Skip to main content
×
Home
    • Aa
    • Aa

Delayed responses and stability in two-species systems

  • K. Gopalsamy (a1)
Abstract
Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Delayed responses and stability in two-species systems
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Delayed responses and stability in two-species systems
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Delayed responses and stability in two-species systems
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] J. R. Beddington and R. M. May , “Time delays are not necessarily destabilising”, Math. Biosci. 27 (1975), 109117.

[3] R. Bellman and K. L. Cooke , Differential difference equations (Academic Press, New York, 1963).

[4] F. Brauer , “Stability of population models with delay”, Math. Biosci. 33 (1977), 345358.

[5] F. Brauer , “Decay rates for solutions of a class of differential difference equations”, SIAM J. Math. Anal. 10 (1979), 783788.

[6] M. Brelot , “Sur le problème biologique héréditaire de deux especès devorante et dévore”, Ann. Mat. Pura Appl. 9 (1931), 5874.

[7] J. B. Calhaun , “A method for self control of population growth among mammals living in wild”, Science 109 (1949), 333335.

[8] J. Caperon , “Time lag in population growth response of Isochrysis galbana to a variable nitrate environment”, Ecology 50 (1969), 188192.

[9] H. Caswell , “A simulation study of a time lag population model”, J. Theoret. Biol. 34 (1972), 419439.

[10] D. S. Cohen , E. Coutsias and J. C. Neu , “Stable oscillations in single species growth models with hereditary effects”, Math. Biosci. 44 (1979), 255268.

[12] J. M. Cushing , “Integrodifferential equations with delay models in population dynamics”, Lecture Notes in Biomathematics 20 (Springer-Verlag, Berlin, 1977).

[14] J. M. Cushing and M. Saleem , “A predator prey model with age structure”, J. Math. Biol. 14 (1982), 231250.

[17] J. K. Hale , Theory of functional differential equations (Springer-Verlag, New York, 1977).

[19] G. E. Hutchinson , “Circular causal systems in ecology”, Ann. N. Y. Acad. Sci. 50 (1948), 221246.

[20] G. S. Ladde , “Stability of model ecosystems with time delay”, J. Theoret. Biol. 61 (1976), 113.

[21] R. M. Lewis and B. D. O. Anderson , “Necessary and sufficient conditions for delay independent stability of linear autonomous systems”, IEEE Trans. Automat. Control 25 (1980), 735739.

[22] R. M. Lewis and B. D. O. Anderson , “Insensitivity of a class of nonlinear compartmental systems to the introduction of arbitrary time delays”, IEEE Trans. Circuits and Systems (1980), 604612.

[23] R. M. May , “Time delay versus stability in population models with two and three trophic levels”, Ecology 54 (1973), 315325.

[26] A. Rescigno and I. W. Richardson , “The struggle for life. I: Two species”, Bull. Math. Biophys. 29 (1967), 377388.

[28] A. Shibata and N. Saito , “Time delays and chaos in two competing species”, Math. Biosci. 51 (1980), 199211.

[29] D. D. Siljak , “When is a complex ecosystem stable?”, Math. Biosci. 25 (1975), 2550.

[33] P. J. Wangersky and W. J. Cunningham , “Time lag in population models”, Cold Spring Harbor Symp. Qual. Biol. 22 (1957), 329338.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 80 *
Loading metrics...

Abstract views

Total abstract views: 37 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 29th May 2017. This data will be updated every 24 hours.