沈林, 周紅玲

(黃淮學(xué)院數(shù)學(xué)科學(xué)系,河南 駐馬店463000)

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一類脈沖反應(yīng)擴(kuò)散系統(tǒng)的行波和傳播速度

沈林, 周紅玲*

(黃淮學(xué)院數(shù)學(xué)科學(xué)系,河南 駐馬店463000)

討論了開放環(huán)境中一類具有固定脈沖時刻的反應(yīng)擴(kuò)散系統(tǒng)的傳播速度和行波解。在空間分布均勻條件下， 給出了正常數(shù)解存在和穩(wěn)定的條件。得到了脈沖反應(yīng)擴(kuò)散系統(tǒng)傳播速度的具體表達(dá)形式，當(dāng)滿足一定條件時，傳播速度大于零，該速度也是系統(tǒng)存在行波解的最小速度。以水流速度為參數(shù)對系統(tǒng)進(jìn)行了數(shù)值模擬，結(jié)果表明通過控制擴(kuò)散系數(shù)、水流速度、離散和連續(xù)時間的死亡率和出生率，可實現(xiàn)生物種群的傳播和持久生存。

脈沖反應(yīng)擴(kuò)散系統(tǒng); 傳播速度; 行波

現(xiàn)階段，生物種群的傳播和持久生存?zhèn)涫苎芯空叩那嗖A，大多數(shù)研究者在假設(shè)生物種群的擴(kuò)散、出生和死亡連續(xù)地依賴于時間和空間的前提下，利用反應(yīng)擴(kuò)散方程進(jìn)行建模，并成功地模擬了部分生物種群在開放環(huán)境中的傳播和持久生存。但是，大多數(shù)魚類和小型昆蟲的繁衍是與季節(jié)息息相關(guān)的，而且這些生物的繁衍階段與生長階段相比是極短的[1-2]，因此, 單純利用反應(yīng)擴(kuò)散方程不能很好地描述生物的繁衍。針對生物種群離散的出生率，Lewis 等[3]建立了如下脈沖反應(yīng)擴(kuò)散系統(tǒng)

(1)

其中，d為擴(kuò)散系數(shù)，α<0為種群的死亡率，r反映種群間的競爭，Nn(x)為第n代繁衍前在x點的種群密度，g(Nn)為繁衍后在x點的種群密度，τ為每一代的生長周期，一個周期生物種群只繁衍一次。本文在上面系統(tǒng)中加入水流速度(風(fēng)速)，并假設(shè)g(N)為Beverton-Holt函數(shù)[4]，可得到下面脈沖反應(yīng)擴(kuò)散系統(tǒng)

(2)

1 開放環(huán)境中脈沖ODE系統(tǒng)

計算可得：

定理1

證明

綜上可得：

又因為

以此類推，可得

2 開放環(huán)境中脈沖PDE系統(tǒng)

脈沖反應(yīng)擴(kuò)散系統(tǒng)(2)的線性化系統(tǒng)為

(3)

進(jìn)而可得

通過驗證可知Nn+1(x)=Q[Nn(x)]滿足H1 ～ H5 。

H1：當(dāng)Nn(x)∈[0,β]時，Nn+1(x)=Q[Nn(x)]∈[0,β] 。

所以，Ty{Q[Nn(x)]}=Q{Ty[Nn(x)]} 。

H3：存在區(qū)間[0,β]，當(dāng)u∈[0,β]時，Q(u)>u，且Q(0)=0,Q(β)=β。

H4：由比較原理可知,u≤v時，Q(u)≤Q(v)。

H5：因為g(u)為可微函數(shù)，且Qτ在[0,β]是完備的，所以Q在[0,β]也是完備的。

綜上，利用Weinberger在文獻(xiàn)[5]中的結(jié)論，可得到定理2。

3 數(shù)值模擬

本節(jié)主要研究流水速度對種群傳播的影響，各個參數(shù)的具體取值見表1。

表1 參數(shù)取值

圖1 當(dāng)q=0.5時，行波解的數(shù)值模擬Fig.1 A numerical approximation to traveling waves with q=0.5

圖2 當(dāng)q=1時，行波解的數(shù)值模擬Fig.2 A numerical approximation to traveling waves with q=1

圖3 當(dāng)q=3.3時，行波解的數(shù)值模擬Fig.3 A numerical approximation to traveling waves with q=3.3

圖4 當(dāng)q=10時，行波解的數(shù)值模擬Fig.4 A numerical approximation to traveling waves with q=10

4 結(jié)論

討論了一類具有脈沖出生與連續(xù)死亡的單種群動力學(xué)系統(tǒng)，給出了系統(tǒng)傳播速度的具體表達(dá)形式，借助傳播速度得到了該系統(tǒng)行波解[7-11]存在的條件，同時，研究了水流速度對行波解的影響，所得結(jié)論對于現(xiàn)實的生態(tài)平衡保護(hù)是十分有益的。

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Traveling waves and spreading speed to impulsive reaction-diffusion models

SHEN Lin, ZHOU Hong-ling*

(Department of Mathematics, Huanghuai University, Zhumadian 463000，China)

∶In this paper, an impulsive reaction-diffusion model with fixed moments of impulses in an unbounded domain was proposed, and the existence of spreading speed and traveling wave solutions for the model were established. First, the existence and the stability of the positive constant solutions were proved in ODE system. Second, the explicit formula of spreading speed to impulsive reaction-diffusion model was given. When certain conditions were satisfied, the spreading speed was greater than zero, which was the minimum speed of the traveling wave solutions. Finally, the numerical simulation of the system was carried out with the velocity of the water flow. The results reveal that the spread and persistence dynamics of the biotic population can be realized through the control of diffusion coefficient, flow velocity, mortality and birthrate corresponding to discrete time and continuous time respectively.

∶impulsive reaction-diffusion models; spreading speed; traveling waves

2017-08-03

國家自然科學(xué)基金(11371164)；國家自然科學(xué)基金委員會河南省人民政府人才培養(yǎng)聯(lián)合基金(U1304104)

沈林(1983—), 男, 講師, 研究方向為偏微分方程及其可視化。

*通信作者，周紅玲。E-mail：8210s@163.com

O175.26

A

1002-4026(2017)03-0088-06