全 文 ：第 35 卷第 7 期
2015年 4月
生 态 学 报
ACTA ECOLOGICA SINICA
Vol.35,No.7
Apr.,2015
http: / / www.ecologica.cn
基金项目:国家自然科学基金资助项目(11171199, 61273311); 中央高校基本科研专项基金资助项目(GK201302004, GK201302006)
收稿日期:2013鄄06鄄05; 摇 摇 网络出版日期:2014鄄05鄄08
*通讯作者 Corresponding author.E鄄mail: chsy398@ 126.com
DOI: 10.5846 / stxb201306051340
陈斯养,靳宝.一类具分段常数变量的捕食鄄食饵系统的 Neimark鄄Sacker分支.生态学报,2015,35(7):2339鄄2348.
Chen S Y, Jin B.Neimark鄄Sacker bifurcation behavior of predator鄄prey system with piecewise constant arguments. Acta Ecologica Sinica,2015,35( 7):
2339鄄2348.
一类具分段常数变量的捕食鄄食饵系统的 Neimark鄄
Sacker分支
陈斯养*,靳摇 宝
陕西师范大学, 数学与信息科学学院, 西安摇 710062
摘要:讨论了具时滞与分段常数变量的捕食鄄食饵生态模型的稳定性及 Neimark鄄Sacker分支;通过计算得到连续模型对应的差分
模型,基于特征值理论和 Schur鄄Cohn判据得到正平衡态局部渐进稳定的充分条件;以食饵的内禀增长率为分支参数,运用分支
理论和中心流形定理分析了 Neimark鄄Sacker 分支的存在性与稳定性条件;通过举例和数值模拟验证了理论的正确性。
关键词:分段常数变量;时滞;稳定性;Neimark鄄Sacker分支
Neimark鄄Sacker bifurcation behavior of predator鄄prey system with piecewise
constant arguments
CHEN Siyang*, JIN Bao
College of Mathematics and Information Science, Shaanxi Normal University, Xi忆an 710062,China
Abstract: 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鄄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鄄
Volterra equations. A predator鄄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鄄Volterra equations that we
know today. Almost at the same time (1926), Vito Volterra, an Italian mathematician, independently established the
Lotka鄄Volterra model after analyzing statistical data of fish catches in the Adriatic. The Lotka鄄Volterra equation is one of the
fundamental population models in theoretical biology. Since these early works, prey鄄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鄄
prey systems. Therefore, it is more realistic to employ the functional response with piecewise constant arguments and a time
delay in predator鄄prey models. In this paper, we discuss the stability and bifurcations of predator鄄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鄄Cohn criterion. Second, by choosing a parameter r, the
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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鄄Sacker bifurcation at an exceptive value of r. The results show that 1) the stability
of the predator鄄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鄄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鄄
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.
Key Words: piecewise constant arguments; delay; stability;Neimark鄄Sacker bifurcation
种群生态学是迄今数学在生态学中应用最为广泛、发展最为成熟的生态学的分支。 捕食鄄食饵系统是种
群生态学中生物种群相互之间的基本关系之一,是构成复杂食物链、食物网和生物化学网络结构的基石,从而
引起了广大数学工作者和生物学家的关注。 祁君和苏志勇[1]在经典的捕食鄄食饵系统中考虑到由于捕食效应
对食饵种群带来的正向调节作用后,提出了具有捕食正效应的捕食鄄食饵系统。 从理论上说明了正向调节作
用对系统的影响,并就第一象限内平衡点存在时的相图解释了捕食正效应的作用。 杨立和李维德[2]利用概
率元胞自动机模型对空间隐式的、食饵具 Allee 效应的一类捕食鄄食饵模型进行模拟,发现随着相关参数的变
化,种群的空间扩散前沿由连续的扩散波逐渐转变为一种相互隔离的斑块向外扩散。 Freedman 与 Wolkowicz
在 Rosenzweig鄄MacArthur模型[3]中选取第 4功能反应函数进行了全局范围内的分支情况的研究。 经典的捕
食鄄食饵模型可以被表达成如下的非线性微分方程模型:
dx( t)
dt
= xf(x)-yg(x,y)
dy( t)
dt
= y(-s+h(x,y
ì
î
í
ï
ï
ï
ï ))
该类模型稳定性、Hopf分支、极限环等问题被广泛的给予研究[4鄄12],式中 x( t),y( t),分别表示 t时刻捕食
者和食饵种群的数量,s表示捕食者的自然死亡率,f(x)表示食饵在无捕食者时的相对增长率,捕食者单位时
间内捕获食饵的数量用功能反应函数 g(x,y)表示。 实际上,由于受到气候、周围环境和捕食者所固有的特性
等因素影响,捕食者对食饵的捕获只在一定时间段或整数时刻并且对食饵的捕获具有滞后效应,故可选择更
加符合实际的具分段常数变量功能反应函数:g(x,y)=
ra2x( t)y([ t-1])
y( t)
,h(x,y)= b1x([ t]) -b2y( t)。 本文
考虑食饵在无捕食者时按通常的 Logistic方式增长,f(x)= r(1-a1x( t)),则模型可描述为:
dx( t)
dt
= rx( t)[1-a1x( t)-a2y([ t-1])]
dy( t)
dt
= y( t)[-s+b1x([ t])-b2y( t
ì
î
í
ï
ï
ï
ï )]
(1)
模型(1)满足初始条件:
x(0)= x0>0y( s)= 渍( s)逸0,渍(0)>0,渍沂C([-1,0 ],R+) (2)
式中,r表示食饵的内禀增长率,a1 表示食饵的环境容纳量,a2 表示捕食系数,b1 表示捕食效率常数,[ t]表示
对变量 t沂[0,+¥)取整。
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1摇 正平衡态稳定性分析
由模型(1)可知 b1>a1s时,模型(1)存在惟一的正平衡态:
E(軃x,軃y)= E(
b2+sa2
a1b2+a2b1
,
b1-sa1
a1b2+a2b1
)
定理 1 模型(1)满足初始条件(2)的解为正、全局存在且有界(坌t逸0)。
说明:对定理 1运用反证法和比较原理即可得证,故将其证明略去。
当 n臆tdx( t)
dt
= rx( t)[1-a1x( t)-a2y(n-1)]
dy( t)
dt
= y( t)[-s+b1x(n)-b2y( t
ì
î
í
ï
ï
ï
ï )]
(3)
对(3)由 n到 t积分并令 t寅n+1,即得:
x(n+1)= x(n)
[e-h+
ra1
h
x(n)(1-e-h)]
y(n+1)= y(n)
[e-q+
b2
q
y(n)(1-e-q
ì
î
í
ï
ï
ï
ï
ï
ï
ïï )]
(4)
其中:h= r[1-a2y(n-1)],q= -s+b1x(n),对(4)在平衡态 E(軃x,軃y)处 Taylor展开,令:
x(n)= 鬃(n)+軃x
y(n)= 渍(n)+軃{ y
得(4)式的线性近似系统 自(n+1)= A自(n)+B自(n-1) (5)
其中 自(n)= (鬃(n),渍(n)) T
A=
P 0
-
b1
b2
(e-b2軃y-1) e-b2軃
æ
è
ç
çç
ö
ø
÷
÷÷y
,B=
0
a2
a1
(P-1)
æ
è
ç
çç
ö
ø
÷
÷÷
0 0
,P=e-ra1軃x
则线性系统(5)的特征方程为 姿3+谆1姿2+谆2姿+谆3 = 0 (6)
其中 谆1 = -(P+e
-b2軃y),谆2 =e
-b2軃yP,谆3 =
a2b1
a1b2
(e-b2軃y-1)(P-1)
以下应用 Schur鄄Cohn判据[13]对模型(1)正平衡态稳定性进行分析,给出捕食者和食饵共存且数量保持
稳定的条件。
定理 2模型(1)满足下列 5种情况之一:
(1) 当 M3>1,M4<0,驻>0时,max(P1,
M3-1
M3
)(2)当 M3 = 1时,P5(3)当 M3<1且 驻<0或 M4<0,M5<0,驻>0或 M4>0,M5<0,驻>0,2M3+
a1b2
a2b1
(1-M3)M3<1时,0(4)当 M3<1,M4>0,M5<0,驻>0, 2M3+
a1b2
a2b1
(1-M3)M3>1时, 0(5)当 M3<1,M5>0,驻>0时,P1则正平衡态 E(軃x,軃y)局部渐近稳定。
1432摇 7期 摇 摇 摇 陈斯养摇 等:一类具分段常数变量的捕食鄄食饵系统的 Neimark鄄Sacker分支 摇
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其中:
M1 =
a2b1
a1b2
(1-e-b2軃y)+e-b2軃y+1,M2 =
a2b1
a1b2
(1-e-b2軃y)-e-b2軃y-1
M3 =
a2b1
a1b2
(1-e-b2軃y),M4 =e
-b2軃y+M3-e
-b2軃yM3-2M3 2,M5 =e
-b2軃yM3+M3 2-1
驻=M4 2-4(M3 2-M3)M5,P1 =
-M4- 驻
2(M3 2-M3)
,P2 =
-M4+ 驻
2(M3 2-M3)
证明摇 由 P=e-ra1軃x知 0(拽1) 谆1+谆3 <谆2+1;(拽2) 谆3 <1;(拽3) 谆2-谆1谆3 < 1-谆3 2
则正平衡态 E(軃x,軃y)局部渐近稳定。
首先考虑条件(拽1) 摇 (拽1)圳 -(P+e-b2軃y)+
a2b1
a1b2
(e-b2軃y-1)(P-1) M2
且[
a2b1
a1b2
(1-e-b2軃y)-e-b2軃y+1]P<
a2b1
a1b2
(1-e-b2軃y)-e-b2軃y+1等价于如下条件(i)或(ii):
(i)当 M2<0时,00时,
M2
M1
其次考虑条件(拽2)摇 由 谆3>0 知(拽2)圳
a2b1
a1b2
(1-e-b2軃y) (1-P) <1 等价于如下条件( iii)或( iv):( iii)当
M3臆1时,01时,
M3-1
M3
最后考虑条件(拽3)摇 由 谆1<0,谆2>0,谆3>0知(拽2)和(拽3)圳(M3 2-M3)P2+M4P+M5<0等价于如下条件
(v)或(vi):
(v) 当 M3>1时,知 M5>0,若 驻<0或 驻>0,M4>0(此时 P i<0( i = 1,2)),则其交集为空集;若 驻>0,M4<0,
知 P i>0( i= 1,2),P1f(x)= (M3 2-M3)P2+M4P+M5,由 f(1)= M3 2-M3+M4+M5 =e
-b2軃y-1<0知 P1<1(vi)当 M3 = 1时,知 M2 = -e
-b2軃y,M4 = -1,M5 =e
-b2軃y,则 M5(vii) 当 M3<1时,知 M3 2-M3<0,若 驻<0时,00,M4<0,M5<0,则当 P i<0( i= 1,2)时,0若 驻>0,M4>0,M5>0或 驻>0,M4<0,M5>0时,知 P1>0,P2<0,由 f(1) <0 得 P1<1,则 P10,M4>0,
M5<0,
由 f(1)<0知 1a1b2
a2b1
(1-M3)M3<1,
则 f忆(1)= -(M3-e
-b2軃y+e-b2軃yM3)= -[2M3+(1-e
-b2軃y)-(1-e-b2軃y)M3-1] = -[2M3+
a1b2
a2b1
(1-M3)M3-1]>0,由此
知 1a1b2
a2b1
(1-M3)M3>1,则 f忆(1)<0,
由此知 0交集可得定理中(1)的结论,由(i)、(iii)、(vi)取交集可得定理中(2)的结论,由(i)、(iii)、(vii)取交集可得定
理中(3)、(4)、(5)的结论,证毕。
2摇 Neimark鄄Sacker分支分析
本节以 r作为分支参数,分别讨论模型(1)的 Neimark鄄sacker 分支存在性及其分支方向与稳定性。 因情
况(2)不会产生分支(分支临界值 r0 趋于零或无穷大),故下文对定理 2 中(1)的情况给出产生分支的条件,
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情况(3)的分支条件可同理给出。
定理 3 设(1)中的参数满足 M3>1,M4<0,驻>0,则当 max(P1,
M3-1
M3
)部渐近稳定;当 P=max(P1,
M3-1
M3
)时,(6)存在一对位于单位圆环上共轭的特征根,且不存在 姿 = 依1 的根;当
P>max(P1,
M3-1
M3
)时,(1)的正平衡态 E(軃x,軃y)不稳定,则(1)产生 Neimark鄄sacker分支。
证明:由定理 2可知,当 M3>1,M4<0,驻>0时,若 max(P1,
M3-1
M3
)稳定;当 P=max(P1,
M3-1
M3
)时,假设 姿1,2 =e依i兹1是特征方程(6)的一对共轭纯虚根,则有
cos3兹1+谆1cos2兹1+谆2cos兹1+谆3 = 0
sin3兹1+谆1sin2兹1+谆2sin兹1 ={ 0 ,求解得 cos兹1 =
-谆1+ 谆1 2-4(谆2-1)
4
;
由定理 2中对(拽1)分析和
M2
M1
<
M3-1
M3
知,(6)不存在 姿= 依1的根,证毕。
下面讨论模型(1)的分支方向及其稳定性. 将(4)式写作如下变换形式:
x1
x2
x
æ
è
ç
ç
çç
ö
ø
÷
÷
÷÷
3
寅
x1
[e-h1+ra1(1-e
-h1)x1 / h1]
x2
[e-q1+b2(1-e
-q1)x2 / q1]
x
æ
è
ç
ç
ç
ç
ç
çç
ö
ø
÷
÷
÷
÷
÷
÷÷
2
=
F1(x1,x2,x3)
F2(x1,x2,x3)
F3(x1,x2,x3
æ
è
ç
ç
çç
ö
ø
÷
÷
÷÷)
(7)
其中,h1 = r[1-a2x3],q1 = -s+b1x1, (7)式在平衡态 E(軃x,軃y,軃y)的临界 Jacobi矩阵:
爪0 =爪( r0)=
軈P 0
a2
a1
(軈P-1)
-
b1
b2
(e-b2軃y-1) e-b2軃y 0
æ
è
ç
ç
ç
ç
ç
ç
ö
ø
÷
÷
÷
÷
÷
÷
0 1 0
由定理 3可知 爪0 存在一对共轭纯虚根 姿1,2 =e依i兹0,cos兹0 = =
-軍谆1+ 軍谆1 2-4(軍谆2-1)
4
,軈P=e-r0a1軃x,
軍谆1 = -(軈P+e
-b2軃y),軍谆2 =e
-b2軃y軈P 。 矩阵 爪0 和 爪T0 满足:爪0q= ei兹0q,爪0 Tp= e
-i兹0p,且其特征向量分别为:
q~
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
,ei兹0,
æ
è
ç
ö
ø
÷1
T
,p~
a1
a2(妆-1)
,
a1b2(軈P-e
-i兹0)
a2b1(軈P-1)(e
-b2軃y-1)
,ei兹
æ
è
çç
ö
ø
÷÷
0
T
取 q=
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
,ei兹0,
æ
è
ç
ö
ø
÷1
T
,p= 1
资
a1
a2(軈P-1)
,
a1b2(軈P-e
-i兹0)
a2b1(軈P-1)(e
-b2軃y-1)
,ei兹
æ
è
ç
ö
ø
÷0
T
则 =移
3
i=1
軃qipi = 1,其中,资=
a1b2[(e
-b2軃y-i兹0-e-2i兹0)+(軈Pe-i兹0-e-2i兹0)]
a2b1(e
-b2軃y-1)(軈P-1)
+ei兹0, 軃qi 为 qi( i= 1,2,3)的共轭复数。
将(7)式写为如下形式:
3432摇 7期 摇 摇 摇 陈斯养摇 等:一类具分段常数变量的捕食鄄食饵系统的 Neimark鄄Sacker分支 摇
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x寅Z0x+F(x)=
軈P 0
a2
a1
(軈P-1)
-
b1
b2
(e-b2軃y-1) e-b2軃y 0
æ
è
ç
ç
ç
ç
ç
ç
ö
ø
÷
÷
÷
÷
÷
÷
0 1 0
x1
x2
x
æ
è
ç
ç
ç
ö
ø
÷
÷
÷
3
+
G1(x)
G2(x)
æ
è
ç
ç
ç
ö
ø
÷
÷
÷
0
其中 F(x)= O(椰x椰2)是光滑函数且在平衡态 E(軃x,軃y,軃y)的 Taylor展开式为:
F(x)= 1
2
B(x,x)+ 1
6
C(x,x,x)+O(椰x椰4)
B(x,y)和 C(x,y,z)分量分别为:
B i(x,y)= 移
3
j,k=1
鄣2F i(孜)
鄣孜 j鄣孜k 孜=0
x jyk,C i(x,y,z)= 移
3
j,k,l=1
鄣3F i(孜)
鄣孜 j鄣孜k鄣孜l 孜=0
x jykzl,i= 1,2,...,n
当 eik兹0屹1(k= 1,2,3,4)时,映射(7)经坐标变换可化成:浊寅ei兹0浊(1+灼 浊 2)+O( 浊 4),其中 灼 决定闭不
变曲线的分支方向,可由下面公式计算:
灼= 1
2
Re{e-i兹0[掖p,C(q,q,軃q)业+2掖p,B(q,( I3-爪0)
-1B(q,軃q))业 +掖p,B(軃q,(e2i兹0I3-爪0)
-1B(q,q))业]}
记 z1 =
鄣2F1
鄣x1 2
=F1,x1x1,z2 =F1,x1x3 =F1,x3x1,z3 =F1,x3x3,z4 =F2,x1x1,z5 =F2,x1x2 =F2,x2x1, z6 =F2,x2x2,
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
经计算可知:
B(x,y)= ( z1 (軃x,軃y,軃y) x1y1+z2 (軃x,軃y,軃y)(x1y3+x3y1)+z3 (軃x,軃y,軃y) x3y3,z4 (軃x,軃y,軃y) x1y1+
z5 (軃x,軃y,軃y)(x1y2+x2y1)+z6 (軃x,軃y,軃y) x2y2 ,0)
T (8)
C(x,y,z)= ( z7 (軃x,軃y,軃y) x1y1z1+z8 (軃x,軃y,軃y)(x1y1z3+x3y1z1+x1y3z1)+z9 (軃x,軃y,軃y)(x1y3z3+x3y3z1+x3y1z3)+
z10 (軃x,軃y,軃y) x3y3z3,z11 (軃x,軃y,軃y) x1y1z1+z12 (軃x,軃y,軃y)(x1y1z2+x2y1z1+x1y2z1)+
z13 (軃x,軃y,軃y)(x2y2z1+x1y2z2+x2y1z2)+z14 (軃x,軃y,軃y) x2y2z2,0)
T (9)
将 q,軃q代入(8)和(9)式可得:
B(q,q)= 赘1,赘2,( )0 T,B(q,軃q)= 赘3,赘4,( )0 T,C(q,q,軃q)= 赘5,赘6,( )0 T
其中:
赘1 =[
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
]
2
z1 (軃x,軃y,軃y) +
2b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
z2 (軃x,軃y,軃y) +z3 (軃x,軃y,軃y)
赘2 =[
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
]
2
z4 (軃x,軃y,軃y) +
2b2(e
-b2軃y+2i兹0-e3i兹0)
b1(e
-b2軃y-1)
z5 (軃x,軃y,軃y) +e
2i兹0z6 (軃x,軃y,軃y)
赘3 =
b2 2(e
-2b2軃y-e-b2軃y+i兹0鄄e-b2軃y-i兹0+1)
b1 2 (e
-b2軃y-1) 2
z1 (軃x,軃y,軃y) +
b2 [e
-b2軃y(ei兹0+e-i兹0)-(e2i兹0+e-2i兹0)]
b1(e
-b2軃y-1)
伊 z2 (軃x,軃y,軃y) +z3 (軃x,軃y,軃y)
赘4 =
b2 2(e
-2b2軃y-e-b2軃y+i兹0鄄e-b2軃y-i兹0+1)
b1 2 (e
-b2軃y-1) 2
z4 (軃x,軃y,軃y) +
b2 (e
-b2軃y-e-b2軃y+2i兹0-ei兹0-e3i兹0)
b1(e
-b2軃y-1)
伊 z5 (軃x,軃y,軃y) +z6 (軃x,軃y,軃y)
摇 摇 赘5 =[
b2 3 (e
-b2軃y+i兹0-e2i兹0) 2(e-b2軃y-i兹0-e-2i兹0)
b1 3 (e
-b2軃y-1) 3
z7 (軃x,軃y,軃y) +{
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
] 2+
2b2 2(e
-2b2軃y-e-b2軃y+i兹0鄄e-b2軃y-i兹0+1)
b1 2 (e
-b2軃y-1) 2
z8 (軃x,軃y,軃y) +[
2b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
+(e
-b2軃y-i兹0-e-2i兹0)
b1(e
-b2軃y-1)
] z9 (軃x,軃y,軃y) +z10 (軃x,軃y,軃y)
4432 摇 生摇 态摇 学摇 报摇 摇 摇 35卷摇
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摇 摇 赘6 =[
b2 3 (e
-b2軃y+i兹0-e2i兹0) 2(e-b2軃y-i兹0-e-2i兹0)
b1 3 (e
-b2軃y-1) 3
z11 (軃x,軃y,軃y) +{
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
] 2e-i兹0+
2b2 2(e
-2b2軃y+i兹0-e-b2軃y+2i兹0鄄e-b2軃y+ei兹0)
b1 2 (e
-b2軃y-1) 2
z12 (軃x,軃y,軃y) +[
b2(e
-b2軃y+i兹0-1)
b1(e
-b2軃y-1)
+
2b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
] z13 (軃x,軃y,軃y) +e
i兹0z14 (軃x,軃y,軃y)
经计算:
( I3-Z0)
-1 = 1
追1
(aij) 3伊3,(e2i兹0I3-Z0)
-1 = 1
追2
(bij) 3伊3
其中:
a11 = 1-e
-b2軃y,a12 = b12 =
a2
a1
(軈P-1),a13 =
a2
a1
(1-e-b2軃y)(軈P-1),a22 =(1-軈P),
a21 =a31 = b31 =
b1
b2
(1-e-b2軃y),a23 = b23 =
a2b1
a1b2
(1-軈P)(e-b2軃y-1),a32 =(軈P-1),
a33 =(1-e
-b2軃y)(1-軈P),b11 =e2i兹0(e2i兹0-e
-b2軃y),b13 =
a2
a1
(e2i兹0-e-b2軃y)(軈P-1),
b21 =
b1
b2
e2i兹0(e2i兹0-e-b2軃y),b22 =e2i兹0(e2i兹0-軈P),b32 =(軈P-e2i兹0)b33 =(e2i兹0-e
-b2軃y)(e2i兹0-軈P),
追1 =(1-e
-b2軃y)(1-軈P)(
a2b1
a1b2
+1)
追2 =e2i兹0(e2i兹0-軈P )(e2i兹0-e
-b2軃y)+
a2b1
a1b2
(1-e-b2軃y)(1-軈P)
( I3-爪0)
-1B(q,軃q)=
1
追1
a11赘3+a12赘4,a21赘3+a22赘4,a31赘3+a32赘( )4 T
(e2i兹0I2-爪0)
-1B(q,q)= 1
追2
b11赘1+b12赘2,b21赘1+b22赘2,b31赘1+b32赘( )2 T
B(q,( I3-爪0)
-1B(q,軃q))= (赘7,赘8,0) T,B(軃q,(e2i兹0I3-爪0)
-1B(q,q))= 赘9,赘10, )( 0 T
其中:
赘7 =
1
追1
{
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
(a11赘3+a12赘4) z1 (軃x,軃y,軃y) +[
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
(a31赘3+a32赘4)
(a11赘3+a12赘4)] z2 (軃x,軃y,軃y) +(a31赘3+a32赘4) z3 (軃x,軃y,軃y)}
赘8 =
1
追1
{
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
(a11赘3+a12赘4) z4 (軃x,軃y,軃y) +
b2(e
-b2軃y+i兹0-e2i兹0)
b1(e
-b2軃y-1)
(a21赘3+a22赘4)
(a11赘3+a12赘4)ei兹0] z5 (軃x,軃y,軃y) +(a21赘3+a22赘4)e
i兹0z6 (軃x,軃y,軃y)}
赘9 =
1
追2
{
b2(e
-b2軃y-i兹0-e-2i兹0)
b1(e
-b2軃y-1)
(b11赘1+b12赘2) z1 (軃x,軃y,軃y) +[
b2(e
-b2軃y-i兹0-e-2i兹0)
b1(e
-b2軃y-1)
(b31赘1+b32赘2)
(b11赘1+b12赘2)] z2 (軃x,軃y,軃y) +(b31赘1+b32赘2) z3 (軃x,軃y,軃y)}
摇 摇 赘10 =
1
追2
{
b2(e
-b2軃y-i兹0-e-2i兹0)
b1(e
-b2軃y-1)
(b11赘1+b12赘2) z4 (軃x,軃y,軃y) +
b2(e
-b2軃y-i兹0-e-2i兹0)
b1(e
-b2軃y-1)
(b21赘1+b22赘2)
(b11赘1+b12赘2)e
-i兹0] z5 (軃x,軃y,軃y) +(b21赘1+b22赘2)e
-i兹0z6 (軃x,軃y,軃y)}
5432摇 7期 摇 摇 摇 陈斯养摇 等:一类具分段常数变量的捕食鄄食饵系统的 Neimark鄄Sacker分支 摇
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经计算可知:
掖p,C(q,q,軃q)业 =
1
軈资
[
a1
a2(軈P-1)
赘5+
a1b2(軈P-ei兹0)
a2b1(軈P-1)(e
-b2軃y-1)
赘6]掖p,B(q,( I3-爪0)
-1B(q,軃q))业
= 1
軈资
[
a1
a2(軈P-1)
赘7+
a1b2(軈P-ei兹0)
a2b1(軈P-1)(e
-b2軃y-1)
赘8]
掖p,B(軃q,(e2i兹0I3-爪0)
-1B(q,q))业 = 1
軈资
[
a1
a2(軈P-1)
赘9+
a1b2(軈P-ei兹0)
a2b1(軈P-1)(e
-b2軃y-1)
赘10]
其中 軈资为 资的共轭复数。
由以上计算知 灼的表达式如下:
灼 = 1
2
Re{e-i兹0[掖p,C(q,q,軃q)业+2掖p,B(q,( I3-爪0)
-1B(q,軃q))业 +掖p,B(軃q,(e2i兹0I3-爪0)
-1B(q,q))业]}
= 1
2
Re{e
-i兹0资
资
[
a1
a2(軈P-1)
(赘5+2赘7+赘9)+
a1b2(軈P-ei兹0)
a2b1(軈P-1)(e
-b2軃y-1)
(赘6+2赘8+赘10)]}
由如上分析和推理可得如下定理 4。
定理 4[14]当 r= r0 = -
ln(P)
a1軃x
时(P 由定理 3 确定),模型(1)在正平衡态 E(軃x,軃y)产生 Neimark鄄sacker 分支;
若 灼<0(>0),则模型(1)从正平衡态 E(軃x,軃y)分支出惟一(不)稳定的超(亚)临界 Neimark鄄sacker分支。
3摇 数值计算
本节将通过实例,运用 Matlab软件绘出相应的分支图,验证以上理论的可行性,并通过图形说明该模型
复杂的动力学行为。
例 在模型(1)中,取 a1 = 0.5,a2 = 0.4,b1 = 4,b2 = 2.5,s= 0.1计算可得:
M3 = 1.2400,M4 = -1.8426,驻= 2.7092,灼= -0.4851<0,妆1 = 0.3304>(M3-1) / M3 = 0.1935
则分支参数的临界值 r0 = 2.4852,惟一正平衡态 E(0.8912,1.3860)
对应分支图为图 1。 由图 1可知,当 r时,产生 N鄄S分支(图 4)及像平面图和解图(图 5);当 r> r0 = 2.4852 时,模型(1)出现复杂的动力学行为
(图 1)。
图 1摇 r-x,r-y分支图
Fig.1摇 r-x,r-ybifurcation map
4摇 总结
本文应用 Schur鄄Cohn判据、分支理论及中心流形投影等理论给出了具有时滞与分段常数变量捕食鄄食饵
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图 2摇 x,y稳定解图( r= 2.26Fig.2摇 stability solution map of x and y( r= 2.26图 3摇 稳定图( r= 2.26Fig.3摇 stability map ( r= 2.26图 4摇 N鄄S分支图
Fig.4摇 N鄄S bifurcation map
模型的稳定性及 Neimark鄄sacker分支的存在性以及稳定性条件。 通过模型的分析得到如下两个主要结论:
H1) 在捕食鄄食饵系统中考虑捕食者只在一定时间段或整数时刻且具有滞后效应捕食时,由定理 2 可知,
系统的稳定性(捕食者和食饵共存且数量保持稳定)将会变得非常复杂。
H2) 由实例可知,系统在其它参数不变的情况下,当食饵的内禀增长率 r<2.4852时,由图 1—图 3可知捕
食者和食饵的数量处于稳定状态;当 r= 2.4852,由图 4,图 5 知捕食者和食饵的数量将呈现周期性变化,系统
产生 Neimark鄄sacker分支;当 r>2.4852时,由图 1知系统的正平衡态由稳定到不稳定。
综上所述,在捕食鄄食饵系统中,若考虑捕食者只在一定时间段或整数时刻且具有滞后效应捕食时,模型
7432摇 7期 摇 摇 摇 陈斯养摇 等:一类具分段常数变量的捕食鄄食饵系统的 Neimark鄄Sacker分支 摇
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图 5摇 x,y像平面图和空间解图( r= r0 = 2.4852)
Fig.5摇 phase plane and space solution map of x and y ( r= r0 = 2.4852)
动力学行为将变得更为错综复杂;食饵的内禀增长率达到确定的临界值时,种群数量将失去原有的稳定性,模
型将产生惟一稳定的超临界 Neimark鄄Sacker分支。
参考文献(References):
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