Single-species population models with age structure and psychological effect in a polluted environment

TL;DR

This paper develops a mathematical model of a single population with age structure and psychological effects in polluted environments, analyzing stability and survival under deterministic and stochastic conditions.

Contribution

It introduces a novel age-structured population model incorporating psychological effects and environmental toxins, including stochastic contact rates, and analyzes stability and survival conditions.

Findings

Established stability conditions using Lyapunov methods.

Proved existence of unique positive solutions for stochastic models.

Identified conditions for long-term population survival.

Abstract

This paper considers a single-population model with age structure and psychological effects in a polluted environment. We divide the single population into two stages of larval and adult structure. The model uses Logistic input, and the larvae are converted into adult bodies by constant ratio. We only consider adulthood. The role of psychological effects makes the contact between adult and environmental toxins a functional form, while the contact between larvae and environmental toxins is linear. For the deterministic model embodied as a nonlinear time-varying system, we discuss the asymptotic stability of the system by Lyapunov one-time approximation theory, and give a sufficient condition for stability to be established. Considering that the contact rate between biological and environmental toxins in nature is not always constant, we make the contact rate interfere with white noise, and then modify the contact rate into a stochastic process, thus establishing a corresponding random single-population model. According to It\^o formula and Lyapunov in the function method, we first prove the existence of globally unique positive solutions for stochastic models under arbitrary initial conditions, and then give sufficient conditions for weak average long-term survival and random long-term survival for single populations in the expected sense.