Canard Limit Cycle of the Holling-Tanner Model

By using the singular perturbation theory on canard cycles, we investigate the canard phenomenon for the Holling-Tanner model with the intrinsic growth rate of the predator small enough. The obtained result shows that there may be at most one canard limit cycle, and the range of small parameters is estimated. The phenomenon of outbreak is explained.


Introduction
The predator-prey system is referred to as the Holling-Tanner model [1].In (1), the variables () and () denote the numbers of preys and predators at time , respectively. 1 is the intrinsic growth rate of the prey,  is the carrying capacity of the prey, and function  1 /( +  1 ), known as the functional response of Holling type II, is the rate at which predators consume the prey, where  1 is the maximum number of a predator and  1 is the half saturation value corresponding to the number of preys necessary to achieve one half of the maximum value.It is assumed that the predator grows logistically with intrinsic growth rate  1 and carrying capacity  taking on the role of a prey-dependent carrying capacity for the predator, where  is a measure of the quality of the prey as food for the predator.
From the biological meaning all the parameters of (1) are positive.
It is easy to see that the set {(, ): 0 <  ≤ ,  > 0} is positively invariant for (1), and there is always a unique positive equilibrium.The local stability of the positive equilibrium was analyzed in [2,3].In [4] Hsu and Huang considered the question of global stability of the equilibrium and obtained conditions under which its local stability implies the global stability.In 2003 Braza [5] investigated the existence of at least one limit cycle of (1) in a Hopf bifurcation.Gasull et al. [6] showed that (1) can exhibit two limit cycles by constructing an example.The bifurcation diagram of limit cycles of (1) was described to show the regions of parameters where two limit cycles or a semistable limit cycle appears [7].Applying Andronov-Hopf bifurcation Theorem Hsu and Hwang proved that the Hopf bifurcation is subcritical and thus (1) may have at least two limit cycles for some parameter ranges and explained the phenomenon of outbreaking in [8] obtained by numerical simulations [9].
In the references mentioned above, it was assumed that all the parameters are positive constants, but there is not much consideration for the system with small parameters.It is interesting and important to analyze the bifurcation of limit periodic set of the system in the case when  1 is very small; that is, the intrinsic growth rate of the predator remains at a very low level.For this case when plotting the trajectories of system (1) with the intrinsic growth rate small enough by numerical simulations, it would take much longer time when the trajectory passes by the vertical -axis.This behaves like the usual relaxation oscillation phenomena observed in slowfast systems.By using the singular perturbation theory [10] Li and Zhu [11] studied the canard limit cycles for predator-prey system with functional response of four Holling types and developed criteria for the existence and multiplicity of canard slow-fast cycle (i.e., limit periodic set) in general predatorprey systems.
Although some references have considered the existence of two limit cycles for (1) [6,7,9], it is still interesting to discuss the existence of canard slow-fast cycle, since the canard slow-fast cycle consists of a part of the fast orbit and a part of the slow manifold and is the limit of a family of periodic orbits as the value of a positive parameter tends to zero.In this paper we mainly consider the existence of canard slow-fast cycle of (1) when  1 is very small.Our result shows that there may be at most one canard limit cycle, and the range of small parameters is estimated.In next section we first reduce the number of parameters to three ones by some transformation for (1) and introduce some concepts of canard slow-fast cycle.In Section 3 we provide the complete inferences for the main result of this paper.Finally we give discussion according to the results of numerical simulation.

Simplification of the Model and Canard Limit Cycle
For the case where  1 is very small, let where  1 is positive and  1 may be negative, and we will choose  1 and | 1 | small enough to make  2 > 0 in later discussion; then system (1) becomes For system (3), making the scaling and then dropping their bars yield where Since  > 0 and  > , letting  = (( +  − )/) gives the equivalent system of (5): where If we introduce slow time  = , then system ( 7) is transformed to By lettng  = 0 for ( 7) and ( 9), we obtain the layer equations and the reduced equations In this paper we will always assume that  < 1.The slow manifold  = {(, ):  =  0 ()} has a unique extreme point at and it is a simple maximum point; see Figure 1(a).Besides,  is globally concave for  ≥ 0. The fast movement of the layer equations ( 10) is shown in Figure 1(b), where the orbits outside  are parallel lines { = } with different constant , and the curve  consists of singular points; hence it is also called the critical curve.Note that the points on  left to ( 0 ,  0 ) are normally hyperbolic repelling while the points on  right to ( 0 ,  0 ) are normally hyperbolic attracting.
To study the slow movement on  \ {( 0 ,  0 )}, from the reduced equations (11) we have and hence along  the slow movement is controlled by In order to show the existence of canard slow-fast cycle, from Figure 1(b) it is necessary to illustrate that the right hand side of (14) should be negative near the point ( 0 ,  0 ).
In the next section, we will make some transformations such that the function (, ) in system (7) becomes (, ) = g(, , ), and for  = 0 the right hand side of ( 14) is negative in an interval of , containing  0 ; then we may construct a slow-fast cycle (i.e., limit periodic set) Γ() by combining a part of the fast orbit  =  0 − , where  ∈ (0,  0 − ), and a part of the slow manifold , shown in Figure 1(a).As we mentioned above, on the left and right sides of the point ( 0 ,  0 ) the points on the slow manifold  are normally hyperbolic repelling and attracting, respectively, the point ( 0 ,  0 ) is called a canard point, and the slow-fast cycle Γ() is called a canard type, or simply called a canard slow-fast cycle; see [10,12].

Existence and Uniqueness of Canard Slow-Fast Cycles
To study the local behavior of system ( 7) near ( 0 ,  0 ), we make the change of coordinates For (, V) near (0, 0) system (7) becomes where Notice that  1 =  0  2 0 ( 0 +) > 0 as  0 = 0 (i.e.,  0 =  0 ); then  > 0 for some  and  making | 0 | small enough.For such  and , making one more change of coordinates equation ( 16) becomes In addition, we choose  and  such that and then (19) can be rewritten as where From ( 17) and (20) we have Furthermore,  1 | =0 > 0 implies that the change of coordinates (18) is available.
According to (3.12) and (3.13) of [10], from (24) we have and then Therefore, the canard cycle of ( 7) is asymptotically stable if it exists.
On the other hand, for ( 14 for  < 1 and  < 1.The fact that   0 () ̸ = 0 for  > 0 and  ̸ =  0 implies that system (14) along the slow manifold  is well defined near  0 and that the point ( 0 ,  0 ) is a canard one.

Numerical Simulation and Discussion
We first plot the trajectories of (5) and the corresponding curves of  = () and  = () as  = 3.2,  = 0.15,

Figure 1 :
Figure 1: The limit periodic cycle Γ() and the slow manifold .