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Existence and stability of travelling wave solutions for an evolutionary ecology model

Published online by Cambridge University Press:  14 November 2011

Roger Lui
Affiliation:
Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, U.S.A.

Synopsis

Monotone travelling wave solutions are known to exist for Fisher's equation which models the propagation of an advantageous gene in a single locus, two alleles population genetics model. Fisher's equation assumed that the population size is a constant and that the fitnesses of the individuals in the population depend only on their genotypes. In this paper, we relax these assumptions and allow the fitnesses to depend also on the population size. Under certain assumptions, we prove that in the second heterozygote intermediate case, there exists a constant θ*>0 such that monotone travelling wave solutions for the reaction–diffusion system exist whenever θ > θ*. We also discuss the stability properties of these waves.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Aronson, D. G. and Weinberger, H. F.. Nonlinear diffusion in population genetics, combustion and nerve propatation. In Partial Differential Equations and related topics, ed. Goldstein, J., pp. 549. Lecture Notes in Mathematics 446 (New York: Springer, 1975).CrossRefGoogle Scholar
2Dunbar, S. R.. Travelling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in ℝ4. Trans. Amer. Math. Soc. 286 (1984), 557594.Google Scholar
3Fife, P.. Mathematical aspects of reacting and diffusing systems. Lecture Notes in Biomathematics 28 (New York: Springer, 1979).CrossRefGoogle Scholar
4Fisher, R. A.. The advance of advantageous genes. Ann. Eugenics 7 (1937), 355369.CrossRefGoogle Scholar
5Hartman, P.. Ordinary differential equations (New York: Wiley, 1964).Google Scholar
6Kolmogoroff, A., Petrovsky, I. and Piscounoff, N.. Étude de l'équations de la diffusion avec croissance de la quantité de matieré et son application a un probleme biologique. Bull. Univ. Moscow Ser. Internal. Sect. A 1 (1937), 125.Google Scholar
7Lui, R.. Asymptotic behavior of solutions to an evolutionary ecology model with diffusion. SIAM J. Applied Math. 49 (1989), 14471461.CrossRefGoogle Scholar
8Tang, M. M. and Fife, P.. Propagating fronts for competing species equations with diffusion. Arch. Rational Mech. Anal. 73 (1980), 6977.CrossRefGoogle Scholar
9Terman, D.. Comparison theorems for reaction-diffusion systems defined in an unbounded domain (Madison: University of Wisconsin MRC Technical Report #2374, April, 1982).Google Scholar
10Troy, W. C.. The existence of travelling wave front solutions of a model of the Belousov- Zhabotinsky chemical reaction. J. Differential Equations 36 (1980), 89–98.CrossRefGoogle Scholar