Boundedness and stabilization in an indirect pursuit-evasion model with nonlinear signal-dependent diffusion and sensitivity

Abstract

This paper deals with an indirect pursuit-evasion model with signal-dependent diffusion and sensitivity $\left(\begin{array}{ccc}& u_t=\nabla \cdot \left(D_1\left(z\right)\nabla u\right)-\chi \nabla \cdot \left(S_1\left(z\right)u\nabla z\right)+u\left(\alpha v-a_1-b_{1}u\right),& x\in \Omega ,t>0,\\ & v_t=\nabla \cdot \left(D_2\left(w\right)\nabla v\right)+\xi \nabla \cdot \left(S_2\left(w\right)v\nabla w\right)+v\left(a_2-b_{2}v-u\right),& x\in \Omega ,t>0,\\ & w_t=\Delta w+\beta u-\gamma w,& x\in \Omega ,t>0,\\ & z_t=\Delta z+\delta v-\rho z,& x\in \Omega ,t>0,\end{array}\right)$under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset R^2$, where the parameters $\chi ,\xi ,\alpha ,\beta ,\gamma ,\delta ,\rho ,a_1,a_2,b_1,b_2$ are positive, $D_1\left(z\right)$, $D_2\left(w\right)$ are signal-dependent diffusion coefficients, $S_1\left(z\right),S_2\left(w\right)$ are nonlinear sensitivity functions. Firstly, using the energy estimate and Moser iteration, we demonstrate the existence of a unique globally bounded classical solution for the system. Furthermore, we investigate the asymptotic stabilization of globally bounded solutions. Finally, we provide numerical simulations that validate our theoretical findings.