Trichotomy dynamics of a free boundary model for biological invasion

Abstract

It is well known that the reaction-diffusion equation $u_t=d{u}_{xx}+f(u)$ with compactly supported nonnegative initial functions exhibits trichotomy dynamics for bistable and combustion type $f(u)$ \cite{DM, zlatos}. The same is true for the corresponding Stefan type free boundary problem \cite{DL}. In this paper, we reveal a rather different type of trichotomy for this reaction-diffusion equation under a new set of (free) boundary conditions, arising as a model for biological invasion with $u(t,x)$ representing the density of an invading species over the one dimensional spatial regin $[0,h(t)]$. The evolution of the invading front $x=h(t)$ is governed by $h^{\prime }(t)=-\frac{d}{\delta }{u}_x(t,h(t))$ and $u(t,h(t))=\delta \in ({\hat{\theta }}_f,1)$, with ${\hat{\theta }}_f\in [0,1)$ uniquely determined by $f$; they allow $h(t)$ to advance as well as to retreat when time increases. At the fixed boundary $x=0$, the density is controlled by $u(t,0)={\delta }_0\ge 0$. We completely classify the long-time dynamics of the model when $f(u)$ is a monostable, or bistable, or combustion type nonlinear function. In the biologically interesting case that ${\delta }_0<\delta$, we show that there are exactly three scenarios: (i) successful spreading, (ii) finite-time vanishing, (iii) a transition state characterized by $h(t)\to l_{\ast }\in (0,\infty )$ and $u(t,x)\to w_{\ast }(x)$ as $t\to \infty$, where $(u(t,x),h(t))\equiv (w_{\ast }(x),l_{\ast })$ is the unique stationary solution of the free boundary problem. The model here does not have the usual order-preserving property enjoyed by those considered in \cite{DM, zlatos, DL} and elsewhere (i.e., $u(0,x)\le v(0,x)$ implies $u(t,x)\le v(t,x)$ for all $t>0$ if $u$ and $v$ are two solutions of the problem), which is intrinsically linked to the many novel features of the model.