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Solving Volterra-Lotka systems with diffusion by monotone iteration. (English) Zbl 0735.35060
An abstract theorem concerning a monotone iteration method in a partially ordered linear space is presented first. Then the method is applied to a stationary Volterra-Lotka system in a convex domain \(\Omega\): \[ \alpha\Delta u+au-bu^ 2+cuv=0,\;\beta\Delta v+dv-ev^ 2+fuv=0 \hbox{ in } \Omega \hbox{ and } u=v=0 \hbox{ on } \partial\Omega, \] with already known subsolutions \(\b{u}_ 0\), \(\b{v}_ 0\) and supersolutions \(\bar u_ 0\), \(\bar v_ 0\) as an initial step. As a result, solutions \(u^*,v^*\in C^ 2(\Omega)\cap C(\bar\Omega)\) are obtained in order- intervals \([\b{u}_ 0,\bar u_ 0]\) and \([\b{v}_ 0,\bar v_ 0]\) respectively under appropriate assumptions on the positive constants \(\alpha\), \(\beta\), \(a\), \(b\), \(d\), \(e\) and the real ones \(c\), \(f\). An extended predator-prey system is also considered. One numerical example is given.
MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
92D25 Population dynamics (general)
Software:
MGOO
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