The Study of Population Dynamics by Delay Differential Equations

Abstract

Population dynamics model is one of the important topics in mathematics modeling. The purpose of using these models is to generate a model that can well
describe the population species at any moment. Nowadays, this technique has been well developed and not only effective in the form of ordinary differential equation, but also derived a new branch, a delayed population dynamics model which is a delay differential equation. In delay differential equation, the time delay is main core for the equation and this factor is always represented as the time lag taken between the implementation of control and responding of the system. In this study, there are two population dynamics models that will be investigated correspond to the form of ordinary and delay differential equations. The comparison between these models will be conducted and this results in no significant changes if the value time delay is small enough, but the solution will meet a great change on it which is the rise of oscillation if the time delay keeps increasing. There is a specific name for the phenomenon, called Hopf-bifurcation. Therefore, the determination to obtain the critical value of time delay will be taken in order to know when the Hopf-bifurcation will occur. According to the result, once the parameter achieves the critical value, the initially stable equilibrium will become unstable which is a loss of stability and then lead to the happening of Hopf-bifurcation which is in the form of periodic solution.