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Stability of the Logistic Population Model with Delayed Environmental Response

Authors:	M. M. Kipnis M. Yu. Vagina

Affiliation:	(1) Chelyabinsk State Pedagogical Institute, Chelyabinsk, Russia

Abstract:	The population dynamics model $frac{{dy}}{{dt}} = varepsilon y(t)left( {1 - frac{1}{N}sumlimits_{k = 0}^n {a_k y(t - tau _k )} } right),{text{ }}varepsilon >0,N > 0,{text{ }}a_k geqslant 0,{text{ }}tau _k geqslant 0{text{ }}(0 leqslant k leqslant n),{text{ }}sumlimits_{k = 0}^n {a_k = 1}$ , was considered. For this model with uniform distribution of delays $(tau _k = ktau {text{,}}tau >0)$ 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 and the mean delay $sumlimits_{k = 0}^n {a_k tau _k }$ .

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