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The existence of periodic solutions for some models with delay. (English) Zbl 1095.34549

Consider the system

$\begin{array}{cc}\hfill \frac{d{v}_{1}\left(t\right)}{dt}& =\tau \left({v}_{1}^{*}+{v}_{1}\left(t\right)\right)\left[-{a}_{1}{v}_{1}\left(t\right)+{a}_{2}g\left({v}_{2}\left(t\right)\right)\right],\hfill \\ \hfill \frac{d{v}_{2}\left(t\right)}{dt}& =\tau \left({v}_{2}^{*}+{v}_{2}\left(t\right)\right)\left[-{a}_{1}{v}_{1}\left(t\right)+{a}_{2}g\left({v}_{2}\left(t\right)\right)-{a}_{3}g\left({v}_{2}\left(t-1\right)\right)\right],\hfill \end{array}$

where ${a}_{1},{a}_{2},{a}_{3},{v}_{1}^{*},{v}_{2}^{*}$ are constants and $\tau$ is the bifurcation parameter. The authors derives conditions such that it holds:

(i) There exists a sequence $\left\{{\tau }_{n}\right\}$ with ${\tau }_{n+1}>{\tau }_{n}$ such that (*) has a Hopf bifurcation at ${\tau }_{n}$, $n=0,1,1,\cdots$

(ii) For $\tau >{\tau }_{1}$, system (*) has at least one nonconstant periodic solution.

Finally, the results are applied to a predator-prey model with Michaelis-Menten type functional response.

##### MSC:
 34K18 Bifurcation theory of functional differential equations 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general)
##### Keywords:
Hopf bifurcation; periodic solutions; predator; prey