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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

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Delay-induced periodic phenomenon in a diffusive regulated logistic model.

Kejun Zhuang¹, Gao Jia²

¹Business School, University of Shanghai for Science and Technology, Shanghai, 200093 China ; School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, 233030 China.

|August 23, 2016

Summary

This study analyzes a diffusive logistic growth model with time delay and feedback control, establishing conditions for solution stability and exploring Hopf bifurcation. Numerical simulations validate the theoretical findings on population dynamics.

Keywords:

Delay Feedback control Hopf bifurcation Logistic model Reaction–diffusion system

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Area of Science:

Mathematical Biology
Dynamical Systems Theory
Population Dynamics

Background:

The logistic growth model is fundamental in population dynamics.
Incorporating time delays and feedback control introduces complex behaviors.
Understanding these complexities is crucial for ecological modeling.

Purpose of the Study:

To analyze a diffusive logistic growth model with time delay and feedback control.
To investigate the well-posedness and permanence of solutions.
To determine conditions for stability and the occurrence of Hopf bifurcation.

Main Methods:

Comparison techniques for well-posedness and permanence.
Stability analysis of steady states.
Normal form computation on the center manifold for bifurcation analysis.
Numerical simulations to verify theoretical results.

Main Results:

Sufficient conditions for the stability of nonnegative constant steady states are established.
The occurrence of Hopf bifurcation at positive steady states is demonstrated.
Bifurcation properties are derived using normal form theory.
The study generalizes and supplements existing research in delayed population models.

Conclusions:

The theoretical analysis provides a comprehensive understanding of the diffusive logistic growth model with time delay and feedback.
The findings offer insights into the complex dynamics, including stability and bifurcations.
Numerical simulations confirm the validity and applicability of the developed mathematical framework.