A robust multiplicity result in a generalized diffusive predator-prey model

TL;DR

This paper proves the existence of multiple coexistence states in a generalized predator-prey model with spatial heterogeneity and saturation effects, especially as the saturation amplitude becomes very large.

Contribution

It establishes new multiplicity results for coexistence states in a spatially heterogeneous predator-prey model with saturation, including cases with three solutions.

Findings

Existence of at least two coexistence states when saturation amplitude is large.

Construction of an S-shaped component leading to three coexistence states.

Multiplicity occurs both in stable and unstable semitrivial solutions regions.

Abstract

This paper analyzes the generalized spatially heterogeneous diffusive predator-prey model introduced by the authors in \cite{LGMH20}, whose interaction terms depend on a saturation coefficient $m(x)\gneq0$. As the amplitude of the saturation term, measured by $\|m\|_\infty$, blows up to infinity, the existence of, at least, two coexistence states, is established in the region of the parameters where the semitrivial positive solution is linearly stable, regardless the sizes and the shapes of the remaining function coefficients in the setting of the model. In some further special cases, an $S$-shaped component of coexistence states can be constructed, which causes the existence of, at least, three coexistence states, though this multiplicity occurs within the parameter regions where the semitrivial positive solution is linearly unstable. Therefore, these multiplicity results inherit a rather different nature.