[Paper Review] Noise in an insect outbreak model

Bao-quan Ai, Wei Chen+5 more|ArXiv.org|Jun 25, 2003

Ecosystem dynamics and resilience3 citations

TL;DR

This study investigates how correlated Gaussian white noise in birth and predation rates affects the steady-state dynamics of a spruce budworm outbreak model using a Fokker-Planck approach. It reveals that noise-induced phase transitions occur: birth rate fluctuations suppress population growth and may cause extinction, while predation rate fluctuations and noise correlation promote population persistence by shifting the system from bistable to monostable states.

ABSTRACT

We study the steady state properties of an insect (spruce budworm) outbreak model in the presence of Gaussian white noise. Based on the corresponding Fokker-Planck equation the steady state solution of the probability distribution function and its extrema have been investigated. It was found that fluctuations of the insect birth rate reduces the population of the insects while fluctuations of predation rate and the noise correlation can prevent the population of the insects from going into extinction. Noise in the model can induce a phase transition.

Motivation & Objective

• To examine the impact of correlated Gaussian white noise on the steady-state behavior of an insect outbreak model.
• To determine how fluctuations in birth rate and predation rate influence population persistence and extinction risk.
• To investigate the role of noise correlation in altering the system's dynamical stability and phase structure.
• To identify conditions under which noise induces a phase transition in the spruce budworm population dynamics.

Proposed method

• Formulates a nondimensionalized stochastic differential equation for spruce budworm population dynamics with multiplicative noise in birth and predation rates.
• Applies the Fokker-Planck equation to derive the steady-state probability distribution function (SPDF) of the population.
• Uses the SPDF to analyze extrema and stability, incorporating noise strengths (D, σ) and correlation parameter (λ).
• Derives analytical expressions for A(x) and B(x) in the Fokker-Planck equation to compute the SPDF via integration.
• Numerically evaluates the SPDF and its extrema for varying noise strengths and correlation parameters to observe phase transitions.
• Employs a Langevin-type stochastic model with correlated noise terms Γ(t) and ξ(t), characterized by delta-correlated statistics and a correlation coefficient λ.

Experimental results

Research questions

• RQ1 How does noise in the birth rate affect the steady-state distribution and population persistence in the spruce budworm model?
• RQ2 What is the effect of predation rate fluctuations on the system’s stability and the likelihood of population extinction?
• RQ3 How does the correlation between birth rate and predation rate noise influence the system’s phase structure?
• RQ4 Can noise induce a phase transition in the insect outbreak model, and if so, under what conditions?
• RQ5 What role does noise correlation play in shifting population probability from low to high states?

Key findings

• Fluctuations in the birth rate reduce the overall insect population and can drive it toward extinction, as shown by the suppression of the high-population peak in the SPDF.
• Predation rate fluctuations prevent extinction by reducing the low-population peak and stabilizing the high-population state, especially at higher noise strengths.
• Noise correlation (λ) enhances population persistence by shifting probability mass from low to high population states, with λ = 1.0 resulting in a single dominant peak.
• The system undergoes a noise-induced phase transition: increasing birth rate noise shifts the system from a single to a bistable state, while increasing predation noise shifts it from bistable to monostable.
• At D = 0.7, the SPDF exhibits three extrema, indicating a transition from monostable to bistable behavior due to birth rate noise, confirming a dynamic phase transition.
• For σ = 0.70, the SPDF collapses into a single peak at high x, demonstrating that predation noise suppresses bistability and prevents extinction.