Conclusions and Remarks

In this article, basic principles of bifurcation have been presented. More importantly, bifurcation analysis of logistic model and an impulsive predator-prey system with Watt-type functional response are presented. Furthermore, we give the details of spatial bifurcations analysis, that is, Hopf and Turing bifurcation. The availability of powerful computers is the driving force that give bifurcation theory a practical significance.

In fact, computer simulation for nonlinear systems is carried out for various choices of biologically feasible parameter values and for different sets to initial conditions. In all the cases being considered here, the data are so chosen that persistence conditions are satisfied. Therefore, as the solution to the system is bounded, we expect that the ecosystem has a rich dynamic, including limit cycle, bifurcation, quasi-periodic, or even chaotic dynamic. In this case, numerical analysis for the ecosystem is an assistant instrumentality.

On the other hand, the ecosystem is so complicated that we cannot use a single method to study it. We must use mixed methods, such as analytical or experimental or numerical methods.

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