Some uniqueness and exact multiplicity results for a predator-prey model

Abstract

In this paper, we consider positive solutions of a predator-prey model with diffusion and under homogeneous Dirichlet boundary conditions. It turns out that a certain parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in this model plays a very important role. A good understanding of the existence, stability and number of positive solutions is gained when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is large. In particular, we obtain various results on the exact number of positive solutions. Our results for large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> reveal interesting contrast with that for the well-studied case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals 0"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , i.e., the classical Lotka-Volterra predator-prey model.