Abstract:
This paper investigates the dynamics of a competitive Lotka-Volterra system containing two free boundaries, where each boundary models the propagation front of one of the two competing species. A free boundary problem is considered for a system of quasilinear parabolic equations with nonlinear convective terms. The paper first establishes a priori estimates of the Hölder norms to solve the problem. Based on these a priori estimates, the existence and uniqueness of the solution are proven. Next, an implicit finitedifference scheme is used to find a numerical solution to the problem, which characterizes the densities of the two competing populations. Using the Python programming language, the obtained solutions are visualized, and graphs of the free boundary dynamics are constructed. From an application perspective, the free boundary problem for the Lotka-Volterra diffusion system is a mathematical model describing predatorprey propagation in a population with a dynamic boundary of the domain of existence. This problem arises when one of the populations (for example, a predator) influences the boundaries of the range of its prey, or when the boundaries of the range are formed under the influence of external factors, and the diffusion itself occurs in this system.
Keywords:model, free boundaries, system of quasilinear parabolic equations, a priori estimates, existence and uniqueness of solutions, numerical algorithm, Python.
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