<p>This paper investigates the asymptotic behavior of an age-structured population model with delayed birth process in a polluted environment. The model integrates nonlinear partial differential equations and integral equations to describe the dynamics of population density and toxicant concentration, incorporating age-dependent growth rate, mortality, and fertility influenced by both population size and pollution levels. The original model is first linearized around the positive stationary solution, and the resulting model is then transformed into an abstract Cauchy problem, for which well-posedness is established using semigroup theory. Under biologically reasonable assumptions, the associated semigroup is shown to be eventually compact and positive, and it satisfies the spectrum-determined growth property. Moreover, spectral analysis and eigenvalue techniques are employed to investigate the linear asymptotic stability and instability of the linearized model. Finally, several numerical examples are presented to demonstrate the theoretical results. This work provides new insights into the stability of structured populations under environmental pollution stress.</p>

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Stability analysis of an age-structured population model with delayed birth process in a polluted environment

  • An Ma,
  • Jing Hu,
  • Ming Ye,
  • Qimin Zhang

摘要

This paper investigates the asymptotic behavior of an age-structured population model with delayed birth process in a polluted environment. The model integrates nonlinear partial differential equations and integral equations to describe the dynamics of population density and toxicant concentration, incorporating age-dependent growth rate, mortality, and fertility influenced by both population size and pollution levels. The original model is first linearized around the positive stationary solution, and the resulting model is then transformed into an abstract Cauchy problem, for which well-posedness is established using semigroup theory. Under biologically reasonable assumptions, the associated semigroup is shown to be eventually compact and positive, and it satisfies the spectrum-determined growth property. Moreover, spectral analysis and eigenvalue techniques are employed to investigate the linear asymptotic stability and instability of the linearized model. Finally, several numerical examples are presented to demonstrate the theoretical results. This work provides new insights into the stability of structured populations under environmental pollution stress.