Stability and Hopf Bifurcation Analysis of the Delay Logistic Equation

TL;DR

This paper analyzes how delays affect the stability and bifurcation behavior of the logistic population growth model, providing insights into population fluctuations and stability conditions.

Contribution

It offers a detailed local stability and bifurcation analysis of the logistic equation with one and two delays, extending understanding to multiple delays.

Findings

Delays can induce oscillations or stabilize populations.

Conditions for stability and bifurcation are derived.

Analysis applicable to biological and marketing models.

Abstract

Logistic functions are good models of biological population growth. They are also popular in marketing in modelling demand-supply curves and in a different context, to chart the sales of new products over time. Delays being inherent in any biological system, we seek to analyse the effect of delays on the growth of populations governed by the logistic equation. In this paper, the local stability analysis, rate of convergence and local bifurcation analysis of the logistic equation with one and two delays is carried out and it can be extended to a system with multiple delays. Since fluctuating populations are susceptible to extinction due to sudden and unforeseen environmental disturbances, a knowledge of the conditions in which the population density is fluctuating or stable is of great interest in planning and designing control as well as management strategies.