Profile of the unique limit cycle in a class of general predator–prey systems

Applied Mathematics and Computation 242 (2014) 397–406

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Profile of the unique limit cycle in a class of general predator–prey systems Jinfeng Wang a,1, Xin Zhang a,b, Junping Shi c,⇑,2, Yuwen Wang a,3 a

Y.Y. Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang 150025, PR China Department of Mathematics, Qiqihar University, Qiqihar, Heilongjiang 161006, PR China c Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA b

a r t i c l e

i n f o

Keywords: Limit cycle Predator–prey system Profile

a b s t r a c t Many predator–prey systems with oscillatory behavior possess a unique limit cycle which is globally asymptotically stable. For a class of general predator–prey system, we show that the solution orbit of the limit cycle exhibits the temporal pattern of a relaxation oscillator, when a certain parameter is small. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction With the wide applications in the natural world, the predator–prey systems has been one of the important topics in ecology and mathematical biology. Along with the development of modern mathematics, the predator–prey systems have been by using qualitative analysis and stability theory. For the research of limit cycles, in 1975 Freedman and Waltman [4,5] used various techniques for establishing the existence of limit cycles. In 1981 Cheng [3] (see also Liou and Cheng [14]) published a result giving a criterion for the uniqueness of limit cycles for a special class of predator–prey models. In 1950’s, Zhang proved a uniqueness theorem of limit cycles of generalized Liénard equations, which was later recorded in [25] in 1986. Zhang’s result was used by Kuang and Freedman [13] to consider a Gause type predator–prey system:



x_ ¼ xgðxÞ  nðyÞpðxÞ; y_ ¼ gðyÞðc þ qðxÞÞ:

in [13], they converted this predator–prey model to a Liénard equation, then showed that the new model satisfies the conditions in [25], consequently proved the uniqueness of limit cycle of this predator–prey system. Models of this type were introduced by Gause et al. [6], and since then, variations of this model have been utilized in Armstrong [1], Hassell [7], Hassell and May [10], and Rosenzweig [19], Alberecht et al. [2], May [16], Rosenzweig [18]. In 2009, Hsu and Shi [11] studied a predator–prey system in the particular form:

(

du dt dv dt

v; ¼ uð1  uÞ  mu aþu v; ¼ dv þ mu aþu

ð1:1Þ

⇑ Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (X. Zhang), [email protected] (J. Shi), wangyuwen1950@yahoo. com.cn (Y. Wang). 1 Partially supported by NSFC (No. 11201101) and Scientific Research Project of Heilongjiang Provincial Department of Education of China (No. 12521152 and No. 1254G034). 2 Partially supported by DMS-1022648. 3 Partially supported by Natural Science Foundation of Heilongjiang Province of China (A201106). http://dx.doi.org/10.1016/j.amc.2014.05.020 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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J. Wang et al. / Applied Mathematics and Computation 242 (2014) 397–406

where a; m; d > 0. The system (1.1) is often called Rosenzweig–MacArthur predator–prey system from the seminar work of Rosenzweig and MacArthur [20]. Hsu and Shi [11] considered the relaxation oscillator profile of the limit cycle of (1.1) by a careful phase portrait analysis and estimates. It is important to know whether such profile is special only to (1.1) or it holds for a more general class of predator–prey systems. Hence the objective of this paper is to study the dynamical properties of a general predator–prey systems, in particular, the asymptotic behavior of the limit cycle. In this paper, we consider a class of more general predator–prey systems in the form

8 du > < dt ¼ ugðuÞ  v pðuÞ; dv ¼ v ðd þ pðuÞÞ; > : dt uð0Þ  0; v ð0Þ  0;

ð1:2Þ

where d > 0, the functions gðuÞ and pðuÞ are sufficiently smooth so that the existence, uniqueness, and continuous dependence on parameters of solutions to the initial-value problem are satisfied. The functions uðtÞ and v ðtÞ represent the prey and predator populations, respectively, at a given time t P 0. In this paper, we assume that the functions gðuÞ and pðuÞ in (1.2) satisfy (H1): g 2 C 2 ðRþ Þ; gð0Þ > 0, there exists k > 0, such that for any u > 0; u – k; gðuÞðu  kÞ < 0 and gðkÞ ¼ 0. (H2): p 2 C 2 ðRþ Þ; pð0Þ ¼ 0; p0 ðuÞ > 0 for any u P 0, and there exists a 2 ð0; kÞ such that pðaÞ ¼ d. 2 þ if u > 0 and Fð0Þ ¼ pgð0Þ (H3): Define FðuÞ ¼ ugðuÞ 0 ð0Þ. Then F 2 C ðR Þ. We assume there exists a 2 ð0; kÞ, such that for any pðuÞ 0 0 u > 0; u – a ; F ðuÞðu  a Þ < 0 and F ða Þ ¼ 0.

It is known that (see Hsu [8]) if (H1)–(H3) are satisfied, then (1.2) possesses a unique coexistence equilibrium point ða; FðaÞÞ. The local stability of ða; FðaÞÞ depends on the sign of F 0 ðaÞ: when a < a < k, then F 0 ðaÞ < 0 and ða; FðaÞÞ is locally asymptotically stable; and when 0 < a < a , then F 0 ðaÞ > 0 and ða; FðaÞÞ is unstable. Moreover the global stability of ða; FðaÞÞ when a < a < k can be established through a Lyapunov functional or Dulac criterion under some extra conditions (see [8,9]). On the other hand, when 0 < a < a , the instability of ða; FðaÞÞ implies the existence of a periodic orbit from the Poincaré–Bendixon theory. The uniqueness of the periodic orbit will make the periodic orbit a limit cycle—the attractor for the predator–prey system. Since the work of Cheng [3], the uniqueness of the limit cycle in (1.2) has been proved under some extra conditions [12,24]. Here we site a result of Kuang and Freedman [13]: if (H1)–(H3) are satisfied, and also (H4): for all 0 6 u 6 k; u – a, we have

d du



pðuÞF 0 ðuÞ dþpðuÞ



6 0,

then the limit cycle of (1.2) is unique and is global asymptotically orbital stable. Moreover, we can verify the uniqueness of limit cycle holds if (H1)–(H3) are satisfied, and also (H4)0 : F 2 C 3 ðRþ Þ, and uF 000 ðuÞ þ 2F 00 ðuÞ 6 0 for 0 6 u 6 k, which can be obtained from results in [22,23]. We also recall that the growth rate of the prey is of logistic type if gðuÞ is strictly decreasing, and it is of weak Allee effect type if gðuÞ is increasing for 0 < u < c and is decreasing for c < u < k. Conditions (H1) and (H3) allow for either type of growth. For example, gðuÞ ¼ k  u is a logistic growth; for gðuÞ ¼ ðk  uÞðu þ aÞ, it is weak Allee effect type when 0 6 a < k, and it is logistic type when a > k. Some examples of pðuÞ are Holling type II functional response pðuÞ ¼ mu=ðb þ uÞ, or Ivlev type as pðuÞ ¼ mð1  ebu Þ. Our result here generalizes the one in [11], in which the relaxation oscillation profile of the limit cycle in a predator–prey model was first studied. An earlier work for relaxation oscillator in predator–prey model appeared in [15]. For many other mathematical models with limit cycle behavior and small parameters, such relaxation oscillation have been well-documented in, for example, [17,21]. For such relaxation oscillation profile, the prey population uðtÞ is near zero for a very long period when d is small (see Figs. 2 and 3 for illustration). Biologically this means the prey population is vulnerable to extinction even with small stochastic perturbations. We prove our main results in Section 2 for the case d ! 0. We will use di and C i , (i 2 N), to denote various positive constants. These constants are independent of d in Section 2. We give an example and some numerical simulations to illustrate our results in Section 3. 2. Asymptotic behavior of the limit cycle for d small In this section, we consider the asymptotical profile of the limit cycle of (1.2). We assume that 0 < a < a and the conditions (H1)–(H4) (or (H1)–(H4)0 ) hold. We define

f ðu; v Þ ¼ ugðuÞ  v pðuÞ;

gðu; v Þ ¼ v ðd þ pðuÞÞ:

ð2:1Þ

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J. Wang et al. / Applied Mathematics and Computation 242 (2014) 397–406

8

8

7

7

6

6

5

5

4

4

v

v

Fig. 1. Illustration of the phase portrait (not up to scale) and the limit cycle in the proof. The isoclines are the thin solid curves: u ¼ 0; v ¼ 0; u ¼ a and the parabola v ¼ FðuÞ; the limit cycle is the thick solid curve O1 O2 O3 O4 ; the boundary of the invariant region R3 : v ¼ v 4 ðuÞ is the outer boundary (together with u ¼ 0 and v ¼ 0); v ¼ v 5 ðuÞ and v ¼ v 6 ðuÞ are the upper and lower portions of inner boundary respectively; the line u ¼ b satisfies FðbÞ ¼ FðaÞ. (This graph is essentially from [11].)

3

3

2

2

1

1

0

0 0

0.2

0.4

0.6

0.8

1 u

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1 u

1.2

1.4

1.6

1.8

2

Fig. 2. Phase portraits of (3.1). Left: d ¼ 0:45 with initial value ðu; v Þ ¼ ð0:8; 3:1Þ; Right: d ¼ 0:3 with initial value ðu; v Þ ¼ ð1:8; 1:4Þ.

9

8

8

7

7

6

6

5

5

v

10

9

v

10

4

4

3

3

2

2

1

1

0

0 0

0.2

0.4

0.6

0.8

1 u

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1 u

1.2

1.4

1.6

1.8

2

Fig. 3. Phase portraits of (3.1). Left: d ¼ 0:1 with initial value ðu; v Þ ¼ ð1; 0:8Þ; Right: d ¼ 0:01 with initial value ðu; v Þ ¼ ð0:6; 1:6Þ.

First we construct an invariant region in which the limit cycle is located. To achieve that, we give an estimate of the unstable manifold U ¼ fðu1 ðtÞ; v 1 ðtÞÞ : t 2 Rg of the saddle point ðk; 0Þ. From the phase portrait, it satisfies 0 < u1 ðtÞ < k for all t 2 R; U is above the isocline v ¼ FðuÞ when a < u < k. Since it is monotone for a < u < k, we denote this portion by fðu; v 1 ðuÞÞ : a 6 u 6 kg with v 1 ðkÞ ¼ 0. We define

v 2 ðuÞ ¼



 1  F 0 ðkÞ ðk  uÞ;

and

v 3 ðuÞ ¼

Z k

u



 d  1 ds: pðsÞ

ð2:2Þ

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J. Wang et al. / Applied Mathematics and Computation 242 (2014) 397–406

Lemma 2.1. Suppose that F also satisfies (H5): For any u P 0; F 0 ðuÞ is monotonically decreasing. Then the unstable manifold U of ðk; 0Þ satisfies

v 3 ðuÞ 6 v 1 ðuÞ 6 v 2 ðuÞ;

a 6 u 6 k:

ð2:3Þ

Proof. From Eq. (1.2), we have

dv v ¼  ðd þ pðuÞÞ: du ugðuÞ  v pðuÞ Since the unstable manifold satisfies 0 < u1 ðtÞ < k for all t 2 R, then along U, we have

dv v d  1: 6  ðd þ pðuÞÞ ¼ pðuÞ du v pðuÞ Integrating along the portion of U from u ¼ k to some u 2 ða; kÞ, we obtain

vP

Z

u



k

 d  1 ds ¼ v 3 ðuÞ; pðsÞ

if ðu; v Þ 2 U and a 6 u 6 k. For the upper bound, we notice that the tangent line of the unstable manifold is

v¼

    0 kg ðkÞ þ d d ðk  uÞ ¼ 1  F 0 ðkÞ  ðk  uÞ; 1 pðkÞ pðkÞ

which is below v ¼ v 2 ðuÞ. Hence we only need to show that the vector field ðf ðu; v Þ; gðu; v ÞÞ points towards the region below the line v ¼ v 2 ðuÞ when ðu; v Þ ¼ ðu; v 2 ðuÞÞ and a < u < k. That is equivalent to

dv 6 1  F 0 ðkÞ; du

ðu; v Þ ¼ ðu; v 2 ðuÞÞ:

Let l ¼ 1  F 0 ðkÞ, then for ðu; v Þ ¼ ðu; v 2 ðuÞÞ; a 6 u < k,

dv lðk  uÞ ¼ lðk  uÞðpðuÞ  dÞ 6 du jFðuÞ  lðk  uÞjpðuÞ jFðuÞ  lðk  uÞj :

From the mean-value theorem, (H1) and (H5), we have

FðuÞ ¼ FðuÞ  FðkÞ ¼ F 0 ðnÞðu  kÞ P F 0 ðkÞðu  kÞ ¼ ð1  lÞðu  kÞ; for some n 2 ðu; kÞ. Hence

dv lðk  uÞ 6 du jð1  lÞðu  kÞ þ lðu  kÞj ¼ l; which proves the upper bound

v 1 ðuÞ 6 v 2 ðuÞ.

h

From Lemma 2.1, the unstable manifold reaches its maximum when u ¼ a, and the maximum value v  can be estimated as

Z k

a



 d  1 ds 6 v  6 ð1  F 0 ðkÞÞðk  aÞ: pðsÞ

ð2:4Þ

From the phase portrait of the system, the limit cycle is below the unstable manifold U, then we also have the following estimate of the outer boundary of the limit cycle. Lemma 2.2. Let

v 4 ðuÞ ¼



v 2 ðuÞ be defined as in (2.2). Define

v 2 ðuÞ; v 2 ðaÞ;

a 6 u 6 k; 0 6 u 6 a:

ð2:5Þ

Then the orbit of the limit cycle R ¼ fðuðtÞ; v ðtÞÞ : 0 6 t 6 Tg satisfies

R  fðu; v Þ : 0 < u < k; 0 < v < v 4 ðuÞg  R1 : By constructing a more precise region R2  R1 containing R, we prove that for a sub-region R3 containing ða; FðaÞÞ; R \ R3 ¼ ;. From (H3), there exists a unique b ¼ bðaÞ 2 ða ; kÞ such that FðbÞ ¼ FðaÞ. Define

J. Wang et al. / Applied Mathematics and Computation 242 (2014) 397–406

R3 ¼ fðu; v Þ 2 R2þ : Wðu; v Þ 6 Wðb; FðbÞÞg;

401

ð2:6Þ

where Wðu; v Þ is defined by the well-known Lyapunov function for (1.2) (when ða; FðaÞÞ is locally asymptotically stable)

Z

Wðu; v Þ ¼

u

a

pðnÞ  d dn þ pðnÞ

Z v g  FðaÞ

g

FðaÞ

dg:

Lemma 2.3. Let R3 be defined as in (2.6). Then R3 is a bounded convex subset of R2þ containing ða; FðaÞÞ, and R \ R3 ¼ ;. In particular R  R2  R1 n R3 . Proof. From the definition in Wðu; v Þ; Wðu; v Þ ¼ W 1 ðuÞ þ W 2 ðv Þ, where

W 1 ðuÞ ¼

Z

u

a

pðnÞ  d dn; pðnÞ

and W 2 ðv Þ ¼

Z v g  FðaÞ

g

FðaÞ

dg:

Since W 01 ðuÞ ¼ ðpðuÞ  dÞ=pðuÞ, then W 1 ðuÞ is strictly decreasing in ð0; aÞ and is strictly increasing in ða; 1Þ; similarly since W 02 ðv Þ ¼ 1  ðFðaÞ=v Þ, then W 2 ðv Þ is strictly decreasing in ð0; FðaÞÞ and is strictly increasing in ðFðaÞ; 1Þ. Hence W achieves the global minimum at the unique critical point ða; FðaÞÞ, and every level curve of Wðu; v Þ is a bounded closed curve. The level curves have convex boundary since W 1 and W 2 are both convex one-variable functions. For R3 defined in (2.6), ðb; FðaÞÞ is the right-most point of R3 . Thus for any solution orbit ðuðtÞ; v ðtÞÞ passing through ðu; v Þ 2 R3 n fðb; FðaÞÞg,

W 0 ðuðtÞ; v ðtÞÞ ¼ pðuÞd ðugðuÞ  v pðuÞÞ þ v FðaÞ pðuÞ v ðpðuÞ  dÞv ¼ ðpðuÞ  pðaÞÞðFðuÞ  FðaÞÞ > 0: In particular, for ðu; v Þ 2 @R3 n fðb; FðaÞÞg, the vector field ðf ðu; v Þ; gðu; v ÞÞ points outwards. Hence from the properties of periodic orbit, R \ R3 ¼ ;. h From Lemmas 2.2 and 2.3, we obtain an invariant region R2 where the limit cycle is located in. Next we give some estimates for the extremal points on the orbit of limit cycle as d ! 0þ . Since a ¼ p1 ðdÞ ! 0 when d ! 0þ , where p1 is the inverse function of p, hence d and a are two equivalent parameters which tend to zero. Define

ua;

¼ minfuðtÞ : ðuðtÞ; v ðtÞÞ 2 Rg;

ua;þ ¼ maxfuðtÞ : ðuðtÞ; v ðtÞÞ 2 Rg;

v a;

¼ minfv ðtÞ : ðuðtÞ; v ðtÞÞ 2 Rg;

v a;þ ¼ maxfv ðtÞ : ðuðtÞ; v ðtÞÞ 2 Rg:

ð2:7Þ

Notice that both the upper and lower portions of the limit cycle are monotone functions, thus we define



R ¼ ðu; v þ ða; uÞÞ : ua; 6 u 6 ua;þ

[

 ðu; v  ða; uÞÞ : ua; 6 u 6 ua;þ ;

ð2:8Þ

such that v  ða; uÞ < FðuÞ < v þ ða; uÞ for ua; < u < ua;þ . That is, fðu; v þ ða; uÞÞg is the upper portion of the limit cycle R, and fðu; v  ða; uÞÞg is the lower portion. From the equations, it is easy to see that ua; and ua;þ are achieved when R intersects with the isocline v ¼ FðuÞ, and v a; ; v a;þ are achieved when R intersects with the line u ¼ a. Our estimates are mainly based on the inner boundary of the region R2 , i.e. the level curve R1 ¼ fðu; v Þ : Wðu; v Þ ¼ Wðb; FðbÞÞg. Hence we also define

u1;a

¼ min fu : ðu; v Þ 2 R1 g;

v 1;a

¼ minfv : ðu; v Þ 2 R1 g;

u2;a ¼ maxfu : ðu; v Þ 2 R1 g; v 2;a ¼ maxfv : ðu; v Þ 2 R1 g;

ð2:9Þ

and



R1 ¼ ðu; v 5 ðuÞÞ : u1;a 6 u 6 u2;a such that

[

 ðu; v 6 ðuÞÞ : u1;a 6 u 6 u2;a ;

ð2:10Þ

v 6 ðuÞ < FðuÞ < v 5 ðuÞ for u1;a < u < u2;a . Notice that

rW ¼



 pðuÞ  d v  FðaÞ ; ; pðuÞ v

hence v 1;a and v 2;a are the two intersection points of Wðu; v Þ ¼ Wðb; FðbÞÞ with the line u ¼ a. Also u2;a ¼ b, and u1;a satisfies Wðu1;a ; FðaÞÞ ¼ Wðb; FðbÞÞ with u1;a < a. We define

h1 ðu; aÞ ¼

Z a

u

pðnÞ  d dn; pðnÞ

and h2 ðv ; FðaÞÞ ¼

Z v g  FðaÞ FðaÞ

g

dg:

ð2:11Þ

From the monotonicity of pðuÞ; h1 ð; aÞ achieves its global minimum 0 at u ¼ a. Similarly,

@h2 ðv ; FðaÞÞ FðaÞ ¼1 ; @v v

ð2:12Þ

hence h2 ð; FðaÞÞ achieves its global minimum 0 at v ¼ FðaÞ. Since h2 ðv ; FðaÞÞ ¼ v  FðaÞ  FðaÞ lnðv =FðaÞÞ; limv !0þ h2 ðv ; FðaÞÞ ¼ limv !1 h2 ðv ; FðaÞÞ ¼ 1. Thus h2 ðv ; FðaÞÞ ¼ C has exactly two roots for any C > 0.

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J. Wang et al. / Applied Mathematics and Computation 242 (2014) 397–406

Lemma 2.4. Let v 1;a and v 2;a be defined as in (2.9), and let v 1 and v 2 be the two roots of h2 ðv ; Fð0ÞÞ ¼ bð0Þ such that v 1 < Fð0Þ < v 2 , where bð0Þ is the unique point in ð0; kÞ such that Fðbð0ÞÞ ¼ Fð0Þ. Then

lim v 1;a ¼ v 1 ;

lim v 2;a ¼ v 2 :

and

a!0þ

ð2:13Þ

a!0þ

Proof. From Lemma 2.3, v a; < v 1;a and v 2;a < v a;þ . By definition form of Wðu; v Þ and (2.11), v ¼ v i;a (i ¼ 1; 2) satisfy

v ¼ v i;a

(i ¼ 1; 2) satisfy Wða; v Þ ¼ Wðb; FðbÞÞ. From the

h2 ðv ; FðaÞÞ ¼ h1 ðbðaÞ; aÞ:

ð2:14Þ

We prove that

lim h1 ðbðaÞ; aÞ ¼ bð0Þ:

ð2:15Þ

a!0þ

Since p0 ð0Þ exists, for any

e > 0, there exists d > 0, such that for 0 < n < d, we have

0

ðp ð0Þ  eÞn 6 pðnÞ 6 ðp0 ð0Þ þ eÞn; Then

h1 ðbðaÞ; aÞ ¼

Z

bðaÞ

 1

a

 Z d Z bðaÞ pðaÞ dn dn dn ¼ bðaÞ  a  pðaÞ  pðaÞ : pðnÞ pðnÞ a pðnÞ d

Since

Z

0<

bðaÞ

d

then pðaÞ

Z

d

a

R bðaÞ d

dn bðaÞ  d < ; pðnÞ pðdÞ

dn pðnÞ

dn 6 pðnÞ

! 0 as a ! 0þ from (H2). On the other hand,

Z a

d

dn 1 ¼ ðln d  ln aÞ: ðp0 ð0Þ  eÞn p0 ð0Þ  e

From L’Hospital’s rule, we have lima!0 pðaÞ  ln a ¼ 0. Then all estimates together imply that lima!0þ h1 ðbðaÞ; aÞ ¼ bð0Þ. Since v ¼ v i;a is continuously differentiable in a, and differentiating (2.14) with respect to a at v ¼ v i;a , and from (2.12), we obtain

    FðaÞ @ v i;a v i;a 0 pðbÞ  d @b F ðaÞ ¼ 1  ln : pðbÞ @a v i;a @a FðaÞ

ð2:16Þ

When a ! 0; d ¼ pðaÞ ! 0, and the right hand side of (2.16) tends to a negative limit @b ð0Þ. On the left hand side of (2.16), @a F 0 ðaÞ > 0 when 0 < a < a , and since v 1;a < FðaÞ;  lnðv 1;a =FðaÞÞF 0 ðaÞ > 0, thus @ v 1;a =@a > 0. In particular, lima!0þ v 1;a exists. On the other hand, for a near 0; v 2;a satisfies

½1  F 0 ðkÞk > v 2 ðaÞ > v 2;a > FðaÞ > Fð0Þ:

ð2:17Þ

Hence fv 2;a : 0 < a < dg for some small d > 0 is uniformly bounded, and there exists a decreasing sequence an ! 0 such that v 2;a converges to a limit v 1 , and v 1 must satisfy h2 ðv 1 ; Fð0ÞÞ ¼ bð0Þ from (2.14) and (2.15). From the bound of v 2;a in (2.17), v 1 must be the larger root v 2 of h2 ðv ; Fð0ÞÞ ¼ bð0Þ. Since each subsequence of fv 2;a g converges to v 2 , then lima!0 v 2;a ¼ v 2 . The same argument yields that lima!0 v 1;a ¼ v 1 , which is the smaller root of h2 ðv ; Fð0ÞÞ ¼ bð0Þ. h n

þ

þ

To obtain the global asymptotical behavior of the limit cycle R, we divide the orbit R into four segments by four reference points (see Fig. 1):

O1

¼ ða; v a;þ Þ;

O3

¼ ða ; v  ða; a ÞÞ;

O2 ¼ ða; v a; Þ;

ð2:18Þ

O4 ¼ ða ; v þ ða; a ÞÞ:

Let T ¼ TðaÞ be the period of R. Then T ¼ T 1 þ T 2 þ T 3 þ T 4 , where T i is the time taken from Oi to Oiþ1 (with O5 ¼ O1 ). We also assume that uð0Þ ¼ a and v ð0Þ ¼ v a;þ , i.e. the orbit starts from the highest point of v ðtÞ. Our main result in this section is Theorem 2.5. Assume that the condition (H1)–(H5) are satisfied. Let R ¼ fðuðtÞ; v ðtÞÞ : t 2 Rg be the orbit of the unique periodic solution of (1.2) when 0 < a < a , the extremal points of R are defined as in (2.7), and Oi ; T i ði ¼ 1; 2; 3; 4Þ and the period T are defined as above. When a > 0 is sufficiently small (or equivalently d > 0 is small), then there exists constants C 2 ; C 3 > 0 independent of a, such that C 3 =pðaÞ P T P C 2 =pðaÞ. Moreover, for a > 0 sufficiently small, there exists C 4 > 0, such that

C3 C4 P T1 P ; pðaÞ pðaÞ as a ! 0þ .

T 2 ¼ Oðj ln ajÞ;

T 3 ¼ Oð1Þ;

T 4 ¼ Oðj ln ajÞ:

ð2:19Þ

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Proof. We prove the theorem in several steps. Step 1. We show that

T1 P

1 ln d  pðua; Þ



v a;þ v a;

 ð2:20Þ

:

Define pðua; Þ ¼ dð1  d2 Þ for some 0 < d2 < 1. Then for 0 < t < T 1 ; ua; 6 uðtÞ 6 a, and from the equation of

v

0

v ðtÞ,

¼ v ðd þ pðuÞÞ P v ðd þ pðua; ÞÞ ¼ dd2 v :

Hence v ðtÞ P v ð0Þ expðdd2 tÞ, which leads to

T1 P

1 ln d2 d



v a;þ v a;



¼

1 ln d  pðua; Þ



v a;þ v a;

 :

ð2:21Þ

Step 2. We show there exist constants d3 ; d4 > 0 such that when 0 < a < d4 ,

0 < T2 6

1 d3

Z

a a

du : pðuÞ

ð2:22Þ

For T 1 6 t 6 T 1 þ T 2 , we have a 6 uðtÞ 6 a . From the equation of uðtÞ,

u0 ¼ pðuÞðv 0 ðuÞ  v Þ P pðuÞðv 0 ðuÞ  v 6 ðuÞÞ;

ð2:23Þ

which follows from Lemma 2.3 that the limit cycle is below the level curve ðu; v 6 ðuÞÞ in this portion. Since v 0 ðuÞ is concave while v 6 ðuÞ is convex, then the minimum of v 7 ðuÞ ¼ v 0 ðuÞ  v 6 ðuÞ on the interval ½a; a  must achieve at either u ¼ a or u ¼ a . From the proof of Lemma 2.4, v 6 ðaÞ ! v 1 , the smaller root of h2 ðv ; Fð0ÞÞ ¼ bð0Þ, and v 0 ðaÞ ¼ FðaÞ ! Fð0Þ as a ! 0þ . Thus v 7 ðaÞ ! Fð0Þ  v 1 > 0 as a ! 0þ . Similarly as a ! 0þ ; v 0 ða Þ ! Fða Þ, and v 6 ða Þ ! v~1 , which is the smaller root of h2 ðv ; Fð0ÞÞ ¼ bð0Þ  a , as we take the limit of a ! 0þ in

Wða ; v 6 ða ÞÞ ¼ h1 ða ; aÞ þ h2 ðv 6 ða Þ; FðaÞÞ ¼ h1 ðbðaÞ; aÞ: Similar to the proof of lima!0þ h1 ðbðaÞ; aÞ ¼ bð0Þ, we have lima!0þ h1 ða ; aÞ ¼ a , hence h2 ðv 6 ða Þ; FðaÞÞ ! bð0Þ  a , i.e.

v 6 ða Þ ! v~1 .

Thus there exists d3 ; d4 > 0 such that when 0 < a < d4 ,

v 0 ðuÞ  v 6 ðuÞ P min fv 0 ðaÞ  v 6 ðaÞ; v 0 ða Þ  v 6 ða Þg P d3 > 0:

ð2:24Þ

Now from (2.23) and (2.24), for T 1 6 t 6 T 1 þ T 2 we have

Z

du P d3 dt; pðuÞ

a a

du P d3 T 2 ; pðuÞ

ð2:25Þ

which implies (2.22). Step 3. We show that

0 < T3 6

1 ln pða Þ  d



v þ ða; a Þ v  ða; a Þ

 :

ð2:26Þ

For this portion, uðtÞ > a . From the equation of v, we have

v 0 ¼ v ðd þ pðuÞÞ P v ðd þ pða ÞÞ: Hence v ðtÞ P v ðT 1 þ T 2 Þ exp ð½pða Þ  dt Þ, and in particular

v þ ða; a Þ P v  ða; a Þ exp ð½pða Þ  dT 3 Þ; which implies (2.26). Step 4. We show there exist constants d5 ; d6 > 0 such that when 0 < a < d6 ,

0 < T4 6

1 d5

Z

a a

du : pðuÞ

ð2:27Þ

This is similar to Step 2. Now we have

u0 ¼ pðuÞðv 0 ðuÞ  v Þ 6 pðuÞðv 0 ðuÞ  v 5 ðuÞÞ 6 pðuÞðv 0 ða Þ  v 5 ða ÞÞ:

ð2:28Þ

Here the first inequality is from Lemma 2.3, and the second inequality is from the fact that v 0 ðuÞ is increasing while v 5 ðuÞ is decreasing in ½a; a Þ, and v 0 ðuÞ < v 5 ðuÞ. Similar to the proof of Step 2, we obtain that when 0 < a < d6 ,

ju0 j P d5 pðuÞ: The remaining part is same as Step 2.

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Step 5. We show that for any 0 < d7 < 1, when a > 0 is sufficiently small, there exists a constant C 1 > 0 such that

T1 6

1 ln d7 pðaÞ



v a;þ v a;



þ C1:

ð2:29Þ

We reconsider the portion of R in ð0; T 1 Þ again. Notice that when a > 0 is sufficiently small, ua; < a and lima!0 ua; ¼ 0. Hence for any 0 < d7 < 1; að1  d7 Þ < a, thus the orbit does reach u ¼ að1  d7 Þ. We write T 1 ¼ T 11 þ T 12 þ T 13 , so that uðT 11 Þ ¼ að1  d7 Þ, and uðT 12 Þ ¼ að1  d7 Þ. That is, T 11 and T 11 þ T 12 are the times that R reaches u ¼ að1  d7 Þ. We also define v 11 ¼ v ðT 11 Þ and v 12 ¼ v ðT 11 þ T 12 Þ. For t 2 ðT 11 ; T 11 þ T 12 Þ, when a > 0 is sufficiently small, similar to Step 1,

v 0 ¼ v ðd þ pðuÞÞ 6 v ðd þ p½að1  d7 ÞÞ ¼ v Hence

  d þ p½p1 ðdÞð1  d7 Þ ¼ dd7 v :

v 12 6 v 11 expðdd7 T 12 Þ, and T 12 6

1 ln dd7



v 11 v 12

 6

1 ln pðaÞd7



v a;þ v a;

 ð2:30Þ

:

Next we estimate T 11 . Similar to Step 4, for a > 0 small, ju0 j P d8 pðuÞ for some d8 > 0, if 0 < a < d9 . Here the estimate of v 0 ðuÞ  v 5 ðuÞ can be obtained using the same proof of Lemma 2.4. Indeed we can replace (2.14) by

h1 ðð1  dÞa; aÞ þ h2 ðv 5 ðð1  dÞaÞ; FðaÞÞ ¼ h1 ðbðaÞ; aÞ;

ð2:31Þ

for 0 < d < d7 . Then the same arguments yield ju0 j P d8 pðuÞ, and an integration gives

Z

a

ð1d7 Þa

du P d8 T 11 : pðuÞ

Hence T 11 is bounded by a constant independent of a. Similarly we can prove T 13 is bounded. Step 6. We show that there exist constants v 3 ; v 4 > 0 such that v a;þ < v 3 and v 4 < v a; for all small a > 0. From Lemma 2.1 and (2.4), we obtain the estimate of upper bound of v a;þ by letting v 3 ¼ 1  F 0 ðkÞ. For the estimate of v 4 , we notice that any solution orbit satisfies

du pðuÞ v 0 ðuÞ  v ¼  : dv pðuÞ  d v

ð2:32Þ

Recall that O1 ¼ ða; v a;þ Þ and O2 ¼ ða; v a; Þ are the highest and lowest points on the orbit of the limit cycle R. Let the leftmost point on R be O5 ¼ ðua; ; v  Þ. Then from (2.32), we obtain that

Z v v a;

v 0 ðu2 ðv ÞÞ  v dv ¼ v

Z

ua;

a

Z v pðuÞ  d du ¼ pðuÞ v a;þ

v 0 ðu1 ðv ÞÞ  v dv ; v

where ðu1 ðv Þ; v Þ; v  6 v 6 v a;þ , represents the orbit O1 O5 , and ðu2 ðv Þ; v Þ; last integral in (2.33)

Z v v a;þ

v 0 ðuÞ  v dv ¼ v

Z v a;þ v

v  v 0 ðuÞ dv 6 v

Z v a;þ v

ð2:33Þ

v a; 6 v 6 v  , represents the orbit O5 O2 . For the

v  v dv ¼ v a;þ  v   v  ln v a;þ þ v  ln v  : v

ð2:34Þ

Since v 2  d0 < v a;þ < v 3 for small a, then the right hand side of (2.34) is bounded. On the other hand, for the first integral in (2.33),

Z v v a;

v 0 ðuÞ  v dv P v

Z v v a;

v  v dv ¼ v a;  v   v  ln v a; þ v  ln v  : v

ð2:35Þ

Thus  ln v a; is bounded from above from (2.33)–(2.35), and consequently v a; is bounded from below by some v 4 > 0 for all small a > 0. Step 7. The completion of the proof. h From Lemma 2.4 and Step 6, when a > 0 is small, v 4 < v a; < v 1 þ d0 and v 2  d0 < v a;þ < v 3 , where v 1 and v 2 are the two roots of h2 ðv ; Fð0ÞÞ ¼ bð0Þ such that v 1 < Fð0Þ < v 2 , since ua; < a and lima!0 a1 ua; =0. Thus from Step 1 and Step 5, for any 0 < d9 < 1, as long as a > 0 is sufficiently small,

1 ln d7 pðaÞ



v3 v4

 þ C1 P T 1 P

1  d9 ln pðaÞ



v 2  d0 v 1 þ d0

 :

ð2:36Þ

Hence we obtain the estimate for T 1 in the theorem, since all constants except a are independent of a. The estimate for T 3 can also be obtained from Step 3 and Step 6 since v þ ða; a Þ < v a;þ < v 3 and v  ða; a Þ > v a; > v 4 , The estimates of T i for i ¼ 2; 4 P are clear from Steps 2 and 4, and T ¼ T i ¼ Oða1 Þ. This completes the proof. h

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Remark. 1. Our construction of an invariant region in Lemma 2.3 does not require the smallness of a—it holds as long as 0 < a < a . This gives a direct proof of the existence of periodic orbit. 2. When defining O3 and O4 , the choice of u ¼ a can be replaced by any fixed u ¼ b 2 ð0; a , and the results of Theorem 2.5 still hold with this change. 3. Examples To visualize the relaxation oscillator profile of the limit cycle, we consider some specific examples of the model (1.2). The existence and uniqueness of the limit cycle have been proved in [8,13,22]. The Lotka–Volterra predator–prey systems with Holling type II functional response and Logistic growth in prey has been discussed completely in [11]. Here we consider a Lotka–Volterra predator–prey system with Holling type II functional response and weak Allee effect growth in prey: (here e; m; d > 0; 0 6 c < k)

8 du muv > < dt ¼ uðk  uÞðu þ cÞ  eþu ; dv v; ¼ dv þ mu eþu > : dt uð0Þ  0; v ð0Þ  0:

ð3:1Þ

Then contraposing the model (1.2), we have gðuÞ ¼ ðk  uÞðu þ cÞ, pðuÞ ¼ mu=ðe þ uÞ and

FðuÞ ¼

ugðuÞ ðk  uÞðu þ cÞðe þ uÞ ¼ : pðuÞ m

Then Fð0Þ ¼ kec > 0. If in addition, m (A1) kðe þ cÞ > ec, holds, then we can get a unique a ¼ ðkecÞþ

F 0 ðuÞðu  a Þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þe2 þc2 þkeþkcec 3

2 ð0; kÞ, which satisfies F 0 ða Þ ¼ 0, and

3u2 þ 2ðk  e  cÞu þ ke þ kc  ec ðu  a Þ < 0; m

for u > 0; u – a . Then (H1)–(H3) are satisfied. Furthermore,

F 00 ðuÞ ¼

6u þ 2ðk  e  cÞ ; m

uF 000 ðuÞ þ 2F 00 ðuÞ ¼

18u þ 4ðk  e  cÞ : m

Therefore, (H4)0 and (H5) are satisfied if (A2) k < e þ c. Thus when (A1) and (A2) are satisfied, the system (3.1) has a unique limit cycle which is globally asymptotically orbital stable when 0 < a < a , and the asymptotic behavior of the limit cycle of (3.1) can be obtained from Theorem 2.5. In particular we have that the period T of Eq. (3.1) satisfies C 5 a1 P T P C 4 a1 , where C 4 ; C 5 > 0 are the constants independent of a. To visualize the asymptotic behavior of the limit cycle of (3.1), we show the phase portraits with fixed k ¼ 2; e ¼ 1; m ¼ 1; c ¼ 1:5 and varying d in Figs. 2 and 3. References [1] R.A. Armstrong, The effects of predator functional response and prey productivity on predator–prey stability: a graphical approach, Ecology 57 (1976) 609–612. [2] F. Albrecht, H. Gatzke, N. Wax, Stable limit cycles in prey–predator populations, Science 181 (1973) 1073–1074. [3] K.S. Cheng, Uniqueness of a limit cycle for a predator–prey system, SIAM J. Math. Anal. 12 (4) (1981) 541–548. [4] H.I. Freedman, Deterministic Mathematical Models in Population ecology, Marcel Dekker, New York, 1980. [5] H.I. Freedman, P. Waltman, Perturbation of two dimensional predator–prey equations, SIAM J. Appl. Math. 28 (1975) 1–10. [6] G.F. Gause, N.P. Smaragdova, A.A. Witt, Further studies of interaction between predator and prey, J. Anim. Ecol. 5 (1936) 1–18. [7] M.P. Hassell, The Dynamics of Arthropod Predator–Prey Systems, Princeton, U.P., Princeton, 1978. [8] S.B. Hsu, On global stability of a predator–prey system, Math. Biosci. 39 (1–2) (1978) 1–10. [9] S.B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese J. Math. 9 (2) (2005) 151–173. [10] M.P. Hassell, R.M. May, Stability in insect host-parasite models, J. Anim. Ecol. 42 (1973) 693–726. [11] S.B. Hsu, J. Shi, Relaxation oscillator profile of limit cycle in predator–prey system, Discrete Contin. Dyn. Syst. Ser. B 11 (4) (2009) 893–911. [12] T.W. Hwang, Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response, J. Math. Anal. Appl. 290 (1) (2004) 113–122. [13] Y. Kuang, H.I. Freedman, Uniqueness of limit cycles in Gause-type models of predator–prey systems, Math. Biosci. 88 (1) (1988) 67–84. [14] L.P. Liou, K.S. Cheng, On the uniqueness of a limit cycle for a predator–prey system, SIMA J. Math. Anal. 19 (1988) 867–878. [15] W.S. Liu, D.M. Xiao, Y.F. Yi, Relaxation oscillations in a class of predator–prey systems, J. Differ. Equ. 188 (2003) 306–331. [16] R.M. May, Limit cycles in predator–prey communities, Science 177 (1972) 900–902. [17] J.D. Murray, Mathematical biology. I. An introduction, Interdisciplinary Applied Mathematics, third ed., vol. 17, Springer-Verlag, New York, 2002. [18] M.L. Rosenzweig, Paradox of enrichment: destabilization of exploitation ecosystems in ecological time, Science 171 (3969) (1971) 385–387. [19] M.L. Rosenzweig, Evolution of the predator isocline, Evolution 27 (1973) 84–94. [20] M.L. Rosenzweig, R. MacArthur, Graphical representation and stability conditions of predator–prey interactions, Am. Nat. 97 (1963) 209–223.

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