Hostname: page-component-54dcc4c588-mz6gc Total loading time: 0 Render date: 2025-10-02T09:14:40.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonnegative solutions for a heterogeneous degenerate competition model

Published online by Cambridge University Press:  17 February 2009

Antonio Suárez
Antonio Suárez
Affiliation:
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Fac. Matemáticas, C/ Tarfia s/n, C.P. 41012, Univ. Sevilla, Sevilla, Spain; e-mail: suarez@us.es.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper deals with the existence, uniqueness and qualitative properties of nonnegative and nontrivial solutions of a spatially heterogeneous Lotka-Volterra competition model with nonlinear diffusion. We give conditions in terms of the coefficients involved in the setting of the problem which assure the existence of nonnegative solutions as well as the uniqueness of a positive solution. In order to obtain these results we employ monotonicity methods, singular spectral theory and a fixed point index.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 2 , October 2004 , pp. 273 - 297
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Amann, H., “On the existence of positive solutions of nonlinear elliptic boundary value problems”, Indiana Univ. Math. J. 21 (1971) 125–146.CrossRefGoogle Scholar
[2]Amann, H., “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces”, SIAM Rev. 18 (1976) 620–709.CrossRefGoogle Scholar
[3]Bandle, C., Pozio, M. A. and Tesei, A., “The asymptotic behaviour of the solutions of degenerate parabolic equations”, Trans. Amer. Math. Soc. 303 (1987)487–501.CrossRefGoogle Scholar
[4]Berestycki, H. and Lions, P. L., “Some applications of the method of sub and supersolutions”, in Bifurcation and nonlinear eigenvalue problem, Springer Lectures Notes 782, (Springer, 1980) 16–41.CrossRefGoogle Scholar
[5]Bertsch, M. and Rostamian, R., “The principle of linearized stability for a class of degenerate diffusion equations”, J. Differential Equations 57 (1985) 373–405.CrossRefGoogle Scholar
[6]Blat, J. and Brown, K. J., “Bifurcation of steady-state solutions in predator-prey and competition systems”, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984) 21–34.CrossRefGoogle Scholar
[7]Cantrell, R. S. and Cosner, C., “On the steady-state problem for the Volterra-Lotka competition model with diffusion”, Houston J. Math. 13 (1987) 337–352.Google Scholar
[8]Cantrell, R. S. and Cosner, C., “Should a park be an island?”, SIAM J. Math. Anal. 53 (1993)219–252.CrossRefGoogle Scholar
[9]Cosner, C. and Lazer, A. C., “Stable coexistence states in the Volterra-Lotka competition model with diffusion”, SIAM J. Appl. Math. 44 (1984)1112–1132.CrossRefGoogle Scholar
[10]Dancer, E. N., “On the indices of fixed points of mappings in cones and applications”, J. Math. Anal. Appl. 91 (1983) 131–151.CrossRefGoogle Scholar
[11]Dancer, E. N., “On positive solutions of some pairs of differential equations, part I”, Trans. Amer. Math. Soc. 284 (1984) 729–743.CrossRefGoogle Scholar
[12]Dancer, E. N., “On the existence and uniqueness of positive solutions for competing species models with diffusion”, Trans. Amer. Math. Soc. 326 (1991) 828–859.CrossRefGoogle Scholar
[13]Delgado, M. and Suárez, A., “On the existence of dead cores for degenerate Lotka-Volterra models”, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 743–766.CrossRefGoogle Scholar
[14]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.CrossRefGoogle Scholar
[15]Delgado, M. and Suárez, A., “Stability and uniqueness for cooperative degenerate Lotka-Volterra model”, Nonlinear Anal. 49 (2002) 757–778.CrossRefGoogle Scholar
[16]Díaz, J. I. and Hernández, J., “On the existence of a free boundary for a class of reaction-diffusion systems”, SIAM. J. Math. Anal. 15 (1984) 670–685.CrossRefGoogle Scholar
[17]Du, Y., “Effects of a degeneracy in the competition model: parts I and II”, J. Differential Equations 181 (2002) 92–132 and 133–164.CrossRefGoogle Scholar
[18]Du, Y. and Brown, K. J., “Bifurcation and monotonicity in competition reaction-diffusion systems”, Nonlinear Anal. 23 (1994) 1–13.CrossRefGoogle Scholar
[19]Eilbeck, J. C., Furter, J. E. and López-Gómez, J., “Coexistence in the competition model with diffusion”, J. Differential Equations 107 (1994) 96–139.CrossRefGoogle Scholar
[20]Furter, J. E. and López-Gómez, J., “On the existence and uniqueness of coexistence states for the Lotka-Volterra competition model with diffusion and spatially dependent coefficients”, Nonlinear Anal. 25 (1995) 363–398.CrossRefGoogle Scholar
[21]Furter, J. E. and López-Gómez, J., “Diffusion-mediated permanence problem for an heterogeneous Lotka-Volterra competition model”, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997) 281–336.CrossRefGoogle Scholar
[22]Gurtin, M. E. and MacCamy, R. C., “On the diffusion of biological populations”, Math. Biosci. 33 (1977) 35–49.CrossRefGoogle Scholar
[23]Leung, A. W. and Fan, G., “Existence of positive solutions for elliptic systems—degenerate and nondegenerate ecological models”, J. Math. Anal. Appl. 151 (1990) 512–531.CrossRefGoogle Scholar
[24]Li, L. and Logan, R., “Positive solutions to general elliptic competition models”, Differential Integral Equations 4 (1991) 817–834.CrossRefGoogle Scholar
[25]López-Gómez, J., “On the structure of the permanence region for the competing species models with general diffusivities and transports effects”, Discrete Contin. Dynam. Systems 2 (1996) 525–542.CrossRefGoogle Scholar
[26]López-Gómez, J., “Coexistence and metacoexistence states in competing species model”, Houston J. Math. 29 (2003) 483–536.Google Scholar
[27]López-Gómez, J. and Sabina De Lis, J. C., “Coexistence states and global attractivity for some convective diffusive competing species models”, Trans. Amer Math. Soc. 347 (1995) 3797–3833.CrossRefGoogle Scholar
[28]Cañada, A. and Gámez, J. L., “Elliptic systems with nonlinear diffusion in population dynamics”, Differential Equations Dynam. Systems 3 (1995) 189–204.Google Scholar
[29]Namba, T., “Density-dependent dispersal and spatial distribution of a population”, J. Theor. Biol. 86 (1980) 351–363.CrossRefGoogle ScholarPubMed
[30]Pozio, M. A. and Tesei, A., “Degenerate parabolic problems in population dynamics”, Japan J. Appl. Math. 2 (1985) 351–380.CrossRefGoogle Scholar
[31]Pozio, M. A. and Tesei, A., “Support properties of solutions for a class of degenerate parabolic problems”, Comm. Partial Differential Equations 12 (1987) 47–75.CrossRefGoogle Scholar
[32]Ruan, W. H. and Pao, C. V., “Positive steady-state solutions of a competing reaction-diffusion system”, J. Differential Equations 117 (1995) 401–427.CrossRefGoogle Scholar