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On nonlinear dynamics of predator-prey models with discrete delay. (English) Zbl 1172.34046

This survey paper presents a review of some studies made by the author and collaborators on predator-prey models with discrete delays. The work starts with a short description of the models whose behaviour is the subject of the paper: Lotka-Volterra predator-prey models with discrete delays that can be included in the class of Kolmogorov-type models, as well as modified versions of non-Kolmogorov type; Gause-type predator-prey models with discrete delays and with harvesting and predator-prey delayed models with non-monotonic functional response. This introductory section contains an interesting and hepful list of references that provides a good overview of the existing work on these models.

Then, a good review of results on stability of steady-states of general nonlinear delayed differential equations through the corresponding linearized system, is presented. This analysis uses the distribution of zeros of some transcendental functions, mainly regarding the roots of second degree transcendental polynomial equations.

These theoretical results are applied in the rest of the work to analyze the stability and existence of bifurcations in terms of the values of the delays for the above-mentioned delayed predator-prey models.

For different versions of Kolmogorov-type models, existence of a positive equilibrium is established and a Hopf bifurcation occurs for a critical value of the delay. In some cases, the model exhibits switch stability for a sequence of critical values of the delay.

The behaviour of delayed predator-prey models with non-monotonic functional response is more complex. Depending on the values of the parameters of the model and of the values of the delay, Bogdanov-Takens bifurcations or Hopf bifurcations can occur.

Finally, the combined effect of constant-rate harvesting and delays on the dynamics of predator-prey systems is analyzed, for some versions of Gause-type models, demonstrating that the harvesting could induce more complex dynamics depending on which species is harvested. In the cases of prey harvesting, these models exhibit Hopf bifurcations, while in some cases of predator harvesting, Bogdanov-Takens bifurcations appear.

MSC:
34K18Bifurcation theory of functional differential equations
92D25Population dynamics (general)
34K20Stability theory of functional-differential equations
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