On the dynamic consistency of a two-species competitive discrete system with toxicity: Local and global analysis

Abstract

Discretization of a highly nonlinear system is inevitable for good approximation of its solutions due to lack of its analytical solution. However, a discrete model constructed by standard discretization technique frequently produces spurious dynamics which are not observed in its constituent continuous model. It is therefore important to construct a discrete model that produces the exact dynamics of its continuous counterpart. Here we transform a continuous-time two species nonlinear competition model with the effect of toxic substances into a discrete-time system to show its dynamic consistencies with the continuous system by employing nonstandard finite difference scheme. Linear stability property of each fixed point of the derived model has been proven to be identical with those of the continuous system and the dynamics has been shown to be independent of the step-size. Moreover, we prove the global stability of the coexisting equilibrium point of our discrete model. Comparative simulation results demonstrate that the standard discrete system shows complex dynamics, like flip bifurcation and chaos depending on the step-size, but the proposed discrete model does not show any such spurious dynamics.