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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]

In this paper, a predator–prey model with prey-stage structure and prey-taxis is proposed and studied. Firstly, the local stability of non-negative constant equilibria is analyzed. It is shown that non-negative equilibria have the same stability between ODE system and self-diffusion system, and self-diffusion does not have a destabilization effect. We find that there exists a threshold value ξ 0 such that the positive equilibrium point of the model becomes unstable when the prey-taxis rate ξ < ξ 0 , hence the taxis-driven Turing instability occurs. Furthermore, by applying Crandall–Rabinowitz bifurcation theory, the existence, the stability and instability, and the turning direction of bifurcating steady state are investigated in detail. Finally, numerical simulations are provided to support the mathematical analysis. [ABSTRACT FROM AUTHOR]

Abstract The local existence and stability of non-constant positive steady states for the Holling–Tanner predator–prey model with prey-taxis was discussed by Wang et al. (2017) through local bifurcation analysis. It is our purpose in this paper to make a detailed description for the structure of the set of the non-constant steady states. We will prove a global bifurcation theorem by treating prey-taxis as a bifurcation parameter, which gives the existence of non-constant steady states under a rather natural condition. A priori estimates for steady state solutions will play a key role in the proof. In addition, the stability results for the homogeneous equilibrium will extend to the case that the derivative of prey's functional response with prey is positive, and it is found that attractive prey-taxis can stabilize homogeneous equilibrium even diffusion-driven instability has occurred. [ABSTRACT FROM AUTHOR]

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]

Abstract: In this paper, the global behavior of solutions is investigated for a Lotka–Volterra predator–prey system with prey-stage structure. First, we can see that the stability properties of nonnegative equilibria for the weakly coupled reaction–diffusion system are similar to that for the corresponding ODE system, that is, linear self-diffusions do not drive instability. Second, using Sobolev embedding theorems and bootstrap arguments, the existence and uniqueness of nonnegative global classical solution for the strongly coupled cross-diffusion system are proved when the space dimension is less than 10. Finally, the existence and uniform boundedness of global solutions and the stability of the positive equilibrium point for the cross-diffusion system are studied when the space dimension is one. It is found that the cross-diffusion system is dissipative if the diffusion matrix is positive definite. Furthermore, cross diffusions cannot induce pattern formation if the linear diffusion rates are sufficiently large. [Copyright &y& Elsevier]

We consider a strongly coupled predator-prey model with one resource and two consumers, in which the first consumer species feeds on the resource according to the Holling II functional response, while the second consumer species feeds on the resource following the Beddington- DeAngelis functional response, and they compete for the common resource. Using the energy estimates and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for the model are proved. Meanwhile, the sufficient conditions for global asymptotic stability of the positive equilibrium for this model are given by constructing a Lyapunov function. [ABSTRACT FROM AUTHOR]