Stability analysis and optimal control of the logistic equation

TL;DR

This paper analyzes the stability of the logistic equation's equilibrium points using Lyapunov functions, introduces a new dynamic logistic model on arbitrary time scales, and applies optimal control techniques to maximize biomass yield.

Contribution

It presents a novel dynamic logistic equation valid on arbitrary time scales and applies optimal control methods to enhance biomass yield.

Findings

Stability of equilibrium points established via Lyapunov functions.

Proposed a new logistic model with biological significance.

Applied optimal control to improve biomass yield.

Abstract

We consider the unexploited/exploited logistic equation and study the stability of equilibrium points through Lyapunov functions. Then, we apply first and second order optimality conditions for the optimal control of the total biomass yield. Finally, we note that the time-scale logistic equation present in the literature lacks biological significance and we propose a new version of a dynamic logistic equation, valid on an arbitrary time scale, for which any trajectory of the system, beginning with a positive initial condition, remains nonnegative.