A Nonlinear Logistic Model for Age-Structured Populations: Analysis of Long-Term Dynamics and Equilibria

TL;DR

This work develops a nonlinear logistic model for age-structured populations by coupling density-dependent fertility with an increasing mortality function via a PDE for the age density $p(a,t)$ and a boundary renewal condition. The authors derive a finite-dimensional ODE system for the total population $P(t)$ and its age moments $P_i(t)$, and analyze equilibria through the net reproduction rate $R_n(x)$, proving that a unique nontrivial equilibrium exists if and only if $R_0>1$, with extinction when $R_0<1$. They establish equivalence between the PDE/renewal formulation and the ODE moment system, prove positivity and global existence, and provide an illustrative example that highlights extinction under certain parameter choices. The paper also discusses generalizations, numerical implementation, and future directions toward fractional, stochastic, and higher-complexity models, emphasizing practical impact for demographic and ecological settings.

Abstract

This paper investigates a nonlinear logistic model for age-structured population dynamics. The model incorporates interdependent fertility and mortality functions within a logistic framework, offering insights into stationary solutions and asymptotic behavior. Theoretical findings establish conditions for the existence and uniqueness of equilibrium solutions and explore long-term population dynamics. This study provides valuable tools for demographic modeling and opens avenues for further mathematical exploration.

Figures (1)

• Figure 1: Population Trajectories and Net Reproduction Rate under System (\ref{['stud']})

Theorems & Definitions (16)

• Lemma 3.1
• proof : Proof of Lemma \ref{['inv']}
• Lemma 3.2
• proof : Proof of Lemma \ref{['ineq']}
• Remark 3.3: Comparison Principle
• Theorem 4.1
• proof : Proof of Theorem \ref{['mp1']}
• Theorem 4.2
• proof : Proof of Theorem \ref{['mp3']}
• Theorem 4.3