Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T19:16:57.308Z Has data issue: false hasContentIssue false

Delayed responses and stability in two-species systems

Published online by Cambridge University Press:  17 February 2009

K. Gopalsamy
Affiliation:
School of Mathematical Sciences, Flinders University, Bedford Park, S.A. 5042.
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.

It is shown that if intraspecific self-regulating negative feedback effects are strong enough such that a nontrivial steady state of a two species system is locally asymptotically stable, then time delays in the positive feedback as well as in other interspecific interactions cannot destabilise the system and hence delay induced instability leading to persistent oscillations is impossible whatever the magnitude of the time delays. A method is also proposed for an estimate of decay rate of perturbations.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Anderson, B. D. O., “Time delays in large scale systems”, Proc. 18th Conf on Decision and Control, (1979).Google Scholar
[2]Beddington, J. R. and May, R. M., “Time delays are not necessarily destabilising”, Math. Biosci. 27 (1975), 109117.Google Scholar
[3]Bellman, R. and Cooke, K. L., Differential difference equations (Academic Press, New York, 1963).Google Scholar
[4]Brauer, F., “Stability of population models with delay”, Math. Biosci. 33 (1977), 345358.Google Scholar
[5]Brauer, F., “Decay rates for solutions of a class of differential difference equations”, SIAM J. Math. Anal. 10 (1979), 783788.Google Scholar
[6]Brelot, M., “Sur le problème biologique héréditaire de deux especès devorante et dévore”, Ann. Mat. Pura Appl. 9 (1931), 5874.Google Scholar
[7]Calhaun, J. B., “A method for self control of population growth among mammals living in wild”, Science 109 (1949), 333335.Google Scholar
[8]Caperon, J., “Time lag in population growth response of Isochrysis galbana to a variable nitrate environment”, Ecology 50 (1969), 188192.Google Scholar
[9]Caswell, H., “A simulation study of a time lag population model”, J. Theoret. Biol. 34 (1972), 419439.Google Scholar
[10]Cohen, D. S., Coutsias, E. and Neu, J. C., “Stable oscillations in single species growth models with hereditary effects”, Math. Biosci. 44 (1979), 255268.Google Scholar
[11]Coppel, W. A., Stability and asymptotic behaviour of differential equations (D. C. Heath and Company, Boston, Mass., 1965).Google Scholar
[12]Cushing, J. M., “Integrodifferential equations with delay models in population dynamics”, Lecture Notes in Biomathematics 20 (Springer-Verlag, Berlin, 1977).Google Scholar
[13]Cushing, J. M., “Volterra integrodifferential equations in population dynamics”, Proc. Centro Internazionale Matematico Estivo Summer Session on Mathematics in Biology (1979).Google Scholar
[14]Cushing, J. M. and Saleem, M., “A predator prey model with age structure”, J. Math. Biol. 14 (1982), 231250.Google Scholar
[15]Gopalsamy, K. and Aggarwala, B. D., “Limit cycles in two species competition models with time delays”, J. Austral. Math. Soc. Ser. B 22 (1980), 148160.CrossRefGoogle Scholar
[16]Gopalsanw., K.A delay induced bifurcation to oscillations”. J. Math. Phys. Sci. 16 (1982), 469488.Google Scholar
[17]Hale, J. K., Theory of functional differential equations (Springer-Verlag, New York, 1977).Google Scholar
[18]Hale, J. K., “Nonlinear oscillations in equations with delays”, in Lectures in Applied Math. 17 (Amer. Math. Soc., Providence, RI., 1979), 157185.Google Scholar
[19]Hutchinson, G. E., “Circular causal systems in ecology”, Ann. N. Y. Acad. Sci. 50 (1948), 221246.Google Scholar
[20]Ladde, G. S., “Stability of model ecosystems with time delay”, J. Theoret. Biol. 61 (1976), 113.Google Scholar
[21]Lewis, R. M. and Anderson, B. D. O., “Necessary and sufficient conditions for delay independent stability of linear autonomous systems”, IEEE Trans. Automat. Control 25 (1980), 735739.Google Scholar
[22]Lewis, R. M. and Anderson, B. D. O., “Insensitivity of a class of nonlinear compartmental systems to the introduction of arbitrary time delays”, IEEE Trans. Circuits and Systems (1980), 604612.Google Scholar
[23]May, R. M., “Time delay versus stability in population models with two and three trophic levels”, Ecology 54 (1973), 315325.CrossRefGoogle Scholar
[24]May, R. M., Stability and complexity in model ecosystems (Princeton University Press, Princeton, N.J., 1974).Google Scholar
[25]Nicholson, A. J., “The balance of animal populations”, J. Anim. Ecol. (1933), 132178.Google Scholar
[26]Rescigno, A. and Richardson, I. W., “The struggle for life. I: Two species”, Bull. Math. Biophys. 29 (1967), 377388.Google Scholar
[27]Ricklefs, R. E., Ecology (Chiron Press, Newton, Mass., 1974).Google Scholar
[28]Shibata, A. and Saito, N., “Time delays and chaos in two competing species”, Math. Biosci. 51 (1980), 199211.CrossRefGoogle Scholar
[29]Siljak, D. D., “When is a complex ecosystem stable?”, Math. Biosci. 25 (1975), 2550.Google Scholar
[30]Tamarin, R. H. (ed.), Population regulation (Dowden, Hutchinson and Ross., Inc., Stroudsburg, Pa., 1978).Google Scholar
[31]Vidyasager, M., Nonlinear system analysis(Prentice-Hall, Englewood Cliffs, N. J., 1978).Google Scholar
[32]Volterra, V., Lecons sur Ia théorie mathématique de Ia lutte pour la vie (Gauthier-Villars, Paris, 1931).Google Scholar
[33]Wangersky, P. J. and Cunningham, W. J., “Time lag in population models”, Cold Spring Harbor Symp. Qual. Biol. 22 (1957), 329338.Google Scholar