S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension

Abstract

We study exact multiplicity and bifurcation diagrams of positive solutions for a multiparameter spruce budworm population steady-state problem in one space dimension{u″(x)+λ(ru(1−uq)−u21+u2)=0,−1<x<1,u(−1)=u(1)=0, where u is the population density of the spruce budworm, q,r are two 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)=2a3a2−1,r(a)=2a3(a2+1)2,1<a<3} or r⩽η2q for some constants η1≈0.0939 and η2≈0.0766. Then on the (λ,‖u‖∞)-plane, we give a classification of three qualitatively different bifurcation diagrams: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Our results settle rigorously a long-standing open problem in Ludwig, Aronson and Weinberger [Spatial patterning of the spruce budworm, J. Math. Biol. 8 (1979) 217–258].