Steady State Distribution and Stability Analysis of Random Differential Equations with Uncertainties and Superpositions: Application to a Predator Prey Model

TL;DR

This paper introduces a computational framework for analyzing steady state distributions and stability in random differential equations with uncertainties, demonstrated on a predator-prey model with complex parameter distributions.

Contribution

It develops a method to characterize equilibrium distributions and stability in stochastic systems with multi-modal parameter uncertainties, extending existing techniques.

Findings

Multi-modal steady state distributions observed in predator-prey model.

Stability regions identified through eigenvalue distribution analysis.

Demonstrated applicability of the Monte Carlo scheme for uncertainty quantification.

Abstract

We present a computational framework to investigate steady state distributions and perform stability analysis for random ordinary differential equations driven by parameter uncertainty. Using the nonlinear Rosenzweig McArthur predator prey model as a case study, we characterize the non-trivial equilibrium steady state of the system and investigate its complex distribution when the parameter probability densities are multi-modal mixture models with partially overlapping or separated components. In consequence, this application includes both, uncertainties and superpositions, of the system parameters. In addition, we present the stability analysis of steady states based on the eigenvalue distribution of the system's Jacobian matrix in this stochastic regime. The steady state posterior density and stability metrics are computed with a recently published Monte Carlo based numerical scheme specifically designed for random equation systems (Hoegele, 2026). Particularly, the simplicity of this stochastic extension of dynamic systems combined with a broadly applicable computational approach is demonstrated. Numerical experiments show the emergence of multi-modal steady state distributions of the predator prey model and we calculate their stability regions, illustrating the method's applicability to uncertainty quantification in dynamical systems.