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

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.

Figures (9)

• Figure 1: Left: Mono-modal parameter densities containing broad uncertainties about the parameters. Right: $\pi_{\text{steady}}(\boldsymbol{x})$ as intenstiy plot. The three expected steady states (the top two are the trivial solutions) and the red encircled density area is the non-trivial equilibrium state. The observed shape of this density shows how the parameter uncertainties propagate to the equilibrium steady state and is non-trivial due to the nonlinearity of the equations.
• Figure 2: Left: The parameters $m$ and $c$ (second and third row) show distinct and even separated multi-modal patterns. Right: $\pi_{\text{steady}}(\boldsymbol{x})$ as intenstiy plot. This leads to the non-trivial steady state distribution with $12$ peaks which comes from $3$ modes for $m$ and $4$ modes for $c$ ($3\times 4=12$ parameter combinations).
• Figure 3: Left: The parameters $m$ and $c$ are again multi-modal but not dominant leading to broad non-distinct parameter densities. Right: $\pi_{\text{steady}}(\boldsymbol{x})$ as intenstiy plot. The non-trivial steady state distribution shows a large blurred region of characteristic shape where individual modes are hardly recognizable.
• Figure 4: Left: The parameter $m$ shows distinct modes and $c$ only a broad non-distinct probability distribution which is build up with four Gaussians. Right: $\pi_{\text{steady}}(\boldsymbol{x})$ as intenstiy plot. The non-trivial steady state distribution shows how this mixture parameters propagate to the steady states, again showing non-trivial patterns due to nonlinearity.
• Figure 5: 2D Histograms of the verification calculation of simulation examples presented in Figure \ref{['fig:Res:sim01']} (left) and in Figure \ref{['fig:Res:sim03']} (right) utilizing $N=240000$ numerical solutions of the sampled deterministic steady state equations as described in Section \ref{['sec:IndepVerif']}. Matlab's fsolve iterative solver was utilized with the constant start value $x=[0.3,0.9]$.