Exact probability distribution for the Bernoulli-Malthus-Verhulst model driven by a multiplicative colored noise

Area of Science:

• * Stochastic processes
• * Mathematical modeling
• * Statistical physics

Background:

• * The Bernoulli-Malthus-Verhulst model describes population dynamics.
• * Understanding its probability distribution under noise is crucial.
• * Colored noise introduces temporal correlations not present in white noise.

Purpose of the Study:

• * To derive an exact result for the probability distribution of the Bernoulli-Malthus-Verhulst model with multiplicative colored noise.
• * To investigate the conditions for a unimodal to bimodal transition in the distribution.
• * To analyze the asymptotic behavior of the model's mean value.

Main Methods:

• * Exact calculation of the probability distribution.
• * Analysis of the model's parameter space.
• * Asymptotic analysis of the mean value.

Main Results:

• * An exact probability distribution for the specified model was obtained.
• * Conditions for a critical parameter-dependent transition from unimodal to bimodal distributions were identified.
• * The mean value of the population size x(t) was shown to asymptotically approach 1.

Conclusions:

• * The study provides an exact analytical solution for a complex stochastic population model.
• * The findings reveal a critical transition in the probability distribution's shape.
• * The model exhibits stable long-term population dynamics with a mean of 1.