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Analysis of Discrete Stochastic Population Models with Normal Distribution

Haiyan Wang

TL;DR

This work studies a discrete-time stochastic logistic model with $X_t ~ N(μ, σ^2)$ under a small perturbation $ε_t$ with $E[ε_t]=1$, and defines a uniform structural growth rate $r$ that enforces $E[X_{t+1}] = α E[X_t]$. The main result expresses $r$ as positive roots of a cubic in $r$ whose coefficients depend on distributional parameters and the perturbation variance, and shows that two positive roots (dual growth states) exist under a sufficent condition, while there are infeasible ranges of $μ$ where no such $r$ exists. Biologically, the two branches correspond to high- and low-growth population states under stochasticity, with α, β, and Var(ε_t) modulating the required growth rate and system stability. The findings provide a theoretical framework for understanding how distributional structure and stochastic perturbations shape growth dynamics, and motivate extensions to other distributions and empirical validation.

Abstract

This paper analyzes a stochastic logistic difference equation under the assumption that the population distribution follows a normal distribution. Our focus is on the mathematical relationship between the average growth rate and a newly introduced concept, the uniform structural growth rate, which captures how growth is influenced by the internal distributional structure of the population. We derive explicit relationships linking the uniform structural growth rate to the parameters of the normal distribution and the variance of a small stochastic perturbation. The analysis reveals the existence of two distinct branches of the uniform structural growth rate, corresponding to alternative population states characterized by higher and lower growth rates. This duality provides deeper insights into the dynamics of population growth under stochastic influences. A sufficient condition for the existence of two uniform structural growth rates is established and rigorously proved, demonstrating that there exist infeasible intervals where no uniform structural growth rate can be defined. We also explore the biological significance of these findings, emphasizing the role of stochastic perturbations and the distribution in shaping population dynamics.

Analysis of Discrete Stochastic Population Models with Normal Distribution

TL;DR

This work studies a discrete-time stochastic logistic model with under a small perturbation with , and defines a uniform structural growth rate that enforces . The main result expresses as positive roots of a cubic in whose coefficients depend on distributional parameters and the perturbation variance, and shows that two positive roots (dual growth states) exist under a sufficent condition, while there are infeasible ranges of where no such exists. Biologically, the two branches correspond to high- and low-growth population states under stochasticity, with α, β, and Var(ε_t) modulating the required growth rate and system stability. The findings provide a theoretical framework for understanding how distributional structure and stochastic perturbations shape growth dynamics, and motivate extensions to other distributions and empirical validation.

Abstract

This paper analyzes a stochastic logistic difference equation under the assumption that the population distribution follows a normal distribution. Our focus is on the mathematical relationship between the average growth rate and a newly introduced concept, the uniform structural growth rate, which captures how growth is influenced by the internal distributional structure of the population. We derive explicit relationships linking the uniform structural growth rate to the parameters of the normal distribution and the variance of a small stochastic perturbation. The analysis reveals the existence of two distinct branches of the uniform structural growth rate, corresponding to alternative population states characterized by higher and lower growth rates. This duality provides deeper insights into the dynamics of population growth under stochastic influences. A sufficient condition for the existence of two uniform structural growth rates is established and rigorously proved, demonstrating that there exist infeasible intervals where no uniform structural growth rate can be defined. We also explore the biological significance of these findings, emphasizing the role of stochastic perturbations and the distribution in shaping population dynamics.

Paper Structure

This paper contains 10 sections, 2 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical formulation
  3. Conditions for two positive solutions
  4. Biological Interpretation
  5. Impact of $\alpha$
  6. Impact of $\beta$
  7. Impact of $var[\epsilon_t]$
  8. Discussion
  9. Appendix: Moments formula
  10. Data Availability

Key Result

Theorem 1

For stochastic logistic equation eq:logistic3, assume that $X_t$ follows a normal distribution $N(\mu, \sigma^2)$, and $E[\epsilon_t] = 1, E[\epsilon_t^2] = v$ with parameters $v > 0$, $\sigma^2>0$, $0 < \mu < 1$. In addition, we assume that where $\beta > 0$, $\alpha > 0$. Then $r$ in eq:logistic3 are positive solutions of the following polynomial in terms of $r$:

Figures (5)

  • Figure 1: Plots of $p(r)$ with two positive real solutions
  • Figure 2: Plots of $p(r)$ with no positive real solution
  • Figure 3: Plots of $r$ in terms of $\mu$. Impact of $\alpha$
  • Figure 4: Plots of $r$ in terms of $\mu$. Impact of $\beta$
  • Figure 5: Plots of $r$ in terms of $\mu$. Impact of $E[\epsilon_t^2]=Var[\epsilon_t]+1$

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof