Rafael Company

This paper considers a model describing population dynamics where diffusion is nonlocal and habitat boundaries expand based on population flux. The population distribution is governed by a nonlocal Fisher–KPP equation with free boundaries. We propose a Front-Tracking method coupled with a fourth-order Runge–Kutta scheme for time integration, which adaptively manages the computational grid. For comparison, we outline the Front-Fixing approach, which transforms the problem into a fixed domain. Numerical simulations are performed for different scenarios, comparing the accuracy and behavior of both proposed methods. Furthermore, a variance-based global sensitivity analysis using Sobol’ indices is conducted to identify the influence of key model parameters on the final habitat length. Results show that the initial habitat size and the boundary expansion rate are the most influential parameters affecting the final domain length, with their relative importance depending on the distributions of the chosen input parameters.

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.

This paper deals with the construction of numerical stable solutions of random mean square Fisher-KPP models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear unhomogeneous system of random differential equations. Then, by extending to the random framework the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity the results are illustrated by comparing the results with a test problem where the exact solution is known.

A free boundary diffusive logistic model finds application in many different fields from biological invasion to wildfire propagation. However, many of these processes show a random nature and contain uncertainties in the parameters. In this paper we extend the diffusive logistic model with unknown moving front to the random scenario by assuming that the involved parameters have a finite degree of randomness. The resulting mathematical model becomes a random free boundary partial differential problem and it is addressed numerically combining the finite difference method with two approaches for the treatment of the moving front. Firstly, we propose a front-fixing transformation, reshaping the original random free boundary domain into a fixed deterministic one. A second approach is using the front-tracking method to capture the evolution of the moving front adapted to the random framework. Statistical moments of the approximating solution stochastic process and the stochastic moving boundary solution are calculated by the Monte Carlo technique. Qualitative numerical analysis establishes the stability and positivity conditions. Numerical examples are provided to compare both approaches, study the spreading-vanishing dichotomy, prove qualitative properties of the schemes and show the numerical convergence.

In this paper a discrete logistic–like dynamic model for the immigration arriving in Spain is constructed. The immigration population is split in two categories: regular and irregular. Taking into account economic, regulation, and political factors the model, is built in a vector form. For the sake of reliability the period of time is intentionally short, because the economic behavior and legal regulation are not stable.

This paper introduces the exponential time differencing (ETD) technique as a numerical method to efficiently solve vulnerable American options pricing. We address several challenges, including removing cross-derivative terms through appropriate transformations, treating early-exercise opportunities using the penalty method, and substituting fixed boundary conditions with corresponding one-sided finite differences. The proposed method is shown to be both accurate and efficient through numerical experiments, which also compare the results with existing methods and analyze the numerical stability and convergence rate.

This paper introduces the exponential time differencing (ETD) technique as a numerical method to solve efficiently American vulnerable options pricing. We address several challenges, including removing cross-derivative terms through appropriate transformations, treating early-exercise opportunities using the penalty method, and substituting fixed boundary conditions with corresponding one-sided finite differences. The proposed method is shown to be both accurate and efficient through numerical experiments, which also compare the results with existing methods and analyse the numerical stability and convergence rate.

A two-dimensional free-boundary diffusive logistic model with radial symmetry is considered. This model is used in various fields to describe the dynamics of spreading in different media: fire propagation, spreading of population or biological invasions. Due to the radial symmetry, the free boundary can be treated by a front-fixing approach resulting in a fixed-domain non-linear problem, which is solved by an explicit finite difference method. Qualitative numerical analysis establishes the stability, positivity and monotonicity conditions. Special attention is paid to the spreading–vanishing dichotomy and a numerical algorithm for the spreading–vanishing boundary is proposed. Theoretical statements are illustrated by numerical tests.

In this paper, we propose a numerical method for American multi‐asset options under jump‐diffusion model based on the combination of the exponential time differencing (ETD) technique for the differential operator and Gauss–Hermite quadrature for the integral term. In order to simplify the computational stencil and improve characteristics of the ETD‐scheme mixed derivative eliminating transformation is applied. The results are compared with recently proposed methods.