The nature of Hopf bifurcation for the Gompertz model with delays. (English) Zbl 1235.34191
Summary: We study the influence of time delays on the dynamics of the classical Gompertz model. We consider the models with one discrete delay introduced in two different ways and the model with two delays which generalise those with one delay. We study asymptotic behaviour and bifurcations with respect to the ratio of delays \(\tau=\tau_1/\tau_2\). Our results show that in such model with two delays there is only one stability switch and for a threshold value of bifurcation parameter, Hopf bifurcation (HB) occurs. However, the type of HB, and therefore its stability (i.e. stability of periodic orbits arising due to it), strongly depends on the magnitude of \(\tau\). The function describing stability of HB is periodic with respect to \(\tau\). Within one period of length 4 five changes of HB stability are observed.
MSC:
| 34K18 | Bifurcation theory of functional-differential equations |
| 37N25 | Dynamical systems in biology |
| 92C37 | Cell biology |
| 92D25 | Population dynamics (general) |
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