Global periodicity in a class of reaction-diffusion systems with time delays
The authors study a class of reaction-diffusion systems modelling the dynamics of food-limited populations with periodic environmental data and time delays. The following two generalized periodic-parabolic systems are considered: \[ {\partial u(x,t)\over \partial t}-Au(t,x)=r(t,x)u(t,x){K(t,x)-u(t,x)\over K(t,x)+c(t,x)u(t,x)}\tag{1} \] in \([0,\infty )\times \Omega \) subject to the boundary condition \(B[u](t,x)=0\) on \((0,\infty )\times \partial \Omega \) and initial data \(u(0,x)=u_0(x)\) on \(\overline\Omega \), \[ {\partial w(x,t)\over \partial t}-Aw(t,x)=r(t,x)w(t,x){K(t,x)-a(t,x)w(t,x)-b(t,x)w(t-mT,x)\over K(t,x)+c(t,x)[a(t,x)w(t,x)+b(t,x)w(t-mT,x)]}\tag{2} \] in \([0,\infty )\times \Omega \) subject to the boundary condition \(B[w](t,x)=0\) on \((0,\infty )\times \partial \Omega \) and initial data \(w(s,x)=\eta (s,x)\) on \([-mT,0]\times \overline\Omega \) (\(T>0\)), \(\Omega \) is a bounded domain with smooth boundary \(\partial\Omega \). Here the boundary operator is either \(B[u]=u\) or \(B[u]={\partial u\over \partial \nu }+\gamma (x)u\) with \(\gamma\in C^{1+\alpha }(\partial \Omega )\), \(\gamma (x)\geq 0\) on \(\partial \Omega \), \(u_0\) and \(\eta \) are Hölder continuous functions, the functions \(K,r,c\) are positive in \(\Omega\times (0,\infty )\) and \(T\)-periodic in \(t\) and Hölder continuous on \([0,\infty )\times \overline\Omega \), \(a+b=1\) (\(a,b\) are nonnegative numbers), \(A\) is a linear differential operator with coefficients depending on \(t\) and \(x\), which are Hölder continuous in \(x\) and \(T\)-periodic in \(t\). The main result is that there exists a unique positive \(T\)-periodic solution \(U(t,x)\) for the problem (1). It depends on the growth rate \(r(t,x)\). It turns out that \(U(t,x)\) is the global attractor for the considered problem with any positive initial density function \(u_0(x)\). For the model (2) with time delays, it is shown that the \(T\)-periodic function \(U(t,x)\) is still a positive global attractor, provided that the magnitude of delay effects \(b\) is smaller than the magnitude of instantaneous effects \(a\). Two interesting numerical examples are discussed. Comparison of numerical simulations for both models is given through finite-difference approximations and monotone iteration method.
scientific article; zbMATH DE number 1894897