Well-Posedness for the Rosenzweig-MacArthur Model with Internal Stochasticity

TL;DR

This work develops a stochastic Rosenzweig–MacArthur predator–prey model driven by internal demographic noise and establishes well-posedness on constrained state spaces by proving a general existence–uniqueness criterion for autonomous SDEs on open submanifolds using a Lyapunov function and proving domain invariance of the positive quadrant. For the stochastic RM system, the authors obtain a unique strong solution and derive $p$-th moment bounds along with an almost-sure Lyapunov exponent bound, showing at most exponential growth and persistence of populations under internal noise. Nondimensionalization yields a dimensionless form that clarifies scaling relations before applying numerical methods. Euler–Maruyama simulations reveal substantial qualitative differences between stochastic and deterministic dynamics, including parameter-dependent asymptotics and persistence behavior within constrained ecological states. Overall, the paper provides a rigorous mathematical framework for analyzing constrained stochastic predator–prey systems with internal noise and demonstrates practical implications for persistence and extinction analyses in ecology.

Abstract

In this work, we propose a stochastic version of the Rosenzweig-MacArthur model solely driven by internal demographic noise, extending classical Lotka-Volterra-type systems focused on external noise. We give a criterion for the existence and uniqueness of autonomous stochastic differential equations (SDEs) on an open submanifold of $\mathbb{R}^{n}$, and the framework allows for a wider choice of Lyapunov functions. In the meantime, the invariance of open submanifolds, which is a biologically feasible result and has been implicitly incorporated into many biological and ecological models, facilitates the application of analytic tools typically suited to $\mathbb{R}^{d}$ and indicates the persistence of predator and prey populations, thus providing a criterion for determining whether a population will become extinct. We apply the well-posedness criterion to our stochastic Rosenzweig-MacArthur model and show the existence and uniqueness of solutions. Furthermore, the asymptotic estimates of solutions are obtained, indicating the at most exponential growth of the population with internal stochasticity. Some numerical experiments are performed, which illustrate the discrepancy between the deterministic and stochastic models. Overall, this work demonstrates the broad applicability of our results to ecological models with constrained dynamics, offering a foundation for analyzing extinction, persistence, and well-posedness in systems where internal randomness dominates. This paper not only promotes the development of stochastic modeling and stochastic differential equations in theoretical ecology but also proposes a rigorous mathematical methodology for studying the predator-prey system with internal stochasticity.

Figures (2)

• Figure 1: Figure (a) is the phase portrait of the system (\ref{['eq1.3']}) with $m=3,c=1,k=3$, and Figure (b) is the phase portrait of the system (\ref{['eq1.3']}) with $m=3,c=1,k=1.5$. The short arrows are approximations of the vector field of the system (\ref{['eq1.3']}). For the stable limit cycle, another trajectory with this cycle as its omega limit set is shown in Figure (a); while two additional trajectories with the positive equilibrium as their omega limit set are shown in Figure (b).
• Figure 1: Simulation of estimated population density with error bars and the phase trajectory in the phase space. Figure (a), (b), and (e) is the source equilibrium case with parameter value $\mathrm{Para}_{1}$ in (\ref{['eq5.3']}), and a unique stable limit cycle exists for the system (\ref{['eq1.3']}) in this parameter setting. Figure (a) is about $\hat{\mathrm{E}(N(t)) }$ as a function of $t$ (middle curve); the upper and lower curve represent, respectively, $\hat{\mathrm{E} }(N(t))\pm \dfrac{1}{2}\left[\hat{\mathrm{Var} }(N(t)) \right]^{1/2}$; figure (b) is the same thing for $P(t)$; figure (c) is the trajectory $(\hat{\mathrm{E}}(N(t)),\hat{\mathrm{E} }(P(t)) )$ as a function of $t$. Figure (c), (d), and (f) is the sink equilibrium case with parameter value $\mathrm{Para}_{2}$ in (\ref{['eq5.3']}), and a stable positive equilibrium exists for the system (\ref{['eq1.3']}) in this parameter setting. Figure (c) is about $\hat{\mathrm{E}(N(t)) }$ as a function of $t$ (middle curve); the upper and lower curve represent, respectively, $\hat{\mathrm{E} }(N(t))\pm \dfrac{1}{2}\left[\hat{\mathrm{Var} }(N(t)) \right]^{1/2}$; figure (d) is the same thing for $P(t)$; figure (f) is the trajectory $(\hat{\mathrm{E}}(N(t)),\hat{\mathrm{E} }(P(t)) )$ as a function of $t$.

Theorems & Definitions (20)

• Definition 3.1: Lyapunov Stability
• Theorem 3.2: Lyapunov Direct Method
• Theorem 3.3: Lyapunov Direct Method
• Theorem 3.4: Lyapunov Indirect Method
• Lemma 3.5: Boundedness of Solutions
• Lemma 3.6: Stability of Positive Equilibrium
• Theorem 3.7: Stable Limit Cycle
• Theorem 3.8: Stable Positive Equilibrium
• Lemma 4.1: Existence and Uniqueness of Strong Solution, Domain Invariance
• Lemma 4.2: Extension Lemma for Continuous Functions