STOCHASTIC CHAOS IN A CLASS OF FOKKER-PLANCK EQUATIONS DERIVED FROM POPULATION DYNAMICS

Abstract

In this work, we obtain results by using a physical potential Φ m ( x , y ) whose parameters have biological significance [5] to explain the interaction between two species in population dynamics. In using the degenerated parameters [8], this potential is reduced to the form of Φ m ( x , y ) , where m is the coupling constant. Consequently, we study the effect of this constant on the potential Φ m ( x , y ) . The deterministic chaos results are obtained in Figures 2 and 3. An interesting result of our theoretical model resides in the fact that after many manifestations of the deterministic chaos, the physical potential Φ m ( x , y ) remained unchanged above a critical value m = 2 . This situation corresponds without any doubt to the Hopf bifurcation in the nonlinear system, where the stationary effect changes to the unstable to stable and leads to a limit cycle. Then we studied the manifestations of the stochastic chaos by considering the transformed potential ˆ Φ ( x , y ) , where ε is the noise intensity. In such a case the m , ε combined effect of the noise and the coupling constant, gives results as illustrated in Figures 4 and 5. The second model [5] leads to a potential Φ m , l ( x , y ) with two coupling constants ( m , l ) which indicate that the use of degenerated parameter is strictly forbidden. The results obtained show the chaotic behavior of the potential Φ m , l ( x , y ) for the arbitraries values of coupling constants ( m , l ) , Figures 6 and 7. The stochastic manifestations are also shown by the ˆ transformed potential Φ m , l , ε ( x , y ) , Figures 8 and 9.