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

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 aa, the linear growth rate of the population, is below λ2+δλ2+δ. Here λ2λ2 is the second eigenvalue of the Dirichlet Laplacian on the domain and δ>0δ>0. Such curves have been obtained before, but only for aa in a right neighborhood of the first eigenvalue. Our analysis provides the exact number of solutions of the equation for a≤λ2a≤λ2 and new information on the number of solutions for a>λ2a>λ2.