhile the theorems extend the corresponding conclusions in which there have no two stage structures.