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

We study exact multiplicity and bifurcation diagrams of positive solutions for a multiparameter diffusive logistic problem with Holling type-IV functional response { u ″ ( x ) + λ f ( u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where the growth rate function f ( u ) = r u ( 1 − u q ) − u 1 + u 2 , q, r are positive dimensionless parameters, and λ > 0 is a bifurcation parameter. Assume that either r ⩽ η 1 q and ( q , r ) lies above the curve Γ 1 = { ( q , r ) : q ( a ) = 1 + 3 a 2 2 a , r ( a ) = 1 + 3 a 2 ( 1 + a 2 ) 2 , 0 < a < 1 / 3 } or r ⩽ η 2 q for some constants η 1 ≈ 0.618 and η 2 ≈ 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.