Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-20T00:24:10.678Z Has data issue: false hasContentIssue false
  • English
  • Français

Time delays in n-species competition – I: Global stability in constant environments

Published online by Cambridge University Press:  17 April 2009

K. Gopalsamy  and
R.A. Ahlip
K. Gopalsamy
Affiliation:
School of Mathematical Sciences, Flinders University of South Australia, Bedford Park, South Australia 5042, Australia.
R.A. Ahlip
Affiliation:
School of Mathematical Sciences, Flinders University of South Australia, Bedford Park, South Australia 5042, Australia.
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.

Sufficient conditions which are verifiable in a finite number of arithmetical steps are derived for the existence and global asymptotic stability of a feasible steady state in an integro-differential system modelling the dynamics of n competing species in a constant environment with delayed interspecific interactions. A novel method involving a nested sequence of “asymptotic” upper and lower bounds is developed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 27 , Issue 3 , June 1983 , pp. 427 - 441
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Busenberg, S.N. and Travis, C.C., “On the use of reducible-functional differential equations in biological models”, J. Math. Anal. Appl. 89 (1982), 46–66.CrossRefGoogle Scholar
[2]Coste, J., Peyraud, J. and Coullet, P., “Does complexity favor the existence of persistent ecosystems?”, J. Theoret. Biol. 73 (1978), 359–362.CrossRefGoogle ScholarPubMed
[3]Cushing, James M., Integrodifferential equations and delay models in population dynamics (Lecture Notes in Biomathematics, 20. Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[4]Gard, T.C. and Hallam, T.G., “Persistence in food webs – I: Lotka-Volterra food chains”, Bull. Math. Biol. 41 (1979), 877–891.Google Scholar
[5]Gard, Thomas C., “Persistence in food chains with general interactions”, Math. Biosci. 51 (1980), 165–174.CrossRefGoogle Scholar
[6]Gard, Thomas C., “Persistence in food webs: Holling-type food chains”, Math. Biosci. 49 (1980), 61–67.CrossRefGoogle Scholar
[7]Goh, Bean San, “Sector stability of a complex ecosystem model”, Math. Biosci. 40 (1978), 157–166.CrossRefGoogle Scholar
[8]Gopalsamy, K. and Aggarwala, B.D., “The logistic equation with a diffusionally coupled delay”, Bull. Math. Biol. 43 (1981),CrossRefGoogle Scholar
[9]Gopalsamy, K. and Aggarwala, B.D., “Limit cycles in two species competition with time delays”, J. Austral. Math. Soc. Ser. B 22 (1980/1981), 148–160.CrossRefGoogle Scholar
[10]Hallam, Thomas G., Svoboda, Linda J. and Gard, Thomas C., “Persistence and extinction in three species Lotka-Volterra competitive systems”, Math. Biosci. 46 (1979), 117–124.CrossRefGoogle Scholar
[11]Harrison, G.W., “Persistent sets via Lyapunov functions”, Nonlinear Anal. 3 (1979), 73–80.CrossRefGoogle Scholar
[12]Ikeda, M. and Šiljak, D.D., “Lotka-Volterra equations: decomposition, stability and structure”, J. Math. Biol. 9 (1980), 65–83.CrossRefGoogle Scholar
[13]Post, Wilfred M. and Travis, Curtis C., “Global stability in ecological models with continuous time delays”, Integral and functional differential equations, 241–250 (Lecture Notes in Pure and Applied Mathematics, 67. Marcel Dekker, New York and Basel, 1981).Google Scholar
[14]Volterra, V., Lecons sur la theorie mathematique de la lutte pour la vie (Gauthier-Villars, Paris, 1931).Google Scholar
[15]Wörz-Busekros, Angelika, “Global stability in ecological systems with continuous time delay”, SIAM J. Appl. Math. 35 (1978), 123–134.CrossRefGoogle Scholar