阶段结构捕食竞争时滞模型分析
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摘要
对于生物数学和其它相关学科而言，种群的持续生存一直是学者们倍受关注的有趣问题。对于这方面的研究，已经取得了很多令人振奋的结果。比如说，对于标准的Lotka-Volterra型捕食被捕食模型(参见[1]～[3])，在这些模型中考虑的平均捕食率一般基于食饵种群的密度。但是近来，越来越多的研究结果(参见[4]～[7])表明，平均捕食率应该考虑所谓的“基于比率”理论。文献[8]中又提出，捕食者应按照一定的比率去捕食不同的食饵种群。在[5，6，9，10]中，已验证出当被捕食者种群十分强壮或者相对于捕食者比例相差很悬殊的情况下，捕食者往往会出现结伴捕食的情况，并在捕食之后，共同分享猎物。为了使生物数学模型更加真实，近年来许多的生物数学家和数学家已经考虑种群的阶段结构对整个生物群落的影响(参见[11～19，31～40])。另一方面，考虑到实际生物意义，时滞现象往往不能被忽略，并且往往较大的时滞会破坏解的稳定性(参见[20，21])；考虑时滞对模型解的渐近性态的影响的文章见[21]～[24]。
在文[8]中提出了一个捕食者按照不同的比例捕食的模型，如下：其中，捕食者x_3按照比率α_1，α_2分别捕食x_1和x_2；φ为功能性反应函数。
文[25]提出了一个新的阶段结构捕食被捕食模型，其中考虑捕食者的成年种群通过捕食减少了自身的死亡率，从而加大了种群持续生存的可能。所示模型如下：其中，x_m是捕食者的成年种群，x_i是捕食者的幼年种群，且幼年不能捕食；y是被捕食者；e~(-γτ)是由幼年到成年的转化的概率；τ是成熟期。
在本文中，在(***)的基础上结合(*)的因素，考虑阶段结构对种群持续生存方面以及在周期和概周期环境下，模型的周期和概周期解的存在唯一性及其相应稳定性方面的影响。论文分为三章，第一章仅考虑含有时滞和基于比率的Holling Ⅱ功能性反应的模型(*)，即模型(1．1．1)；第二章在模型(1．1．1)中引入了阶段结构，即模型(2．1．1)；在两章中分别给出系统的一致持久性、保证周期解和概周期解的存在唯一性及其相应稳定性的条件，并给出了保证其可行性的数学例子；第三章通过比较模型(1．1．1)和(2．1．1)相应结果的条件，得到了十分有趣的比较结果，并进一步验证了考虑阶段结构的必要性。比如，虽然引进了阶段结构，使考虑因素增多但对于保证种群的一致持久和全局稳
定性方面而言，反而更加容易保证其实现．本文有明显的生物意义，例如大刀螳螂的生长分为嫖峭
一 和成虫两个阶段，成虫为肉食性且食忖大，在农、林区主婆以蝶类和蜂类等为食，而撅峭无捕食和
生育能力；其中，蝶类以植物的花为食和蜂类的采蜜相互影响，从而两者产生竞争．从此例子可以
～看出，本文的数学结论和生物意义是相符合的．
在第一商中，考虑如下的模型：
】ti厂）＝一川t）。rl（t）
【oI】川I 了1TIll）卜宫．OJ一T】JJJ 一T） 十门／J一了kqOI一了】【A
I 十丁了r一二二卜人一X一一二《二一上一斗二二上了了一＞卜二L二了一上子7二一二子一一一二子f二上二丁一一一一一一下＝矿l。，
I．．、I．、．．、．、．．、．、尸什】tr，11v’！J 凸
（幻川＝x。O）卜小）一山川。。（t）一山（t）。：；（t）一百了了刀n下六仟宁石譬X于7了百瞩7刀1三枷，
I．、。l．、．．、．、．．、．、Pltltylftlte】Iiil 凸
0 f：j（O＝X。川I hh（O一山川0。川一b（0X。川一百穴7厂厂下只仟宁芳譬节子7了x厂7不I三hx，
【。t（川＝人（川，7＝＝1，2，3．o〔卜丁，川．
（1．1二）
Z寸厂任何正的连续有界函数／（t），记了＝ l。11。；〔；；f（t），z二 illft〔严；f（t）．
记x（J）＝（。l厂），x。（一，x；；（L》，一＝（办，人，办；）．
初始条件为
。V）一…E卜”，IE卜，，01，叫（‘）＞0．（1·1幻
上蛮结果为
引理1．2．fo系统O二·l）满足初始条件（1．1．2），则I。。tR＋是（1．1＊）的正不变集．
定理1．2．2 设门．二．巨）满足（豆．1川以及下列条件
《h12 厂”O！，。。。。
7I＞＞上干 －－－－，（1．2．1）
MZT！r’《12，、。n、
o＞y！－－＋h＝，（1．2．2）
htb，It
kCH＞dijll，（?
The prohlc-m of persistence of the biological population is very important with regard to biomath-ematics and other related courses. There have been a fair amount of previous works on the standard Lotka - Volterra type prey-predator model (see [1]-[3]),in which the mean predation rate was also assumed to be dependent on the density of the prey. But recently more and more papers have been proved that,the mean predation rate should be dependent on the theory of " ratio-dependent " by the proofs of the biology and experiments (see [4]-[7]). Nowadays it has been provided that the predator hunting for the preys according to the different rates of different preys by [8]. Because in many environments,especially the predators have to search for food or maybe the prey is so big or strong that the predators must hunt for the prey with the help of the other predators,then they must share the food with each other. These facts have been proved by the [5,C,9,10). In order to make the biological model more practical,many b
iomathematicians and biologists have been working on the stage-structured models (see[11-19,31-40j).
On the other hand,more and more people have begun to find out that the time-delay to biological
sense can't be ignored and the too longer time-delay would destabilize the stability of the solutions of the model (see [20,21]). Recently many papers have been shown the influence of time-delay on the asymptotic properties of the solutions (see [21]-[24]).
In [8],it is provided that the functional response is dependent on the different rates of different preys. Such model is as follows
where xj,x denotes the density of the two competitive populations A'i,A'2 respectively;13 denotes the density of the predator population X3. a1,a2 denotes the rate of AS hunting fot the populations X1,X2,f is the functional response;.
In [25],a new prey-predator modle with stage-structured has been provided in which the predator might reduce the death rate by feeding on the preys. Such model is as follows
where Xi(t),xm(t) denotes the density of the immature and the mature population predator A' at time t respectively;y(t) denotes the density of the prey population Y at time t. The mature predator of X can reduce tins death rate;by hunting and feeding on the prey }'.
In this paper,( ) combined with the factor of () is considered. It show the influence of the stage structure on the uniform persistence and the existence and uniqueness of the positive positive periodic and almost periodic solution and their corresponding stability. In Chapter one,model (1.1.1) is considered ;In Chapter two,model (2.1.1) is considered,then mathematical examples are given to show their practical respectively;In Chapter three,through comparison of the conditions of (1.1.1) and (2.1.1),which gurantee the same results,many interesting results are obtained. Such as,in spite of the consideration of stage-structure,the conditions,which can gurantee the uniform persistence and the properties of the same results of the positive periodic and almost periodic solution,are much simpler. So considering the stage-structure is very essential. The paper lias obvious biological background,for example,Chinese mantid preys on the bees and butterflies in the forest and field. But the immature Chinese mantid,i. e. its spawn,can not prey on them. Butterflies feed on the flowers and bees also need them,so they affect and compete with each other. From the right model,we can see that our theoretical results accord with the biological facts.
The model in Chapter 1 is defined as follows
Let J(t) is a strictly positive bounded continous function,set Note x(t) = (xi(t),xa(t), The initial conditions are as follows,
The main results are as follows,
Lemma 1.2.1 Assume that every solution of system (1.1.1) satisfies the initial condition (1.1.2),then is the positive invariant set,of the system (1.1.1).
Theorem 1.2.2 Assume that the system (1.1.1) with the initial condition (1.1.2) satisfies the following conditions,
then system (1.1.1) is uniformly persis
在文[8]中提出了一个捕食者按照不同的比例捕食的模型，如下：其中，捕食者x_3按照比率α_1，α_2分别捕食x_1和x_2；φ为功能性反应函数。
文[25]提出了一个新的阶段结构捕食被捕食模型，其中考虑捕食者的成年种群通过捕食减少了自身的死亡率，从而加大了种群持续生存的可能。所示模型如下：其中，x_m是捕食者的成年种群，x_i是捕食者的幼年种群，且幼年不能捕食；y是被捕食者；e~(-γτ)是由幼年到成年的转化的概率；τ是成熟期。
在本文中，在(***)的基础上结合(*)的因素，考虑阶段结构对种群持续生存方面以及在周期和概周期环境下，模型的周期和概周期解的存在唯一性及其相应稳定性方面的影响。论文分为三章，第一章仅考虑含有时滞和基于比率的Holling Ⅱ功能性反应的模型(*)，即模型(1．1．1)；第二章在模型(1．1．1)中引入了阶段结构，即模型(2．1．1)；在两章中分别给出系统的一致持久性、保证周期解和概周期解的存在唯一性及其相应稳定性的条件，并给出了保证其可行性的数学例子；第三章通过比较模型(1．1．1)和(2．1．1)相应结果的条件，得到了十分有趣的比较结果，并进一步验证了考虑阶段结构的必要性。比如，虽然引进了阶段结构，使考虑因素增多但对于保证种群的一致持久和全局稳
定性方面而言，反而更加容易保证其实现．本文有明显的生物意义，例如大刀螳螂的生长分为嫖峭
一 和成虫两个阶段，成虫为肉食性且食忖大，在农、林区主婆以蝶类和蜂类等为食，而撅峭无捕食和
生育能力；其中，蝶类以植物的花为食和蜂类的采蜜相互影响，从而两者产生竞争．从此例子可以
～看出，本文的数学结论和生物意义是相符合的．
在第一商中，考虑如下的模型：
】ti厂）＝一川t）。rl（t）
【oI】川I 了1TIll）卜宫．OJ一T】JJJ 一T） 十门／J一了kqOI一了】【A
I 十丁了r一二二卜人一X一一二《二一上一斗二二上了了一＞卜二L二了一上子7二一二子一一一二子f二上二丁一一一一一一下＝矿l。，
I．．、I．、．．、．、．．、．、尸什】tr，11v’！J 凸
（幻川＝x。O）卜小）一山川。。（t）一山（t）。：；（t）一百了了刀n下六仟宁石譬X于7了百瞩7刀1三枷，
I．、。l．、．．、．、．．、．、Pltltylftlte】Iiil 凸
0 f：j（O＝X。川I hh（O一山川0。川一b（0X。川一百穴7厂厂下只仟宁芳譬节子7了x厂7不I三hx，
【。t（川＝人（川，7＝＝1，2，3．o〔卜丁，川．
（1．1二）
Z寸厂任何正的连续有界函数／（t），记了＝ l。11。；〔；；f（t），z二 illft〔严；f（t）．
记x（J）＝（。l厂），x。（一，x；；（L》，一＝（办，人，办；）．
初始条件为
。V）一…E卜”，IE卜，，01，叫（‘）＞0．（1·1幻
上蛮结果为
引理1．2．fo系统O二·l）满足初始条件（1．1．2），则I。。tR＋是（1．1＊）的正不变集．
定理1．2．2 设门．二．巨）满足（豆．1川以及下列条件
《h12 厂”O！，。。。。
7I＞＞上干 －－－－，（1．2．1）
MZT！r’《12，、。n、
o＞y！－－＋h＝，（1．2．2）
htb，It
kCH＞dijll，（?
The prohlc-m of persistence of the biological population is very important with regard to biomath-ematics and other related courses. There have been a fair amount of previous works on the standard Lotka - Volterra type prey-predator model (see [1]-[3]),in which the mean predation rate was also assumed to be dependent on the density of the prey. But recently more and more papers have been proved that,the mean predation rate should be dependent on the theory of " ratio-dependent " by the proofs of the biology and experiments (see [4]-[7]). Nowadays it has been provided that the predator hunting for the preys according to the different rates of different preys by [8]. Because in many environments,especially the predators have to search for food or maybe the prey is so big or strong that the predators must hunt for the prey with the help of the other predators,then they must share the food with each other. These facts have been proved by the [5,C,9,10). In order to make the biological model more practical,many b
iomathematicians and biologists have been working on the stage-structured models (see[11-19,31-40j).
On the other hand,more and more people have begun to find out that the time-delay to biological
sense can't be ignored and the too longer time-delay would destabilize the stability of the solutions of the model (see [20,21]). Recently many papers have been shown the influence of time-delay on the asymptotic properties of the solutions (see [21]-[24]).
In [8],it is provided that the functional response is dependent on the different rates of different preys. Such model is as follows
where xj,x denotes the density of the two competitive populations A'i,A'2 respectively;13 denotes the density of the predator population X3. a1,a2 denotes the rate of AS hunting fot the populations X1,X2,f is the functional response;.
In [25],a new prey-predator modle with stage-structured has been provided in which the predator might reduce the death rate by feeding on the preys. Such model is as follows
where Xi(t),xm(t) denotes the density of the immature and the mature population predator A' at time t respectively;y(t) denotes the density of the prey population Y at time t. The mature predator of X can reduce tins death rate;by hunting and feeding on the prey }'.
In this paper,( ) combined with the factor of () is considered. It show the influence of the stage structure on the uniform persistence and the existence and uniqueness of the positive positive periodic and almost periodic solution and their corresponding stability. In Chapter one,model (1.1.1) is considered ;In Chapter two,model (2.1.1) is considered,then mathematical examples are given to show their practical respectively;In Chapter three,through comparison of the conditions of (1.1.1) and (2.1.1),which gurantee the same results,many interesting results are obtained. Such as,in spite of the consideration of stage-structure,the conditions,which can gurantee the uniform persistence and the properties of the same results of the positive periodic and almost periodic solution,are much simpler. So considering the stage-structure is very essential. The paper lias obvious biological background,for example,Chinese mantid preys on the bees and butterflies in the forest and field. But the immature Chinese mantid,i. e. its spawn,can not prey on them. Butterflies feed on the flowers and bees also need them,so they affect and compete with each other. From the right model,we can see that our theoretical results accord with the biological facts.
The model in Chapter 1 is defined as follows
Let J(t) is a strictly positive bounded continous function,set Note x(t) = (xi(t),xa(t), The initial conditions are as follows,
The main results are as follows,
Lemma 1.2.1 Assume that every solution of system (1.1.1) satisfies the initial condition (1.1.2),then is the positive invariant set,of the system (1.1.1).
Theorem 1.2.2 Assume that the system (1.1.1) with the initial condition (1.1.2) satisfies the following conditions,
then system (1.1.1) is uniformly persis
引文
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[3] Hey Z. Stability and delays in a predator-prey system [J]. J. Math. Anal. Appl. , 1996, 198: 355～370.
[4] Arditi R. and Saiah H. Empirical evidence of the role of heterogeneity in ratio-dependent consumption [J]. Ecology. 1992, 73: 1544～1551.
[5] Arditi R, Ginzburg L. R. and Akcakaya H. R. Variation in plankton densities among lakes: A Case for ratio-dependent models [J]. American Nut. 1991, 138: 1287～1296.
[6] Arditi R, Perrin N. and Saiah H. Functional response and heterogeneities: An experiment test with cladocerans [M]. OIKOS. 1991, 60: 69～75.
[7] Gutierrez A. P. The physiological basis of ratio-dependent predator-prey theory: a methbolic pool model of Nicholson's blowflies as an example [J]. Ecology, 1992, 73: 1552～1563.
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[24] Kuang Y, and Beretta, Global qualitative analysis of a ratio-dependent predator-prey system [J]. J. Math. Biol. , 1998, 36: 386～406.
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[27] Zhidong Teng, Lansun Chen, The positive periodic solutious of periodic KOLMOGOROVE type systems with delays [J]. Acta Mathcmaticae Applicatae Sinica, 1999, 22(3): 446～456.
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[32] Shengqing Liu, A nonautonomous stage-struetured single species model with diffusion [J], Ann. of Diff. Eqs. , 2000, 2(16): 153-168.
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[39] Z. Lu, and Y. Takeuchi, Permanence and global attractivity for competitive Lotka-Volterra systems with delay [J]. Nonal, Anal. , 1994, 22: 847～856.
[40] A. Shibata and N. Satio, Time delays and chaos in two competition system [J], Math. Biosci., 1980, 51: 156～168.
[2] Hastings A. Global stability in two species systems [J]. J. Math. Biol. ,1978, 5: 399～403.
[3] Hey Z. Stability and delays in a predator-prey system [J]. J. Math. Anal. Appl. , 1996, 198: 355～370.
[4] Arditi R. and Saiah H. Empirical evidence of the role of heterogeneity in ratio-dependent consumption [J]. Ecology. 1992, 73: 1544～1551.
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