Abstract
We study exact multiplicity and bifurcation diagrams of positive solutions for a multiparameter spruce budworm population steady-state problem in one space dimension, 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≤η <sub>1</sub> q and (q, r) lies above the curve Γ1={(q,r):q(a)=2a <sup>3</sup> /a <sup>2-1</sup> , r(a)=2a <sup>3</sup> /(a <sup>2</sup> +1) <sup>2</sup> , 1<a<√3} or r≤η <sub>2</sub> q for some constants η <sub>1</sub> ≈0.0939 and η <sub>2</sub> ≈0.0766. Then on the (λ, ||u|| <sub>∞</sub> )-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]. © 2013 Elsevier Inc.