Large deviations for stochastic predator-prey model with Lévy noise (Q6599957)

The paper explores a stochastic predator-prey model incorporating Lévy noise to account for random disturbances. The study employs Galerkin approximation to prove the existence and uniqueness of solutions for the model. It further applies the large deviation principle (LDP) to analyze the behavior of rare events with significant ecological impacts.\N\NStarting from traditional (deterministic) predator-prey models of Lotka-Volterra type\N\begin{align*} \N& \frac{\partial u_{1}}{\partial t}-\eta_{1} \Delta u_{1}=u_{1}\left(\alpha-u_{1}\right)-\frac{\beta u_{1}^{2} u_{2}}{1+u_{1}^{2}} \\\N& \frac{\partial u_{2}}{\partial t}-\eta_{2} \Delta u_{2}=\frac{\gamma u_{1}^{2} u_{2}}{1+u_{1}^{2}}-\delta u_{2}, \N\end{align*} \Nthe authors integrate stochastic elements to capture environmental randomness, including both continuous (e.g., Brownian motion) and discrete jumps (e.g., Lévy noise). More precisely, the stochastic model uses partial differential equations influenced by Lévy noise, including nonlinear functional responses and boundary conditions of Neumann type. The authors propose an abstract formulation of the stochastic predator-prey model driven by Lévy noise written in the form\N\[ \N\mathrm{d} u^{\varepsilon}+A u^{\varepsilon} \mathrm{d} t=f\left(u^{\varepsilon}\right) \mathrm{d} t+\sqrt{\varepsilon} \sigma\left(t, u^{\varepsilon}\right) \mathrm{d} W(t)+\varepsilon \int_{Z} g\left(u^{\varepsilon}, z\right) \tilde{N}(\mathrm{d} t, \mathrm{d} z) \N\]\N\[ \Nu^{\varepsilon}(0)=\left(u_{1}^{\varepsilon}(0), u_{2}^{\varepsilon}(0)\right)=u_{0}, \quad \frac{\partial u^{\varepsilon}}{\partial \nu}=0,\N\]\Nwhere\N\begin{itemize}\N\item \(W(t)=\left(W_{1}(t), W_{2}(t)\right)\) - is a Brownian motion with two independent components;\N\item \(\tilde{N}=\left(\tilde{N}_{1}, \tilde{N}_{2}\right)\) - is a compensated Poisson random measure;\N\item The linear operator \(A\) and nonlinear functional response \(f\) are given by\N\[ \NA=\left(\begin{array}{cc} -\eta_{1} \Delta-\alpha & 0 \\\N0 & -\eta_{2} \Delta+\delta \end{array}\right), \quad f(u)=\binom{-u_{1}^{2}-\frac{\beta u_{1}^{2} u_{2}}{1+u_{1}^{2}}}{\frac{\gamma u_{1}^{2} u_{2}}{1+u_{1}^{2}}}, \quad u=\left(u_{1}, u_{2}\right) \N\]\N\item The functions \(\sigma\left(t, u^{\varepsilon}\right)=\left(\sigma_{1}\left(t, u_{1}^{\varepsilon}\right), \sigma_{2}\left(t, u_{2}^{\varepsilon}\right)\right)\) and \(g\left(u^{\varepsilon}, z\right)=\left(g_{1}\left(u_{1}^{\varepsilon}, z\right), g_{1}\left(u_{1}^{\varepsilon}, z\right)\right)\) are noise coefficients subject to regularity conditions defined in assumptions (A1) and (A2) of the paper.\N\end{itemize}\N\NTo prove the existence and uniqueness of solutions to the stochastic integro-differential equations so obtained, they use a Galerkin approximation method. In addition, the weak convergence view on large deviation theory allows the authors to obtain an LDP result for their model. This important principle helps to evaluate the probability of rare events and their potential ecological impacts, such as sudden population changes due to extreme events. As a result, the proposed model can be useful in predictions and management strategies in ecological systems, especially in understanding the dynamics of predator-prey populations under stochastic disturbances. The LDP framework can be employed to assess risks and devise strategies to mitigate the effects of rare and potentially catastrophic events on predator-prey populations. While this study lays the theoretical groundwork, it leaves room for numerical analysis and simulations to further validate and explore the model's applications in ecological contexts.