Stability of the Logistic Population Model with Delayed Environmental Response

Abstract

The population dynamics model $\frac{{dy}}{{dt}} = \varepsilon y(t)\left( {1 - \frac{1}{N}\sum\limits_{k = 0}^n {a_k y(t - \tau _k )} } \right),{\text{ }}\varepsilon >0,N > 0,{\text{ }}a_k \geqslant 0,{\text{ }}\tau _k \geqslant 0{\text{ }}(0 \leqslant k \leqslant n),{\text{ }}\sum\limits_{k = 0}^n {a_k = 1}$, was considered. For this model with uniform distribution of delays $(\tau _k = k\tau {\text{,}}\tau >0)$ and a _n = 0, nonnegativeness and convexity of the sequence a _k (0 ≤ k ≤ n) was shown to be the sufficient stability condition. Therefore, there is no need to constrain the reproduction rate ɛ and the mean delay $\sum\limits_{k = 0}^n {a_k \tau _k }$.