This paper studies systematically a differentialalgebraic preypredator 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 differentialalgebraic system to a steady state. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results.
In recent years, the growing human needs for more food and more energy have led to increased exploitation of these resources. The problems related to many fields like fishery, forestry, and wildlife. Therefore, mankind is facing the dual problems of resource shortages and environmental degradation. Concerning the conservation for the longterm benefits of humanity, there is a widerange of interest in analysis and modelling of bioeconomic systems. In many earlier studies, it has been shown that harvesting has a strong impact on population dynamics, ranging from rapid depletion to complete preservation of biological populations. Two main kinds of harvesting were focused on nonzero constant harvesting [
In power systems, neural networks, and genetic networks [
On the other hand, since reproduction of predator after consuming prey is not instantaneous in most cases, some time lag for gestation is required. Therefore, in this paper, we consider a differentialalgebraic preypredator model with time delay and the Allee effect on the growth of the prey population. We analyze the stability properties and bifurcation behavior of this model. A state delayed feedback control method is also proposed, which can eliminate Hopf bifurcation and drive the differentialalgebraic preypredator model to stay at a steady state.
The general predatorprey model in its classical form is represented by
In 1954, Gordon [
Let the fertility rate
When the prey population is subject to Allee effect above, the predatorprey model (
Assume that the per capita maximum fertility rate of prey must exceed its death rate, that is,
We nondimensionalize the model (
From the assumption mentioned above, the following reasonable condition must be satisfied:
For simplicity of computation, we consider the above model (
Let
Considering zero economic revenue, the model (
(1) The model (
(2) There exist two boundary equilibrium points
(3) There exists another boundary equilibrium point
(4) The model (
The Jacobian matrix of the model (
Hence, the Jacobian matrix of the model (
From Theorem
Note that
Since
Assume that
Next, we analyze the stability of the equilibrium point
For
Assume that
Based on Theorem
Assume that
When
According to the literature [
Transcritical bifurcation implies that the equilibrium point
The Jacobian matrix of the model (
The corresponding characteristic equation is
In the presence of delay, assume that a purely imaginary solution of the form
Assume that the condition (
In succession, we discuss the bifurcation behavior regarding
If
We define a new variable as
For the model (
After simple computation, it can be seen that the Jacobian matrix
When the economic profit is positive, the model (
The Jacobian matrix of the model (
For the sake of simplicity, let
The characteristic polynomial for the model (
According to the Jacobian matrix
Assume that a purely imaginary solution of the form
Assume that conditions (
From Theorem
Under the state delayed feedback control, the Jacobian matrix at the positive equilibrium point
For the simplicity of computation, the feedback gain term
Assume that (
Under the conditions of Theorem
In fact, the two differential equations of the differentialalgebraic system (
From the practical point of view,
In this section, we firstly assign some parameter values of the model (
The differentialalgebraic model (
The existence of singularity induced bifurcation of the model (
Equilibrium points and eigenvalues of the model (
Allee effect  Economic profit  Equilibrium point  Eigenvalues 



(0.957, 0.0284, 1.424) 
−0.930, −201.571 




(0.874, 0.0285, 2.367) 
−0.775, −559.217 




(0.442, 0.0265, 0.136) 
−0.203, −1.684 
The maximum eigenvalue of the model (
For the fixed parameter
The critical curve of the delay
When
When
Dynamical responses of the controlled system (
Nowadays, much attention has been paid to preserving biological resources with the aim of stemming the damage and ensuring the balance of ecosystems, which inspires the introduction of harvesting in the biological system. In this paper, we analyze the dynamical behavior of a delayed predatorprey model with Allee effect and harvesting by using differential algebraic systems theory. From the analysis of the proposed model, we have obtained some interesting and useful results, which extend the work done in [
In the first part, we consider a delayed differentialalgebraic predatorprey model with zero economic revenue. It is observed that transcritical bifurcation and singular induced bifurcation phenomena take place when handling time of predator and economic revenue are regarded as bifurcation parameters, respectively.
As the handling time decreases through the critical point
Singular induced bifurcation may cause impulsive phenomenon due to the variation of the economic revenues of harvesting. From a biological point of view, singular induced bifurcation implies rapid expansion of biological population, which may cause ecosystem unbalance and hamper the sustainable development. Hence, it is necessary to investigate the singular induced bifurcation in the presence of a reserve depending on the variation of economic revenue. The analysis of singular induced bifurcation can provide more information of forecasting so that ecological managers can lay down better management strategy. Furthermore, numerical simulation shows that the impulsive phenomenon can be much weaker by increasing Allee effect constant, which implies that Allee effect has an impact on the dynamical behavior of the proposed model. Therefore, ecological managers need to consider the inherent characters of the biological population and some external factors comprehensively.
In general, an individual prey killed does not contribute instantaneously to the growth of predator population. And differential equations with time delay always exhibit much more complicated dynamics than ordinary differential equations. Hence, in the second part of this paper, the effect of time delay on dynamical behavior of the differentialalgebraic model is discussed. It shows that time delay plays an important role in the dynamical behavior of the differentialalgebraic model. Hopf bifurcation occurs as time delay increases through a certain threshold, and time delay switches the stability of the proposed model. Furthermore, a state delayed feedback controller is designed to eliminate bifurcational phenomenon and keep population density at steady state. From numerical simulation, it is noted that the stronger Allee effect is not beneficial to the stability of biological species.
It should be noted that almost the existing bioeconomic models (see [
The authors do not have a conflict of interests with any commercial identities.
The authors gratefully thank the anonymous authors whose work largely constitutes this paper. This work was supported by the National Science Foundation of China (61273008), Doctor Startup Fund of Liaoning Province (20131026), and Fundamental Research Funds for the Central University (N120405009).