Hostname: page-component-7dd5485656-j7khd Total loading time: 0 Render date: 2025-10-31T00:30:54.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Equations for biological evolution*

Published online by Cambridge University Press:  14 November 2011

Angel Calsina  and
Carles Perelló
Angel Calsina
Affiliation:
Departament de Matematiques, Universitat Autonoma de Barcelona, Edifici C, 08193 Bellaterra (Barcelona), Spain
Carles Perelló
Affiliation:
Departament de Matematiques, Universitat Autonoma de Barcelona, Edifici C, 08193 Bellaterra (Barcelona), Spain
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we consider mathematical models inspired by the mechanisms of biological evolution. We take populations which are subject to interaction and mutation. In the cases we consider, the interaction is through competition or through a prey-predator relationship. The models consider the specific characteristics as taking values in real intervals and the equations are of the integro—differential type. In the case of competition, thanks to the fact that some of the equations have solutions which are quite explicit, we succeed in proving the existence of attracting stationary solutions. In the case of prey and predator, using techniques of dynamical systems in infinite-dimensional spaces, we succeed in showing the existence of a global attractor, which in some instances reduces to a point. Our analysis takes into account having δ distributions, corresponding to all individuals having the same characteristics, as possible populations.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 5 , 1995 , pp. 939 - 958
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Calsina, A., Perelló, C. and Saldaña, J.. Non-local reaction diffusion equations modelling predatorprey coevolution. Publ. Mat. 38 (1994), 315–25.CrossRefGoogle Scholar
2Chan, W. L. and Guo, B.-Z.. Global behaviour of age dependent logistic population models. J. Math. Biol. 28(1990), 225–35.Google Scholar
3Fiedler, B. and Polácik, P.. Complicated dynamics of scalar diffusion equations with a nonlocal term. Proc. Roy. Soc. Edinburgh Sect. A 125 (1990), 167–92.CrossRefGoogle Scholar
4Freitas, P.. A non-local Sturm-Liouville eigenvalue problem. Proc. Roy. Soc. Edinburgh Sect. A (to appear).Google Scholar
5Freitas, P.. Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations (preprint).Google Scholar
6Friedman, A.. Partial Differential Equations (New York: Holt-Rinehart-Wilson, 1969).Google Scholar
7Furter, J. and Grinfeld, M.. Local vs. nonlocal interactions in Population dynamics. J. Math. Biol. 27 (1989), 65–80.CrossRefGoogle Scholar
8Hale, J. K.. Asymptotic Behaviour of Dissipative Systems. Math. Surveys Monographs 25 (Providence, R.I.: American Mathematical Society, 1988).Google Scholar
9Henry, D.. Geometric Theory of Semilinear Parabolic Equations (Berlin: Springer, 1981).Google Scholar
10Küppers, B.. Molecular Theory of Evolution (Berlin: Springer, 1983).CrossRefGoogle Scholar
11Lawlor, L. R. and Smith, J. Maynard. The coevolution and stability of competing species. Amer. Nat. 110(1976), 79–99.CrossRefGoogle Scholar
12Lions, J. L. and Magènes, E.. Problèmes aux Limites Non Homogènes et Applications, vol. 1 (Paris: Dunod, 1968).Google Scholar
13Smith, J. Maynard and Price, G. R.. The logic of animal conflict. Nature 246 (1973), 15–18.CrossRefGoogle Scholar
14Metz, J. A. J. and Diekman, O.. IV Age dependence, The dynamics of Physiologically structured populations (Berlin: Springer, 1986).CrossRefGoogle Scholar
15Miklavcic, M.. Stability for semilinear parabolic equations with non-invertible linear operator. Pacific J. Math. 118 (1985), 199–214.CrossRefGoogle Scholar
16Miletta, P. D.. An evolution equation with nonlocal nonlinearities: Existence, uniqueness and asymptotic behaviour. Math. Methods Appl. Sci. 10 (1988), 407–25.Google Scholar
17Mora, X.. Sistemes dinamics determinats per equacions diferencials semilineals sobre espais de Banach (Doctoral Thesis, Universitat Autònoma de Barcelona, 1982).Google Scholar
18Pazy, A.. Semigroups of linear operators and applìcations to partial differential equations (Berlin: Springer, 1983).CrossRefGoogle Scholar
19Rand, D. A., Wilson, H. B. and McGlade, J. M.. Dynamics and evolution: evolutionary stable attractor, invasion exponents and phenotype dynamics (Warwick Preprints: 53/1992).Google Scholar
20Rubinstein, J. and Sternberg, P.. Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48(1992), 249–64.CrossRefGoogle Scholar
21Temam, R.. Infinite Dimensional Dynamical Systems in Mechanics and Physics (Berlin: Springer, 1988).CrossRefGoogle Scholar
22Webb, G. P.. Logistic models of structured population growth. Comput. Math. Anal. 12 (1986), 527–39.Google Scholar