The diffusive logistic model with a free boundary and seasonal succession
The authors study one-phase free boundary problem with unknown functions \(u(t,x)\), \(h(t)\) \[ u_t = - \delta u, \;m\omega <t \leq m\omega + (1-\phi)\omega, \;0< x < h(t), \] \[ u_t - d u_{xx} = u\,(a - bu),\;m\omega + (1-\phi)\omega <t \leq (m+1)\omega, \;0< x < h(t), \] \[ u_x(t,0) = 0, \;u(t,h(t)) = 0, \;t >0, \] \[ h(t)=h(m\omega), \;m\omega <t \leq m\omega + (1-\phi)\omega, \] \[ h'(t) = -\mu u_x(t,h(t)) = 0, \;m\omega + (1-\phi)\omega <t \leq (m+1)\omega, \] \[ h(0) = h_0, \;u(0,x) = u_0(x), \;0 \leq x \leq h_0, \] where \(\omega, \;0 < \phi < 1, \delta\), \ \(d, \;a, \;b, \mu, \;h_0\) are positive constants, \(m=0,1,2,\dots\). The parameters \(\omega\) and \(\phi\) represent the period of seasonal succession and the duration of the good season, respectively. Function \(u\) is the density of a species. Let \( \lim_{t\to\infty}h(t) =: h_\infty\). In the bad season (\(m\omega <t \leq m\omega + (1-\phi)\omega\)) the species vanishes, i.e., does not migrate and stays within the region \([0,\,h(m\omega)]\) (\(h_\infty < \infty\)), during the good season (\(m\omega + (1-\phi)\omega <t \leq (m+1)\omega\)) the spreading of the species to the infinity is governed by the Stefan problem in a domain \((0,\,h(t))\) (\(h_\infty = \infty\)). The authors prove that {\parindent=6mm\begin{itemize}\item[-] if \(a\phi - (1-\phi)\delta \leq 0,\) then \(h_\infty < \infty\) and \( \lim_{t\to\infty} \|u(t,\cdot)\|_{C([0,h(t)]})=0\); \item[-] if \(a\phi - (1-\phi)\delta > 0,\) then one of two alternatives is valid: 1.\,\(h_\infty = \infty\) and \( \lim_{n\to\infty} u(t+n\omega,x)=u^*(t)\) locally uniformly for \((t,x)\in[0,\omega]\times [0,\infty]\), 2.\, \(h_\infty < h^*=: \frac{\pi}{2}\sqrt{\frac{d\phi}{a\phi - (1-\phi)\delta}}\) and \(\lim_{t\to\infty} \|u(t,\cdot)\|_{C([0,h(t)]})=0\); moreover, if \(h_0 \geq h^*,\) then \(h_\infty = \infty\); \item[-] if \(h_\infty = \infty\), then \[ \lim_{n\to\infty} \frac{h(t)}{t}=\frac{1}{\omega}\int_{(1-\phi)\omega}^{\omega} k_0(t)dt, \] where \(u^*(t),\;k_0(t)\) are entirely definite functions. \end{itemize}} The cases of spreading or vanishing of species in dependence on the parameter \(\mu\) are considered also.
scientific article; zbMATH DE number 6157291