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Some uniqueness and exact multiplicity results for a predator-prey model. (English) Zbl 0965.35041

The authors study the predator-prey model \[ \Delta u+u(a-u-bv(1+mu))=0,\;\Delta v+v(d-v-cu(1+mu))=0 \text{ in }D, \quad u=v=0 \text{ on }\partial D, \] where \(D\) is a smoothly bounded domain in \(\mathbb R^n\), \(a,b,c>0\), \(m\geq 0\) and \(d\) may change sign. If \(m>0\) is large then this problem can be viewed as a perturbation of a simpler limiting problem and this is used in order to prove various existence, uniqueness, multiplicity and stability results.

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B35 Stability in context of PDEs
35B20 Perturbations in context of PDEs
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