Stability and Instability in Age- and Sex-Structured Population Models with Discrete Time Delays

Abstract

The objective of this study is to explore the dynamics of human population growth through the use of time delay mathematical models. Traditional population models often assume instantaneous interactions and transitions between different stages of growth, which can oversimplify real-world population behaviour. In contrast, our study introduces delay differential equations (DDEs) that incorporate time delays to more accurately reflect the biological and social processes that influence population development over time. We focus particularly on models that account for the age structure of the population, recognizing that individuals pass through distinct stages—such as infancy, adolescence, adulthood, and old age—each with different rates of reproduction and mortality. Additionally, we examine cases where time delays are introduced based on sex-specific factors, acknowledging that males and females may contribute differently to population dynamics due to biological and sociocultural roles. Several model scenarios are considered, including discrete delays corresponding to reproduction and maturation times. For each case, we derive equilibrium solutions and perform a comprehensive stability analysis. Our results demonstrate that introducing discrete time delays can significantly alter the behaviour of the system, often leading to instability in population growth. This instability may manifest as population oscillations or divergence from equilibrium, highlighting the critical role that time delays play in shaping long-term demographic trends.