Spiraling bifurcation diagrams in superlinear indefinite problems

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April 2015, 35(4): 1561-1588. doi: 10.3934/dcds.2015.35.1561

Spiraling bifurcation diagrams in superlinear indefinite problems

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

Received August 2013 Revised April 2014 Published November 2014

Abstract
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This paper computes and discusses a series of intricate global bifurcation diagrams for a class of one-dimensional superlinear indefinite boundary value problems arising in population dynamics, under non-homogeneous boundary conditions, measured by $M>0$; the main bifurcation parameter being the amplitude $b$ of the superlinear terms.

Keywords: pseudo-spectral methods, path following, symmetry breaking, bifurcation and turning points, superlinear indefinite problems, Global bifurcation diagrams, isolas..

Mathematics Subject Classification: 65N35, 65P30, 65N2.

Citation: Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Spiraling bifurcation diagrams in superlinear indefinite problems. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1561-1588. doi: 10.3934/dcds.2015.35.1561

References:

[1]	S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215. doi: 10.1006/jfan.1996.0125.
[2]	E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, SIAM Classics in Applied Mathematics 45, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898719154.
[3]	H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns., 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.
[4]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Top. Meth. Nonl. Anal., 4 (1994), 59-78.
[5]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, Nonl. Diff. Eqns. Appns., 2 (1995), 553-572. doi: 10.1007/BF01210623.
[6]	M. D. Bertness and R. M. Callaway, Positive interactions in communities, Trends in Ecology and Evolution, 9 (1994), 191-193. doi: 10.1016/0169-5347(94)90088-4.
[7]	F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part I: Branches of Nonsingular Solutions, Numer. Math., 36 (1980), 1-25. doi: 10.1007/BF01395985.
[8]	F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part II: Limit Points, Numer. Math., 37 (1981), 1-28. doi: 10.1007/BF01396184.
[9]	F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part III: Simple bifurcation points, Numer. Math., 38 (1981), 1-30. doi: 10.1007/BF01395805.
[10]	R. M. Callaway and L. R. Walker, Competition and facilitation: A synthetic approach to interactions in plant communities, Ecology, 78 (1997), 1958-1965. doi: 10.2307/2265936.
[11]	C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods. Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.
[12]	A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Diff. Int. Eqns., 7 (1994), 411-439.
[13]	J. H. Connell, On the prevalence and relative importance of interspecific competition: evidence from field experiments, Amer. Natur., 122 (1983), 661-696. doi: 10.1086/284165.
[14]	M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.
[15]	M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation from simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180.
[16]	M. Crouzeix and J. Rappaz, On Numerical Approximation in Bifurcation Theory, Recherches en Mathématiques Appliquées 13. Masson, Paris, 1990.
[17]	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. doi: 10.1137/0907040.
[18]	J. García-Melián, Multiplicity of positive solutions to boundary blow up elliptic problems with sign-changing weights, J. Funct. Anal., 261 (2011), 1775-1798. doi: 10.1016/j.jfa.2011.05.018.
[19]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin-Heidelberg, 2001.
[20]	M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., 32 (1979), 21-98. doi: 10.1002/cpa.3160320103.
[21]	M. Golubitsky and D. G. Shaeffer, Singularity and Groups in Bifurcation Theory, Springer, 1985.
[22]	R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations, J. Diff. Eqns., 167 (2000), 36-72. doi: 10.1006/jdeq.2000.3772.
[23]	R. Gómez-Reñasco and J. López-Gómez, The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations, Diff. Int. Eqns., 14 (2001), 751-768.
[24]	G. E. Hutchinson, The Ecological Theater and the Evolutionary Play, Yale University Press, New Haven, Connecticut, 1965.
[25]	T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, 1995.
[26]	H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Tata Insitute of Fundamental Research, Springer, Berlin, 1987.
[27]	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.
[28]	J. López-Gómez, On the existence of positive solutions for some indefinite superlinear elliptic problems, Comm. Part. Diff. Eqns., 22 (1997), 1787-1804. doi: 10.1080/03605309708821320.
[29]	J. López-Gómez, Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems, Trans. Amer. Math. Soc., 352 (2000), 1825-1858. doi: 10.1090/S0002-9947-99-02352-1.
[30]	J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sci. Math. Jpn., 61 (2005), 493-516.
[31]	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, II (2005), 211-309. doi: 10.1016/S1874-5733(05)80012-9.
[32]	J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Diff. Eqns., 224 (2006), 385-439. doi: 10.1016/j.jde.2005.08.008.
[33]	J. López-Gómez, J. C. Eilbeck, K. Duncan and M. Molina-Meyer, 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 (1992), 405-428. doi: 10.1093/imanum/12.3.405.
[34]	J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Diff. Int. Eqns., 7 (1994), 383-398.
[35]	J. López-Gómez and M. Molina-Meyer, Superlinear indefinite systems: Beyond Lotka-Volterra models, J. Diff. Eqns., 221 (2006), 343-411. doi: 10.1016/j.jde.2005.05.009.
[36]	J. López-Gómez and M. Molina-Meyer, The competitive exclusion principle versus biodiversity through segregation and further adaptation to spatial heterogeneities, Theoretical Population Biology, 69 (2006), 94-109.
[37]	J. López-Gómez and M. Molina-Meyer, Biodiversity through co-opetition, Discrete and Continuous Dynamical Systems B, 8 (2007), 187-205. doi: 10.3934/dcdsb.2007.8.187.
[38]	J. López-Gómez and M. Molina-Meyer, Modeling coopetition, Mathematics and Computers in Simulation, 76 (2007), 132-140. doi: 10.1016/j.matcom.2007.01.035.
[39]	J. López-Gómez, M. Molina-Meyer and A. Tellini, The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions, J. Diff. Eqns., 255 (2013), 503-523. doi: 10.1016/j.jde.2013.04.019.
[40]	J. López-Gómez, M. Molina-Meyer and M. Villareal, Numerical computation of coexistence states, SIAM J. Numer. Anal., 29 (1992), 1074-1092. doi: 10.1137/0729065.
[41]	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. doi: 10.3934/cpaa.2014.13.1.
[42]	J. Mawhin, D. Papini and F. Zanolin, Boundary blow-up for differential equations with indefinite weight, J. Diff. Eqns., 188 (2003), 33-51. doi: 10.1016/S0022-0396(02)00073-6.
[43]	L. Ping, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.
[44]	F. I. Pugnaire (Editor), Positive Plant Interactions and Community Dynamics, Fundación BBVA, CRC Press, Boca Raton, 2010.
[45]	M. B. Saffo, Invertebrates in endosymbiotic associations, Amer. Zool., 32 (1992), 557-565. doi: 10.1093/icb/32.4.557.
[46]	T. W. Shoener, Field experiments on interspecific competition, Amer. Natur., 122 (1983), 240-285. doi: 10.1086/284133.
[47]	J. L. Wulff, Clonal organisms and the evolution of mutualism. In Jackson, J.B.C., Buss, L.W., Cook, R.E. (Eds.), Population Biology and Evolution of Clonal Organisms, 437-466, Yale University Press, New Haven, Connecticut, 1985.

show all references

References:

[1]	S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215. doi: 10.1006/jfan.1996.0125.
[2]	E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, SIAM Classics in Applied Mathematics 45, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898719154.
[3]	H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns., 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.
[4]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Top. Meth. Nonl. Anal., 4 (1994), 59-78.
[5]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, Nonl. Diff. Eqns. Appns., 2 (1995), 553-572. doi: 10.1007/BF01210623.
[6]	M. D. Bertness and R. M. Callaway, Positive interactions in communities, Trends in Ecology and Evolution, 9 (1994), 191-193. doi: 10.1016/0169-5347(94)90088-4.
[7]	F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part I: Branches of Nonsingular Solutions, Numer. Math., 36 (1980), 1-25. doi: 10.1007/BF01395985.
[8]	F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part II: Limit Points, Numer. Math., 37 (1981), 1-28. doi: 10.1007/BF01396184.
[9]	F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part III: Simple bifurcation points, Numer. Math., 38 (1981), 1-30. doi: 10.1007/BF01395805.
[10]	R. M. Callaway and L. R. Walker, Competition and facilitation: A synthetic approach to interactions in plant communities, Ecology, 78 (1997), 1958-1965. doi: 10.2307/2265936.
[11]	C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods. Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.
[12]	A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Diff. Int. Eqns., 7 (1994), 411-439.
[13]	J. H. Connell, On the prevalence and relative importance of interspecific competition: evidence from field experiments, Amer. Natur., 122 (1983), 661-696. doi: 10.1086/284165.
[14]	M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.
[15]	M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation from simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180.
[16]	M. Crouzeix and J. Rappaz, On Numerical Approximation in Bifurcation Theory, Recherches en Mathématiques Appliquées 13. Masson, Paris, 1990.
[17]	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. doi: 10.1137/0907040.
[18]	J. García-Melián, Multiplicity of positive solutions to boundary blow up elliptic problems with sign-changing weights, J. Funct. Anal., 261 (2011), 1775-1798. doi: 10.1016/j.jfa.2011.05.018.
[19]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin-Heidelberg, 2001.
[20]	M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., 32 (1979), 21-98. doi: 10.1002/cpa.3160320103.
[21]	M. Golubitsky and D. G. Shaeffer, Singularity and Groups in Bifurcation Theory, Springer, 1985.
[22]	R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations, J. Diff. Eqns., 167 (2000), 36-72. doi: 10.1006/jdeq.2000.3772.
[23]	R. Gómez-Reñasco and J. López-Gómez, The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations, Diff. Int. Eqns., 14 (2001), 751-768.
[24]	G. E. Hutchinson, The Ecological Theater and the Evolutionary Play, Yale University Press, New Haven, Connecticut, 1965.
[25]	T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, 1995.
[26]	H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Tata Insitute of Fundamental Research, Springer, Berlin, 1987.
[27]	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.
[28]	J. López-Gómez, On the existence of positive solutions for some indefinite superlinear elliptic problems, Comm. Part. Diff. Eqns., 22 (1997), 1787-1804. doi: 10.1080/03605309708821320.
[29]	J. López-Gómez, Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems, Trans. Amer. Math. Soc., 352 (2000), 1825-1858. doi: 10.1090/S0002-9947-99-02352-1.
[30]	J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sci. Math. Jpn., 61 (2005), 493-516.
[31]	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, II (2005), 211-309. doi: 10.1016/S1874-5733(05)80012-9.
[32]	J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Diff. Eqns., 224 (2006), 385-439. doi: 10.1016/j.jde.2005.08.008.
[33]	J. López-Gómez, J. C. Eilbeck, K. Duncan and M. Molina-Meyer, 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 (1992), 405-428. doi: 10.1093/imanum/12.3.405.
[34]	J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Diff. Int. Eqns., 7 (1994), 383-398.
[35]	J. López-Gómez and M. Molina-Meyer, Superlinear indefinite systems: Beyond Lotka-Volterra models, J. Diff. Eqns., 221 (2006), 343-411. doi: 10.1016/j.jde.2005.05.009.
[36]	J. López-Gómez and M. Molina-Meyer, The competitive exclusion principle versus biodiversity through segregation and further adaptation to spatial heterogeneities, Theoretical Population Biology, 69 (2006), 94-109.
[37]	J. López-Gómez and M. Molina-Meyer, Biodiversity through co-opetition, Discrete and Continuous Dynamical Systems B, 8 (2007), 187-205. doi: 10.3934/dcdsb.2007.8.187.
[38]	J. López-Gómez and M. Molina-Meyer, Modeling coopetition, Mathematics and Computers in Simulation, 76 (2007), 132-140. doi: 10.1016/j.matcom.2007.01.035.
[39]	J. López-Gómez, M. Molina-Meyer and A. Tellini, The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions, J. Diff. Eqns., 255 (2013), 503-523. doi: 10.1016/j.jde.2013.04.019.
[40]	J. López-Gómez, M. Molina-Meyer and M. Villareal, Numerical computation of coexistence states, SIAM J. Numer. Anal., 29 (1992), 1074-1092. doi: 10.1137/0729065.
[41]	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. doi: 10.3934/cpaa.2014.13.1.
[42]	J. Mawhin, D. Papini and F. Zanolin, Boundary blow-up for differential equations with indefinite weight, J. Diff. Eqns., 188 (2003), 33-51. doi: 10.1016/S0022-0396(02)00073-6.
[43]	L. Ping, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.
[44]	F. I. Pugnaire (Editor), Positive Plant Interactions and Community Dynamics, Fundación BBVA, CRC Press, Boca Raton, 2010.
[45]	M. B. Saffo, Invertebrates in endosymbiotic associations, Amer. Zool., 32 (1992), 557-565. doi: 10.1093/icb/32.4.557.
[46]	T. W. Shoener, Field experiments on interspecific competition, Amer. Natur., 122 (1983), 240-285. doi: 10.1086/284133.
[47]	J. L. Wulff, Clonal organisms and the evolution of mutualism. In Jackson, J.B.C., Buss, L.W., Cook, R.E. (Eds.), Population Biology and Evolution of Clonal Organisms, 437-466, Yale University Press, New Haven, Connecticut, 1985.

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