李建東
(呂梁學(xué)院 數(shù)學(xué)系,山西 呂梁 033000)
LI Jiandong
(Department of Mathematics,Lüliang University,Lüliang 033000,China)
一類中立型捕食者-食餌系統(tǒng)的正周期解
李建東
(呂梁學(xué)院 數(shù)學(xué)系,山西 呂梁 033000)
捕食者-食餌是種群生態(tài)學(xué)中一類重要的模型,考慮了一類帶有Watt型功能反應(yīng)的中立型捕食者-食餌系統(tǒng),利用重合度理論中的延拓定理和一些數(shù)學(xué)分析方法,得到該系統(tǒng)正周期解的存在性,并推廣了有關(guān)結(jié)果.
捕食-食餌系統(tǒng);中立型;延拓定理
捕食者-食餌是一類重要的種群模型,其定性研究受到了很多國(guó)內(nèi)外專家和學(xué)者的關(guān)注,并產(chǎn)生了許多具有很強(qiáng)實(shí)際背景的研究課題.
根據(jù)生物實(shí)驗(yàn)以及捕食者與食餌的內(nèi)在變化規(guī)律,國(guó)內(nèi)外許多學(xué)者研究了具有不同功能反應(yīng)的捕食者-食餌系統(tǒng),如Monod型功能反應(yīng)[1]、Holling型功能反應(yīng)[2]、Bedington型功能反應(yīng)[3]等.
文獻(xiàn)[4]建立了具有Watt型功能反應(yīng)的捕食者-食餌系統(tǒng):
(1)
由于Watt型功能反應(yīng)能夠合理地刻畫一些捕食者與食餌之間的關(guān)系,受到許多生物數(shù)學(xué)專家的重視,并做了一些很好的研究工作[5-7].
2001年,F(xiàn)an 和Wang[8]研究了下列比率依賴型捕食者-食餌系統(tǒng):
(2)
的周期解與穩(wěn)定性.
本文將利用延拓定理研究下列具有Watt型功能反應(yīng)的中立型捕食者-食餌系統(tǒng):
(3)
正周期解的存在性.
對(duì)于連續(xù)的正的ω-周期函數(shù)p(t), 定義:
在系統(tǒng)(3)中,假設(shè):
(H1)b,ρ,n∈(0,+);m∈(0,1);σ∈R;a∈C(R,R);c,d,f,k∈C(R,[0,+])是ω-周期函數(shù);
定理1 若(H1)~(H4)成立,則系統(tǒng)(3)至少存在一個(gè)正ω-周期解.
證明令x(t)=eu1(t),y(t)=eu2(t),系統(tǒng)(3)可寫為:
(4)
X={u=(u1(t),u2(t))T∈C1(R,R2):ui(t+ω)=ui(t),t∈R,i=1,2},
Z={u=(u1(t),u2(t))T∈C(R,R2):ui(t+ω)=ui(t),t∈R,i=1,2},
其中X的范數(shù)為|u|{|u1(t)|+|u2(t)|},Z的范數(shù)為‖u‖=|u|+|u′|.令L:X→Z為:
L(u1(t),u2(t))T=(u1′(t),u2′(t))T.
(1-exp(-neu1(t)/emu2(t)))-d(t)+f(t)(1-exp(-neu1(t)/emu2(t)))].
顯然,KerL=ImP=R2,KerQ=ImL=Im(1-Q). 此外,dimKerL=codim ImL=2,因此L為一個(gè)指標(biāo)為零的Fredholm算子.LP的逆KP:ImL→DomL∩KerP為:
另外,QN:X→Z為:
對(duì)于任意λ∈(0,1),考慮方程Lu=λNu,即:
(5)
若(u1(t),u2(t))T∈X是系統(tǒng)(5)的解,則:
(6)
(7)
根據(jù)式(6),
(8)
根據(jù)式(7),
(9)
根據(jù)式(5)~(8)及條件(H1),
(10)
由式(6),
(11)
另外,
(12)
根據(jù)式(11)與式(12)得:
(13)
所以,
(14)
(15)
根據(jù)式(15)與條件(H1),
(16)
(17)
根據(jù)式(10),式(16)~(17),任意t∈[0,ω],
所以對(duì)任意t∈[0,ω],
u1(t)≤B.
(18)
根據(jù)式(5),式(8),式(18),
c(t)eu2(t)-u1(t)(1-exp(-neu1(t)/emu2(t)))|dt≤
利用(H2),
(19)
根據(jù)式(5)與式(9),
(20)
任意(u1(t),u2(t))T∈X,有ξi,ηi∈[0,ω],i=1,2使:
(21)
根據(jù)式(8),式(21)和條件(H3),
所以,
(22)
根據(jù)式(19),式(22),任意t∈[0,ω],
(23)
根據(jù)式(18),式(23),
(24)
利用條件(H3),
由于:
|u2′(t)|=|λ[-d(t)+f(t)(1-exp(-neu1(t)/emu2(t)))]|≤|d(t)|+|f(t)|≤|d|0+|f|0,
所以:
(25)
根據(jù)式(9),式(24)及條件(H3),
(26)
根據(jù)式(5),式(18),式(24)及式(26),
通過條件(H2),
(27)
因?yàn)閨u2′(t)|≤|d|0+|f|0,所以:
(28)
根據(jù)式(24),式(26)~(28),
‖u‖=|u|+|u′|≤β2+β6+β7+β8.
令β=β2+β6+β7+β8+β0.顯然β與λ無關(guān).令:Ω={(u1(t),u2(t))T∈X:‖(u1(t),u2(t))T‖<β}.
顯然引理1中的條件(1)滿足.當(dāng)(u1(t),u2(t))T∈?Ω∩KerL=?Ω∩R2時(shí),(u1(t),u2(t))T∈R2滿足|u1|+|u2|=β.根據(jù)條件(H4),QN(u1,u2)T≠(0,0)T,即,引理1中的條件(2)滿足.
令J=I:ImQ→KerL,(u1,u2)T→(u1,u2)T.記:
根據(jù)條件(H1), deg{JQN,Ω∩KerL,0}=
從而引理1中的條件(3)滿足.因此系統(tǒng)(4)至少存在一個(gè)ω-周期解.即系統(tǒng)(3)至少存在一個(gè)正ω-周期解.
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責(zé)任編輯:時(shí)凌
ThePositivePeriodicSolutionsofaClassofNeutralPredator-preySystem
Predator-prey is a kind of important model in population ecology.In this paper, a class of neutral predator- prey system with Watt-type functional response is considered. Using the continuation theorem of coincidence degree theory and some mathematical analysis method, we obtain the existence of positive periodic solutions of this system and generalize the results in the existing literature.
predator-prey system;neutral;continuation theorem
2014-11-04.
李建東(1978- ),男,講師,主要從事微分方程及其應(yīng)用的研究.
O175.14
A
1008-8423(2014)04-0402-06
LI Jiandong
(Department of Mathematics,Lüliang University,Lüliang 033000,China)