Threshold convergence to steady states for nonlocal reaction-diffusion equations with time delay in bounded domain

Abstract

In this paper, we aim at studying the asymptotic behavior for the time-delayed nonlocal reaction-diffusion equation for population dynamics with Dirichlet boundary condition in $\Omega \subset R^N$. We recognize that there are threshold convergence results of the solutions which depend on the ecological parameters: the spatial diffusion coefficient $D>0$, the death rate coefficient $\delta >0$, the birth rate coefficient $p>0$, and two principal eigenvalues $0<{\lambda }_i<1$ ($i=1,2$) of the linear nonlocal dispersion operators induced by the two different kernels with Dirichlet boundaries, respectively. Precisely, when $0<\frac{(1-{\lambda }_2)p}{D{\lambda }_1+\delta }<1$, we prove that the solution globally converges to the trivial steady state 0 at the exponential rate. When $1<\frac{(1-{\lambda }_2)p}{D{\lambda }_1+\delta }\le e$, we further prove that the solution globally converges to the non-trivial steady state $\varphi \left(x\right)$ at the exponential rate, and yet this convergence locally holds if $e<\frac{(1-{\lambda }_2)p}{D{\lambda }_1+\delta }<e^2$. The convergence rates are also time-exponential. The proof is based on the Fourier transform and the energy method involving the eigenvalue problems for nonlocal dispersion equations. Some new techniques and skills for treating the nonlocality and non-monotonicity with restriction in bounded domain are also proposed. Finally, a number of numerical simulations are carried out, which confirm our theoretical results. For $\frac{(1-{\lambda }_2)p}{D{\lambda }_1+\delta }>e^2$, the solutions are numerically tested to be oscillating.