Global boundedness of a quasilinear predator-prey model with nonlinear indirect signal production mechanism

Abstract

This paper studies the following Neumann initial-boundary problem of a quasilinear predator-prey model with nonlinear indirect signal production mechanism:

$\left\{\begin{array}{ll} u_{1 t}=\nabla \cdot\left(\phi_{1}\left(u_{1}\right) \nabla u_{1}+\varphi_{1}\left(u_{1}\right) \nabla v\right)+\alpha_{1} u_{1}\left(1-u_{1}^{k_{1}-1}-\beta_{1} u_{2}\right), & x \in \Omega, t>0, \\ u_{2 t}=\nabla \cdot\left(\phi_{2}\left(u_{2}\right) \nabla u_{2}-\varphi_{2}\left(u_{2}\right) \nabla v\right)+\alpha_{2} u_{2}\left(1-u_{2}^{k_{2}-1}+\beta_{2} u_{1}\right), & x \in \Omega, t>0, \\ v_{t}=\Delta v-v+w^{\rho}, & x \in \Omega, t>0, \\ 0=\Delta w-w+u_{1}^{\vartheta}+u_{2}^{\eta}, & x \in \Omega, t>0, \end{array}\right.$

in a smoothly bounded domain $\Omega \subset \mathbb{R}^{n}$(n ≥ 1). Assume that the parameters $\alpha_{i}, \beta_{i}, \rho, \vartheta,$ η > 0, $\phi_{i}$ and $\varphi_{i}$ (i=1, 2) are nonlinear functions satisfying $\phi_{i}(s) \geq \chi(1+s)^{\delta}$ and $\left|\varphi_{i}(s)\right| \leq \xi s(1+s)^{\sigma-1}$, respectively, for alls ≥ 0 with χ, ξ > 0 and $\delta, \sigma \in \mathbb{R}$. The main result shows that for all nonnegative initial data with appropriate regularity, if the power exponents satisfy k₁ > 1, k₂ > 2 and $\sigma+\rho(\vartheta+\eta)＜\frac{n+2}{n}+\delta$, the model possesses a bounded global classical solution.

Keywords

indirect signal / boundedness / predator-prey model / density-dependent diffusion

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