Classification of bifurcation diagrams for a multiparameter diffusive logistic problem with Holling type-IV functional response

Abstract

We study exact multiplicity and bifurcation diagrams of positive solutions for a multiparameter diffusive logistic problem with Holling type-IV functional response{u″(x)+λf(u)=0,−1<x<1,u(−1)=u(1)=0, where the growth rate function f(u)=ru(1−uq)−u1+u2, q, r are positive dimensionless parameters, and λ>0 is a bifurcation parameter. Assume that either r⩽η1q and (q,r) lies above the curve Γ1={(q,r):q(a)=1+3a22a,r(a)=1+3a2(1+a2)2,0<a<1/3} or r⩽η2q for some constants η1≈0.618 and η2≈0.601. Then on the (λ,‖u‖∞)-plane, we give a classification of four qualitatively different bifurcation diagrams: an S-shaped curve, a broken S-shaped curve, a ⊂-shaped curve and a monotone increasing curve.