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Positive solutions of a class of biological models in a heterogeneous environment

Published online by Cambridge University Press:  17 April 2009

Afshin Ghoreishi  and
Roger Logan
Afshin Ghoreishi
Affiliation:
Department of Mathematics, Bowdoin College Brunswick, ME 04011, United States of America
Roger Logan
Affiliation:
Department of Mathematics, The College of Charleston, Charleston SC 29424, United States of America
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Abstract

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In this paper we discuss existence of positive solutions to a general nonlinear elliptic system of reaction-diffusion equations representing a predator-prey or competition model of interaction between two species, in a heterogeneous environment. We also consider homogeneous Dirichlet and/or Robin boundary conditions. In the predator-prey case we give necessary and sufficient conditions for the existence of positive solutions, while in the competition case we give sufficient conditions. We use index theory in a positive cone to attack our problem and characterise our results by the sign of the first eigenvalues of certain Schrodinger type operators.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 1 , August 1991 , pp. 79 - 94
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Amann, H., ‘Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces’, SIAM Rev. 18 (1976), 620–709.CrossRefGoogle Scholar
[2]Beretyski, H. and Lions, P.L., ‘Some applications of the method of super and subsolutions’, in Lecture Notes in Math. 782, pp. 16–42 (Springer-Verlag, Berlin, Heidelberg, New York, 1980).Google Scholar
[3]Blat, J. and Brown, K.J., ‘Bifurcation of steady-state solutions in predator-prey and competition systems’, Proc. Roy. Soc. Edinburgh 97A (1984), 21–34.CrossRefGoogle Scholar
[4]Brown, P.N., ‘Decay to uniform states in ecological interactions’, SIAM J. Appl. Math. 38 (1980), 22–37.CrossRefGoogle Scholar
[5]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
[6]Cantrell, R. S. and Cosner, C., ‘On the uniqueness and stability of positive solutions in the Lotka-Volterra competition model with diffusion’, Houston J. Math, (to appear).Google Scholar
[7]Conway, E. D., Gardner, R. and Smoller, J., ‘Stability and bifurcation of steady-state solutions for predator-prey equations’, Adv. Appl. Math. 3 (1982), 288–334.CrossRefGoogle Scholar
[8]Dancer, E. N., ‘On positive solutions of some pairs of differential equations II,’, J. Differential Equations 60 (1985), 236–258.CrossRefGoogle Scholar
[9]Dancer, E.N., ‘On the existence and uniqueness of positive solutions for competing species models with diffusion’, (preprint).Google 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]Keller, C. and Lui, R., ‘Existence of steady-state solutions to predator-prey equations in a heterogeneous enviroment’, Math. Anal. Appl. 123 (1987), 306–323.CrossRefGoogle Scholar
[12]Korman, P. and Leung, A., ‘A general monotone scheme for elliptic systems with applications to ecological models’, Proc. Roy. Soc. Edinburgh 102A (1986), 315–325.CrossRefGoogle Scholar
[13]Korman, P. and Leung, A., ‘On the existence and uniqueness of positive steady-states in the Volterra-Lotka ecological model with diffusion’, Appl. Anal. 26 (1987), 145–160.CrossRefGoogle Scholar
[14]Li, L., Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1988), 143–166.CrossRefGoogle Scholar
[15]Li, L., On uniqueness of positive solution of a nonlinear elliptic system, J. Differential Equations (to appear).Google Scholar
[16]Li, L. and Ghoreishi, A., ‘On positive solutions of general nonlinear elliptic symbiotic interacting systems’, Appl. Anal, (to appear).Google Scholar
[18]McKenna, P. J. and Walter, W., ‘On the Dirichlet problem for elliptic systems’, Appl. Anal. 21 (1986), 207–224.CrossRefGoogle Scholar
[19]Pao, C. V., ‘On nonlinear reaction-diffusion systems’, Math. Anal. Appl. 87 (1982), 165–198.CrossRefGoogle Scholar
[20]Sattinger, D. H., ‘Monotone methods in nonlinear elliptic and parabolic boundary problems’, Indiana Univ. Math. J. 21 (1971), 979–1000.CrossRefGoogle Scholar
[21]Zygourakis, K. and Avis, R., ‘Weakly coupled systems of nonlinear elliptic boundary value problems’, Nonlinear Anal. 6 (1982), 553–569.CrossRefGoogle Scholar