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A degree theoretic approach to N-species periodic competition systems on the whole ℝN

Published online by Cambridge University Press:  14 November 2011

Yihong Du
Yihong Du
Affiliation:
School of Mathematical and Computer Sciences, University of New England, Armidale, NSW 2351, Australia
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Abstract

Previous fixed-point index calculation results (see [4] and [7]) exploited in the study of population systems on bounded domains are no longer applicable to systems on the whole ℝn, due mainly to the lack of compactness. In this paper, we develop fixed-point index calculation techniques for non-compact operators so that they are applicable to systems on the whole ℝn. We illustrate the use of our fixed-point index calculation results by a simple representative model, namely, the Lotka–Volterra N-species periodic competition system on the whole ℝn.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 2 , 1999 , pp. 295 - 318
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Ali, S. W. and Cosner, C.. On the uniqueness of the positive steady state for Lotka–Volterra models with diffusion. J. Math. Analysis Applic. 168 (1992), 329–341.CrossRefGoogle Scholar
2Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620–709.CrossRefGoogle Scholar
3Brown, K. J. and Du, Y.. Bifurcation and monotonicity in competition reaction–diffusion systems. Nonlinear Analysis 23 (1994), 1–13.Google Scholar
4Dancer, E. N.. On the indices of fixed points of mappings in cones and applications. J. Math. Analysis Applic. 91 (1983), 131–151.CrossRefGoogle Scholar
5Dancer, E. N.. Multiple fixed points of positive mappings. J. Reine. Angew. Math. 371 (1986), 46–66.Google Scholar
6Dancer, E. N.. On positive solutions of some pairs of differential equations. II. J. Diffl Eqns 60 (1985), 236–258.CrossRefGoogle Scholar
7Dancer, E. N. and Du, Y.. Positive solutions for a three-species competition system with diffussion. I. General existence results. Nonlinear Analysis 24 (1995), 337–357.CrossRefGoogle Scholar
8Daners, D. and Koch-Medina, P.. Abstract evolution equations, periodic problems and applications (Harlow, Essex: Longman, 1992).Google Scholar
9Daners, D. and Koch-Medina, P.. Exponential stability, change of stability and eigenvalue problems for linear time periodic parabolic equations on ℝN. Diffl Integral Eqns 7 (1994), 1265–1284.Google Scholar
10Deimling, K.. Nonlinear functional analysis (Springer, 1985).CrossRefGoogle Scholar
11Du, Y.. Bifurcation from semi-trivial solution bundles and applications to certain equation systems. Nonlinear Analysis 27 (1996), 1407–1435.CrossRefGoogle Scholar
12Du, Y.. On Dancer's fixed point index formulas. Diffl Integral Eqns. (In the press.)Google Scholar
13Du, Y.. Positive periodic solutions of a competitor-competitor-mutualist model. Diffl Integral Eqns 9 (1996), 1043–1066.Google Scholar
14Eilbeck, J. C. and López-Gómez, J.. On the periodic Lotka–Volterra competition model. J. Math. Analysis Applic. 210 (1997), 58–87.CrossRefGoogle Scholar
15Gopalsamy, K.. Global asymptotic stability in a periodic Lotka-Volterra system. J. Austral. Math. Soc. B27 (1985), 66–72.CrossRefGoogle Scholar
16Hess, P.. Periodic-parabolic boundary value problems and positivity (Harlow, Essex: Longman, 1991).Google Scholar
17Hess, P. and Lazer, A. C.. On an abstract competition model and applications. Nonlinear Analysis 11 (1991), 917–940.CrossRefGoogle Scholar
18Hieber, M., Koch-Medina, P. and Merino, S.. Linear and semilinear parabolic equations on BUC(ℝN). Math. Nachrichten 179 (1996), 107–118.CrossRefGoogle Scholar
19Hieber, M., Koch-Madina, P. and Merino, S.. Diffusive logistic growth on ℝN. Nonlinear Analysis 27 (1996), 879–894.CrossRefGoogle Scholar
20Kato, T.. Perturbation theory for linear operators (New York: Springer, 1966).Google Scholar
21Medina, P. Koch and Schatti, G.. Long-time behaviour for reaction-diffusion equations on ℝN. Nonlinear Analysis 25 (1995), 831–870.CrossRefGoogle Scholar
22Krasnoselskii, M. A.. Positive solutions of operator equations (Groningen: Noordhoff, 1964).Google Scholar
23López-Gómez, L.. Positive periodic solutions of Volterra–Lotka reaction diffusion systems. Diffl Integral Eqns 1 (1992), 55–72.Google Scholar
24Merino, S.. Positive periodic solutions for semilinear reaction diffusion systems on ℝN. Adv. Diffl Eqns 1 (1996), 579–609.Google Scholar
25Mottoni, P. de and Schiafflno, A.. Competition systems with periodic soefficients: A geometric approach. J. Math. Biol. 11 (1981), 319–335.CrossRefGoogle Scholar
26Nussbaum, R. D.. The radius of the essential spectrum. Duke Math. J. 38 (1970), 473–478.Google Scholar
27Smith, H.. Periodic competitive differential equations and the discrete dynamics of competitive maps. J. Diffl Eqns 64 (1986), 165–194.CrossRefGoogle Scholar
28Tineo, A. R.. An iterative scheme for the N-competing species problem. J. Diffl Eqns 116 (1995), 1–15.CrossRefGoogle Scholar