Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-18T13:23:26.234Z Has data issue: false hasContentIssue false
  • English
  • Français

Intricate dynamics caused by facilitation in competitive environments within polluted habitat patches

Published online by Cambridge University Press:  07 January 2014

JULIÁN LÓPEZ-GÓMEZ ,
MARCELA MOLINA-MEYER  and
ANDREA TELLINI
JULIÁN LÓPEZ-GÓMEZ
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain emails: Lopez_Gomez@mat.ucm.es, andrea.tellini@mat.ucm.es
MARCELA MOLINA-MEYER
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain email: mmolinam@math.uc3m.es
ANDREA TELLINI
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain emails: Lopez_Gomez@mat.ucm.es, andrea.tellini@mat.ucm.es
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper analyses a canonical class of one-dimensional superlinear indefinite boundary value problems of great interest in population dynamics under non-homogeneous boundary conditions; the main bifurcation parameter in our analysis is the amplitude of the superlinear term. Essentially, it continues the analysis of López-Gómez et al. (López-Gómez, J., Tellini, A. & Zanolin, F. (2014) High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Comm. Pure Appl. Anal. 13(1), 1–73) with empty overlapping, by computing the bifurcation diagrams of positive steady states of the model and by proving analytically a number of significant features, which have been observed from the numerical experiments carried out here. The numerics of this paper, besides being very challenging from the mathematical point of view, are imperative from the point of view of population dynamics, in order to ascertain the dimensions of the unstable manifolds of the multiple equilibria of the problem, which measure their degree of instability. From that point of view, our results establish that under facilitative effects in competitive media, the harsher the environmental conditions, the richer the dynamics of the species, in the sense discussed in Section 1.

Keywords

Facilitation in polluted patchesBifurcation diagramsSuperlinear indefinite problemsUnstable manifoldsDegree of instabilityUniqueness of local attractor
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 2 , April 2014 , pp. 213 - 229
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alama, S. & Tarantello, G. (1996) Elliptic problems with nonlinearities indefinite in sign. J. Funct. Anal. 141, 159–215.CrossRefGoogle Scholar
[2]Allgower, E. L. & Georg, K. (2003) Introduction to Numerical Continuation Methods, SIAM Classics in Applied Mathematics 45, SIAM, Philadelphia, PA.Google Scholar
[3]Amann, H. (1974) Multiple positive fixed points of asymptotically linear maps. J. Funct. Anal. 17, 174–213.Google Scholar
[4]Amann, H. (1976) Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709.Google Scholar
[5]Amann, H. & López-Gómez, J. (1998) A priori bounds and multiple solutions for superlinear indefinite elliptic problems. J. Diff. Equ. 146, 336–374.Google Scholar
[6]Berestycki, H., Capuzzo-Dolcetta, I. & Nirenberg, L. (1994) Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Top. Meth. Nonl. Anal. 4, 59–78.Google Scholar
[7]Berestycki, H., Capuzzo-Dolcetta, I. & Nirenberg, L. (1995) Variational methods for indefinite superlinear homogeneous elliptic problems. Nonl. Diff. Equ. Appns. 2, 553–572.Google Scholar
[8]Bertness, M. D. & Callaway, R. M. (1994) Positive interactions in communities. Trends Ecol. Evol. 9, 191–193.Google Scholar
[9]Brezzi, F., Rappaz, J. & Raviart, P. A. (1980) Finite dimensional approximation of nonlinear problems, part I: Branches of nonsingular solutions. Numer. Math. 36, 1–25.Google Scholar
[10]Brezzi, F., Rappaz, J. & Raviart, P. A. (1981) Finite dimensional approximation of nonlinear problems, part II: Limit points. Numer. Math. 37, 1–28.Google Scholar
[11]Brezzi, F., Rappaz, J. & Raviart, P. A. (1981) Finite dimensional approximation of nonlinear problems, part III: Simple bifurcation points. Numer. Math. 38, 1–30.CrossRefGoogle Scholar
[12]Callaway, R. M. & Walker, L. R. (1997) Competition and facilitation: A synthetic approach to interactions in plant communities. Ecology 78, 1958–1965.Google Scholar
[13]Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. (1988) Spectral Methods in Fluid Mechanics, Springer, Berlin, Germany.Google Scholar
[14]Connell, J. H. (1983) On the prevalence and relative importance of interspecific competition: Evidence from field experiments. Amer. Natur. 122, 661–696.Google Scholar
[15]Crouzeix, M. & Rappaz, J. (1990) On Numerical Approximation in Bifurcation Theory, Recherches en Mathématiques Appliquées 13. Masson, Paris, France.Google Scholar
[16]Doedel, E. J. & Oldeman, B. E. (2012) AUTO-07P: Continuation and bifurcation software for ordinary differential equations [online]. Available at: http://www.dam.brown.edu/people/sandsted/auto/auto07p.pdf. Accessed on 7 October 2013.Google Scholar
[17]Eilbeck, J. C. (1986) The pseudo-spectral method and path-following in reaction–diffusion bifurcation studies. SIAM J. Sci. Stat. Comput. 7, 599–610.CrossRefGoogle Scholar
[18]García-Melián, J. (2011) Multiplicity of positive solutions to boundary blow up elliptic problems with sign-changing weights. J. Funct. Anal. 261, 1775–1798.Google Scholar
[19]Gilbarg, D. & Trudinger, N. S. (2001) Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, Germany.Google Scholar
[20]Golubitsky, M. & Shaeffer, D. G. (1985) Singularity and Groups in Bifurcation Theory, Springer, Berlin, Germany.CrossRefGoogle Scholar
[21]Gómez-Reñasco, R. & López-Gómez, J. (2000) The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations. J. Diff. Equ. 167, 36–72.Google Scholar
[22]Gómez-Reñasco, R. & López-Gómez, J. (2001) The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations. Diff. Int. Equ. 14, 751–768.Google Scholar
[23]Gómez-Reñasco, R. & López-Gómez, J. (2002) On the existence and numerical computation of classical and non-classical solutions for a family of elliptic boundary value problems. Nonlinear Anal. Theory Methods Appl. 48, 567–605.Google Scholar
[24]Hutchinson, G. E. (1965) The Ecological Theater and the Evolutionary Play, Yale University Press, New Haven, CT.Google Scholar
[25]Keller, H. B. (1986) Lectures on Numerical Methods in Bifurcation Problems, Tata Insitute of Fundamental Research, Springer, Berlin, Germany.Google Scholar
[26]López-Gómez, J. (1988) Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéricos, Cuadernos de Matemática y Mecánica, Serie Cursos y Seminarios N° 4, Santa Fe.Google Scholar
[27]López-Gómez, J. (1997) On the existence of positive solutions for some indefinite superlinear elliptic problems. Comm. Part. Diff. Equ. 22, 1787–1804.CrossRefGoogle Scholar
[28]López-Gómez, J. (1999) Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems. Trans. Amer. Math. Soc. 352, 1825–1858.Google Scholar
[29]López-Gómez, J. (2005) Global existence versus blow-up in superlinear indefinite parabolic problems. Sci. Math. Jpn. 61, 493–516.Google Scholar
[30]López-Gómez, J. (2005) Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra. In: Chipot, M. & Quittner, P. (editors), Handbook of Differential Equations ‘Stationary Partial Differential Equations’, Elsevier Science, B. V. North Holland, Amsterdam, Netherlanda, Chapter 4, pp. 211–309.CrossRefGoogle Scholar
[31]López-Gómez, J., Eilbeck, J. C., Duncan, K. & Molina-Meyer, M. (1992) Structure of solution manifolds in a strongly coupled elliptic system, IMA Conference on Dynamics of Numerics and Numerics of Dynamics (Bristol, 1990). IMA J. Numer. Anal. 12, 405–428.CrossRefGoogle Scholar
[32]López-Gómez, J. & Molina-Meyer, M. (1994) The maximum principle for cooperative weakly coupled elliptic systems and some applications. Diff. Int. Equ. 7, 383–398.Google Scholar
[33]López-Gómez, J. & Molina-Meyer, M. (2006) Superlinear indefinite systems: Beyond Lotka–Volterra models. J. Differ. Equ. 221, 343–411.Google Scholar
[34]López-Gómez, J. & Molina-Meyer, M. (2006) The competitive exclusion principle versus biodiversity through segregation and further adaptation to spatial heterogeneities. Theor. Popul. Biol. 69, 94–109.CrossRefGoogle ScholarPubMed
[35]López-Gómez, J. & Molina-Meyer, M. (2007) Biodiversity through co-opetition. Discrete Contin. Dyn. Syst. B 8, 187–205.Google Scholar
[36]López-Gómez, J. & Molina-Meyer, M. (2007) Modeling coopetition. Math. Comput. Simul. 76, 132–140.Google Scholar
[37]López-Gómez, J., Molina, M. & Villareal, M. (1992) Numerical coexistence of coexistence states. SIAM J. Numer. Anal. 29, 1074–1092.Google Scholar
[38]López-Gómez, J., Tellini, A. & Zanolin, F. (2014) High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Comm. Pure Appl. Anal. 13 (1), 1–73.Google Scholar
[39]Mawhin, J., Papini, D. & Zanolin, F. (2003) Boundary blow-up for differential equations with indefinite weight. J. Diff. Equ. 188, 33–51.Google Scholar
[40]Pugnaire, F. I. (Editor) (2010) Positive Plant Interactions and Community Dynamics, Fundación BBVA, CRC Press, Boca Raton, FL.Google Scholar
[41]Saffo, M. B. (1992) Invertebrates in endosymbiotic associations. Amer. Zool. 32, 557–565.Google Scholar
[42]Shoener, T. W. (1983) Field experiments on interspecific competition. Amer. Natur. 122, 240–285.Google Scholar
[43]Wulff, J. L. (1985) Clonal organisms and the evolution of mutualism. In: Jackson, J. B. C., Buss, L. W. & Cook, R. E. (editors), Population Biology and Evolution of Clonal Organisms, Yale University Press, New Haven, CT, pp. 437–466.Google Scholar