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Positive periodic solutions of predator-prey reaction-diffusion systems. (English) Zbl 0743.35030

The paper is concerned with $T$-periodic nonnegative solutions of a system of reaction-diffusion equations (modelling a predator-prey situation) of the form

${u}_{t}\left(x,t\right)-{d}_{1}\left(t\right){\Delta }u=a\left(x,t\right)u-b\left(x,t\right){u}^{2}-c\left(x,t\right)uv,$
${v}_{t}\left(x,t\right)-{d}_{2}\left(t\right){\Delta }v=-e\left(x,t\right)v-f\left(x,t\right){v}^{2}+g\left(x,t\right)uv\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}x\in D,t>0,$

with $u\left(x,0\right)=u\left(x,T\right)$, $v\left(x,0\right)=v\left(x,T\right)$ for $x\in D$ and $Bu\left(x,t\right)=0$, $Bv\left(x,t\right)=0$ for $x\in \partial D$, $t\ge 0$. Here, $D$ is a bounded domain in ${R}^{n}$ with smooth boundary $\partial D$, the boundary operator $B$ is of the form $Bu=u$ or $Bu=\partial u/\partial n+{b}_{0}\left(x\right)u\phantom{\rule{4pt}{0ex}}\left({b}_{0}\ge 0\right)$, and the coefficient functions are assumed as smooth on $D×R$, $T$- periodic with respect to $t$ and, in the cases of ${d}_{1}$ and ${d}_{2}$, strictly positive.

Generalizing results by the first author [Nonlinear Anal., Theory Methods Appl. 11, 685-689 (1987; Zbl 0631.92014)] on steady-state solutions in the case of constant coefficients, in the present paper necessary and sufficient conditions are derived for the existence of solutions of the above problem. This is undertaken by utilizing the theory of periodic parabolic operators as developed by A. Beltramo and the second author [Commun. Partial Differ. Equations 9, 919-941 (1984; Zbl 0563.35033)] and by A. Castro and A. C. Lazer [Boll. Unione Mat. Ital., VI. Ser., B1, 1089-1104 (1982; Zbl 0501.35005)].

In a concluding section, an outline is given how, by similar arguments, theorems on the existence of steady-state solutions can be obtained in the time-independent (elliptic) case.

##### MSC:
 35K57 Reaction-diffusion equations 35J65 Nonlinear boundary value problems for linear elliptic equations 92D25 Population dynamics (general) 35B10 Periodic solutions of PDE