Structures and evolution of bifurcation diagrams for a one-dimensional diffusive generalized logistic problem with constant yield harvesting. II. Convex-concave and convex-concave-convex nonlinearities

Abstract

We study the diffusive generalized logistic problem with constant yield harvesting:{u″(x)+λg(u)−μ=0,−1<x<1,u(−1)=u(1)=0,where λ,μ>0. We assume that g satisfies g(0)=g(1)=0, g(u)>0 on (0,1), and g is either convex-concave or convex-concave-convex on (0,1) and satisfies certain conditions. We prove that, for any fixed μ>0, on the (λ,‖u‖∞)-plane, the bifurcation diagram always consists of a continuous, ⊂-shaped curve, and we study the structures and evolution of bifurcation diagrams for varying μ>0. We also prove that, for any fixed λ>λ0⁎ for some λ0⁎>0, on the (μ,‖u‖∞)-plane, the bifurcation diagram consists of a reversed ⊂-shaped curve with possibly the disjoint union of a strictly increasing curve, and we study the structures and evolution of bifurcation diagrams for varying λ>λ0⁎. Our results complement those of Hung, Suen, and Wang (2020) [10] in which problem g is either concave or concave-convex on (0,1).