Stability Analysis of a Diffusive Predator-prey Model with Hunting Cooperation

In this paper, we are concerned with the dynamics of a diffusive predator-prey model that incorporates the functional response concerning hunting cooperation. First, we investigate the stability of the semi-trivial steady state. Then, we investigate the influence of the diffusive rates on the stability of the positive constant steady state. It is shown that there exists diffusion-driven Turing instability when the diffusive rate of the predator is smaller than the critical value, which is dependent on the diffusive rate of the prey, and the semi-trivial steady state and the positive constant steady state are both locally asymptotically stable when the diffusive rate of the predator is larger than the critical value. Finally, the nonexistence of nonconstant steady states is discussed.     

Dynamical Property Analysis of a Delayed Diffusive Predator-prey Model with Fear Effect

In this paper, we study a delayed diffusive predator-prey model with fear effect and Holling II functional response. The stability of the positive equilibrium is investigated. We find that time delay can destabilize the stable equilibrium and induce Hopf bifurcation. Diffusion may lead to Turing instability and inhomogeneous periodic solutions. Through the theory of center manifold and normal form, some detailed formulas for determining the of Hopf bifurcation are presented. Some numerical simulations are also provided.     

一类捕食与被捕食系统的稳定性与分岔分析

本文利用Schur—Cohn—Jury引理及分岔理论讨论了一类捕食与被捕食系统的动力学性质,分析了其正平衡点的稳定性,并讨论了Neimark—Sacker分岔稳定性与方向。通过数值模拟验证了所得结果的正确性。     

Turing and Hopf Bifurcation in a Diffusive Tumor-immune Model

In order to understand the effect of the diffusion reaction on the interaction between tumor cells and immune cells, we establish a tumor-immune reaction diffusion model with homogeneous Neumann boundary conditions. Firstly, we investigate the existence condition and the stability condition of the coexistence equilibrium solution. Secondly, we obtain the sufficient and necessary conditions for the occurrence of Turing bifurcation and Hopf bifurcation. Thirdly, we perform some numerical simulations to illustrate the complex spatiotemporal patterns near the bifurcation curves. Finally, we explain spatiotemporal patterns in the diffusion action of tumor cells and immune cells.     

Analysis of the Dynamics of a Predator-prey Model with Holling Functional Response

A diffusive predator-prey system with Holling functional response is considered. Firstly, existence of positive equilibrium of this reaction diffusion model under Neumann boundary condition is obtained. Meanwhile, the existence conditions for Turing instability and Hopf bifurcations of a system with Holling \uppercase\expandafter{\romannumeral2} functional response are established. Next, the existence of the hydra effect is demonstrated, when the system is undergoing non-homogeneous steady-state solutions. Finally, numerical simulations are illustrated to support our theory results.     

一类具有阶段结构和时滞的捕食模型的稳定性及Hopf分支(英文)

研究一类具有阶段结构和时滞的捕食模型.通过特征方程分别分析了正平衡点和边界平衡点的局部稳定性,到了系统Hopf分支存在的充分条件.通过规范型理论和中心流型定理,给出了确定Hopf分支方向和分支周期解的稳定性的计算公式.     

一类自催化可逆生化反应模型的Hopf分支及其稳定性

在齐次Neumann边界条件下,研究一类自催化可逆三分子生化反应模型.首先对常微分系统,给出Hopf分支的存在性及稳定性.其次对偏微分系统,建立由扩散系数引起的Turing不稳定性,同时给出Hopf分支的存在性,并利用规范型理论和中心流形定理建立Hopf分支的方向和稳定性.最后,借助Matlab软件进行数值模拟,验证补...     

具有时滞的生态流行病模型的稳定性和Hopf分支

该文考虑一类食饵染病的时滞捕食被捕食模型. 作者分析了系统的非负不变性, 边界平衡点的性质和全局稳定性. 证明了当时滞τ=τ\-1+τ\-2适当小时, 正平衡点是局部渐近稳定的,随着时滞的增加, 正平衡点由稳定变为不稳定, 系统在正平衡点附近发生Hopf分支.     

Spatial Dynamics of a Diffusive Predator-prey Model with Leslie-Gower Functional Response and Strong Allee Effect

In this paper, spatial dynamics of a diffusive predator-prey model with Leslie-Gower functional response and strong Allee effect is studied. Firstly, we obtain the critical condition of Hopf bifurcation and Turing bifurcation of the PDE model. Secondly, taking self-diffusion coefficient of the prey as bi- furcation parameter, the amplitude equations are derived by using multi-scale analysis methods. Finally, numerical simulations are carried out to verify our theoretical results. The simulations show that with the decrease of self- diffusion coefficient of the prey, the preys present three pattern structures: spot pattern, mixed pattern, and stripe pattern. We also observe the transi- tion from spot patterns to stripe patterns of the prey by changing the intrinsic growth rate of the predator. Our results reveal that both diffusion and the intrinsic growth rate play important roles in the spatial distribution of species.     

Hopf Bifurcation Analysis of a Host-generalist Parasitoid Model with Diffusion Term and Time Delay

In this paper, we studied a delayed host-generalist parasitoid model with Holling II functional response and diffusion term. The Turing instability and local stability are studied. The existence of Hopf bifurcation is investigated, and some explicit formulas for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived by the theory of center manifold and normal form method. Some numerical simulations are carried out.     

含两时滞捕食-被捕食系统的稳定性及分歧

本文研究一类含两相异时滞的捕食－被捕食系统的稳定性及分歧。首先，我们讨论两相异时滞对系统唯一正平衡点的稳定性的影响，通过对系数与时滞有关的特征方程的分析，建立了一种稳定性判别性。其次，将一个时滞看成分歧参数，而另一个看作固定参数，我们证明了该系统具有HOPF分歧特性。最后，我们讨论了分歧解的稳定性。     

Nonlocal Interaction Induces the Self-organized Mussel Beds

Mussel beds are important habitats and food sources for biodiversity in coastal ecosystems. The predation of mussel on algae depends not only on the current time and location, but also on the quantity of algae at other spatial location and time. To know the impacts of such predation behavior on the dynamics of mussel beds well, we pose a reaction-diffusion mussel-algae model coupling nonlocal interaction with kernel function. By calculating the critical conditions of Hopf bifurcation and Turing bifurcation, the conditions for the generation of Turing pattern are obtained. We find that the diffusion rate and predation rate of mussels have effect on the structure and density of spatial pattern of mussels under the nonlocal interaction, and the predation rate of mussels can produce different pattern types, while the diffusion rate plays a more important role on the pattern density. Moreover, the nonlocal interaction promotes the stability of the mussel beds. These results suggest that the nonlocal interaction between mussels and algaes is one of the important mechanisms for the formation of the spatial structure of mussel beds.     

Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response

We consider a delayed predator-prey system with Beddington-DeAngelis functional response. The stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. An example is given and numerical simulations are performed to illustrate the obtained results.     

Dynamics of a Predator-prey Model with Delay and Fear Effect

Recent manipulations on vertebrates showed that the fear of preda- tors, caused by prey after they perceived predation risk, could reduce the prey''s reproduction greatly. And it''s known that predator-prey systems with fear ef- fect exhibit very rich dynamics. On the other hand, incorporating the time delay into predator-prey models could also induce instability and oscillations via Hopf bifurcation. In this paper, we are interested in studying the com- bined effects of the fear effect and time delay on the dynamics of the classic Lotka-Volterra predator-prey model. It''s shown that the time delay can cause the stable equilibrium to become unstable, while the fear effect has a stabi- lizing effect on the equilibrium. In particular, the model loses stability when the delay varies and then regains its stability when the fear effect is stronger. At last, by using the normal form theory and center manifold argument, we derive explicit formulas which determine the stability and direction of periodic solutions bifurcating from Hopf bifurcation. Numerical simulations are carried to explain the mathematical conclusions.     

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.     

Gause型食物链模型Hopf分支的存在性分析

考虑了一类三维时滞Gause型食物链模型.首先分析了共存平衡点稳定的条件,然后利用多项式理论分析了特征方程特征根的分布,得到了Hopf分支存在的条件,最后给出了几组数值模拟验证文中得到的结论,进一步预测了Hopf分支的全局存在性.     

Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect

A reaction-diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior. The stability of the positive spatially nonhomogeneous positive equilibrium and associated Hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially homogeneous dispersal kernel.     

Bifurcation analysis for a three-species predator-prey system with two delays

In this paper, a three-species predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Spatiotemporal dynamics in a predator-prey model with a functional response increasing in both predator and prey densities

In this paper, we studied a diffusive predator-prey model with a functional response increasing in both predator and prey densities. The Turing instability and local stability are studied by analyzing the eigenvalue spectrum. Delay induced Hopf bifurcation is investigated by using time delay as bifurcation parameter. Some conditions for determining the property of Hopf bifurcation are obtained by utilizing the normal form method and center manifold reduction for partial functional differential equation.     

具阶段结构的多时滞SIR模型的稳定性分析

考虑了一类具有两个阶段结构的SIR模型,得到了解的正性和有界性,通过分析特征方程根的分布,以(?)_1和(?)_2为参数分析了平衡点的稳定性和局部Hopf分支存在性.进一步地,利用规范型和中心流型理论,给出了决定Hopf分支方向和分支周期解的稳定性的隐式算法.最后利用一些数值模拟来支持所得到的理论分析结果.