[Paper Review] Analysis of bacterial population growth using extended logistic growth model with distributed delay

Tahani Ali Omer|arXiv (Cornell University)|Jul 21, 2018

Mathematical and Theoretical Epidemiology and Ecology Models7 references3 citations

TL;DR

This paper proposes an extended logistic growth model with distributed delay to analyze bacterial population dynamics under decontamination pressure. Using linear stability analysis and numerical simulations, it demonstrates that increased decontamination can drive populations to extinction or stabilize at a positive equilibrium, depending on delay and decay parameters.

ABSTRACT

In the present work, we develop a delayed Logistic growth model to study the effects of decontamination on the bacterial population in the ambient environment. Using the linear stability analysis, we study different case scenarios, where bacterial population may establish at the positive equilibrium or go extinct due to increased decontamination. The results are verified using numerical simulation of the model.

Motivation & Objective

• To model bacterial population growth under environmental decontamination using a delayed logistic framework.
• To investigate how distributed delays in population response affect long-term stability and extinction dynamics.
• To determine conditions under which bacterial populations stabilize at a positive equilibrium or go extinct due to decontamination.
• To validate analytical results through numerical simulations of the model under varying decontamination intensities.
• To provide a theoretical basis for predicting bacterial persistence or collapse in controlled or natural environments.

Proposed method

• Formulates a delayed logistic growth model with distributed delay to represent time-lagged responses in bacterial population dynamics.
• Incorporates decontamination as a time-dependent decay term affecting the population growth rate.
• Applies linear stability analysis to the equilibrium solutions of the delay differential equation system.
• Uses the characteristic equation of the delay model to determine stability conditions based on delay distribution and decontamination rate.
• Performs numerical simulations using specific parameter sets to visualize population trajectories under different decontamination scenarios.
• Analyzes the impact of varying delay distributions (e.g., uniform or gamma-distributed) on population outcomes.

Experimental results

Research questions

• RQ1 Under what conditions does the bacterial population stabilize at a positive equilibrium despite decontamination?
• RQ2 How does the distribution of delays in population response influence the likelihood of extinction?
• RQ3 What role does the intensity of decontamination play in shifting the system from persistence to extinction?
• RQ4 How do different delay distribution types affect the stability of the equilibrium solution?
• RQ5 Can numerical simulations confirm the stability predictions derived from linear analysis?

Key findings

• The model predicts that increased decontamination intensity can shift the system from a stable positive equilibrium to extinction.
• Distributed delays significantly affect the stability of the positive equilibrium, with certain delay distributions leading to oscillatory or unstable behavior.
• Linear stability analysis identifies critical thresholds for decontamination rate and delay parameters that determine population persistence or collapse.
• Numerical simulations confirm analytical predictions, showing that high decontamination rates lead to population extinction even with moderate delays.
• The system exhibits a bifurcation-like behavior at critical decontamination levels, where small increases can trigger a transition from persistence to extinction.
• The choice of delay distribution (e.g., uniform vs. gamma) alters the stability boundary, indicating that the shape of the delay distribution is biologically significant.