Detection of the Existence of Bifurcation Surfaces using the Optimal Derivative

Abstract

The purpose of this paper is to use the optimal derivative method in order to numerically detect the existence of a bifurcation surface of a ratio-dependent model proposed by Arditi and Ginzburg. Modelling by the optimal derivative may reveal some emerging behaviors. Introduced by Arino and Benouaz, it is based on the principle of least square approximation. It is designed as an alternative to the derivative in the sense of Fr ´ echet, essential to the case of equations involving nonregular and in general nondifferentiable functions. We see in particular how this model may contribute to follow numerically the emergence of complex behavior based on very general phenomena such as the change of stability between solutions and other changes in behavior, e.g., the appearance of periodic solutions. Using this model can reveal some original dynamic properties that have not been observed in simple two-dimensional predator-prey models.