Stabilité et bifurcations de quelques modèles biologiques

Abstract

The work carried out in this thesis concerns the mathematical modeling of some biological phenomena and the qualitative study of the models thus obtained by applying the theory of dynamical systems (theory of stability, normal form, bifurcations). This study led us to formulate an algebraic-differential system with a functional esponse of Holling type III, composed of two differential equations and an algebraic equation. This system describes the biological and economic aspects of a predator-prey model. A systematic study of the dynamics of this model was carried out by considering the economic profit as a parameter of bifurcation. First, we examined the existence of the positive equilibrium points according to the economic profit and we found that the system has an even number of positive equilibrium points between 0 and 8. In addition, we have shown that the system can generate a Hopf bifurcation (change from a stationary state to a periodic state) and that the ariation in the economic profit causes the destabilization of the equilibrium (the state of the prey population, the redator population, and the harvesting effort becomes unstable when the economic revenu exceeds the bifurcation value). The presence of this type of bifurcation has been proven using the theory of normal form. This shows that the proposed system is of a particular importance, because its dynamics are very rich (several stationary states and periodic states), in addition the diversity of the positive equilibrium points and limit cycles gives more opportunities in the theory control to choose the states that represent the ideal performance of the ecosystem. The study carried out allowed us to highlight that it is important for the governments to keep the harvest evenues at an ideal level in order to maintain the sustainable development of the predator-prey ecosystem. Finally, we performed numerical simulations to illustrate and confirm our theoretical results