Analysis of Periodic Solutions for the Nicholson-Bailey Model

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*Department of Mathematical Sciences, Faculty of Science and Engineering, Doshisha University, Kyoto, TEL:+81-774-65-6702, E-mail: [email protected].

Analysis of Periodic Solutions for the Nicholson-Bailey Model

Seiji SAITO*

(Received 20 August, 2012)

In this paper we consider the Nicholson-Bailey model in mathematical biology and prove the existence of periodic solutions for the model. Moreover we deal with periodic nonlinear difference equations and discuss the existence of periodic solutions by applying Liapunov functions under hypotheses that all solutions are equi-ultimate bounded.

Ydifference equationōNicholson-Bailey modelōperiodic solution, equi-ultimate boundedness, Liapunov function

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（　 ）

T^HE SCIENCE AND ENGINEERING R^{EVIEW OF} D^OSHISHA UNIVERSITY, V^OL. 53^, N^O.3 October 2012

K > 0Ŕ2čUDQK4Ā±0PòKRjvr{b

>GHE/&1

Û¸āFĺĹ9E,·Dōr ¬ĺĹ:R·1 +RŎĺĹH÷‹öFı6RD„¶9ō ˆãŀF /5Rr¬ĺĹ:R™¦HoV`y”ÅFÏ-D:

RDōÛ¸āFĺĹ9E,·G™¦f = e^-ap 0P Ë(1)UÐRŎ

·Ŕ

¡ &Èġ &8E3 &×ġN(n) ¡ 1 - f

Û¸āŔ×ġP(n) & Èġ

Fig. 1. Host and parasitoid.

TR*B:AI:

_mï‘n[fxďŀG x(n), n = 0,1,2,…,FŁ:R ĔáČË(2)FA2ĨGåăÓG¶ęUĵJRŎŁ Þ f : Z+ R^m & R^mHķĖōZ+ HŇİßÞGŅ¦

D:RŎßÞ k ) 1 D9BË(2)1ōk ©æąō+

R,Hk©æUNADHō†ÕGn ∈ Z+x ∈ R^mF

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f (n + k,x) = f (n,x)

-ŎS!= { x ∈ R^m : || x || ( ! }D:RŎ||x||H x ∈ R^m GhxqC+RŎ

MAI:

¶ęŏŎ([E.Ult.B]) Ë(2)GĨ1§ČÉēÀåă

(equi-ultimately bounded) C+RDHō+RðÞX > 0

1³®9Bō†ÕG–æã—n0 D†ÕGðÞ ! >

0 F»9ðGßÞ # = #(n0, !)1³®9Bō$∈ S!EPIō–æé…x(n0) = $UL=:Ĩ x(n) = x(n;

n0 ,$)HōßÞn ) n0 +#GD2ō|| x(n) || ( XU L=:Ŏ

¶ ę Ő Ŏ([U.Ult.B]) Ë(2)GĨ1|îēÀåă

(uniformly ultimately bounded) C+RDHō+RðÞ

X > 01³®9Bō†ÕGðÞ!> 0 F»9ðGßÞ

#=#(!)1³®9Bōn0 ∈ Z₊$∈ S!EPIō–æ é…x(n0) = $UL=:Ĩ x(n) = x(n; n0 ,$)Hō†Õ

GßÞn ) n0 +#GD2ō|| x(n) || ( XUL=:Ŏ

*BM.+

_k ©æąÄ”áČË GĨ1§ČÉēÀåă(¶ ę1)C+SIō|îēÀåă(¶ę )N×QĐAŎK

=ōË GĨ1k©æąō+R,Hk©æĨC+R DHōĆĄERkŽGõ0PERŅ¦ Pk = { x0, x1, …, xk-1 }ōxp+1 = f (n, xp) (p = 0,1,…, k-2; f (n, xk-1) = x0 ; n ) 0 H†ÕGßÞ)1³®:R6DU,-Ŏï¶ÿ HļĦC+RŎ

¶ÿő k ©æąÄ”áČË 1§ČÉēÀå ăEPIōË(2)H¾E4DN|AGk©æĨUNAŎ

UR*BM

"#$ *BM g\x`yznWw{sexG©æĨFA,B ĵJR N(n), P(n) > 0 GH;>0PōN(n+1) <

N(n)exp(r(1 – N(n)/K)) D E R Ŏ 6 6 C ōh(N) = Nexp(r(1- N/K)) (N > 0 )ōM = max(h(K/r), N(n0)) D/

4D †ÕGßÞn ) n0 + 1 F»9BōM ) N(n) UÐRŎK=0 < 1 – exp(-aP(n)) 0PōM ) P(n) C +RŎ‚ō

õ ^t(N(n), P(n)) ∈ R² GhxqU ||(N,P)|| = |N|+ |P| D :SIōË(1)H§ČÉēÀåăōO@B|îēÀå ăCō||(N(n), P(n))|| ( 2M (n ) n0 + 1).

ËHĝœđō:ET?ō¤ijGËFHã—n 1 ŃFþSB,E,=MōËH†ÕG k FA2ōk

©æąC+RŎO@Bō¶ÿő0Pōg\x`yz nWw{sexHō†ÕGk©æĨUNAŎ

VR*BMDO01:

DO01:4;:-,:

ËF/,Bōn[fxx NP ∈ R²ō

¤ijUFx Nr% N/KapN ap D9BōËU

42 齋 藤 誠 慈

（　 ） 156

_{x(n+1) F}x(n) D/4Ŏ Hn[fxGIJĘUÕª:RŎ

˜ĵGĚºOQōËHō†ÕGßÞ k ) F Ł9Bk©æĨPk = { x0, x1, …, xk-1 }UNAŎ<GŅ

¦GŏAGõy ∈ Pk U­¶:RŎk¬¦×UGx F^kxD/4DōGy y:ET?õō+R, H

_{z(n+1) G}z(n) zn ∈ R²GÆģõC+RŎdWv{¶ÿ0P Gx Gy D G)yx - y "

UÐRŎ=>9ōD GHt\kĢ•ō"= o(||x y||) as

||x y|| & 0C+RŎ

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³®:RD„¶:RŎR+={ x ∈ R: x ) 0 }Ŏ

(i) VH y FŁ9Bð¶ŕ

(ii) V (z) = V (G (z)) – V (z) Hō V (z) < 0 (z ≠ y)ŕ (iii) V(z) &+' as ||z|| &+'Ŏ

6GD2ōË(5)GÆģõz Ŗ yHō²¯ôĴµ¶C +RŎE/ōé…(i),(ii)UL=:VULiapunovŁÞ D,-Ŏ

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|ĞGŇėÌĔáČË UĚ.RŎàü C HōïG¶ÿD§îE G¶ÿUĵJB,RŎ CI(R+) ={ f ∈ C(R+): f(r) )0, increasing and f(∞) = ∞ }Ŏ

V (n,x) = V (n+1, f (n,x)) – V (n,x)D:RŎ

¶ÿœŎïGé…(i), (ii)UL=:V : Z+ R^m &

R+ 1³®:RD„¶:RŎ

BHc = { x ∈ R^m : ||x|| ) H }(H > 0)ōD:RŎ (i) +Ra, b ∈ CI(R+) 1³®9Bō

a(||x||) (V(n, x) (b(||x||) (n ∈ Z^+, x ∈ BHc )ŕ (ii) +Rc ∈ CI(R+)ō¶ÞMŗ0DŁÞP: R+ & R+

1³®9BōP(u) > u (u > 0)D:RŎ

N9P(V(n, x)) > V(n, x)ōn ∈ Z+, x ∈ BHc EPIō V (n,x) ( M – c(||x||)Ŏ

6GD2ōk©æąEË(2)H|îēÀåăC+RŎ

}ĩG¶ÿGé…UÜğ9é…F/5R ŁÞ P UĂ,E,áóHåĂC+RŎïG«ʼnGĨ ò1æÍ8SRŎ

«ʼnŎïGé…(i), (ii)UL=: V : Z+ R^m &

R+ 1³®:RD„¶:RŎ

BHc = { x ∈ R^m : ||x|| ) H }(H > 0)ōD:RŎ

(i) +R a ∈ CI(R+) 1³®9Bōa(||x||) (V(n, x) (n ∈ Z^+, x ∈ BHc )ŕ

(ii) +R c ∈ CI(R+)D¶Þ Mŗ0 1³®9ōN9

V(n+1, f(n,x)) > V(n, x)ōn ∈ Z^+, x ∈ BHc EPIō V (n,x) ( M – c(||x||)Ŏ

6GD2ōk©æąEË(2)H|îēÀåăC+RŎ

}ĩ«ʼnGĔĭD¶ÿő0Pōk ©æąÄ”áČ Ë(2)Hō¾E4DN|AGk©æĨUNAŎ

XRNQ

ç°¨CHōg\x`yznWw{sex K > 0Ŕ2čUDQK4Ā±0PòKRjvr{b

>GHE/&1

Û¸āFĺĹ9E,·Dōr ¬ĺĹ:R·1 +RŎĺĹH÷‹öFı6RD„¶9ō ˆãŀF /5Rr ¬ĺĹ:R™¦HoV`y”ÅFÏ-D:

RDōÛ¸āFĺĹ9E,·G™¦f = e^-ap 0P Ë(1)UÐRŎ

·Ŕ

¡ &Èġ &8E3 &×ġN(n) ¡ 1 - f

Û¸āŔ×ġP(n) & Èġ

Fig. 1. Host and parasitoid.

TR*B:AI:

_mï‘n[fxďŀG x(n), n = 0,1,2,…,FŁ:R ĔáČË(2)FA2ĨGåăÓG¶ęUĵJRŎŁ Þ f : Z+ R^m & R^mHķĖōZ+ HŇİßÞGŅ¦

D:RŎßÞ k ) 1 D9BË(2)1ōk ©æąō+

R,Hk©æUNADHō†ÕGn ∈ Z+x ∈ R^mF

»9B

€

f (n + k,x) = f (n,x)

-ŎS!= { x ∈ R^m : || x || ( ! }D:RŎ||x||H x ∈ R^m GhxqC+RŎ

MAI:

¶ęŏŎ([E.Ult.B]) Ë(2)GĨ1§ČÉēÀåă

(equi-ultimately bounded) C+RDHō+RðÞX > 0

1³®9Bō†ÕG–æã—n0 D†ÕGðÞ ! >

0 F»9ðGßÞ # = #(n0, !)1³®9Bō$∈ S!EPIō–æé…x(n0) = $UL=:Ĩ x(n) = x(n;

n0 ,$)HōßÞn ) n0 +#GD2ō|| x(n) || ( XU L=:Ŏ

¶ ę Ő Ŏ([U.Ult.B]) Ë(2)GĨ1|îēÀåă

(uniformly ultimately bounded) C+RDHō+RðÞ

X > 01³®9Bō†ÕGðÞ!> 0 F»9ðGßÞ

#=#(!)1³®9Bōn0 ∈ Z₊$∈ S!EPIō–æ é…x(n0) = $UL=:Ĩ x(n) = x(n; n0 ,$)Hō†Õ

GßÞn ) n0 +#GD2ō|| x(n) || ( XUL=:Ŏ

*BM.+

_k ©æąÄ”áČË GĨ1§ČÉēÀåă(¶ ę1)C+SIō|îēÀåă(¶ę )N×QĐAŎK

=ōË GĨ1k©æąō+R,Hk©æĨC+R DHōĆĄERkŽGõ0PERŅ¦ Pk = { x0, x1, …, xk-1 }ōxp+1 = f (n, xp) (p = 0,1,…, k-2; f (n, xk-1) = x0 ; n ) 0 H†ÕGßÞ)1³®:R6DU,-Ŏï¶ÿ HļĦC+RŎ

¶ÿő k ©æąÄ”áČË 1§ČÉēÀå ăEPIōË(2)H¾E4DN|AGk©æĨUNAŎ

UR*BM

"#$ *BM g\x`yznWw{sexG©æĨFA,B ĵJR N(n), P(n) > 0 GH;>0PōN(n+1) <

N(n)exp(r(1 – N(n)/K)) D E R Ŏ 6 6 C ōh(N) = Nexp(r(1- N/K)) (N > 0 )ōM = max(h(K/r), N(n0)) D/

4D †ÕGßÞn ) n0 + 1 F»9BōM ) N(n) UÐRŎK=0 < 1 – exp(-aP(n)) 0PōM ) P(n) C +RŎ‚ō

õ ^t(N(n), P(n)) ∈ R² GhxqU ||(N,P)|| = |N|+ |P| D :SIōË(1)H§ČÉēÀåăōO@B|îēÀå ăCō||(N(n), P(n))|| ( 2M (n ) n0 + 1).

ËHĝœđō:ET?ō¤ijGËFHã—n 1 ŃFþSB,E,=MōËH†ÕG k FA2ōk

©æąC+RŎO@Bō¶ÿő0Pōg\x`yz nWw{sexHō†ÕGk©æĨUNAŎ

VR*BMDO01:

DO01:4;:-,:

ËF/,Bōn[fxx NP ∈ R²ō

¤ijUFx Nr% N/KapN ap D9BōËU

43

ニコルソン・ベイリーモデルの周期解に対する解析

（　 ）

157

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N(n + 1) = N(n)e

⎛

⎝ ⎜ ⎞

⎠ ⎟ −aP(n)

,

€

P(n + 1) = N(n) 1 ( − e

)

G¼“D©æĨG³®ĪâUĉ9ō©æĨ1ôĴµ

¶C+R=MG‚ÎGÁĿUĵJ=Ŏ 8PFō|ĞGŇėÌĔáČË

€

x(n + 1) = f (n,x(n))

F»9BōwVmhlŁÞGÒĂFOR©æĨG³

®¶ÿUįĭ9ōÏêGĔìUÜğ:R6DUĬʼn D9B,RŎ

çĈĎG|ĻH ÇɧÑĊ²´ÿôĈĎ ØĈϛץFO@BĸĢ8S=Ŏ66Fĩ9BĮ ÕUĤ:RŎ

)L?F

L. Edelstein-Keshet, Mathematical Models in Biology, Random House (1988), p79.

2) S.N. Elaydi, An Introduction to Difference Equations 3^rd Edition, Springer (2000), p232.

‰ÊōÞÿāû´’ľō “úÀ J. D. Logan, W. R. Wolesensky, Mathematical Methods in Biology, Wiley(2009), p117.

T. Furumochi and M. Muraoka, Periodic solutions of periodic difference equations,Advanced Studies in Pure Mathematics, 53, 51-57 (2009).

)‡Ġńäōðg\x`yznWw{sexG©æĨō §ÑĊ²´Ã´ĻWydWw^Yyf԰ôċŸí ĭà

A. Halanay, V. Rasvan Stability and Stable Oscillations in Discrete Time Systems, Gordon and Breach Sci. Publ.(2000), p62.

S.N. Elaydi, Discrete Chaos 2^rd Edition, Chapman & Hall /CRC (2008), p209.

9) S. Zhang, Boundedness of finite delay difference systems, Ann. Differential equations, 9 (1993), 107-115.

44 齋 藤 誠 慈

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