一類具分段常數變量的捕食-食餌系統的Neimark-Sacker分支

陳斯養,靳 寶

陜西師范大學, 數學與信息科學學院, 西安 710062

一類具分段常數變量的捕食-食餌系統的Neimark-Sacker分支

陳斯養*,靳 寶

陜西師范大學, 數學與信息科學學院, 西安 710062

討論了具時滯與分段常數變量的捕食-食餌生態模型的穩定性及Neimark-Sacker分支；通過計算得到連續模型對應的差分模型，基于特征值理論和Schur-Cohn判據得到正平衡態局部漸進穩定的充分條件；以食餌的內稟增長率為分支參數，運用分支理論和中心流形定理分析了Neimark-Sacker 分支的存在性與穩定性條件；通過舉例和數值模擬驗證了理論的正確性。

分段常數變量；時滯；穩定性；Neimark-Sacker分支

種群生態學是迄今數學在生態學中應用最為廣泛、發展最為成熟的生態學的分支。捕食-食餌系統是種群生態學中生物種群相互之間的基本關系之一，是構成復雜食物鏈、食物網和生物化學網絡結構的基石,從而引起了廣大數學工作者和生物學家的關注。祁君和蘇志勇[1]在經典的捕食-食餌系統中考慮到由于捕食效應對食餌種群帶來的正向調節作用后，提出了具有捕食正效應的捕食-食餌系統。從理論上說明了正向調節作用對系統的影響，并就第一象限內平衡點存在時的相圖解釋了捕食正效應的作用。楊立和李維德[2]利用概率元胞自動機模型對空間隱式的、食餌具Allee 效應的一類捕食-食餌模型進行模擬，發現隨著相關參數的變化，種群的空間擴散前沿由連續的擴散波逐漸轉變為一種相互隔離的斑塊向外擴散。Freedman 與 Wolkowicz 在Rosenzweig-MacArthur模型[3]中選取第4功能反應函數進行了全局范圍內的分支情況的研究。經典的捕食-食餌模型可以被表達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表示食餌的內稟增長率，a1表示食餌的環境容納量，a2表示捕食系數，b1表示捕食效率常數,[t]表示對變量t∈[0,+)取整。

1 正平衡態穩定性分析

由模型(1)可知b1>a1s時，模型(1)存在惟一的正平衡態：

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：對定理1運用反證法和比較原理即可得證，故將其證明略去。

當n≤t

(3)

對(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

υ(n+1)=Aυ(n)+Bυ(n-1)

(5)

其中

υ(n)=(ψ(n),φ(n))T

則線性系統(5)的特征方程為

λ3+?1λ2+?2λ+?3=0

(6)

其中

以下應用Schur-Cohn判據[13]對模型(1)正平衡態穩定性進行分析，給出捕食者和食餌共存且數量保持穩定的條件。

定理2模型(1)滿足下列5種情況之一：

(2)當M3=1時，P5

(5)當M3<1，M5>0，Δ>0時，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等價于如下條件(v)或(vi)：

(v) 當M3>1時，知M5>0，若Δ<0或Δ>0，M4>0(此時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M3<1時，知M32-M3<0，若Δ<0時，00，M4<0,M5<0,則當Pi<0(i=1,2)時，00，M4>0,M5>0或Δ>0，M4<0,M5>0時,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

本節以r作為分支參數，分別討論模型(1)的Neimark-sacker分支存在性及其分支方向與穩定性。因情況(2)不會產生分支(分支臨界值r0趨于零或無窮大)，故下文對定理2中(1)的情況給出產生分支的條件，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靜互襯，也就是既描寫運動的事物又描寫靜止的事物，使一方襯托另一方，特點更為突出。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

經計算可知:

(8)

(9)

其中:

經計算:

其中:

經計算可知:

由以上計算知ζ的表達式如下：

由如上分析和推理可得如下定理4。

3 數值計算

本節將通過實例，運用Matlab軟件繪出相應的分支圖，驗證以上理論的可行性，并通過圖形說明該模型復雜的動力學行為。

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

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

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

對應分支圖為圖1。由圖1可知，當rr0=2.4852時，模型(1)出現復雜的動力學行為(圖1)。

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

圖2 x,y穩定解圖(r=2.26

圖3 穩定圖(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

本文應用Schur-Cohn判據、分支理論及中心流形投影等理論給出了具有時滯與分段常數變量捕食-食餌模型的穩定性及Neimark-sacker分支的存在性以及穩定性條件。通過模型的分析得到如下兩個主要結論：

H1) 在捕食-食餌系統中考慮捕食者只在一定時間段或整數時刻且具有滯后效應捕食時，由定理2可知，系統的穩定性(捕食者和食餌共存且數量保持穩定)將會變得非常復雜。

H2) 由實例可知，系統在其它參數不變的情況下，當食餌的內稟增長率r<2.4852時，由圖1—圖3可知捕食者和食餌的數量處于穩定狀態；當r=2.4852，由圖4,圖5知捕食者和食餌的數量將呈現周期性變化，系統產生Neimark-sacker分支；當r>2.4852時，由圖1知系統的正平衡態由穩定到不穩定。

綜上所述，在捕食-食餌系統中，若考慮捕食者只在一定時間段或整數時刻且具有滯后效應捕食時，模型動力學行為將變得更為錯綜復雜；食餌的內稟增長率達到確定的臨界值時，種群數量將失去原有的穩定性，模型將產生惟一穩定的超臨界Neimark-Sacker分支。

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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

國家自然科學基金資助項目(11171199， 61273311); 中央高?；究蒲袑ｍ椈鹳Y助項目(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,靳寶.一類具分段常數變量的捕食-食餌系統的Neimark-Sacker分支.生態學報,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.