Bifurcation curves of a diffusive logistic equation with harvesting orthogonal to the first eigenfunction

TL;DR

This paper analyzes the global bifurcation structure of a diffusive logistic equation with harvesting orthogonal to the first eigenfunction, revealing new behaviors as the growth rate crosses eigenvalues.

Contribution

It provides new insights into the bifurcation curves' behavior, including their transformation at the first eigenvalue and the structure of solution sets.

Findings

Bifurcation curves transform at the first eigenvalue.

Solutions are compact above the first eigenvalue.

Two families of degenerate solutions with Morse index zero exist.

Abstract

We study the global bifurcation curves of a diffusive logistic equation, when harvesting is orthogonal to the first eigenfunction of the Laplacian, for values of the linear growth up to $\lambda_2+\delta$, examining in detail their behavior as the linear growth rate crosses the first two eigenvalues. We observe some new behavior with regard to earlier works concerning this equation. Namely, the bifurcation curves suffer a transformation at $\lambda_1$, they are compact above $\lambda_1$, there are precisely two families of degenerate solutions with Morse index equal to zero, and the whole set of solutions below $\lambda_2$ is not a two dimensional manifold.