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Stability and bifurcation of difference equations from stochastic logistic models
1School of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069, USA.
Mathematical Biosciences and Engineering : MBE
|January 29, 2026
Summary
This study analyzes a stochastic logistic model for population dynamics. It found that extinction is stable under certain conditions, while positive populations emerge with stability depending on random environmental fluctuations.
Area of Science:
- Ecology
- Mathematical Biology
- Stochastic Processes
Background:
- Population dynamics are often modeled using the logistic equation.
- Environmental stochasticity significantly impacts population persistence and stability.
- Understanding the interplay between deterministic growth and random perturbations is crucial for ecological modeling.
Purpose of the Study:
- To investigate the behavior of a stochastic logistic difference equation with multiplicative random perturbations.
- To analyze the conditions for population extinction versus stable positive populations.
- To explore the role of stochastic intensity in population dynamics and bifurcations.
Main Methods:
- Developed a Gaussian moment-closure approximation for the stochastic logistic difference equation.
- Derived a closed system of difference equations for the mean and variance of the population size.
- Analyzed the stability of trivial (extinction) and nontrivial (positive population) equilibria.
- Investigated saddle-node bifurcations induced by stochastic intensity variations.
- Validated analytical findings using Monte Carlo simulations.
Main Results:
- Identified conditions for the stability of the extinction equilibrium ($r^2v < 1$).
- Determined conditions for the emergence of nontrivial equilibria ($r > 1$), with stability contingent on stochastic intensity ($v$).
- Characterized the saddle-node bifurcation as a function of stochastic intensity.
- Confirmed analytical predictions through numerical simulations.
Conclusions:
- The stochastic logistic model exhibits complex dynamics influenced by both intrinsic growth rates and environmental randomness.
- Extinction is a stable outcome under low stochasticity, while positive populations can emerge and persist under higher stochasticity, depending on the growth rate.
- Stochastic intensity plays a critical role in shaping population dynamics, leading to bifurcations between extinction and persistence states.
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