Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics

Pages: 515 - 524, Issue special, November 2013

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Julián López-Gómez - Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040-Madrid, Spain (email)
Marcela Molina-Meyer - Departamento de Matemáticas, Universidad Carlos III de Madrid Campus de Leganés, Avda. Universidad 30, 28911 Leganés, Madrid, Spain (email)
Andrea Tellini - Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain (email)

Abstract: It has been recently shown in [10] that Problem (1), for the special choice (2), admits an arbitrarily large number of positive solutions, provided that $\lambda$ is sufficiently negative. Moreover, using $b$ as the main bifurcation parameter, some fundamental qualitative properties of the associated global bifurcation diagrams have been established. Based on them, the authors computed such bifurcation diagrams by coupling some adaptation of the classical path-following solvers with spectral methods and collocation (see [9]). In this paper, we complete our original program by computing the global bifurcation diagrams for a wider relevant class of weight functions $a(x)$'s. The numerics suggests that the analytical results of [10] should be true for general symmetric weight functions, whereas some of them can fail if $a(x)$ becomes asymmetric around $0.5$. In any of these circumstances, the more negative $\lambda$, the larger the number of positive solutions of Problem (1). As an astonishing ecological consequence, facilitation in competitive environments within polluted habitat patches causes complex dynamics.

Keywords:  Global bifurcation diagrams, isolas, path following, pseudo-spectral methods, large solutions, superlinear indefinite problems.
Mathematics Subject Classification:  65N35, 65P30, 65N22.

Received: August 2012; Published: November 2013.