ON A PREDATOR-PREY SYSTEM OF HOLLING TYPE

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 7, July 1997, Pages S ) ON A PREDATOR-PREY SYSTEM OF HOLLING TYPE JITSURO SUGIE, RIE KOHNO, AND RINKO MIYAZAKI Communicated by Hal L. Smith) Abstract. We consider the redator-rey system with a fairly general functional resonse of Holling tye and give a necessary and sufficient condition under which this system has exactly one stable limit cycle. Our result extends revious results and is an answer to a conjecture which was recently resented by Sugie, Miyamoto and Morino. 1. Introduction The urose of this aer is to give a necessary and sufficient condition for the uniqueness of limit cycles of a redator-rey system of the form ẋ = rx 1 x ) x y a + x, 1.1) ) µx ẏ = y a + x D, where = d/dt; x and y reresent the rey oulation or density) and the redator oulation or density), resectively; r,, a, µ, D and are ositive arameters is not always an integer). The arameters are as follows: i) r and are the intrinsic rate of increase and the carrying caacity for the rey oulation, resectively; ii) a is the half-saturation constant for the redator; iii) µ and D are the birth rate and the death rate for the redator, resectively. x The function a + x is often called a functional resonse of Holling tye when =1or= 2. System 1.1) is an imortant model on oulation dynamics refer to [3], [5], [8] [10] and references therein). If 1.2) def ad µ>d and >λ = µ D, Received by the editors January 25, Mathematics Subject Classification. Primary 34C05, 92D25; Secondary 58F21, 70K10. Key words and hrases. Limit cycles, global asymtotic stability, redator-rey system, functional resonse. The first author was suorted in art by Grant-in-Aid for Scientific Research c 1997 American Mathematical Society

2 2042 JITSURO SUGIE, RIE KOHNO, AND RINKO MIYAZAKI then system 1.1) has the only critical oint λ,ν ) in the first quadrant x, y): x>0andy>0 },where ν = rµ D 1 λ ) λ. It is clear that system 1.1) has no limit cycles when assumtion 1.2) fails. Many attemts have been made to give sufficient conditions and necessary conditions to guarantee the existence and the uniqueness of limit cycles of 1.1). For examle, see [1], [2], [4], [6]. The following theorems are well-nown. Results in [1], [2], [4], [6] were stated in a slightly different form.) Theorem A. Let =1. Then, under the assumtion 1.2), system1.1) has a unique stable limit cycle if and only if D + µ)λ 1 <D. Theorem B. Let =2. Then, under the assumtion 1.2), system1.1) has a unique stable limit cycle if and only if 2Dλ 2 < 2D µ). In a recent aer [12], Sugie, Miyamoto and Morino gave a necessary condition for the existence of limit cycles of 1.1) with = 3 and made the following conjecture. Conjecture. Let be any ositive integer. Then, under the assumtion 1.2), system 1.1) has no limit cycles if and only if ) 1.3) D 2)µ λ D 1)µ ). In this aer we rove that the conjecture is true and extend any ositive integer in the conjecture to any ositive real number satisfying a certain condition Theorems 2.1 and 3.1). To be exact, if condition 1.3) holds, then system 1.1) has no limit cycles; otherwise, system 1.1) has a unique limit cycle. Since the solutions of 1.1) are ositive and bounded for all future time, from the Poincaré-Bendixson theorem, we see that under the assumtion 1.2), condition 1.3) is necessary and sufficient for the equilibrium λ,ν ) to be globally asymtotically stable. 2. Uniqueness of limit cycles Kuang and Freedman [8] gave the following result on the uniqueness of limit cycles of the system: 2.1) ẋ = xρx) yφx), ẏ = y ν + ψx) ), where ν>0; all the functions are sufficiently smooth on [0, ) and satisfy 2.2) φ0) = ψ0) = 0 and φ x) > 0, ψ x) > 0 for x>0.

3 ON A PREDATOR-PREY SYSTEM OF HOLLING TYPE 2043 Theorem C. Assume 2.2). If there exist constants x and m with 0 <x <m such that 2.3) ψx )=ν and x m)ρx) < 0 for x m; ) d xρx) 2.4) > 0; dx φx) x=x d xρ x)+ρx) xρx) φ x) φx) 2.5) 0 for x x, dx ν + ψx) then system 2.1) has exactly one limit cycle which is globally asymtotically stable. By means of Theorem C, we will show that only if -art of Conjecture in Section 1 is correct. Theorem 2.1. Let be a ositive number with 1 or 2. If1.2) and ) 2.6) D 2)µ λ < D 1)µ ) are satisfied, then system 1.1) has a unique limit cycle. Proof of Theorem 2.1. We can rewrite system 1.1) as system 2.1) with ν = D, ρx) =r 1 x ) 2.7), φx)= x a+x and ψx) =µφx). It is clear that φx) andψx) satisfy assumtion 2.2). Let x = λ and m =. Then by 1.2) we see that x <m. Assumtion 2.3) is satisfied. In fact, we have ψλ )=D; ρx)>0 if 0 <x< and ρx) < 0 if x>. For the sae of convenience, let Hx) =xρ x)+ρx) xρx) φ x) φx) and W x) = Hx) ψx) D for x λ. 2.8) Since d dx ) xρx) = Hx) φx) φx) = r 1 ) 1 x ) a + x x xa + x ) x + 1 x ) }, we get ) d xρx) = r 1 ) 1 λ ) µ dx φx) x=λ D λ µ D + 1 λ )} = r D ) ) } 1)µ D 2)µ λ > 0 D by 2.6). Hence, assumtion 2.4) holds.

4 2044 JITSURO SUGIE, RIE KOHNO, AND RINKO MIYAZAKI From 2.7) and 2.8) it turns out that r W x) = µx Da + x 1 ) 1 x ) a + x ) xa + x ) + 1 x ) } x ) r = µ D)x ad ) a1 )+a 2)x + x 2x +1}. Differentiating this equality and using the fact that ad =µ D)λ,weobtain W x)= rµ D) µ D)x ad ) 2 2x )λ a 1) 2) ) x + λ a 1) ) x 1 + a 2)λ Taing 2.6) into account, we have λ a 1) ) = a ) D 1)µ µ D > a ) D 2)µ λ µ D = λ 2λ a 2) ). Hence W x) > rµ D) µ D)x ad ) 2 2x )λ a 1) 2)) x + λ 2λ a 2) ) x 1 + a 2)λ }. } where = rµ D) µ D)x ad ) 2 2x 1 Ux)+a 2)V x) ), Ux) =x +1 +1)λ x+λ+1 and V x) = 1)x λ x 1 + λ. Since U x) =+1)x λ )andv x)= 1)x 2 x λ ), it follows that for x>0 Ux) Uλ )=0; Vx) Vλ )=0 if 1 and Vx) Vλ )=0 if >1. We therefore conclude that if 1or 2, then 2x 1 Ux)+a 2)V x) 0 for x>0. Thus, assumtion 2.5) is also satisfied. Using Theorem C, we see that system 1.1) has a unique stable limit cycle. The roof is comlete. Remar 2.1. In the above roof, if 1 <<2, then W x) < 0forxsufficiently small and so assumtion 2.5) fails. Hence, we cannot use Theorem C.

5 ON A PREDATOR-PREY SYSTEM OF HOLLING TYPE Non-existence of limit cycles In this section we will show that if -art of Conjecture in Section 1 is true. To see this, we need Theorem D below which was stated in [11] we can find a similar result to Theorem D in [7]). By changing variables u = x λ, v =logy log ν and dτ = x a + x dt, system 1.1) can be transformed into the system du dτ = ν e v 1) F u), 3.1) dv dτ = gu), where F u) =r 1 u+λ ) a+u+λ ) ad u+λ ) 1 ν and gu) =µ D u + λ ). Note that F u) andgu) are defined for u> λ and satisfy F 0) = 0 and ugu) > 0 for u 0. Define u Gu) = gs)ds. Then the inverse function of w = Gu)sgnu exists. Let G 1 w) betheinverse. Theorem D. Suose that 3.2) F G 1 w) ) F G 1 w) ) for 0 <w<m, where M = G λ +0).Then3.1) has no limit cycles and neither has 1.1). Unfortunately, however, it is difficult to construct exlicitly the inverse function G 1 w) because µ D)u+ ad 1 Gu) = 1 u + λ ) 1 1 ) λ 1 if 1, µ D)u ad ) logu + λ ) log λ if =1. Hence, to rove the following result, we intend to chec 3.2) without calculating G 1 w) directly. Theorem 3.1. Let be a ositive number with 1 2 or 1. If1.2) and ) 3.3) D 2)µ λ D 1)µ ) are satisfied, then system 1.1) has no limit cycles. Proof of Theorem 3.1. When = 1, the theorem is an immediate consequence of Theorem A. We thus consider only the case 1. By virtue of Theorem D, it is enough to rove that 3.3) imlies 3.2). We will show this by way of contradiction. Suose that F G 1 w 0 ) ) = F G 1 w 0 ) ) 0

6 2046 JITSURO SUGIE, RIE KOHNO, AND RINKO MIYAZAKI for some w 0 0,M). Then we have 3.4) F α) =Fβ) and G α)=gβ) where α = G 1 w 0 )andβ=g 1 w 0 ). Here we note that λ < α <0<β. For simlicity, let 3.5) γ = λ α and δ = λ + β. Then 3.4) becomes 3.6) 3.7) γ) a δ 1 = ) a a ) γ 1 + γ = δ) δ 1 +δ ; a γ 1)α + β)µ D). 1 D Substituting 3.7) into the right-hand side of 3.6) and using the fact that α + β = δ γ, weobtain a D 1)µ 3.8) +γ+ δ ) =0. γ 1 D Similarly, we get a D 1)µ 3.9) + δ + γ ) =0. δ 1 D Hence, from 1.2) and 3.5), it turns out that 3.10) D 1)µ >0. Now, we define the function hz) = D D 1)µ a z 1 + z ) + for z>0 and consider two curves η = hξ) andξ=hη)intheξ,η)-lane. Then the curves are symmetric with resect to the straight line η = ξ. The equalities 3.8) and 3.9) show that the curves intersect each other at the oint γ,δ) which is in the region R = ξ,η): 0 <ξ<λ <η }. We divide our argument into two cases. Case i) >1. From 3.10) it follows that hz) has the maximum value hz ) with z = a 1) because ) h D a 1) z) = 1 D 1)µ z. It is clear that hz) as z 0 +. By 3.3) and 3.10) we also see that z <λ and hλ ) λ. In fact, we have λ z = ad a ) a 1) = D 1)µ > 0; µ D µ D

7 ON A PREDATOR-PREY SYSTEM OF HOLLING TYPE 2047 ) Dλ a λ hλ )=λ + D 1)µ λ +1 λ µ =λ + D 1)µ ) D 2)µ λ D 1)µ ) = 0. D 1)µ Thus, the curve η = hξ) crosses the straight line η = ξ at two oints P z 1,z 1 )and Qz 2,z 2 ) with 3.11) 0 <z 2 <z <z 1 λ. Because of the symmetry of η = hξ) andξ=hη), the curve ξ = hη) also asses through the oints P and Q see Figure 1). Figure 1. The arameters =4,a=1,D=25,= 9 2,µ=26 and the function hz) = z + z ) Since hz) is strictly decreasing for z z 1,thecurveξ=hη)η z 1 )canbe rewritten as η = h 1 ξ) ξ z 1 ). Taing notice of 3.10), we obtain 3.12) h z) < 0 <h z) for z>0. Hence, it follows from 3.11) that 3.13) h z 1 ) h λ )= D D 1)µ 1 a 1) λ ) = 1,

8 2048 JITSURO SUGIE, RIE KOHNO, AND RINKO MIYAZAKI and therefore, 3.14) h 1 ) z1 )= 1 h z 1 ) 1. Next, consider the curvature Kξ) = h ξ) 1+ h ξ) ) 2}3 2 at a oint ξ,hξ) ) on the curve η = hξ). Then we have K ξ) = 1 1+ h ξ) ) 2}5 2 h ξ) 1+ h ξ) ) 2} 3h ξ) h ξ) ) 2 }. Consequently, by 3.12), the curvature Kξ) is negative and strictly increasing for ξ z. This fact means that the absolute value of curvature of η = hξ) is larger than that of η = h 1 ξ) in the interval [z,z 1 ). Hence, together with 3.13) and 3.14), we see that the curve η = hξ) lies below the curve η = h 1 ξ) inthis interval. It is obvious that the curves η = hξ) andη=h 1 ξ) do not meet for 0 <ξ z. We therefore conclude that the curves η = hξ) andξ=hη)haveno common oint in the region R. This is a contradiction. Case ii) 1 2. As in the roof of the case i), we have 3.15) hλ ) λ. The curve η = hξ) intersects the straight line η = ξ at a oint because h z) < 0 for z>0; hz) as z 0 + and hz) as z. Let P z 1,z 1 ) be the oint of intersection. Then by 3.15) we see 3.16) 0 <z 1 λ. The curve ξ = hη) also asses through the oint P see Figure 2). Rewrite ξ = hη) asη=h 1 ξ). Since 3.17) h z) < 0 <h z) for z>0, it follows from 3.16) that 3.18) h z 1 ) h λ )= 1 and h 1 ) z 1 )= 1 h z 1 ) 1. Consider again the curvature Kξ) ataoint ξ,hξ) ) on the curve η = hξ). By 3.17), the curvature Kξ) is ositive for ξ>0. Also, a simle calculation yields K ξ) < a1 )D3 ξ + a1 ) ) +1)ξ +a1 )1 2) ) 1+ h ξ) ) 2}5 2 D +1 )µ ) < 0 3 ξ 3+2 for ξ>0. Hence, taing into account 3.18), we see that the curves η = hξ) and ξ=hη) fails to cross in the region R. This contradicts the fact that γ,δ) isa common oint in R. Thus, we find a contradiction in both cases i) andii). The roof is now comlete.

9 ON A PREDATOR-PREY SYSTEM OF HOLLING TYPE 2049 Figure 2. The arameters = 1 2,a=2,D =1,=2,µ=3 and the function hz) = z 1 2 z+2. Remar 3.1. By examining the sloe and the curvature of η = hξ), where hξ) = ad ) log a ξ) log ξ + ξ for 0 <ξ< a, µ D the roof of the case = 1 can be carried out in the same way. References 1. K.-S. Cheng, Uniqueness of a limit cycle for a redator-rey system, SIAMJ. Math. Anal ), MR 82h: S.-H. Ding, On a ind of redator-rey system, SIAMJ. Math. Anal ), MR 91f: H. I. Freedman, Deterministic Mathematical Models in Poulation Ecology, Marcel Deer, New Yor, MR 83h: A. Gasull and A. Guillamon, Non-existence of limit cycles for some redator-rey systems, Proceedings of Equadiff 91, , World Scientific, Singaore, CMP 94:02 5. C. S. Holling, The functional resonse of redators to rey density and its role in mimicry and oulation regulation, Mem. Ent. Soc. Can ), X.-C. Huang, Uniqueness of limit cycles of generalised Liénard systems and redator-rey systems, J. Phys. A: Math. Gen ), L685 L691. MR 89i: Y. Kuang, Global stability of Gause-tye redator-rey systems, J. Math. Biol ), MR 91g: Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-tye models of redatorrey system, Math. Biosci ), MR 89g: R. May, Stability and Comlexity in Model Ecosystems, 2nd ed., Princeton Univ. Press, Princeton, 1974.

10 2050 JITSURO SUGIE, RIE KOHNO, AND RINKO MIYAZAKI 10. L. A. Real, Ecological determinants of functional resonse, Ecology ), J. Sugie and T. Hara, Non-existence of eriodic solutions of the Liénard system, J.Math. Anal. Al ), MR 92m: J. Sugie, K. Miyamoto and K. Morino, Absence of limit cycles of a redator-rey system with a sigmoid functional resonse, Al. Math. Lett ), CMP 97:03 Deartment of Mathematics, Faculty of Science, Shinshu University, Matsumoto 390, Jaan Current address: Deartment of Mathematics and Comuter Science, Shimane University Matsue 690, Jaan address: jsugie@rio.shimane-u.ac.j Deartment of Mathematics, Faculty of Science, Shinshu University, Matsumoto 390, Jaan Deartment of Mathematical Sciences, Osaa Prefecture University, Saai 593, Jaan address: rino@ms.osaafu-u.ac.j