Research and Innovation Articles
Published 2010-06-09
Keywords
- Population dynamics,
- non-local terms,
- non-linear diffusion
How to Cite
Corrêa, F. J., Delgado, M., & Suárez, A. (2010). Some Non-Local Population Models with Non-Linear Diffusion. Revista Integración, Temas De matemáticas, 28(1), 37–49. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/2058
Abstract
In this paper we present some theoretical results concerning to anon-local elliptic equation with non-linear diffusion arising from populationdynamics.
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References
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[2] Ambrosetti A., Brezis H., and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
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[8] Corrêa F.J.S.A., Delgado M., and Suárez A., Some non-local models with non-linear diffusion, to appear in Mathematical and Computer Modelling.
[9] Corrêa F.J.S.A., Delgado M., and Suárez A., Some non-local heterogeneous problems with non-linear diffusion, Advances in Differential Equations, 16 (2011), 622-641.
[10] Corrêa F.J.S.A., Delgado M., and Suárez A., A variational approach to a nonlocal elliptic problem with sign-changing nonlinearity, Advanced Nonlinear Studies, 11 (2011), 361-375.
[11] Davidson F.A., and Dodds N., Existence of positive solutions due to non-local interactions in a class of nonlinear boundary value problems, Methods Appl. Anal., 14 (2007), 15–27.
[12] Delgado M., and Suárez A., On the existence of dead cores for degenerate LotkaVolterra models, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 743-766.
[13] Delgado M., and Suárez A., On the structure of the positive solutions of the logistic equation with nonlinear diffusion, J. Math. Anal. Appl., 268 (2002), 200–216.
[14] Delgado M., and Suárez A., Nonnegative solutions for the degenerate logistic indefinite sublinear equation, Nonlinear Analysis, 52 (2003), 127-141.
[15] Delgado M., and Suárez A., Positive solutions for the degenerate logistic indefinite superlinear problem: the slow diffusion case, Houston Journal of Mathematics, 29 (2003), 801-823.
[16] Freitas P., Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations, J. Dynam. Differential Equations, 6 (1994), 613–629.
[17] Freitas P., A nonlocal Sturm-Liouville eigenvalue problem, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 169–188.
[18] Freitas P., Nonlocal reaction-diffusion equations, Differential equations with applicationsto biology (Halifax, NS, 1997), 187–204, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999.
[19] Freitas P., and Sweers G., Positivity results for a nonlocal elliptic equation, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 697–715.
[20] Freitas P., and Vishnevskii M.P., Stability of stationary solutions of nonlocal reaction-diffusion equations in m-dimensional space, Differential Integral Equations, 13 (2000), 265–288.
[21] Furter J., and Grinfeld M., Local vs. nonlocal interactions in population dynamics, J. Math. Biol., 27 (1989), 65–80.
[22] Gurtin M.E., and MacCamy R.C., On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49.
[23] Hernández J., Mancebo F., and Vega de Prada J.M., On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincare Anal.Non-Lineaire, 19 (2002), 777-813.
[24] López-Gómez J., On the structure and stability of the set of solutions of a nonlocal problem modeling Ohmic heating, J. Dynam. Differential Equations, 10 (1998), 537–566.
[25] Namba T., Density-dependent dispersal and spatial distribution of a population, J. Theor. Biol., 86 (1980), 351-363.
[26] Pozio M.A., and Tesei A., Support properties of solutions for a class of degenerate parabolic problems, Commun. Partial Differential Eqns., 12 (1987), 47-75.
[27] Rabinowitz P., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
[2] Ambrosetti A., Brezis H., and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
[3] Alama S., Semilinear elliptic equations with sublinear indefinite nonlinearities, Adv. in Differential Equations, 4 (1999), 813-842.
[4] Allegretto W., and Barabanova A., Positivity of solutions of elliptic equations with nonlocal terms, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 643–663.
[5] Allegretto W., and Barabanova A., Existence of positive solutions of semilinear elliptic equations with nonlocal terms, Funkcial. Ekvac., 40 (1997), 395–409.
[6] Arcoya D., Carmona J., and Pellacci B., Bifurcation for some quasi-linear operators, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 733–765.
[7] Bouguima S.M., Kada M., and Montero J.A., Bifurcation of positive solutions for a population dynamics model with nonlocal terms, preprint.
[8] Corrêa F.J.S.A., Delgado M., and Suárez A., Some non-local models with non-linear diffusion, to appear in Mathematical and Computer Modelling.
[9] Corrêa F.J.S.A., Delgado M., and Suárez A., Some non-local heterogeneous problems with non-linear diffusion, Advances in Differential Equations, 16 (2011), 622-641.
[10] Corrêa F.J.S.A., Delgado M., and Suárez A., A variational approach to a nonlocal elliptic problem with sign-changing nonlinearity, Advanced Nonlinear Studies, 11 (2011), 361-375.
[11] Davidson F.A., and Dodds N., Existence of positive solutions due to non-local interactions in a class of nonlinear boundary value problems, Methods Appl. Anal., 14 (2007), 15–27.
[12] Delgado M., and Suárez A., On the existence of dead cores for degenerate LotkaVolterra models, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 743-766.
[13] Delgado M., and Suárez A., On the structure of the positive solutions of the logistic equation with nonlinear diffusion, J. Math. Anal. Appl., 268 (2002), 200–216.
[14] Delgado M., and Suárez A., Nonnegative solutions for the degenerate logistic indefinite sublinear equation, Nonlinear Analysis, 52 (2003), 127-141.
[15] Delgado M., and Suárez A., Positive solutions for the degenerate logistic indefinite superlinear problem: the slow diffusion case, Houston Journal of Mathematics, 29 (2003), 801-823.
[16] Freitas P., Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations, J. Dynam. Differential Equations, 6 (1994), 613–629.
[17] Freitas P., A nonlocal Sturm-Liouville eigenvalue problem, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 169–188.
[18] Freitas P., Nonlocal reaction-diffusion equations, Differential equations with applicationsto biology (Halifax, NS, 1997), 187–204, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999.
[19] Freitas P., and Sweers G., Positivity results for a nonlocal elliptic equation, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 697–715.
[20] Freitas P., and Vishnevskii M.P., Stability of stationary solutions of nonlocal reaction-diffusion equations in m-dimensional space, Differential Integral Equations, 13 (2000), 265–288.
[21] Furter J., and Grinfeld M., Local vs. nonlocal interactions in population dynamics, J. Math. Biol., 27 (1989), 65–80.
[22] Gurtin M.E., and MacCamy R.C., On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49.
[23] Hernández J., Mancebo F., and Vega de Prada J.M., On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincare Anal.Non-Lineaire, 19 (2002), 777-813.
[24] López-Gómez J., On the structure and stability of the set of solutions of a nonlocal problem modeling Ohmic heating, J. Dynam. Differential Equations, 10 (1998), 537–566.
[25] Namba T., Density-dependent dispersal and spatial distribution of a population, J. Theor. Biol., 86 (1980), 351-363.
[26] Pozio M.A., and Tesei A., Support properties of solutions for a class of degenerate parabolic problems, Commun. Partial Differential Eqns., 12 (1987), 47-75.
[27] Rabinowitz P., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.