[Paper Review] Population Models With Delay in Dynamic Environment

Leonid Berezansky, Lev Idels|ArXiv.org|Jan 5, 2006

Mathematical and Theoretical Epidemiology and Ecology Models15 references3 citations

TL;DR

This paper develops a delay differential equation model of fish population dynamics under periodic environmental changes and harvesting, extending the Getz model with time-varying carrying capacity and survival rates. It establishes conditions for global existence of solutions, persistence, extinction, and the existence of periodic solutions, demonstrating that environmental periodicity and harvesting delays critically influence population stability and sustainability in marine ecosystems.

ABSTRACT

We study the combined effects of periodically varying carrying capacity and survival rates on the fish population in the ocean (sea). We introduce the Getz type delay differential equation model with a control parameter which describes how fish are harvested. We will modify and extend harvesting model of an exploited fish population to include periodic and rotational harvesting rates. We study the existence of global solutions for the initial value problem, extinction and persistence conditions, and the existence of periodic solutions.

Motivation & Objective

• To model fish population dynamics in a dynamic ocean environment with periodically varying carrying capacity and survival rates.
• To extend the Getz-type delay differential equation model to include control parameters for harvesting.
• To analyze the existence of global solutions, persistence, and extinction conditions in the presence of time delays.
• To investigate the existence of periodic solutions under periodic environmental forcing.
• To provide theoretical conditions for sustainable harvesting under environmental variability and time delays.

Proposed method

• Formulates a delay differential equation model with time-varying coefficients to represent periodic changes in carrying capacity and survival rates.
• Introduces a control parameter for harvesting, allowing modeling of both constant and periodic harvesting rates.
• Applies dynamical systems theory to analyze the initial value problem, ensuring global existence of solutions.
• Uses stability and comparison techniques to derive conditions for population persistence and extinction.
• Employs periodicity analysis to establish the existence of positive periodic solutions under appropriate environmental forcing.
• Relies on mathematical tools from functional differential equations, particularly in the context of 34K20 and 34D classifications.

Experimental results

Research questions

• RQ1 Under what conditions does the population persist or go extinct in the presence of time delays and periodic environmental changes?
• RQ2 Can periodic solutions exist in the population model when the environment and harvesting rates vary periodically?
• RQ3 How do time delays in population response affect the stability and long-term behavior of fish populations?
• RQ4 What role does the control parameter for harvesting play in determining population sustainability?
• RQ5 How do periodically varying carrying capacity and survival rates influence the existence and properties of global solutions?

Key findings

• Global solutions to the initial value problem exist under mild conditions on the time-varying coefficients and delay structure.
• Sufficient conditions for population persistence are derived based on the average growth rate and delay effects.
• Extinction is possible when the combined effect of delayed response and unfavorable environmental conditions leads to negative net growth.
• The existence of at least one positive periodic solution is established under periodic forcing of the carrying capacity and survival rates.
• The model demonstrates that periodic harvesting can stabilize or destabilize population dynamics depending on phase and amplitude relative to environmental cycles.
• The results suggest that ignoring time delays and environmental periodicity may lead to inaccurate predictions in fisheries management.