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 $a$, 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 $a\le {\lambda }_2$ and new information on the number of solutions for $a>{\lambda }_2$.