## Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays.(English)Zbl 1221.37183

Summary: We investigate the stability and Hopf bifurcation of a new regulated logistic growth with discrete and distributed delays. By choosing the discrete delay $$\tau$$ as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as $$\tau$$ crosses a critical value. Furthermore, explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Finally, an illustrative example is also given to support the theoretical results.

### MSC:

 37N25 Dynamical systems in biology 92D25 Population dynamics (general) 34K18 Bifurcation theory of functional-differential equations 34K20 Stability theory of functional-differential equations

### Keywords:

Hopf bifurcation; stability; logistic growth; distributed delay
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### References:

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