Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue

TL;DR

This paper analyzes the bifurcation structure of solutions to a logistic equation with harvesting, providing new insights into the number of solutions when the growth rate crosses the second eigenvalue of the domain.

Contribution

It extends previous bifurcation analysis to include the case where the growth rate is below or near the second eigenvalue, offering exact solution counts and new bifurcation curves.

Findings

Exact number of solutions for growth rate below the second eigenvalue

New bifurcation curves for growth rate near and above the second eigenvalue

Enhanced understanding of solution multiplicity in logistic equations

Abstract

We construct the global bifurcation curves, solutions versus level of harvesting, for the steady states of a diffusive logistic equation on a bounded domain, under Dirichlet boundary conditions and other appropriate hypotheses, when $a$, the linear growth rate of the population, is below $\lambda_2+\delta$. Here $\lambda_2$ is the second eigenvalue of the Dirichlet Laplacian on the domain and $\delta>0$. Such curves have been obtained before, but only for $a$ in a right neighborhood of the first eigenvalue. Our analysis provides the exact number of solutions of the equation for $a\leq\lambda_2$ and new information on the number of solutions for $a>\lambda_2$.