An indefinite nonlinear problem in population dynamics: high multiplicity of positive solutions

TL;DR

This paper investigates a one-dimensional reaction-diffusion model in population dynamics with heterogeneous habitats, demonstrating the existence of multiple positive solutions under certain conditions and exploring their implications through numerical simulations.

Contribution

It establishes the existence of eight positive solutions for a parameter-dependent nonlinear equation modeling population dynamics with heterogeneous environments.

Findings

Existence of eight positive solutions for large parameter values

Multiple solutions depend on habitat heterogeneity and boundary conditions

Numerical simulations support theoretical results and suggest open problems

Abstract

Reaction-diffusion equations have several applications in the field of population dynamics and some of them are characterized by the presence of a factor which describes different types of food sources in a heterogeneous habitat. In this context, to study persistence or extinction of populations it is relevant the search of nontrivial steady states. Our paper focuses on a one-dimensional model given by a parameter-dependent equation of the form $u'' + \bigl{(} \lambda a^{+}(t)-\mu a^{-}(t) \bigr{)}g(u) = 0$, where $g\colon \mathopen{[}0,1\mathclose{]} \to \mathbb{R}$ is a continuous function such that $g(0)=g(1)=0$, $g(s) > 0$ for every $0<s<1$ and $\lim_{s\to0^{+}}g(s)/s=0$, and the weight $a(t)$ has two positive humps separated by a negative one. In this manner, we consider bounded habitats which include two favorable food sources and an unfavorable one. We deal with various boundary conditions, including the Dirichlet and Neumann ones, and we prove the existence of eight positive solutions when $\lambda$ and $\mu$ are positive and sufficiently large. Throughout the paper, numerical simulations are exploited to discuss the results and to explore some open problems.