Abstract
This paper studies systematically a differential-algebraic prey-predator model with time delay and Allee effect. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. This model also exhibits singular induced bifurcation as the economic revenue increases through zero, which causes impulsive phenomenon. It can be noted that the impulsive phenomenon can be much weaker by strengthening Allee effect in numerical simulation. On the other hand, at a critical value of time delay, the model undergoes a Hopf bifurcation; that is, the increase of time delay destabilizes the model and bifurcates into small amplitude periodic solution. Moreover, a state delayed feedback control method, which can be implemented by adjusting the harvesting effort for biological populations, is proposed to drive the differential-algebraic system to a steady state. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results.
Highlights
In recent years, the growing human needs for more food and more energy have led to increased exploitation of these resources
It has been shown that harvesting has a strong impact on population dynamics, ranging from rapid depletion to complete preservation of biological populations
Xiao et al [2] have investigated the dynamical properties of a ratio-dependent predator-prey model with nonzero constant rate predator harvesting
Summary
The growing human needs for more food and more energy have led to increased exploitation of these resources. Xiao et al [2] have investigated the dynamical properties of a ratio-dependent predator-prey model with nonzero constant rate predator harvesting. Xiang et al [7] consider a Lotka-Volterra model with impulsive harvest for the prey and investigate globally attractive periodic solution. Most of these discussions are only based on differential equations or difference equations. In this paper, we consider a differential-algebraic prey-predator model with time delay and the Allee effect on the growth of the prey population. A state delayed feedback control method is proposed, which can eliminate Hopf bifurcation and drive the differential-algebraic prey-predator model to stay at a steady state
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