The Dynamical Behavior of a Predator-Prey System with Holling Type II Functional Response and Allee Effect

In this paper, we mainly considered the dynamical behavior of a predator-prey system with Holling type II functional response and Allee-like effect on predator, including stability analysis of equilibria and Hopf bifurcation. Firstly, we gave some sufficient conditions to guarantee the existence, the local and global stability of equilibria as well as non-existence of limit cycles. By using the cobweb model, some cases about the existence of interior equilibrium are also illustrated with numerical outcomes. These existence and stability conclusions of interior equilibrium are also suitable in corresponding homogeneous reaction-diffusion system subject to the Neumann boundary conditions. Secondly, we theoretically deduced that our system has saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation under certain conditions. Finally, for the Hopf bifurcation, we choose d as the bifurcation parameter and presented some numerical simulations to verify feasibility and effectiveness of the theoretical derivation corresponding to the existence of k y , respectively. The Hopf bifurcations are supercritical and limit cycles generated by the critical points are stable.


Introduction
In this paper, we consider a predator-prey system with Holling type II functional response and Allee effect on predator, which is described by the following nonlinear ordinary differential equations (ODEs) ( ) 2  : , e qxy y y y r y m y g x y K y e a x subject to initial conditions ( ) ( ) ( ) y y t = are the prey and predator densities at time t, respectively. All above positive constants have biological considerations. Parameters 1 r and 2 r denote the intrinsic growth rate of prey and predator, respectively; 1 K and 2 K represent the carrying capacity of the environment for prey and predator, respectively; a is the half-saturation constant; q is the search efficiency of predator for prey; 1 m and 2 m are the mortality rate of prey and predator species, respectively; 1 e is the biomass conversion; d is the intra-specific competition coefficient; e is the Allee effect constant. For convenience, we denote 1 e q as c.
The specific growth term 1 The term y y e + on predator is called the Allee effect. The Allee effect, named by Warder Clyde Alee, plays a significant role in determining dynamical behavior of predator-prey systems, even ecological and social models [5] [6] [7].
These phenomena occur in small or sparse populations and are widely accepted to be common in nature. They are related to a positive correlation between population size and fitness at very low population size and another phenomenological feature that may induce extinction of populations [8] [9] [10]. The Allee effects are called strong if they cause critical population sizes, while they are called weak if they do not result in critical sizes [7]. The strong Allee effect causes the so-called Allee threshold that populations need to surpass it in order to avoid extinction and survive permanently [9] [11]. On the contrary, a population with weak Allee effect does not have such above threshold [12]. Other names are positive density dependence (in contrast to classical negative density dependence) and depensatory dynamics (in contrast to classical compensatory dynamics) [7].  [18]. For Allee effect incorporating into the prey population with multiply form, in reference [13], the authors extended a predator-prey system with strong Allee effect and a functional response of the ratio of prey to predator.
By means of bifurcation analysis and advance numerical techniques for the calculation of invariant manifolds of equilibria, they stated the consequences of the (dis)appearance of limit cycles, homoclinic orbits, and heterclinic connections in the global arrangement of phase plane near a Bogdanov-Takens bifurcation. The reference [14] examined global behavior of a Gause-type predator-prey system with Holling type III functional response and weak Allee effect on the prey growth. They also proved that the origin is a saddle point and obtained existence of two limit cycles surround a stable interior equilibria: just like that with strong Allee effect. In [15], Zu et al. studied a predator-prey system with Allee effect on prey and investigated local asymptotic stability of equilibria. Meanwhile, the authors found that Allee effect of prey population can bring about unstable or stable periodic fluctuations. Based on above reference, Zu and Mimura [16] considered the Holling type II functional response and local asymptotic stability of equilibria. Compared with the model without Allee effect, they found that the effect on prey increases extinction risk of predator and prey. An explicit algorithm was also obtained to determine the direction of Hopf bifurcation as well as the stability of periodic solutions. The phenomenon of periodicity in this reference is similar to that in [15].
While the reference [17] showed that, when the Leslie-Gower predator-prey model with additive Allee effect on prey has two positive equilibria, there exists a separatrix curve that separates the behavior of trajectories, i.e. the model is highly sensitive to initial conditions. In [18], the authors began with local population dynamics and then constructed a model including both local population and metapopulation dynamics. Their results indicated that the Allee-like effect in a metapopulation may emerge from the imposed Allee effect at the local population level and severe demographic stochasticity may compound the metapopulation extinction risk posed by the Allee effect.
The Allee effect was also researched in predator-prey systems with delay, impulse or diffusion. Xiao et al. [19]  This paper mainly concentrate on the dynamical behavior analysis of a complicated and realistic predator-prey system (1) with Allee-like effect [18] [22] on the specific growth term of predator with multiply form and an intrinsic decrease term on prey, which is different from above references involving ODEs systems with the additive Allee effect on prey or Allee-like effect on prey. The rest of this paper is organized as follows. Preliminaries, such as boundedness and permanence, are given in Section 2. In section 3, we give sufficient conditions for stability analysis of equilibria by using linearization technique and non-existence of limit cycles. In Section 4, bifurcation analysis and numerical simulations of Hopf bifurcation are presented. In Section 5, we present summary and some remarks.

Preliminaries
In this section, we devote to give priori foundations for our system. Firstly, we denote the first quadrant as 2 R + , and its closure is denoted as 2 For biological consideration, the system (1) is defined on the domain 2 R + and all the solutions are nonnegative with initial conditions ( ) ( ) is an invariant set. Thus, any trajectory starting from 2 R + cannot cross the coordinate axes. Furthermore, all the solutions are bounded. Now we will prove following theorems.
Theorem 1 (Uniform boundedness) All the solutions of system (1) are uniformly bounded in 2 R + with initial conditions ( ) ( ) By applying the theory of differential inequality [23], we obtain All the solutions of the system (1) are confined in the region Thus we complete the proof.
For the Equation (1a), we have 2 0 M > and sufficiently large 0 T , such that By using above lemmas again, we complete the proof.

Equilibria
In this section we will discuss equilibria of system (1) with their existence conditions and stability analysis. It is obvious that the system (1) has equilibria: : collide with each other and the system (1) has an unique boundary equilibrium when 0 y > . In this special case, we denote it as ( )

Existence of Interior Equilibria
Here we firstly denote interior equilibrium as ( ) The Equation (6a) implies that 1 1 r m > , and we have: K ac y e y x r y y e K e a x hold, then an interior equilibrium exist. For instance, let 1 (9) holds, then an interior equilibrium exist. For instance, let 1 1 r = , 2 1 r = , hold, then an interior equilibrium exist. For instance, let 1 1 r = , 2 6 r = , < < < (12) hold, then an interior equilibrium exist. For instance, let 1 1 r = , 2 0.8 r = , Case 6. If k y not exist and following condition holds, then an interior equilibrium exist. For instance, let 1 0.4 r = , 2 0.2 r = ,  From above cases, we will also give some cases to illustrate the phenomena that two interior equilibria exist at the same time.

Stability Analysis
In this subsection, we use the Routh-Hurwitz criterion and the Perron's theorems to analyze local stability of above equilibria. Recall the system (1) again, its Jacobian matrix takes the following form Notice that Jacobian matrix at the axial equilibrium 1 E takes the form ( ) Combining the existence condition of 1 E , we have: In the case that two axial equilibrium ( ) ( ) Due to the further consideration of stability at the point * E , we will give following theorem to explain its global stability.  (20) then * E is globally asymptotically stable. Proof of theorem 3. Here we take an unbounded positive definite Lyapunov

Closed Orbits and Limit Cycles
In this subsection, we consider non-existence of closed orbits and limit cycles of system (1 1 . y y Q x y r y a x cxy y e m y a x y e K Notice that we still denote u, v and τ as x, y and t. Above system is a C ∞ -qualitatively equivalent polynomial extension of the system (1) to 2 R + and more convenient to study limit cycles throughout [13].
Theorem 4 (Non-existence of limit cycles) If ad c > , 1 1 m r > and 2 2 m r > , then for system (22), there are no closed orbits and limit cycles in Proof of theorem 4. Here we take a Dulac function ( ) 1 , B x y xy = and calculate following partial derivative:

Bifurcations
In this section, we will consider and give sufficient conditions to show the existence of saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation of the system (1). Firstly, we denote the system (1) as the following form for simplicity and convenient: , .
Theorem 5 (Saddle-node bifurcation) Suppose the point 2 E exist, if

Transcritical Bifurcation
Keeping in mind that there is an exchange of stability between the points ( ) J E c ≥ − , then the system (23) undergoes a transcritical bifurcation around the point ( ) 1 2 E with respect to the bifurcation parameter 1 r .
Theorem 7 (Hopf bifurcation) Assume that the equilibrium * E exist and parameters satisfy conditions (27)  , respectively. This implies that the Hopf bifurcation occurs in the system (1). The first Lyapunov number 0 σ < , thus the Hopf bifurcation is supercritical and a limit cycle generated by the critical point is stable. Example 4.2 We consider the case 6 in subsection 3.1 again and notice the critical value [ ] H d . Figure 3 and Figure  , respectively. The first Lyapunov number is also found to be negative. Example 4.3 We consider the case 7 in subsection 3.1 again and notice the critical value [ ] H d . Figure 5 and Figure  , respectively. The first Lyapunov number is also negative.

Summary and Remarks
In summary, we consider a predator-prey system with Holling type II functional response and Allee effect on the specific growth term of predator with multiply form. In Subsection 3.1, some cases about the existence of interior equilibria * E are derived with the help of the cobweb model but we neglect the monotonicity