Periodicity in a logistic type system with several delays.

*(English)*Zbl 1061.34050Summary: With the help of a continuation theorem based on Gaines and Mawhin’s coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following logistic-type system with several delays
\[
\frac{du(t)}{dt}=u(t)\left[a(t)-\sum^m_{j=1}b_j(t) \biggl(u\bigl(t-\tau_j(t)\bigr)\biggr)^{\theta_j} \right],
\]
where \(a(t)\), \(b_j(t)\) are positive periodic continuous functions with periodic \(\omega>0\), \(\tau_j(t)\) are nonnegative continuous periodic functions with periodic \(\omega>0\). After that, by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the stability of the positive periodic solutions are obtained.

##### MSC:

34K13 | Periodic solutions to functional-differential equations |

34K20 | Stability theory of functional-differential equations |

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\textit{F. Chen} and \textit{J. Shi}, Comput. Math. Appl. 48, No. 1--2, 35--44 (2004; Zbl 1061.34050)

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##### References:

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