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Analysis of propagation for impulsive reaction-diffusion models
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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 \begin{equation}\label{} u^{(m)}_t = \text{div}(A\nabla u^{(m)}-q u^{(m)}) + f(u^{(m)}) \quad \text{for} \ \ (x,t)\in\mathbb R^n \times (0,1] , \end{equation} for some function $f$, a drift $q$ and a diffusion matrix $A$. When the discrete-time map is local in space we use $Nm(x)$ 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 $um(x)$. The local discrete-time map is \begin{eqnarray}\label{}\left{ \begin{array}{lcl} u^{(m)}(x,0) = g(Nm(x)) \quad \text{for} \ \ x\in \mathbb R^n , \ N{m+1}(x):=u^{(m)}(x,1) \quad \text{for} \ \ x\in \mathbb R^n , \end{array}\right. \end{eqnarray} for some function $g$. The nonlocal discrete time map is \begin{eqnarray}\label{}\left{ \begin{array}{lcl} u^{(m)}(x,0) = u{m}(x) \quad \text{for} \ \ x\in \mathbb R^n , \ \label{mainb2} u{m+1}(x) := g\left(\int_{\mathbb R^n} K(x-y)u^{(m)}(y,1) dy\right) \quad \text{for} \ \ x\in \mathbb R^n, \end{array}\right. \end{eqnarray} when $K$ is a nonnegative normalized kernel.
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- Date created
- 2019-01-01
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- Article (Published)