Numerical bifurcation and hybrid control of a Gompertz model with time delay

Abstract

In order to study the stability of species, the biological systems are required to reduce or expand the stable region. For a Gompertz model with time delay, a hybrid control Euler method is proposed in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. The local stability of the equilibria is discussed according to Hopf bifurcation theory. For controlling Neimark-Sacker bifurcation, the hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired bifurcation point. The explicit algorithms for determining the direction of the bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form method and center manifold theorem. Numerical examples are provided to illustrate the theoretical results. The research results show that the branch point can be in advance or delay for the delay Gompertz model system through choosing appropriate control parameters. The algorithm has obtained good results both in theory and numerical performance, which provides a new method and has certain theoretical significance for its application in many control problems.

Keywords

• numerical solution of ordinary differential equation
• Gompertz model
• hybrid control
• Euler method
• delay
• Neimark-Sacker bifurcation