On impulsive reaction-diffusion models in higher dimensions

Abstract

Assume that $N_m(x)$ denotes the density of the population at a point $x$ at the beginning of the reproductive season in the $m$th year. We study the following impulsive reaction-diffusion model for any $m\in \mathbb Z^+$ \begin{eqnarray*}\label{} \ \ \ \ \ \left\{ \begin{array}{lcl} u^{(m)}_t = div(A\nabla u^{(m)}-a u^{(m)}) + f(u^{(m)}) \quad \text{for} \ \ (x,t)\in\Omega\times (0,1] u^{(m)}(x,0)=g(N_m(x)) \quad \text{for} \ \ x\in \Omega N_{m+1}(x):=u^{(m)}(x,1) \quad \text{for} \ \ x\in \Omega \end{array}\right. \end{eqnarray*} for functions $f,g$, a drift $a$ and a diffusion matrix $A$ and $\Omega\subset \mathbb R^n$. Study of this model requires a simultaneous analysis of the differential equation and the recurrence relation. When boundary conditions are hostile we provide critical domain results showing how extinction versus persistence of the species arises, depending on the size and geometry of the domain. We show that there exists an {\it extreme volume size} such that if $|\Omega|$ falls below this size the species is driven extinct, regardless of the geometry of the domain. To construct such extreme volume sizes and critical domain sizes, we apply Schwarz symmetrization rearrangement arguments, the classical Rayleigh-Faber-Krahn inequality and the spectrum of uniformly elliptic operators. The critical domain results provide qualitative insight regarding long-term dynamics for the model. Lastly, we provide applications of our main results to certain biological reaction-diffusion models regarding marine reserve, terrestrial reserve, insect pest outbreak and population subject to climate change.