# Numerical Study of Nonlinear Dynamics of a Population System with Time Delay

## TL;DR

This paper investigates how introducing time delay into a nonlinear population model affects system dynamics, revealing richer behaviors and new orbits through numerical simulations based on generalized Volterra equations.

## Contribution

It generalizes a population interaction model to include time delay and demonstrates its impact on dynamics using numerical solutions with a modified Adams method.

## Key findings

- Time delay introduces new orbits in the phase space.
- The dynamics become more complex with delay, depending on parameters.
- Numerical solutions show significant impact of delay on population behavior.

## Abstract

Mathematical models of interacting populations are often constructed as systems of differential equations, which describe how populations change with time. Below we study one such model connected to the nonlinear dynamics of a system of populations in presence of time delay. The consequence of the presence of the time delay is that the nonlinear dynamics of the studied system become more rich, e.g., new orbits in the phase space of the system arise which are dependent on the time-delay parameters. In more detail we introduce a time delay and generalize the model system of differential equations for the interaction of three populations based on generalized Volterra equations in which the growth rates and competition coefficients of populations depend on the number of members of all populations \cite{Dimitrova2001a},\cite{Dimitrova2001b} and then numerically solve the system with and without time delay. We use a modification of the method of Adams for the numerical solution of the system of model equations with time delay. By appropriate selection of the parameters and initial conditions we show the impact of the delay time on the dynamics of the studied population system.