| Popis: |
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 ( r u ( 1 − u q ) − u 2 1 + u 2 ) = 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 ⩽ η 1 q and ( q , r ) lies above the curve Γ 1 = { ( q , r ) : q ( a ) = 2 a 3 a 2 − 1 , r ( a ) = 2 a 3 ( a 2 + 1 ) 2 , 1 a 3 } or r ⩽ η 2 q for some constants η 1 ≈ 0.0939 and η 2 ≈ 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]. |
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