Positive periodic solutions for semilinear reaction diffusion systems on $\bf {R}^N$

Abstract

The existence and nonexistence of positive time-periodic solutions of semilinear reaction diffusion systems of the type $$ \begin{cases} \partial_{t}u +\mathcal{A}_{1}u = au-bg_{1}(u)u - h_{1}(u,v)u\\ \partial_{t}v +\mathcal{A}_{2}v = dv-fg_{2}(v)v + h_{2}(u)v \end{cases} \quad \text{ in } \ \mathbb{r}^{N}\times (0,\infty ), $$ is discussed. Here the coefficients $a,b,d,f,$ and the coupling terms $g_{1},g_{2},h_{1},h_{2}$ depend on space and time and satisfy suitable conditions. The differential operators $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are elliptic with space- and time-dependent coefficients. In particular, the time dependence is always assumed to be periodic with a fixed common period. A topological method, based on the Leray-Schauder degree, is used to obtain the existence of positive time-periodic solutions. Finally, we apply our results to various population models.