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

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 λ2+δ, 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 λ1, they are compact above λ1, there are precisely two families of degenerate solutions with Morse index equal to zero, and the whole set of solutions below λ2 is not a two dimensional manifold.