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Zbl 0743.35030
Brown, K.J.; Hess, P.
Positive periodic solutions of predator-prey reaction-diffusion systems.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 16, No.12, 1147-1158 (1991). ISSN 0362-546X

The paper is concerned with $T$-periodic nonnegative solutions of a system of reaction-diffusion equations (modelling a predator-prey situation) of the form $$u\sb t(x,t)-d\sb 1(t)\Delta u=a(x,t)u-b(x,t)u\sp 2-c(x,t)uv,$$ $$v\sb t(x,t)-d\sb 2(t)\Delta v=-e(x,t)v-f(x,t)v\sp 2+g(x,t)uv \text{ for } x\in D, t>0,$$ with $u(x,0)=u(x,T)$, $v(x,0)=v(x,T)$ for $x\in D$ and $Bu(x,t)=0$, $Bv(x,t)=0$ for $x\in\partial D$, $t\ge 0$. Here, $D$ is a bounded domain in $R\sp n$ with smooth boundary $\partial D$, the boundary operator $B$ is of the form $Bu=u$ or $Bu=\partial u/\partial n+b\sb 0(x)u\ (b\sb 0\ge 0)$, and the coefficient functions are assumed as smooth on $D\times R$, $T$- periodic with respect to $t$ and, in the cases of $d\sb 1$ and $d\sb 2$, strictly positive.\par 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 {\it A. Beltramo} and the second author [Commun. Partial Differ. Equations 9, 919-941 (1984; Zbl 0563.35033)] and by {\it A. Castro} and {\it A. C. Lazer} [Boll. Unione Mat. Ital., VI. Ser., B1, 1089-1104 (1982; Zbl 0501.35005)].\par 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.
[M.Kracht (DÃ¼sseldorf)]
MSC 2000:
*35K57 Reaction-diffusion equations
35J65 (Nonlinear) BVP for (non)linear elliptic equations
92D25 Population dynamics
35B10 Periodic solutions of PDE

Keywords: positive periodic solutions; steady-state solutions; existence

Citations: Zbl 0631.92014; Zbl 0563.35033; Zbl 0501.35005

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