Canard Limit Cycle of the Holling-Tanner Model

Abstract

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.

1. Introduction

The predator-prey system 𝑑𝑥𝑑𝑡=𝑟₁𝑥𝑥1−𝐾−𝑐₁𝑥𝑦𝑥+𝑎₁,𝑑𝑥𝑑𝑡=𝑠₁𝑦𝑦1−𝛾𝑥 (1) is referred to as the Holling-Tanner model [1]. In (1), the variables 𝑥(𝑡) and 𝑦(𝑡) denote the numbers of preys and predators at time 𝑡, respectively. 𝑟₁ is the intrinsic growth rate of the prey, 𝐾 is the carrying capacity of the prey, and function 𝑐₁𝑥/(𝑥+𝑎₁), known as the functional response of Holling type II, is the rate at which predators consume the prey, where 𝑐₁ is the maximum number of a predator and 𝑎₁ 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 𝑠₁ and carrying capacity 𝛾𝑥 taking on the role of a prey-dependent carrying capacity for the predator, where 𝛾 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 {(𝑥,𝑦):0<𝑥≤𝐾,𝑦>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 𝑠₁ 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 𝑦-axis. This behaves like the usual relaxation oscillation phenomena observed in slow-fast 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 predator-prey 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 𝑠₁ 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.

2. Simplification of the Model and Canard Limit Cycle

For the case where 𝑠₁ is very small, let 𝑠₁=𝜀₁𝑠₂,𝑎₁=𝑎₂−𝜀₁𝜆₁, (2) where 𝜀₁ is positive and 𝜆₁ may be negative, and we will choose 𝜀₁ and |𝜆₁| small enough to make 𝑎₂>0 in later discussion; then system (1) becomes 𝑑𝑥𝑑𝑡=𝑟₁𝑥𝑥1−𝐾−𝑐₁𝑥𝑦𝑥+𝑎₂−𝜀₁𝜆₁,𝑑𝑥𝑑𝑡=𝜀₁𝑠₂𝑦𝑦1−.𝛾𝑥 (3)

For system (3), making the scaling 𝑥=𝐾𝑟𝑥,𝑦=¹𝐾𝑐₁1𝑦,𝑡=𝑟₁𝑡, (4) and then dropping their bars yield 𝑑𝑥()−𝑑𝑡=𝑥1−𝑥𝑥𝑦,𝑥+𝑎−𝜀𝜆𝑑𝑦𝑦𝑑𝑡=𝜀𝑦𝛿−𝑥, (5) where 𝜀𝜀=¹𝑠₂𝑐₁𝛾𝑐,𝜆=¹𝛾𝜆₁𝐾𝑠₂𝑎,𝑎=²𝐾𝑐,𝛿=¹𝛾𝑟₁. (6)

Since 𝑥>0 and 𝑎>𝜀𝜆, letting 𝑑𝑡=((𝑥+𝑎−𝜀𝜆)/𝑥)𝑑𝜏 gives the equivalent system of (5): 𝑑𝑥𝑑𝜏=𝑓_𝜀(𝑥)−𝑦,𝑑𝑦,𝑑𝜏=𝜀𝑔𝑥,𝑦 (7) where 𝑓_𝜀(𝑥)=(),𝑔=𝑦1−𝑥)(𝑥+𝑎−𝜀𝜆𝑥,𝑦𝑥²(𝑥+𝑎−𝜀𝜆)𝛿𝑥−𝑦. (8)

If we introduce slow time 𝑡=𝜀𝜏, then system (7) is transformed to 𝜀𝑑𝑥𝑑𝑡=𝑓_𝜀(𝑥)−𝑦,𝑑𝑦.𝑑𝑡=𝑔𝑥,𝑦 (9)

By lettng 𝜀=0 for (7) and (9), we obtain the layer equations 𝑑𝑥𝑑𝜏=𝑓₀(𝑥)−𝑦,𝑑𝑦𝑑𝜏=0, (10) and the reduced equations 𝑦=𝑓₀(𝑥),𝑑𝑦,𝑑𝑡=𝑔𝑥,𝑦 (11) respectively.

In this paper we will always assume that 𝑎<1. The slow manifold 𝑆={(𝑥,𝑦):𝑦=𝑓₀(𝑥)} has a unique extreme point at 𝑥₀,𝑦₀=1−𝑎2,(1+𝑎)²4, (12) and it is a simple maximum point; see Figure 1(a). Besides, 𝑆 is globally concave for 𝑥≥0. The fast movement of the layer equations (10) is shown in Figure 1(b), where the orbits outside 𝑆 are parallel lines {𝑦=𝑐} with different constant 𝑐, and the curve 𝑆 consists of singular points; hence it is also called the critical curve. Note that the points on 𝑆 left to (𝑥₀,𝑦₀) are normally hyperbolic repelling while the points on 𝑆 right to (𝑥₀,𝑦₀) are normally hyperbolic attracting.

(a)

(b)

Figure 1 

The limit periodic cycle Γ(𝑠) and the slow manifold 𝑆.

To study the slow movement on 𝑆⧵{(𝑥₀,𝑦₀)}, from the reduced equations (11) we have 𝑓^′₀(𝑥)𝑑𝑥=𝑑𝑡𝑑𝑦𝑑𝑡_{𝑦=𝑓0(𝑥)}=𝑔𝑥,𝑓₀(𝑥), (13) and hence along 𝑆 the slow movement is controlled by 𝑑𝑥=𝑔𝑑𝑡𝑥,𝑓₀(𝑥)𝑓^′₀(𝑥). (14)

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 (𝑥₀,𝑦₀).

In the next section, we will make some transformations such that the function 𝑔(𝑥,𝑦) in system (7) becomes 𝑔(𝑥,𝑦)=𝑔(𝑥,𝑦,𝜆), and for 𝜆=0 the right hand side of (14) is negative in an interval of 𝑥, containing 𝑥₀; then we may construct a slow-fast cycle (i.e., limit periodic set) Γ(𝑠) by combining a part of the fast orbit 𝑦=𝑦₀−𝑠, where 𝑠∈(0,𝑦₀−𝑎), and a part of the slow manifold 𝑆, shown in Figure 1(a). As we mentioned above, on the left and right sides of the point (𝑥₀,𝑦₀) the points on the slow manifold 𝑆 are normally hyperbolic repelling and attracting, respectively, the point (𝑥₀,𝑦₀) is called a canard point, and the slow-fast cycle Γ(𝑠) is called a canard type, or simply called a canard slow-fast cycle; see [10, 12].

3. Existence and Uniqueness of Canard Slow-Fast Cycles

To study the local behavior of system (7) near (𝑥₀,𝑦₀), we make the change of coordinates 𝑢=𝑥−𝑥₀,𝑣=𝑦−𝑦₀. (15)

For (𝑢,𝑣) near (0,0) system (7) becomes 𝑑𝑢𝑑𝜏=−𝑣−𝑢²+𝜀𝜆𝑢−1+𝑥₀,𝑑𝑣=𝜀𝑑𝜏𝑥⁴₀𝑏₀+𝑏₁𝑢+𝑏₂𝑢²+𝑣𝑑₀+𝑑₁𝑢+𝑑₂𝑣+𝑜|𝑢,𝑣|², (16) where 𝑏₀=𝑏₀−𝜀𝜆𝛿𝑥₀−𝑦₀𝑥²₀𝑦₀,𝑏₀=𝑥₀+𝑎𝛿𝑥₀−𝑦₀𝑥²₀𝑦₀,𝑏₁=𝑏₁+𝜀𝜆𝛿𝑥₀−2𝑦₀𝑥₀𝑦₀,𝑏₁=𝑥₀𝑦₀𝑥₀𝑦₀−𝛿𝑎𝑥₀+2𝑎𝑦₀,𝑏₂=𝑏₂+𝜀𝜆𝑦₀3𝑦₀−𝛿𝑥₀,𝑏₂=𝑦₀𝛿𝑎𝑥₀−𝑥₀𝑦₀−3𝑎𝑦₀,𝑑₀=𝑑₀−𝜀𝜆𝑥²₀𝛿𝑥₀−2𝑦₀,𝑑₀=𝑥₀+𝑎𝑥²₀𝛿𝑥₀−2𝑦₀,𝑑₁=𝑑₁+𝜀𝜆𝑥₀𝛿𝑥₀−4𝑦₀,𝑑₁=𝑥₀2𝑥₀𝑦₀+4𝑎𝑦₀−𝛿𝑎𝑥₀,𝑑₂=𝑑₂+𝜀𝜆𝑥²₀,𝑑₂=−𝑥₀+𝑎𝑥²₀. (17)

Notice that 𝑏₁=𝑥₀𝑦²₀(𝑥₀+𝑎)>0 as 𝑏₀=0 (i.e., 𝛿𝑥₀=𝑦₀); then 𝑏>0 for some 𝛿 and 𝑎 making |𝑏₀| small enough. For such 𝛿 and 𝑎, making one more change of coordinates 𝑢=−𝑏₁𝑥²₀𝑏𝑥,𝑣=−¹𝑥⁴₀𝑥𝑦,𝜏=²⁰𝑏₁𝑡, (18) equation (16) becomes 𝑑𝑥𝑑𝑡=−𝑦+𝑥²𝑥+𝜀𝜆⁴⁰𝑏₁𝑏₁𝑥²₀𝑥+1−𝑥₀,𝑑𝑦𝑑𝑡=𝜀𝑏₁𝑏₁𝑥−𝑏₀𝑥²₀𝑏₁𝑏₁−𝑏₂𝑏₁𝑥²₀𝑥²+𝑦𝑏₁𝑥²₀𝑑₀−𝑑₁𝑏₁𝑥²₀𝑥−𝑑₂𝑏₁𝑥⁴₀𝑦+𝑜𝑥,𝑦². (19) In addition, we choose 𝛿 and 𝑎 such that 𝑏₀𝑥²₀𝑏₁𝑏₁=𝜆, (20) and then (19) can be rewritten as 𝑑𝑥𝑑𝑡=−𝑦ℎ₁𝑥,𝑦,𝜆,𝜀+𝑥²ℎ₂𝑥,𝑦,𝜆,𝜀+𝜀ℎ₃𝑥,𝑦,𝜆,𝜀,𝑑𝑦𝑑𝑡=𝜀𝑥ℎ₄𝑥,𝑦,𝜆,𝜀−𝜆ℎ₅𝑥,𝑦,𝜆,𝜀+𝑦ℎ₆𝑥,𝑦,𝜆,𝜀+𝑜𝑥,𝑦², (21) where ℎ₁ℎ𝑥,𝑦,𝜆,𝜀=1,₂ℎ𝑥,𝑦,𝜆,𝜀=1,₃𝑥𝑥,𝑦,𝜆,𝜀=𝜆⁴⁰𝑏₁𝑏₁𝑥²₀𝑥+1−𝑥₀,ℎ₄𝑥,𝑦,𝜆,𝜀=1+𝑏₁−𝑏₁𝑏₁−𝑏₂𝑏₁𝑥²₀𝑥=1+𝜀𝜆𝛿𝑥₀−2𝑦₀𝑥₀𝑦₀𝑏₁−𝑏₂𝑏₁𝑥²₀ℎ𝑥,₅𝑥,𝑦,𝜆,𝜀=1+𝑏₀−𝑏₀𝑥²₀𝑏₁𝑏₁𝜆𝜀=1−𝛿𝑥₀−𝑦₀𝑥⁴₀𝑦₀𝑏₁𝑏₁,ℎ₆=1𝑥,𝑦,𝜆,𝜀𝑏₁𝑥²₀𝑑₀−𝑑₁𝑏₁𝑥²₀𝑥−𝑑₂𝑏₁𝑥⁴₀𝑦. (22) From (17) and (20) we have 𝜆=0⟺𝑏₀=0⟺𝛿𝑥₀=𝑦₀()⟺𝛿=𝑎+1²2(1−𝑎)=∶𝛿₀,𝑏_1𝜆=0=𝑥₀𝑦²₀𝑥₀𝑏+𝑎>0,_2𝜆=0=−𝑦²₀𝑥₀,𝑑+2𝑎_0𝜆=0=−𝑥₀+𝑎𝑥²₀𝑦₀. (23)(24) Furthermore, 𝑏₁|_𝜆=0>0 implies that the change of coordinates (18) is available.

According to (3.12) and (3.13) of [10], from (24) we have 𝑎₁=𝜕ℎ₃()𝑎𝜕𝑥0,0,0,0=0,₂=𝜕ℎ₁()𝑎𝜕𝑥0,0,0,0=0,₃=𝜕ℎ₂()𝑎𝜕𝑥0,0,0,0=0,₄=𝜕ℎ₄()=()√𝜕𝑥0,0,0,01+𝑎)(1+3𝑎1−𝑎²(1−𝑎)³,𝑎₅=ℎ₆()√0,0,0,0=−1−𝑎²,1−𝑎 (25) and then 𝐴=−𝑎₂+3𝑎₃−2𝑎₄−2𝑎₅()√=−4𝑎𝑎+31−𝑎²(1−𝑎)³<0. (26) Therefore, the canard cycle of (7) is asymptotically stable if it exists.

On the other hand, for (14) we have 𝑑𝑥𝑑𝑡_𝜆=0=𝑔𝑥,𝑓₀(𝑥)𝑓^′₀(𝑥)_𝜆=0()=−1−𝑥)(𝑎+𝑥²[2𝑎+(1−𝑎)𝑥]2(1−𝑎)𝑥²<0 (27) for 𝑎<1 and 𝑥<1. The fact that 𝑓^′₀(𝑥)≠0 for 𝑥>0 and 𝑥≠𝑥₀ implies that system (14) along the slow manifold 𝑆 is well defined near 𝑥₀ and that the point (𝑥₀,𝑦₀) is a canard one.

Further, by Theorems 3.3 of [10], 𝐴<0 implies that a 𝜀-family of canard cycles bifurcates from the limit periodic set Γ(𝑠), for each 𝑠∈(0,𝑦₀−𝑎), and for some small 𝜆=𝜆(𝑠,√𝜀) and 0<𝜀≪1. Moreover, 𝜆(𝑠,√𝜀) satisfies 𝜆𝑠,√𝜀−𝜆_𝑐√𝜀≤𝑒^{−1/𝜀𝜎}, (28) where 𝜎∈(0,1) is a constant and 𝜆_𝑐√𝜀𝑎=−¹+𝑎₅2+𝐴8𝜀+𝑂𝜀^3/2=2𝑎²+𝑎+1√1−𝑎²2(1−𝑎)³𝜀+𝑂𝜀^3/2. (29)

The inequality (28) implies that, when there is a family of canard cycles bifurcating from the canard slow-fast cycle, 𝜆 in (7) depends on 𝜀, and 𝜆→0 as 𝜀→0.

Further, direct computation shows 𝑓^′₀(𝑥)²𝑔𝑥,𝑓₀(𝑥)_𝜆=0=2(1−𝑎)[2𝑥−(1−𝑎)]𝑥²(1−𝑥)(𝑥+𝑎)²)](1[(1−𝑎𝑥+2𝑎=21−𝑎)(𝑎+1)²()+1−𝑥2𝑎+3(𝑎+1)²()−𝑎𝑥+𝑎(𝑎+1)(𝑥+𝑎)²−4()].𝑎+1)[(1−𝑎𝑥+2𝑎 (30) According to the symmetry of the slow manifold 𝑆 about 𝑥=(1−𝑎)/2, for 𝜆=0 (i.e., 𝛿=𝛿₀) we suppose that the interval of 𝑥-coordinate of the slow-fast cycle Γ(𝑠) is [(1−𝑎)/2−𝜂,(1−𝑎)/2+𝜂], where 𝜂∈(0,(1−𝑎)/2) and 𝑠∈(0,𝑦₀−𝑎); see Figure 1(a). By straightforwardly calculating, we have 𝐼𝑠,𝛿₀=^{(1−𝑎)/2−𝜂}_{(1−𝑎)/2+𝜂}(𝑓^′₀(𝑥))²𝑔(𝑥,𝑓₀(𝑥))_𝜆=0𝑑𝑥=2(𝑎−1)ℎ𝜂, (31) where ℎ𝜂=2(𝑎+2)(𝑎+1)²ln1+𝑎+2𝜂−41+𝑎−2𝜂1−𝑎²()ln𝑎+1²+2𝜂(1−𝑎)(𝑎+1)²()−−2𝜂1−𝑎8𝜂𝑎(𝑎+1)(1+𝑎)²−4𝜂². (32)

Note that ℎ^′𝜂=−64𝑎𝜂²4(3−𝑎)𝜂²−1−𝑎²(3+𝑎)(1+𝑎)²−4𝜂²²(𝑎+1)⁴−4𝜂²(1−𝑎)²>0 (33) for 0<𝑎<1 and 0<𝜂<(1−𝑎)/2; then ℎ(0)=0 implies that ℎ(𝜂)>0 for 0<𝜂<(1−𝑎)/2; that is, 𝐼(𝑠,𝛿₀)<0 for 0<𝑠<𝑦₀−𝑎. Hence, it follows form the results in [12] that (7) has at most one canard slow-fast cycle.

Summarizing the above inference, we obtain the following result.

Theorem 1. When 𝑎<1, (7) has at most one canard slow-fast cycle, which is asymptotically stable for 0<𝜀≪1 and 𝜆 satisfying (28).

4. Numerical Simulation and Discussion

We first plot the trajectories of (5) and the corresponding curves of 𝑥=𝑥(𝑡) and 𝑦=𝑦(𝑡) as 𝛿=3.2, 𝑎=0.15, 𝜖=0.005, and 𝜆=0.001 (see Figure 2), in which Figures 2(b) and 2(c) illustrate the existence of the fast stages of 𝑥=𝑥(𝑡) and 𝑦=𝑦(𝑡).

(a)

(b)

(c)

Figure 2 

The trajectories of (5) with the fast stage and the slow stage and the corresponding curves of 𝑥=𝑥(𝑡) and 𝑦=𝑦(𝑡), where 𝛿=3.2, 𝑎=0.15, 𝜖=0.005, and 𝜆=0.001.

Note that the vertical and horizontal isoclines of (5) divide the first quadrant of the (𝑥,𝑦)-plane into four regions: 𝐷₁=𝑥,𝑦∈𝑅²₊∶𝑑𝑥𝑑𝑡>0,𝑑𝑦;𝐷𝑑𝑡>0₂=𝑥,𝑦∈𝑅²₊∶𝑑𝑥𝑑𝑡<0,𝑑𝑦;𝐷𝑑𝑡>0₃=𝑥,𝑦∈𝑅²₊∶𝑑𝑥𝑑𝑡<0,𝑑𝑦;𝐷𝑑𝑡<0₄=𝑥,𝑦∈𝑅²₊∶𝑑𝑥𝑑𝑡>0,𝑑𝑦.𝑑𝑡<0 (34) And from (2) and (6) it follows that 𝑟₁=𝑐₁𝛾/𝛿, 𝑎₁=(𝑎−𝜀𝜆)𝐾, and 𝑠₁=𝑐₁𝛾𝜀, where 𝛼, 𝛿, 𝜀, and 𝜆 are the parameters in (5); then, corresponding to (1), the value of parameter 𝑠₁ is very small relative to other parameters as 𝜀 is very small.

In order to more clearly show the fast stages, we locally enlarge the two curves in Figures 2(b) and 2(c) into Figures 3(a) and 3(b), respectively. From Figures 3(a) and 3(b), we can know that the fast stages of 𝑥=𝑥(𝑡) occur in region 𝐷₁, and those of 𝑦=𝑦(𝑡) occur in region 𝐷₃. Biologically, 𝑥=𝑥(𝑡) and 𝑦=𝑦(𝑡) represent the numbers of preys and predators at time 𝑡, respectively; then, when the intrinsic growth rate of the predator is very small and there exists the periodic solution for (1), according to signs of derivatives of 𝑥(𝑡) and 𝑦(𝑡) with respect to time 𝑡, the number of the preys increases quickly from the number small enough as (𝑥(𝑡),𝑦(𝑡))∈𝐷₁, and the number of the predators decreases quickly as (𝑥(𝑡),𝑦(𝑡))∈𝐷₃. The quick increase of the prey from the small value may explain the phenomenon of outbreak in [8].

(a)

(b)

Figure 3 

The curves of 𝑥=𝑥(𝑡) and 𝑦=𝑦(𝑡) obtained by locally enlarging the curves of 𝑥=𝑥(𝑡) and 𝑦=𝑦(𝑡) in Figure 2, respectively.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Fund of China (nos. 11371369 and 11371313).