SIAM Journal on Applied Mathematics, Volume 80, Issue 1, Page 521-542, January 2020.
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.

中文翻译：

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脉冲反应扩散模型的传播分析

SIAM应用数学杂志，第80卷，第1期，第521-542页，2020年1月。
我们研究了一个由反应-对流-扩散方程给出的混合脉冲反应-对流-扩散模型，该方程由一个离散时间图组成，在空间维数$ n \ in \ mathbb N $中。反应-对流-扩散方程的形式为$ u ^ {（m）} _ t \！= \！{div}（A \ nabla u ^ {（m）}-qu ^ {（m）}）+ f（u ^ {（m）}）{for}（x，t）\ in \ mathbb R ^ n \ （0,1] $，对于某些函数$ f $，漂移$ q $和扩散矩阵$ A $。当离散时间图在空间中是局部的时，我们使用$ N_m（x）$表示在第m个年的繁殖季节开始时$ x $点的人口密度，当地图为非本地地图时，我们使用$ u_m（x）$。本地离散时间地图为$ \ {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 \} $对于某些函数$ g $。非本地离散时间映射为$ \ {u ^ {（m）}（x，0）= u_ {m}（x）{for} x \ in \ mathbb R ^ n，u_ {m + 1}（x）：= g（\ int _ {\ mathbb R ^ n} K（xy）u ^ {（m）}（y，1）dy）{for} x \ in \ mathbb R ^ n \} $，当$ K $是非负归一化内核时。在这里，我们从多种角度分析上述模型，以了解传播现象。我们为在任何方向$ e \ in \ mathbb R ^ n $中的传播传播速度提供了明确的公式。由于模型的结构，我们对微分方程和递归关系进行了同时分析，以建立行波解的存在性。值得注意的是，作为漂移的函数，扩展速度公式的根源恰好是产生临界域尺寸爆炸的值，就像经典的平流费舍尔方程一样。