Tracking Interfaces in a Random Logistic Free-Boundary Diffusion Problems: a Random Level Set Method

Free-boundary diffusive logistic model finds applications in diverse fields associated with population dynamics. These processes often possess stochastic characteristics and involve parameters with uncertainties. This study focuses on enhancing a two-dimensional diffusive logistic partial differential model with free boundary by incorporating randomness in the mean square sense, considering the conditions for well-posedness in the random case, which is crucial for the further analysis. Both unknown stochastic processes the solution and its moving front, and the parameters involved in the random problem as random variables, are constrained by a finite degree of randomness. To tackle this challenge, we propose a random level set method. Given the complexity of the problem, we employ alternating direction explicit methods for the interior solvers, to effectively address computational challenges. Since computing the mean and the standard deviation of both unknown stochastic processes are required, we combine the sample approach of the difference schemes together with Monte Carlo technique avoiding the storage accumulation of symbolic expressions of all the previous levels of the iteration process. Parallel computing is employed to enhance performance. A careful numerical analysis is performed in the mean square context to ensure stability, positivity, and boundedness. The set of presented examples illustrates these qualitative properties, assess numerical convergence and enables us to gain a deeper understanding of the system’s behavior attending to the geometry of the initial habitat. This approach provides valuable tools for analyzing and predicting spreading-vanishing dichotomy.