Search

In this paper, the dynamic behavior of a diffusive Rosenzweig-MacArthur (R-M) predator-prey model with hyperbolic tangent functional response and fear effect was investigated. For the local system, we gave a detailed classification of equilibria and performed bifurcation analysis. It was shown by numerical simulation that both the capture rate and fear factor have a stabilizing effect. Furthermore, the existence of limit cycles was discussed when the prey was in low fear or carrying capacity was sufficiently large. For the reaction-diffusion system, we considered the local stability of a unique positive equilibrium, Turing instability of both positive equilibrium and homogeneous periodic orbits under weak fear effect or strong carrying capacity, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions under small fear cost and large diffusion coefficients, as well as the existence of positive nonconstant steady states. However, in the absence of fear effect, Turing instability of both positive equilibrium and homogeneous periodic orbits did not occur. Meanwhile, numerical examples were given to illustrate the corresponding analytic results. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive prey-predator model with hunting cooperation and prey-taxis is proposed and investigated. First, we prove the uniform boundedness and global existence of time-varying solutions to the model. Then the stability and prey-taxis-driven instability of positive equilibrium are discussed through linearization analysis. It is found that negative prey-taxis may change the spatial stability of the positive equilibrium, implying formation of Turing pattern. Next, we discuss the existence and stability of the spatial pattern by choose the prey-tactic sensitivity coefficient as bifurcation parameter. Finally, some numerical simulations are performed to visualize the impact of prey-taxis on the complex spatial pattern formation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global behavior of classical solutions to the following two-species predator-prey chemotaxis model with signal-dependent diffusion and sensitivity$ \left\{\begin{array}{ll} u_{1t} = \nabla\cdot(d_{1}(v)\nabla u_{1})+ \nabla\cdot(\chi_{1}(v)u_{1}\nabla{v})+\mu_{1}u_{1}(1-u_{1}-e_{1}u_{2}), \\ u_{2t} = \nabla\cdot(d_{2}(v)\nabla u_{2})- \nabla\cdot(\chi_{2}(v)u_{2}\nabla{v})+\mu_{2}u_{2}(1+e_{2}u_{1}-u_{2}), \\ v_{t} = d_{3}\Delta{v}+\alpha u_{1}+\beta u_{2}-\gamma v \end{array}\right. $under homogeneous Neumann boundary condition in a bounded smooth domain \begin{document}$ \Omega\subset\mathbb{R}^{2} $\end{document}, where \begin{document}$ d_{3}, \; \mu_{1}, \; \mu_{2}, \; e_{1}, \; e_{2}, \; \beta, \; \gamma $\end{document} are positive constants, \begin{document}$ \alpha\in \mathbb{R} $\end{document}, and the functions \begin{document}$ d_{i}(v), \chi_{i}(v)(i = 1, 2) $\end{document} satisfy the following assumptions:(H1) \begin{document}$ d_{i}(v), \chi_{i}(v)\in C^{2}[0, \infty) $\end{document} satisfy \begin{document}$ d_{i}(v), \; \chi_{i}(v)>0 $\end{document} for \begin{document}$ v\geq0 $\end{document}, \begin{document}$ d'_{i}(v)<0 $\end{document} and \begin{document}$ \lim\limits_{v\rightarrow \infty}d_{i}(v) = 0 $\end{document}.(H2) The limits \begin{document}$ \lim\limits_{v\rightarrow \infty}\frac{\chi_{i}(v)}{d_{i}(v)} $\end{document} and \begin{document}$ \lim\limits_{v\rightarrow \infty}\frac{d'_{i}(v)}{d_{i}(v)} $\end{document} exist.We first apply the Moser iteration method to prove the uniform boundedness and global existence of solutions to the model. Then, by constructing appropriate Lyapunov functions, we establish the following ctiteria on the global asymptotic stability of the equilibria to the model: (i) If \begin{document}$ e_{1}<1 $\end{document} and both \begin{document}$ \mu_{1} $\end{document} and \begin{document}$ \mu_{2} $\end{document} are sufficiently large, the solution \begin{document}$ (u_1, u_2, v) $\end{document} converges to a unique positive equilibrium of the model. (ii) If \begin{document}$ e_{1}\geq 1 $\end{document} and \begin{document}$ \mu_{2} $\end{document} is large enough, the solution \begin{document}$ (u_1, u_2, v) $\end{document} converges to the semi-trivial equilibrium \begin{document}$ (0, 1, \frac{\beta}{\gamma}) $\end{document}. The respective convergence rates are at least exponential if \begin{document}$ e_1\neq1 $\end{document}, and algebraic if \begin{document}$ e_1 = 1 $\end{document}. [ABSTRACT FROM AUTHOR]

We study a cubic predator-prey system with stage structure for the prey. This system is a generalization of the two-species Lotka-Volterra predator-prey model. Firstly, we consider the asymptotical stability of equilibrium points to the system of ordinary differential equations type. Then, the global existence of solutions and the stability of equilibrium points to the system of weakly coupled reaction-diffusion type are discussed. Finally, the existence of nonnegative classical global solutions to the system of strongly coupled reaction-diffusion type is investigated when the space dimension is less than 6, and the global asymptotic stability of unique positive equilibrium point of the system is proved by constructing Lyapunov functions.

This paper deals with a Holling type III diffusive predator-prey model with stage structure and nonlinear density restriction in the space . We first consider the asymptotical stability of equilibrium points for the model of ODE type. Then, the existence and uniform boundedness of global solutions and stability of the equilibrium points for the model of weakly coupled reaction-diffusion type are discussed. Finally, the global existence and the convergence of solutions for the model of cross-diffusion type are investigated when the space dimension is less than 6.

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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a diffusive epidemic model with nonlinear incidence rate 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 states. And epidemiologically, we find that the system exhibits Turing patterns controlled by diffusion. [ABSTRACT FROM AUTHOR]