Bifurcation analysis and control of a class of hybrid biological economic models

Abstract

This paper systematically studies a hybrid predator–prey economic model, which is formulated by differential-difference-algebraic equations. It shows that this model exhibits two bifurcation phenomena at the intersampling instants. One is saddle–node bifurcation, and the other is singular induced bifurcation which indicates that economic profit may bring impulse at some critical value, i.e., rapid expansion of biological population in terms of ecological implications. On the other hand, for the sampling instants, the system undergoes Neimark–Sacker bifurcation at a critical value of economic profit, i.e., the increase of economic profit destabilizes the system and generates a unique closed invariant curve. Moreover, the state feedback controller is designed so that singular induced bifurcation and Neimark–Sacker bifurcation can be eliminated and the population can be driven to steady states by adjusting harvesting costs and the economic profit. At the same time, by using Matlab software, numerical simulations illustrate the effectiveness of the results obtained here.