Uniqueness of limit cycle in the predator–prey system with symmetric prey isocline

Mathematical Biosciences 164 (2000) 203±215

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Uniqueness of limit cycle in the predator±prey system with symmetric prey isocline Karel Hasõk *,1 Mathematical Institute, Silesian University, Bezrucovo nam. 13, 746 01 Opava, Czech Republic Received 8 April 1999; received in revised form 12 October 1999; accepted 9 December 1999

Abstract We consider a special form of the Gause model of interactions between predator and prey populations. Using the ideas of Cheng, we prove the uniqueness of the limit cycle for more general systems, satisfying some additional conditions. These include also a condition due to Kuang and Freedman. Moreover, in this paper it is shown that the similar generalization of Cheng's uniqueness proof by Conway and Smoller is not correct. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Predator±prey system; Limit cycle; Symmetry of the prey isocline

1. Introduction The existence and the uniqueness of the limit cycle are two important problems which are closely connected with two-dimensional predator±prey models. This question has been completely solved for the well-known system    x m yx 0 x ˆ rx 1 ÿ ÿ ; k a a‡x   …1† mx 0 ÿ D0 ; y ˆy a‡x where x is the prey density, y the predator density, and r; k; m; a; a; D0 are positive constants. The coecient D0 is the relative death rate of the predator, m the maximal relative increase of the *

Tel.: +420-653 684 341; fax: +420-653 215 029. E-mail address: [email protected] (K. HasõÂk). 1 This research was partially supported by the Grant Agency of the Czech Republic, grant no. 201/97/0001.

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 0 7 - 9

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predator and a is the Michaelis±Menten constant. It represents the amount of prey necessary for the reproduction rate p=2 of the predator. We have a < 1, since the whole biomass of the prey is not transformed to the biomass of the predator and the constant k is the carrying capacity of the prey population. In the absence of the predator, the prey population develops according to the logistic equation. Hsu et al. [1] showed that if there exists an asymptotically stable positive equilibrium, then it is also globally stable. For the same model Cheng [2] proved that if such an equilibrium is unstable, then it possesses a unique globally asymptotically stable limit cycle. His proof was extended by Conway and Smoller [3] to systems for which the prey isocline is symmetric with respect to its maximum. Moreover, the existence of a system with at least two limit cycles is proved in [3]. In this paper, we consider the predator±prey system x0 ˆ xg…x† ÿ yp…x†; y 0 ˆ y‰cp…x† ÿ cŠ; …x…0† P 0; y…0† P 0†;

…2†

which is a special case of the model introduced by Gause et al. [4]. The function g…x† represents the relative increase of the prey in terms of its density. For low densities the number of o€spring is greater than the number who have died, and so g…x† is positive. As the density increases, living conditions deteriorate and the death rate is greater than birth rate and hence g…x† is negative. The function cp…x† ÿ c gives the total increase of the predator population. This is negative for low values of prey densities, i.e., the prey population is insucient to sustain the predator. The function p…x†, called trophic function of the predator or functional response, expresses the number of consumed prey by a predator in a unit of time as a function of the density of the prey population [5]. In the next section, we extend Cheng's proof of uniqueness of the limit cycle for this system. This extension cannot be done without several additional assumptions. Keeping the original assumption of symmetry of the prey isocline with respect to its maximum, which was implicitly contained in Cheng's paper, we have to assume two further conditions. First, a condition concerning p…x† is natural since in system (2) but not in (1) p…x† is an unknown function. The latter condition was found by Kuang and Freedman [6]. They investigated a predator±prey system of the Gause type. By transforming to a generalized Lienard system they derived sucient conditions for the uniqueness of limit cycle, which can be applied to system (2). But since this system has certain symmetric properties, we can considerably contract the interval in which this condition must be satis®ed. At the end of this paper, we show that the extension of Cheng's proof due to Conway and Smoller [3] is incorrect. 2. Some known facts on system (2) We study system (2) under the following assumptions: (i) There exists a number k > 0 such that g…x† > 0 for 0 6 x < k; 0

g…k† ˆ 0;

g…x† < 0 for x > k:

p‡0 …0†

> 0. (ii) p…0† ˆ 0; p …x† > 0 for x > 0;   (iii) There is a unique point ‰x ; y Š with 0 < x < k; y  > 0 such that

K. Hasõk / Mathematical Biosciences 164 (2000) 203±215

cp…x † ÿ c ˆ 0;

205

x g…x † ÿ y  p…x † ˆ 0:

(iv) The prey isocline h…x† :ˆ xg…x†=p…x† is a strict concave down function symmetric with respect to its maximum which is attained at a point m > 0. (v) The functions g…x†; p…x† are as smooth as required. All these conditions, except for the condition (iv), are natural in the biological context mentioned above. We suppose (iv) since we are motivated by the fact that this qualitative property occurs in some predator±prey systems, e.g., in (1). Our analysis may also be useful from the mathematical point of view since, possibly, it can be generalized to an asymmetric case, as indicated in [7]. For such models the following results are known [8]. Proposition 2.1. (i) The cone R2‡ ˆ f‰x; yŠ; x P 0; y P 0g is an invariant set of system (2). (ii) The equilibria 0 ˆ …0; 0† and K ˆ …k; 0† of system (2) are saddles. (iii) The positive equilibrium E ˆ ‰x ; y  Š is asymptotically stable if h0 …x † < 0, unstable if h0 …x † > 0, and is a center if h0 …x † ˆ 0 (Rosenzweig and MacArthur criterion). Proposition 2.2. (i) The system (2) has a bounded positively invariant set in the positive quadrant Int R‡ 2.   (ii) If E is unstable then there exists at least one limit cycle surrounding E . (iii) The limit cycles of the system (2), when they exist, must lie inside the strip 0 < x < k; 0 < y < 1. The following theorem is due to Kuang and Freedman [6]. Theorem 2.3. Suppose in system (2)   d xg0 …x† ‡ g…x† ÿ …xg…x†=p…x††p0 …x† 6 0; dx cp…x† ÿ c in 0 6 x < x and x < x 6 k. Then system (2) has exactly one limit cycle which is globally asymptotically stable with respect to the set R2‡ n E . 3. Main result In this section, we give the extension of Cheng's proof from [2]. We follow his method and we mainly devote attention to the places where our proof is more general. When possible, we also adopt his notation. Our main result is the following theorem. Theorem 3.1. Let the following assumptions be satisfied for system (2). (i) x < m, (ii) p…x0 †…cp…x† ÿ c† ‡ p…x†…cp…x0 † ÿ c† 6 0 for x0 2 ‰0; xQ Š, where x0 ˆ 2m ÿ x,

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(iii) d dx



xg0 …x† ‡ g…x† ÿ …xg…x†=p…x††p0 …x† cp…x† ÿ c

 6 0;

for x 2 …2m ÿ x ; kŠ:

Then system (2) possesses a unique limit cycle which is globally asymptotically stable in the positive quadrant. In Theorem 3.1 as well as in Lemma 3.3, we consider the condition (ii) in ‰0; xQ Š regardless of the fact that we do not know where exactly the point xQ is located. Thus, this is an auxiliary point useful only in our proofs. However, when considering the function p…x0 †…cp…x† ÿ c† ‡ p…x†…cp…x0 † ÿ c† in the interval ‰0; x Š, we can provide examples of the class of functional responses satisfying (ii), cf. Note 3.1. Now we need the following modi®cations of Lemmas 1 and 2 from [2] to be able to prove Theorem 3.1, which we give in the Appendix A. Lemma 3.2. Let C be a non-trivial closed orbit of system (2). Then C  f‰x; yŠ; 0 < x < k; 0 < yg. Let L; R; H and J be the left-most, right-most, highest and lowest points of C, respectively. Then L 2 f‰x; yŠ; 0 < x < x ; y ˆ h…x†g;

H 2 f‰x; yŠ; x ˆ x ; y  < yg;

R 2 f‰x; yŠ; x < x < k; y ˆ h…x†g;

J 2 f‰x; yŠ; x ˆ x ; 0 < y < y  g;

where h…x† ˆ …xg…x†=p…x††. The proof is clear and can be omitted here. Now we turn attention to Cheng's Lemma 2. We have to assume more than Cheng. Lemma 3.3. Let C be a non-trivial closed orbit of (2) and let (i) x < m, (ii) p…x0 †…cp…x† ÿ c† ‡ p…x†…cp…x0 † ÿ c† 6 0 for x0 2 ‰0; xQ Š, where x0 ˆ 2m ÿ x. Then the mirror image of arc HLJ with respect to the line x ˆ m intersects arc BRA at two points (see Fig. 1) P ˆ ‰xP ; yP Š and Q ˆ ‰xQ ; yQ Š, with yQ > h…xQ † and yP < h…xP †. Moreover, if P 0 ˆ ‰xP 0 ; yP 0 Š and Q0 ˆ ‰xQ0 ; yQ0 Š are the mirror images of the points P ; Q, respectively, then 0<

p…xQ0 † p…xQ † 6 c ÿ cp…xQ0 † cp…xQ † ÿ c

…3†

0<

p…xP 0 † p…xP † 6 : c ÿ cp…xP 0 † cp…xP † ÿ c

…4†

and

Proof. Consider the function Z x Z cp…n† ÿ c 1 y g ÿ y V …x; y† ˆ dn ‡ dg cp…n† c y g x

…5†

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207

Fig. 1.

and obtain dV 1 ˆ …cp…x† ÿ c†…h…x† ÿ y  †: dt c

…6†

Moreover, if C is a periodic orbit of (2) then Z T Z dV dy 0ˆ dt ˆ …h…x† ÿ y  † : dt y 0 C The proof of the fact that arc H 0 L0 J 0 intersects arc BRA is analogous to that in [2]. Similarly,    0 dy dy P 0> dx Q dx Q implies Eq. (3). Now consider the function G…x0 † (in [2] it is a quadratic function) in the form   p…x0 † p…x† 0 0 ; G…x † ˆ …cp…x† ÿ c†…c ÿ cp…x †† ÿ c ÿ cp…x0 † cp…x† ÿ c

…7†

where x0 ˆ 2m ÿ x. Then (3) and (7) imply that G…xQ † 6 0; G…xQ0 † 6 0. If we want to obtain the same conclusion as Cheng (that arc H 0 L0 J 0 intersects the arc BRA at two points) we have to assume (ii). This assumption ensures that G…x† is negative in the intervals ‰0; xQ Š and ‰2m ÿ xQ ; 2mŠ, since G…x† is symmetric with respect to line x ˆ m. Consequently,     dy dy P …8† 0> dx QR dx QL0

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on arcs QR and QL0 . Thus, arcs H 0 L0 and BR have at most one point in common. A similar conclusion holds for arcs J 0 L0 and AR. This completes our proof.  The fact that the intersection of the limit cycle with its mirror image consists of just four points is used in the proof of Theorem 3.1. However, this fact imposes strong restrictions on the data compatible with the symmetric model. Thus, in some cases, e.g., we can exclude the symmetry and this could be interesting from the biological point of view. Note 3.1. As we already mentioned, condition (ii) in Theorem 3.1 involves the point xQ , the meaning of which is only theoretical. Now we give a short discussion on relations between various types of functional responses p…x† and condition (ii). By the assumptions, p…x† is an increasing function, at least up to a critical point representing the maximum prey consumption within the prescribed time. Holling [9] considered three such responses illustrated by Fig. 2: type I ± the response rises linearly to a plateau, type II ± the response rises at a continually decreasing rate, and type III ± the response is sigmoid. Consider system (2) with a concave down function of functional response (type II). By computing the ®rst derivative of G…x0 † we obtain G0 …x0 † ˆ p0 …x0 †…cp…x† ÿ c† ÿ p0 …x†…cp…x0 † ÿ c† ÿ c…p0 …x†p…x0 † ÿ p…x†p0 …x0 ††: It is easy to see that inequalities p0 …x0 †…cp…x† ÿ c† ÿ p0 …x†…cp…x0 † ÿ c† > 0

for x0 2 ‰0; x Š;

p0 …x0 †…cp…x† ÿ c† ÿ p0 …x†…cp…x0 † ÿ c† < 0

for x0 2 ‰2m ÿ x ; 2mŠ

are satis®ed. And since we assume that p…x† is concave down we have ÿ 0
p …x†p…x0 † ÿ p…x†p0 …x0 † 6 0 for x0 2 ‰0; x Š; ÿ 0
p …x†p…x0 † ÿ p…x†p0 …x0 † P 0 for x0 2 ‰2m ÿ x ; 2mŠ:

Fig. 2.

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209

Indeed, if x0 2 ‰0; x Š, then x 2 ‰2m ÿ x ; 2mŠ and p…x0 † < p…x†, p0 …x† < p0 …x0 † (the latter condition holds analogously). These four inequalities imply that G…x† increases in interval ‰0; x Š and decreases in ‰2m ÿ x ; 2mŠ. Hence, we conclude that condition (ii) is satis®ed, because G…xQ † 6 0; G…xQ0 † 6 0 and xQ0 < x ; 2m ÿ x < xQ . Hence, if the functional response in system (2) is concave down function condition (ii) in Theorem 3.1 and Lemma 3.3 is always satis®ed. It does not hold in case the functional response is sigmoid. In the case of the linear functional response the satis®ed assumptions concerning the ®rst derivative of a function p…x† are not satis®ed (but we can approximate such function by the function of type II). Note 3.2. Conway and Smoller [3] studied the system x0 ˆ x…g…x† ÿ y†; y 0 ˆ y‰x ÿ cŠ;

…x…0† P 0; y…0† P 0†;

…9†

under assumptions (i) g…x† > 0 for x 2 ‰0; k†, g…x† < 0 for x > k. (ii) g00 …x† < 0 for 0 < x < k, g0 …m† ˆ 0 for some m, 0 < m < k. (iii) g…k† ˆ 0; g0 …k† < 0; c < k. They proved the uniqueness of limit cycle for (9) under assumption that g…x† is symmetric about its maximum. They also followed Cheng's proof and they claim that the Eqs. (28)±(32) from Cheng's paper are valid for system (9) too. But if we write, for example, Eq. (28) from [2] for system (9) we obtain (cf. proof of Theorem 3.1 in Appendix A) Z Z Z Z xg0 …x† xg0 …x† xg0 …x† 0 dy ‡ dy dy xg …x† dt ˆ Q0 LP 0 Q0 LP 0 y…x ÿ c† P 0 Q0 y…x ÿ c† Q0 P 0 y…x ÿ c† ! Z Z Z 1 g00 …x†x…x ÿ c† ÿ cg0 …x† xg0 …x† ˆ dy dx dy ‡ 2 …x ÿ c† X y Q0 P 0 y…x ÿ c† Z ? xg0 …x† dy; …10† < Q0 P 0 y…x ÿ c† where Q0 P 0 denotes the line segment. If g…x† is a quadratic function, as in the Cheng's case, then this inequality is true since the integral over X is negative. But the situation is di€erent when considering system (9) with g…x† ˆ …x ‡ 1†…x ÿ 7†…ÿx2 ‡ 6x ÿ 26†. In this case, we have g00 …x†x…x ÿ c† ÿ cg0 …x† ˆ ÿ12x4 ‡ 118:4x3 ÿ 423:2x2 ‡ 638x ÿ 330:6: In Fig. 3 we can see the graph of this function. It is clear that g00 …x†x…x ÿ c† ÿ cg0 …x† > 0 in some subinterval of ‰0; mŠ. Thus, it is not possible to claim that the above quoted integral over region X is negative. It implies that inequality (10) may be violated. Thus, Cheng's proof cannot be generalized without assumption (ii) from our theorem or without proof of negativity of the integral over the region X.

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Fig. 3.

4. Example Consider the system   bx 0 x ˆ xg…x† ÿ y ; a‡x   bx ÿc ; y0 ˆ y c a‡x

…11†

where a; b; c; c are positive constants. Since system (11) is a special case of system (2) we can simplify the form of condition (iii) in Theorem 3.1 (condition (ii) is satis®ed since p…x† ˆ bx= …a ‡ x† is concave down function ± see Note 3.1). Condition (iii) is equivalent [6] to the condition  00  0 xg…x† xg…x† 0 ÿ cp …x† 6 0: …cp…x† ÿ c†p…x† p…x† p…x† In our case for function p…x† ˆ bx=…a ‡ x† we obtain   bx bx 00 ba c h0 …x† 6 0; ÿc h …x† ÿ c 2 a‡x a‡x …a ‡ x†   ac …cb ÿ c† x ÿ xh00 …x† ÿ ach0 …x† 6 0; bc ÿ c

…12†

…x ÿ x †xh00 …x† ÿ x h0 …x† 6 0; where x ˆ ac=…bc ÿ c†. Condition (12) is satis®ed for any quadratic concave down function h…x† in 0 6 x < x and  x < x 6 k under assumption x < m. The proof of this fact is easy and we omit it here since it was made, although not exactly in this form, in [6]. But if h…x† is a polynomial of degree 4, condition

K. Hasõk / Mathematical Biosciences 164 (2000) 203±215

211

(12) may be violated in 0 6 x < x and x < x 6 k as we can see in Note 3.2. Now consider system (11) whose prey isocline is a polynomial of degree 4 symmetric with respect to its maximum. Such a polynomial can be written in the form h…x† ˆ …x ‡ a†…x ÿ k†…ÿx2 ‡ …k ÿ a†x ÿ a†; where a is a positive constant. Since the condition h00 …x† < 0 must be satis®ed in ‰0; kŠ and since h00 …x† ˆ ÿ12x2 ‡ 12…k ÿ a†x ÿ 2……k ÿ a†2 ‡ a ÿ ka†; the function h00 …x† has no real roots. Therefore, we obtain …k ÿ a†2 ÿ 2…a ÿ ka† < 0. Moreover, if we denote H…x† :ˆ …x ÿ x †xh00 …x† ÿ x h0 …x† 6 0; then we obtain

  H 0 …x† ˆ …x ÿ x † 2h00 …x† ‡ xh000 …x† :

It is easy to see that function H…x† can attain extreme values either at x or at roots of polynomial 2h00 …x† ‡ xh000 …x† (if they do not exist then condition (12) is satis®ed since in this case x is unique extreme (maximum) and H…x † < 0). For these latter roots, there holds q ÿ36…k ÿ a†  528…k ÿ a†2 ÿ 768…a ÿ ka† r1;2 ˆ : ÿ96 But condition …k ÿ a†2 ÿ 2…a ÿ ka† < 0 implies q q 528…k ÿ a†2 ÿ 768…a ÿ ka† > 144…k ÿ a†2 ˆ 12…k ÿ a†: Hence the roots of polynomial 2h00 …x† ‡ xh000 …x† lie in the interval

Fig. 4.

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3 1 …k ÿ a†  …k ÿ a†: 8 8 From this fact and from properties of function H …x† it follows that condition (12) can be violated only in the interval …0; …k ÿ a†=2†. This means that system (11) has a unique stable limit cycle if x < …k ÿ a†=2 and h…x† is a polynomial of degree 4. Fig. 4 illustrates the evolution of system (11). In this case g…x† ˆ …x ÿ 3†…ÿx2 ‡ 2x ÿ 10†; c ˆ 0:46; c ˆ 1; a ˆ 1. The initial conditions are x…0† ˆ 1; y…0† ˆ 35 and x…0† ˆ 2; y…0† ˆ 35. 5. Discussion Using the symmetry of the prey isocline with respect to its maximum we proved the uniqueness of the limit cycle. Now we want to discuss the following question. What does the symmetry of the prey isocline mean from the biological or ecological point of view? To make the answer more simple, we ®rst analyze the cases when the symmetry is violated. There are two basic situations as depicted in Figs. 5 and 6 (in the ®rst ®gure is sketched a modi®cation of the prey isocline, in the

Fig. 5.

Fig. 6.

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213

second a modi®cation of function xg…x†, and in the third a modi®cation of function p…x†). In the case A, the right part of the function h…x† lies under the mirror image of the left one, while the case B demonstrates the opposite situation. First consider case A. Here the prey isocline slopes down more rapidly than it would under conditions of symmetry. There are two possibilities: A1. The per capita rate of growth of the prey, g…x†, declines faster than linearly with respect to prey population size. A2. The predator's e€ectiveness at prey consumption continues to increase for arbitrarily large values of prey population size, rather than approaching an asymptote. Case B can be similarly interpreted. The right-hand side of the prey isocline is above the mirror image of the left-hand side. Again, there are two possibilities: B1. The per capita rate of growth of the prey, g…x†, declines more slowly than linearly with increasing prey population size. B2. The consumption e€ectiveness of the predator slows with increasing values of prey population size. Our analysis leads to the following conclusion. When the structure of the predator±prey system obeys the model with a symmetric prey isocline, this is due to a particular combination of forms of the prey per capita growth rate, g…x†, and the functional response, p…x†. So, in a sense, in systems with a symmetric prey isocline, there is a certain sort of balance between these two in¯uences, deviations from which can cause asymmetry. Of course, we are conscious of the fact that a real situation can be more complicated than that of Figs. 5 and 6, and our interpretation of the symmetry of the prey isocline is not the only possible one. It is clear that asymmetry of the prey isocline plays an important role in the systems. However, we are not able to determine what is more typical in nature: symmetry or asymmetry. The answer must be based on observations made by specialists. We can only point out that in the predator± prey models with asymmetric isocline, several limit cycles can appear. This problem has been studied by Wrzosek [10]. Using a modi®cation of the function corresponding to xg…x† in our notation, he proved that there is a system with several limit cycles. Although the symmetry is a rather special property, our result compared with a general case studied by Kuang and Freedman [6] shows that considering such systems can be useful. Naturally, [6] brings a more general result which can be applied to a more general system. On the other hand, there are symmetric functions not satisfying the condition from Theorem 2.3. Comparison with results due to Liou and Cheng [7] is interesting, too. They investigate the predator±prey system x0 ˆ x…g…x† ÿ y†; y 0 ˆ y…p…x† ÿ c†; with non-negative x…0† and y…0†, and generalized the original Cheng's proof to asymmetric functions satisfying Kuang and Freedman's condition in the intervals ‰0; x Š and ‰x ; 0Š, where x is an ``asymmetric'' mirror image of x [7]. In our paper, we apply the same condition only to the decreasing part of the isocline; instead, we assume condition (ii) (cf. Theorem 3.1). Ref. [7] also closes a gap appearing in the original proof [2]. In our paper, though it is based on [2], this incorrectness is also avoided.

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One can see from the above note concerning the paper of Conway and Smoller that it is necessary to be careful when extending Cheng's proof to more general systems. Acknowledgements The author expresses appreciation to Professor K. Smõtalova for stimulating discussions and advice. The author thanks the referees for their helpful remarks and suggestions. Appendix A Proof of Theorem 3.1. Let C be any non-trivial closed orbit of (2). We can again follow Cheng's proof to obtain the equations g…x; y† ˆ xg…x† ÿ yp…x†;

h…x; y† ˆ y…cp…x† ÿ c†;

Div…g; h† ˆ g…x† ‡ xg0 …x† ÿ yp0 …x† ‡ cp…x† ÿ c; Z Z dy ˆ 0; …cp…x† ÿ c† dt ˆ C C y Z Z xg…x† 0 0 p …x† dt; yp …x† dt ˆ C C p…x† xg…x† 0 p …x†: p…x†

p…x†h0 …x† ˆ g…x† ‡ xg0 …x† ÿ The last three conditions imply Z Z Z Div…g; h† dt ˆ ‡ C

AP

PRQ

Z ‡

QB

Z ‡

Z BQ0

‡



Z Q0 LP 0

‡

P 0A

p…x†h0 …x† dt:

Now using the symmetry of h…x† with respect to its maximum we can compute Z Z xA Z xP Z  p…x†h0 …x† p…x†h0 …x† 0 dx ‡ dx ‡ p…x†h …x† dt ˆ P 0A AP xP 0 p…x†…h…x† ÿ y1 …x†† xA p…x†…h…x† ÿ y2 …x†† Z xP Z xP h0 …x† h0 …2m ÿ x† ˆ dx ‡ dx xA …h…x† ÿ y1 …x†† xA …h…2m ÿ x† ÿ y2 …2m ÿ x†† Z xP y2 …x† ÿ y1 …x† h0 …x† dx < 0: ˆ …h…x† ÿ y …2m ÿ x††…h…x† ÿ y2 …x†† 1 xA Similarly, Z QB

Z ‡

BQ0

 p…x†h0 …x† dt < 0:

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215

Now we compute Z Q0 LP 0



Z ‡

PRQ

p…x†h0 …x† dt ˆ

Z Q0 LP 0

Z

Z ‡

PL0 Q

‡

Z QLP 0

‡

PRQ

 p…x†h0 …x† dt:

By the Green theorem Z Z Z  Z  p…x†h0 …x† ‡ p…x†h0 …x† dt ˆ ‡ y…cp…x† ÿ c† QL0 P PQR PQR Z ZQL0 P 0 1 d xg …x† ‡ g…x† ÿ …xg…x†=p…x††p0 …x† ˆ dx dy; cp…x† ÿ c X y dx where X is the region bounded by arcs QL0 P and PQR. Assumption (iii) ensures that this integral is negative. Since Z  Z yQ Z yQ Z p…S1 …y††h0 …S1 …y†† p…2m ÿ S1 …y††h0 …S1 …y†† dx ÿ dx ‡ p…x†h0 …x† dt ˆ y …c ÿ cp…S1 …y††† y …cp…2m ÿ S1 …y†† ÿ c† Q0 LP 0 PL0 Q yP yP Z yQ 0 h …S1 …y†† G…S1 …y†† ˆ …A:1† y …c ÿ cp…S1 …y†††…cp…2m ÿ S1 …y†† ÿ c† yP and from the properties of G…x† we also get that this integral is negative. Consequently, Z Div…g; h† dt < 0; C

i.e., C is orbitally stable and hence, unique.

…A:2†



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