陳斯養(yǎng),靳 寶

陜西師范大學(xué), 數(shù)學(xué)與信息科學(xué)學(xué)院, 西安 710062

一類具分段常數(shù)變量的捕食-食餌系統(tǒng)的Neimark-Sacker分支

陳斯養(yǎng)*,靳 寶

陜西師范大學(xué), 數(shù)學(xué)與信息科學(xué)學(xué)院, 西安 710062

討論了具時(shí)滯與分段常數(shù)變量的捕食-食餌生態(tài)模型的穩(wěn)定性及Neimark-Sacker分支；通過計(jì)算得到連續(xù)模型對(duì)應(yīng)的差分模型，基于特征值理論和Schur-Cohn判據(jù)得到正平衡態(tài)局部漸進(jìn)穩(wěn)定的充分條件；以食餌的內(nèi)稟增長(zhǎng)率為分支參數(shù)，運(yùn)用分支理論和中心流形定理分析了Neimark-Sacker 分支的存在性與穩(wěn)定性條件；通過舉例和數(shù)值模擬驗(yàn)證了理論的正確性。

分段常數(shù)變量；時(shí)滯；穩(wěn)定性；Neimark-Sacker分支

種群生態(tài)學(xué)是迄今數(shù)學(xué)在生態(tài)學(xué)中應(yīng)用最為廣泛、發(fā)展最為成熟的生態(tài)學(xué)的分支。捕食-食餌系統(tǒng)是種群生態(tài)學(xué)中生物種群相互之間的基本關(guān)系之一，是構(gòu)成復(fù)雜食物鏈、食物網(wǎng)和生物化學(xué)網(wǎng)絡(luò)結(jié)構(gòu)的基石,從而引起了廣大數(shù)學(xué)工作者和生物學(xué)家的關(guān)注。祁君和蘇志勇[1]在經(jīng)典的捕食-食餌系統(tǒng)中考慮到由于捕食效應(yīng)對(duì)食餌種群帶來的正向調(diào)節(jié)作用后，提出了具有捕食正效應(yīng)的捕食-食餌系統(tǒng)。從理論上說明了正向調(diào)節(jié)作用對(duì)系統(tǒng)的影響，并就第一象限內(nèi)平衡點(diǎn)存在時(shí)的相圖解釋了捕食正效應(yīng)的作用。楊立和李維德[2]利用概率元胞自動(dòng)機(jī)模型對(duì)空間隱式的、食餌具Allee 效應(yīng)的一類捕食-食餌模型進(jìn)行模擬，發(fā)現(xiàn)隨著相關(guān)參數(shù)的變化，種群的空間擴(kuò)散前沿由連續(xù)的擴(kuò)散波逐漸轉(zhuǎn)變?yōu)橐环N相互隔離的斑塊向外擴(kuò)散。Freedman 與 Wolkowicz 在Rosenzweig-MacArthur模型[3]中選取第4功能反應(yīng)函數(shù)進(jìn)行了全局范圍內(nèi)的分支情況的研究。經(jīng)典的捕食-食餌模型可以被表達(dá)成如下的非線性微分方程模型：

(1)

模型(1)滿足初始條件：

x(0)=x0>0y(s)=φ(s)≥0,φ(0)>0,φ∈C([-1,0],R+)

(2)

式中，r表示食餌的內(nèi)稟增長(zhǎng)率，a1表示食餌的環(huán)境容納量，a2表示捕食系數(shù)，b1表示捕食效率常數(shù),[t]表示對(duì)變量t∈[0,+)取整。

1 正平衡態(tài)穩(wěn)定性分析

由模型(1)可知b1>a1s時(shí)，模型(1)存在惟一的正平衡態(tài)：

定理1 模型(1)滿足初始條件(2)的解為正、全局存在且有界(?t≥0)。

說明：對(duì)定理1運(yùn)用反證法和比較原理即可得證，故將其證明略去。

當(dāng)n≤t

(3)

對(duì)(3)由n到t積分并令t→n+1，即得:

(4)

得(4)式的線性近似系統(tǒng)

υ(n+1)=Aυ(n)+Bυ(n-1)

(5)

其中

υ(n)=(ψ(n),φ(n))T

則線性系統(tǒng)(5)的特征方程為

λ3+?1λ2+?2λ+?3=0

(6)

其中

以下應(yīng)用Schur-Cohn判據(jù)[13]對(duì)模型(1)正平衡態(tài)穩(wěn)定性進(jìn)行分析，給出捕食者和食餌共存且數(shù)量保持穩(wěn)定的條件。

定理2模型(1)滿足下列5種情況之一：

(2)當(dāng)M3=1時(shí)，P5

(5)當(dāng)M3<1，M5>0，Δ>0時(shí)，P1

其中:

最后考慮條件(Η3) 由?1<0，?2>0，?3>0知(Η2)和(Η3)?(M32-M3)P2+M4P+M5<0等價(jià)于如下條件(v)或(vi)：

(v) 當(dāng)M3>1時(shí)，知M5>0，若Δ<0或Δ>0，M4>0(此時(shí)Pi<0(i=1,2))，則其交集為空集；若Δ>0，M4<0，知 Pi>0(i=1,2)，P1

(vii) 當(dāng)M3<1時(shí)，知M32-M3<0，若Δ<0時(shí)，00，M4<0,M5<0,則當(dāng)Pi<0(i=1,2)時(shí)，00，M4>0,M5>0或Δ>0，M4<0,M5>0時(shí),知P1>0，P2<0，由f(1)<0得P1<1，則P10，M4>0,M5<0,

由此知0

2 Neimark-Sacker分支分析

本節(jié)以r作為分支參數(shù)，分別討論模型(1)的Neimark-sacker分支存在性及其分支方向與穩(wěn)定性。因情況(2)不會(huì)產(chǎn)生分支(分支臨界值r0趨于零或無窮大)，故下文對(duì)定理2中(1)的情況給出產(chǎn)生分支的條件，情況(3)的分支條件可同理給出。

下面討論模型(1)的分支方向及其穩(wěn)定性. 將(4)式寫作如下變換形式:

(7)

取

B(x,y)和C(x,y,z)分量分別為：

還有一種情況就是動(dòng)靜互襯，也就是既描寫運(yùn)動(dòng)的事物又描寫靜止的事物，使一方襯托另一方，特點(diǎn)更為突出。如：

記

z7=F1,x1x1x1,z8=F1,x1x1x3=F1,x1x3x1=F1,x3x1x1z9=F1,x3x3x1=F1,x3x1x3=F1,x1x3x3，

z10=F1,x3x3x3,z11=F2,x3x3x3,z12=F2,x1x1x2=F2,x1x2x1=F2,x2x1x1,z13=F2,x2x2x1=F2,x2x1x2=F2,x1x2x2，z14=F2,x2x2x2

經(jīng)計(jì)算可知:

(8)

(9)

其中:

經(jīng)計(jì)算:

其中:

經(jīng)計(jì)算可知:

由以上計(jì)算知ζ的表達(dá)式如下：

由如上分析和推理可得如下定理4。

3 數(shù)值計(jì)算

本節(jié)將通過實(shí)例，運(yùn)用Matlab軟件繪出相應(yīng)的分支圖，驗(yàn)證以上理論的可行性，并通過圖形說明該模型復(fù)雜的動(dòng)力學(xué)行為。

例 在模型(1)中，取a1=0.5，a2=0.4，b1=4，b2=2.5，s=0.1計(jì)算可得:

M3=1.2400,M4=-1.8426,Δ=2.7092,ζ=-0.4851<0,Ρ1=0.3304>(M3-1)/M3=0.1935

則分支參數(shù)的臨界值r0=2.4852，惟一正平衡態(tài)E(0.8912,1.3860)

對(duì)應(yīng)分支圖為圖1。由圖1可知，當(dāng)rr0=2.4852時(shí)，模型(1)出現(xiàn)復(fù)雜的動(dòng)力學(xué)行為(圖1)。

圖1 r-x,r-y分支圖Fig.1 r-x，r-ybifurcation map

圖2 x,y穩(wěn)定解圖(r=2.26

圖3 穩(wěn)定圖(r=2.26

圖4 N-S分支圖Fig.4 N-S bifurcation map

圖5 x,y像平面圖和空間解圖(r=r0=2.4852)Fig.5 phase plane and space solution map of x and y (r=r0=2.4852)

4 總結(jié)

本文應(yīng)用Schur-Cohn判據(jù)、分支理論及中心流形投影等理論給出了具有時(shí)滯與分段常數(shù)變量捕食-食餌模型的穩(wěn)定性及Neimark-sacker分支的存在性以及穩(wěn)定性條件。通過模型的分析得到如下兩個(gè)主要結(jié)論：

H1) 在捕食-食餌系統(tǒng)中考慮捕食者只在一定時(shí)間段或整數(shù)時(shí)刻且具有滯后效應(yīng)捕食時(shí)，由定理2可知，系統(tǒng)的穩(wěn)定性(捕食者和食餌共存且數(shù)量保持穩(wěn)定)將會(huì)變得非常復(fù)雜。

H2) 由實(shí)例可知，系統(tǒng)在其它參數(shù)不變的情況下，當(dāng)食餌的內(nèi)稟增長(zhǎng)率r<2.4852時(shí)，由圖1—圖3可知捕食者和食餌的數(shù)量處于穩(wěn)定狀態(tài)；當(dāng)r=2.4852，由圖4,圖5知捕食者和食餌的數(shù)量將呈現(xiàn)周期性變化，系統(tǒng)產(chǎn)生Neimark-sacker分支；當(dāng)r>2.4852時(shí)，由圖1知系統(tǒng)的正平衡態(tài)由穩(wěn)定到不穩(wěn)定。

綜上所述，在捕食-食餌系統(tǒng)中，若考慮捕食者只在一定時(shí)間段或整數(shù)時(shí)刻且具有滯后效應(yīng)捕食時(shí)，模型動(dòng)力學(xué)行為將變得更為錯(cuò)綜復(fù)雜；食餌的內(nèi)稟增長(zhǎng)率達(dá)到確定的臨界值時(shí)，種群數(shù)量將失去原有的穩(wěn)定性，模型將產(chǎn)生惟一穩(wěn)定的超臨界Neimark-Sacker分支。

[1] 祁君, 蘇志勇. 具有捕食正效應(yīng)的捕食-食餌系統(tǒng). 生態(tài)學(xué)報(bào), 2011, 31(24): 7471- 7478.

[2] 楊立, 李維德. 利用元胞自動(dòng)機(jī)研究一類捕食食餌模型中的斑塊擴(kuò)散現(xiàn)象. 生態(tài)學(xué)報(bào), 2012, 32(6): 1773- 1782.

[3] Kot M. Elements of Mathematical Ecology. London: The Cambridge University Press, 2001: 107- 160.

[4] He Z M, Lai X. Bifurcation and chaotic behavior of a discrete-time predator-prey system. Nonlinear Analysis: Real world Applications, 2011, 12(1): 403- 417.

[5] Wang W M, Ling L. Stability and Hopf bifurcation analysis of a delayed predator-prey model with constant rate harvesting. Journal of Mathematical Biology, 2009, 24(4): 1- 14.

[6] Wang J F, Shi J P, Wei J J. Predator-prey system with strong Allee effect in prey. Journal of Mathematical Biology, 2011, 62(3): 291- 331.

[7] Beretta E, Kuang Y. Global analyses in some delayed ratio-dependent predator-prey systems. Nonlinear Analysis: Theory, Methods & Applications, 1988, 32(3): 381- 408.

[8] Skalski G T, Gilliam J F. Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology, 2001, 82(11): 3083- 3092.

[9] Mischaikow K, Wolkowicz G. A predator-prey system involving group defense: a connection matrix approach. Nonlinear Analysis: Theory, Methods & Applications, 1990, 14(11): 955- 969.

[10] Jost C, Arditi R. From pattern to process: identifying predator-prey models from time-series data. Population Ecology, 2001, 43(3): 229- 243.

[11] Salt G W. Predator and prey densities as controls of the rate of capture by the predatorDidiniumnasutum. Ecology, 1974, 55(2): 434- 439.

[12] Martin A, Ruan S G. Predator-prey models with delay and prey harvesting. Journal of Mathematical Biology, 2001, 43(3): 247- 267.

[13] Jury E I. Theory and Application of the Z-transform Method. New York: Wiley, 1964.

[14] Kuznetsov Y A. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 2004.

Neimark-Sacker bifurcation behavior of predator-prey system with piecewise constant arguments

CHEN Siyang*, JIN Bao

CollegeofMathematicsandInformationScience,ShaanxiNormalUniversity,Xi′an710062,China

The dynamic relationship between prey and predator has long been and will continue to be a dominant theme in ecology because of its universality. The prey-predator interaction, one of the most fundamental interspecies interactions, was first described mathematically by Lotka and Volterra in two independent works, resulting in what are now called the Lotka-Volterra equations. A predator-prey model based on the logistic equation was initially proposed by Alfred J. Lotka in 1910 to describe autocatalytic reactions. He later developed this model and in 1925 arrived at the Lotka-Volterra equations that we know today. Almost at the same time (1926), Vito Volterra, an Italian mathematician, independently established the Lotka-Volterra model after analyzing statistical data of fish catches in the Adriatic. The Lotka-Volterra equation is one of the fundamental population models in theoretical biology. Since these early works, prey-predator interactions have been studied systematically. Much of this work has focused on models with continuous time delay as well as their stability, oscillations, Hopf bifurcations and limit cycles, but no attention has been paid to models with piecewise constant arguments and a time delay. In fact, because of environmental factors or predator characteristics, prey are often captured only during certain times of the season. In addition, there is a time delay before hunting because of predator maturation times in practical predator-prey systems. Therefore, it is more realistic to employ the functional response with piecewise constant arguments and a time delay in predator-prey models. In this paper, we discuss the stability and bifurcations of predator-prey systems with piecewise constant arguments and a time delay. First, a discrete model that can equivalently describe the dynamical behavior of the original differential model is deduced. Sufficient conditions for the local asymptotic stability of the steady state are achieved based on an analysis of the eigenvalues and Schur-Cohn criterion. Second, by choosing a parameter r, the intrinsic growth rate of prey, as the bifurcation parameter and using the bifurcation theory and center manifold, we find that the discrete model undergoes a Neimark-Sacker bifurcation at an exceptive value ofr. The results show that 1) the stability of the predator-prey system is very complex when we consider piecewise constant arguments and a time delay; and 2) the positive equilibrium of the model switches from being stable to unstable as the intrinsic growth rate of prey increases beyond a critical value, at which point the unique supercritical Neimark-Sacker bifurcation will occur. Finally, computer simulations based on the system supported our main results and illustrated them intuitively. The numerical examples also justify the reasonableness of the conditions given in our paper for the loss of equilibrium. The parameters of the predator-prey model come from nature. However, we can still add to the model a feedback control factor and interference from outside to change the equilibrium, bifurcation point, or amplitude of the periodic solution. Study of our model and its ameliorated version can provide a theoretical basis for understanding ecology and protecting the environment.

piecewise constant arguments; delay; stability;Neimark-Sacker bifurcation

國(guó)家自然科學(xué)基金資助項(xiàng)目(11171199， 61273311); 中央高校基本科研專項(xiàng)基金資助項(xiàng)目(GK201302004, GK201302006)

2013- 06- 05;

日期:2014- 05- 08

10.5846/stxb201306051340

*通訊作者Corresponding author.E-mail: chsy398@126.com

陳斯養(yǎng),靳寶.一類具分段常數(shù)變量的捕食-食餌系統(tǒng)的Neimark-Sacker分支.生態(tài)學(xué)報(bào),2015,35(7):2339- 2348.

Chen S Y, Jin B.Neimark-Sacker bifurcation behavior of predator-prey system with piecewise constant arguments.Acta Ecologica Sinica,2015,35(7):2339- 2348.