Analysis of Propagation for Impulsive Reaction-Diffusion Models

Abstract

We study a hybrid impulsive reaction-advection-diffusion model given by a reaction-advection-diffusion equation composed with a discrete-time map in space dimension $n\in\mathbb N$. The reaction-advection-diffusion equation takes the form $u^{(m)}_t \!=\! {div}(A\nabla u^{(m)}-q u^{(m)}) + f(u^{(m)}) {for} (x,t)\in\mathbb R^n \times (0,1]$, for some function $f$, a drift $q$, and a diffusion matrix $A$. When the discrete-time map is local in space we use $N_m(x)$ to denote the density of population at a point $x$ at the beginning of reproductive season in the $m$th year, and when the map is nonlocal we use $u_m(x)$. The local discrete-time map is $\{u^{(m)}(x,0) = g(N_m(x)) {for} x\in \mathbb R^n , N_{m+1}(x):=u^{(m)}(x,1) {for} x\in \mathbb R^n \}$ for some function $g$. The nonlocal discrete time map is $\{u^{(m)}(x,0) = u_{m}(x) {for} x\in \mathbb R^n , u_{m+1}(x) := g(\int_{\mathbb R^n} K(x-y)u^{(m)}(y,1) dy) {for} x\in \mathbb R^n\}$, when $K$ is a nonnegative normalized kernel. Here, we analyze the above model from a variety of perspectives so as to understand the phenomenon of propagation. We provide explicit formulas for the spreading speed of propagation in any direction $e\in\mathbb R^n$. Due to the structure of the model, we apply a simultaneous analysis of the differential equation and the recurrence relation to establish the existence of traveling wave solutions. The remarkable point is that the roots of spreading speed formulas, as a function of drift, are exactly the values that yield blow-up for the critical domain dimensions, just as with the classical Fisher's equation with advection. We provide applications of our main results to impulsive reaction-advection-diffusion models describing periodically reproducing populations subject to climate change, insect populations in a stream environment with yearly reproduction, and grass growing logistically in the savannah with asymmetric seed dispersal and impacted by periodic fires.

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