American Institute of Mathematical Sciences

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

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.
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. [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. [3] F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part II: Limit Points, Numer. Math., 37 (1981), 1-28. [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. [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. [6] H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Tata Insitute of Fundamental Research, Springer, Berlin, 1986. [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. [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. [9] J. López-Gómez, M. Molina-Meyer and A. Tellini, Spiraling bifurcation diagrams in superlinear indefinite problems,, Submitted., (). [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.

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. [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. [3] F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part II: Limit Points, Numer. Math., 37 (1981), 1-28. [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. [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. [6] H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Tata Insitute of Fundamental Research, Springer, Berlin, 1986. [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. [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. [9] J. López-Gómez, M. Molina-Meyer and A. Tellini, Spiraling bifurcation diagrams in superlinear indefinite problems,, Submitted., (). [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.
 [1] Julián López-Góme, Andrea Tellini, F. Zanolin. High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Communications on Pure and Applied Analysis, 2014, 13 (1) : 1-73. doi: 10.3934/cpaa.2014.13.1 [2] 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 [3] Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561 [4] Andrea Tellini. Imperfect bifurcations via topological methods in superlinear indefinite problems. Conference Publications, 2015, 2015 (special) : 1050-1059. doi: 10.3934/proc.2015.1050 [5] Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047 [6] M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure and Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411 [7] Behrouz Kheirfam. A weighted-path-following method for symmetric cone linear complementarity problems. Numerical Algebra, Control and Optimization, 2014, 4 (2) : 141-150. doi: 10.3934/naco.2014.4.141 [8] Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014 [9] Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277 [10] Zeyu Xia, Xiaofeng Yang. A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3749-3763. doi: 10.3934/dcdsb.2020089 [11] Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, 2021, 29 (5) : 2915-2944. doi: 10.3934/era.2021019 [12] Xuemei Zhang, Meiqiang Feng. Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2149-2171. doi: 10.3934/cpaa.2018103 [13] J. F. Toland. Path-connectedness in global bifurcation theory. Electronic Research Archive, 2021, 29 (6) : 4199-4213. doi: 10.3934/era.2021079 [14] D. Motreanu, Donal O'Regan, Nikolaos S. Papageorgiou. A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1791-1816. doi: 10.3934/cpaa.2011.10.1791 [15] Julián López-Gómez, Pavol Quittner. Complete and energy blow-up in indefinite superlinear parabolic problems. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 169-186. doi: 10.3934/dcds.2006.14.169 [16] Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436 [17] Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018 [18] Yuxia Guo, Shaolong Peng. Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1637-1648. doi: 10.3934/cpaa.2022037 [19] Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations and Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1 [20] Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107

Impact Factor:

Metrics

• HTML views (0)
• Cited by (0)

• on AIMS