Stability and Hopf bifurcation for a population delay model with diffusion effects
The system \[ \partial U(t, x)/\partial t= \partial^2 U(t, x)/\partial x^2+ kU(t, x)[1- U(t- r, x)],\quad t> 0, \] \[ U(t, 0)= U(t, \pi)= 0,\quad t\geq 0,\quad U(t, x)= \psi(t, x),\;(t, x)\in [- r, 0]\times [0, \pi], \] is considered, where \(\psi\in C([- r, 0], H^1_0[0, \pi])\), \(k> 1\) (\(k- 1\) sufficiently small), \(r> 0\) are constants. The main results of this paper are the following: 1) There is an \(r(k)> 0\) such that the equilibrium \(U_k\), the unique positive solution of the boundary problem \[ d^2 U(x)/dx^2+ k(1- U(x)) U(x)= 0,\quad U(0)= U(\pi)= 0,\quad U(x)> 0, \] \(x\in (0, \pi)\) is locally stable if \(0\leq r< r(k)\) and unstable if \(r> r(k)\). 2) There exists a sequence \(\{r_{k_n}\}^\infty_{n= 0}\), \(r(k)= r_{k_0}< r_{k_1}<\cdots\) such that there is a Hopf bifurcation arising from \(U_k\) as the delay \(r\) monotonically passes through each \(r_{k_n}\), \(n= 0, 1,2,\dots\). Furthermore, the periodic solution occurring from the Hopf bifurcation point \(r(k)= r_{k_0}\) is stable and those occurring from the Hopf bifurcation points \(r_{k_n}\), \(n= 1, 2,\dots\), are unstable.
scientific article; zbMATH DE number 838711