The analysis of a finite element method for the three-species Lotka–Volterra competition-diffusion with Dirichlet boundary conditions

Abstract

A Galerkin finite element method is developed for the two dimensional/three dimensional nonlinear time-dependent threespecies Lotka-Volterra competition-diffusion equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation are proved. An error estimate for the numerical solution is obtained. Numerical computations are carried out to examine the expected orders of accuracy in the error estimates.

Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system

“…Nevertheless the classical Lotka-Volterra system was independently introduced by A. J. Lotka and V. Volterra about 90 years ago, its different generalizations are widely studied at the present time because of their importance for mathematical modeling many processes in physics, biology, ecology etc. Notably, there are several recent papers devoted to rigorous study of multi-component diffusion Lotka-Volterra system (see [17,18,19] and papers cited therein).…”

RETRACTED: Stability conditions for ecosystem modeling using the fuzzy Lyapunov method

“…This theoretical approach enables the prediction of species competition, ecosystem stability, long term species coexistence, and so on. For example, species competition could be modeled using the Lotka-Volterra equations and different useful mathematical methods (Roughgarden, 1975;Arato´, 2003;Du et al, 2004;Tang, et al, 2006;Batiha et al, 2007;Wong, 2009;Geijzendorffer et al, 2011;Chen and Zhang, 2012). In real world systems, species coexistence is complicated.…”

Global Stability of Traveling Waves for the Lotka–Volterra Competition System with Three Species

“…For the three-species competition model, the studies on the dynamical behaviors are also receiving increased attention. The existence of traveling wave solutions for the three-species system has been extensively studied in [10][11][12][13][14]. In addition, many scholars [15][16][17][18] investigated the speed selection, and for more studies on other aspects of the three-species system, please see [19][20][21].…”