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2013, 2013(special): 515-524. doi: 10.3934/proc.2013.2013.515

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

Julián López-Gómez 1, , Marcela Molina-Meyer 2, and  Andrea Tellini 3,

1. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040-Madrid
2. 

Departamento de Matemáticas, Universidad Carlos III de Madrid Campus de Leganés, Avda. Universidad 30, 28911 Leganés, Madrid, Spain
3. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid

Received  August 2012 Published  November 2013

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: superlinear indefinite problems., large solutions, pseudo-spectral methods, isolas, path following, Global bifurcation diagrams.
Mathematics Subject Classification: 65N35, 65P30, 65N2.
Citation: Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515
References:
[1]

E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, SIAM Classics in Applied Mathematics 45, SIAM, Philadelphia, 2003.  Google Scholar

[2]

F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part I: Branches of Nonsingular Solutions, Numer. Math., 36 (1980), 1-25.  Google Scholar

[3]

F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part II: Limit Points, Numer. Math., 37 (1981), 1-28.  Google Scholar

[4]

F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part III: Simple bifurcation points, Numer. Math., 38 (1981), 1-30.  Google Scholar

[5]

J. C. Eilbeck, The pseudo-spectral method and path-following in Reaction- Diffusion bifurcation studies, SIAM J. of Sci. Stat. Comput., 7 (1986), 599-610.  Google Scholar

[6]

H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Tata Insitute of Fundamental Research, Springer, Berlin, 1986.  Google Scholar

[7]

J. López-Gómez, Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéricos, Cuadernos de Matemática y Mecánica, Serie Cursos y Seminarios No 4, Santa Fe, 1988. Google Scholar

[8]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra. Stationary partial differential equations, in "Handbook of Differential Equations: Stationary partial differential equations. Vol. II'' (eds. M. Chipot and P. Quittner), Elsevier, (2005), 211-309.  Google Scholar

[9]

J. López-Gómez, M. Molina-Meyer and A. Tellini, Spiraling bifurcation diagrams in superlinear indefinite problems,, Submitted., ().   Google Scholar

[10]

J. López-Gómez, A. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems, Comm. Pure Appl. Anal., 13 (2014), 1-73. Google Scholar

show all references

References:
[1]

E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, SIAM Classics in Applied Mathematics 45, SIAM, Philadelphia, 2003.  Google Scholar

[2]

F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part I: Branches of Nonsingular Solutions, Numer. Math., 36 (1980), 1-25.  Google Scholar

[3]

F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part II: Limit Points, Numer. Math., 37 (1981), 1-28.  Google Scholar

[4]

F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part III: Simple bifurcation points, Numer. Math., 38 (1981), 1-30.  Google Scholar

[5]

J. C. Eilbeck, The pseudo-spectral method and path-following in Reaction- Diffusion bifurcation studies, SIAM J. of Sci. Stat. Comput., 7 (1986), 599-610.  Google Scholar

[6]

H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Tata Insitute of Fundamental Research, Springer, Berlin, 1986.  Google Scholar

[7]

J. López-Gómez, Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéricos, Cuadernos de Matemática y Mecánica, Serie Cursos y Seminarios No 4, Santa Fe, 1988. Google Scholar

[8]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra. Stationary partial differential equations, in "Handbook of Differential Equations: Stationary partial differential equations. Vol. II'' (eds. M. Chipot and P. Quittner), Elsevier, (2005), 211-309.  Google Scholar

[9]

J. López-Gómez, M. Molina-Meyer and A. Tellini, Spiraling bifurcation diagrams in superlinear indefinite problems,, Submitted., ().   Google Scholar

[10]

J. López-Gómez, A. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems, Comm. Pure Appl. Anal., 13 (2014), 1-73. Google Scholar

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