阶段结构时滞扩散模型分析
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摘要
在种群动力学的研究过程中，种群的持续生存问题一直倍受关注。为了使模型更加的实用和准确，越来越多的实际因素被考虑到模型中来，比如说，阶段结构(参见[1～10])，扩散(参见[10～15，27～30])，时滞(参见[14～18，31～40])，但是到目前为止，综合考虑诸上因素的模型还没有见到。
在本论文中，分别讨论了两类带有扩散和时滞的阶段结构捕食被捕食模型。在第一章中，讨论了在两个斑块环境下，每个斑块都含有捕食者的幼年和成年种群，且在其中一个斑块上存在另一个被捕食者种群；在第二章中，讨论了仅在一个斑块上含有捕食者的两阶段结构，而成年的捕食者可以在两个斑块之间扩散，且在没有幼年种群的斑块上，存在捕食者的明显的食饵。出于季节的影响，进而考虑了模型的周期和概周期的情况。
在本文中，首先对于这两类模型，分别考虑各种群的一致持久性；通过对独立子系统的讨论得到了，当系统是周期系统时，周期正解的存在唯一和全局吸引性，当系统是概周期系统时，正概周期解的存在性和一致渐近稳定的条件，并对上述两种情况分别给出了说明其可行性的数学例子。
在以上的证明中，考虑了系统的独立子系统，使问题更加简单，条件更加简捷，并且使用了比较原理、Liapunov泛函方法、Razumikhin函数法。
在第一章中，讨论如下模型：其中，x_i(t)，y_i(t)分别表示t时刻捕食者在斑块i(i=1，2)中的幼年和成年种群的密度；z(t)表示在斑块1上被y_1(t)所捕食的种群；成年捕食者种群可以扩散，D_i(t)(i=1，2)表示它们的扩散率。
以某种生活在两个岛屿上的海鸟为例来说明本模型的生物意义，其成年海鸟的飞行能力强，可以在两个岛屿间飞行，而幼年海鸟不具有这种能力；其中一个岛屿有充足的鱼类资源，便成了它们扩散的最充分动力。而另外的岛屿，因为鱼类资源较贫乏，而忽略不计。
初始条件为
φ∈C~+，φ_i(s)＞0，i=2，5，s∈[-τ，0]；φ_i(0)＞0，i=1，3，4． (1．2．2)
对任何定义在[0，+∞)上的有界连续函数ω(t)，令。
为保证初值的连续性，设
x_i(0)=integral from -τ to 0(α_i(s)y_i(s)e~(-integral from (?) to 0 (γ_i(θ)dθ)ds)，i=1，2．(1．2．3)
主要结果为
定理l．3．l如果系统（1．2．l）满足（l．2．3），则系统（．2．l）的解最终有界．
定理l．3．2如果（HI），人）和（H3）成立，其中
（H）t一6M25 ＞ 0；
（H）ue－”’”＞ DI一Rin3；
（H。）aze””＞DZ．
则系统（二．2＊）是一致持久的．
考虑（1．2．及）的独立子系统
f．，．、。。、一I 卞1（a）da．．、。。、、，、．＿．．、．。、。、、。＿．．、
191厂J—ofu一川e“’一gin一丁）一厂l卜）忻u］十UI厂八VZ卜）一山U＊十川【）91厂厂uJ，
ILl41＿＿141＿I4＼＿I4＼＿2I41Dll＼。111＿111
IZ厂I二o3u厂厂I一h u乃“uI一们n91卜乃卜I，
【．I．、I．、一0 7】（日）d刃，、。，、D，、。＿，。、．
【pZV1＝＊2厂一川e“’一 叨u 门一厂2卜川引U十UZ卜八DI厂I’92u川，
【WI LS），ZIS），pZ【S））”I一ZIS），…3IS），…5［S）），S EI一了，UI．
（二．4．二）
定理 1．4二若（HI）～抑）成立，其中，
（兄）ZPgu。＋旦l一DZ一0一RM一了e－工”＞0；
（w2内m。＋丛一页一欧－2”＞0；
（H6）73—RMS＞0．
则。－周期系统（1．4J的每个正解是在 I。tR＋全局吸弓l的·
定理二人2若（HI）～（＆）成立，则。周期系统（二人．1）存在唯一全局吸引的正。周期解．
定理二＊S若w卜（仇）成立，则。周期系统（且J．互）存在唯一全局吸引的正。周期解．
为方便起见，定义系统（二．2川的右端函数为F（t，if，yi。l，xZ，yZ）．
定理二．5．1假设系统门．4二）满足（HI）～收）和（H7）～（Hg），其中，
（＆）2凡。。十凡一马一0一呐R—q了e－y＞0；
（W2内。＋q一马一昨e工”＞0；
（m）＞。h一Rm。ma—qsq－2”＞0，（q＞l为常数）
则系统门．4l）存在一个一致渐近稳定的正概周期僻p（t），且满足 ined…l）C lod（FI）．进而，若系统
（二A．二）是。周期系统，则（二．4．1）存在一个正的。周期解．
定理二石．2假设系统（1．2．二）满足出卜抑）和（H7）一（Hg），则方程（二．2．1）存在一个一致渐近
稳定的概周期解p（t）．如果（1．2＊）是关于t的。周期系统，则（1．2＊）也相应的存在。周期解．
In the study of the population dynamics,the population's persistence is a very interesting and important problem. In order to make the models more practical and accurate,more and more realistic factors have been considered,such as stage-structure (see [1-10]),diffusion (see [10-15,27-30]),time delay(see [14-18,31-40]),but the model with all the factors seemed rarely to be considered.
Considering all these factors,the paper discusses two types of predator-prey models which are two-patch,two-stage and two-population dynamics. The model in Chapter 1 describes that each of the two patches has the immature and mature predator populations,and there is another prey population only in one patch;The model in Chapter 2 describes that there are immature and mature predator populations only in one patch,and the mature predator can disperse between the two patches,and there is a prey population on the patch which has no immature population. Because of the season's influence,it is reasonable to assume all the coefficients are periodic and almost periodic.
Each chapter of the paper reads as follows. Firstly,the uniform persistence of the system is proved;Secondly,by using the independent subsystem method,the more brief sufficient conditions for the existence and global attractivity of the periodic solution and a mathematical example is obtained to show its practicality. Finally,the conditions of the existence of the positive almost periodic solution which is uniformly asymptotically stable are derived by the Razumikhin function method and a mathematical
example is obtained.
In the proof,by the subsystem which makes the discussion more convenient,we get more brief sufficient conditions. We also use the comparison theorem,Liapunov functional method and Razumikhin function method.
In Chapter one,consider the following model
where represent the immature and mature predator populations densities in the patch i at time t respectively. z(t) represents the prey population density in the patch 1 at time t,and yi(t) in patch 1 is its predator at time t. The mature population yt(t) can disperse between the two patches at time t,Di(t) is the diffusion coefficient in the patch i at time t,where,t = 1,2.
The models have their right background,for example,some kind of sea birds which live in two different islands,the adult birds can fly between the two islands,so the sufficient fish supply is one of most important driving forces. The initial conditions are as follows,
For a bounded continuous function w(t) defined on
For the continuity of the initial conditions,set
The main results are as follows,
Theorem 1.3.1 If system(1.2.1) satisfies (1.2.3),then the solution of (1.2.1) is ultimately bounded.
Theorem 1.3.2 If (H1),(H2)and (H3) hold,where
then (1.2.1) is uniformly persistent.
Let's consider the subsystem of (1.2.1)
Theorem 1.4.1 If (H1) - (H6) hold,where
then every positive solution of w-periodic system (1.4.1) is globally attractive in IntR .
Theorem 1.4.2 If (H1) - (H6) hold,then u;-periodic system (1.4.1) has a unique positive w-periodic solution which is globally attractive.
Theorem 1.4.3 If (H1) - (H6) hold,then w-periodic system (1.2.1) has a unique positive w-periodic solution which is globally attractive.
For convenience,define the right part of (1.2.1) as F(t,xi,yi,z,xy,3 2)-
Theorem 1.5.1 Assume system (1.4.1) satisfies (H1) - (H3) and (H7) - (H9),where,
then equation (1.4.1) has an almost periodic solution p(t) which is uniformly asymptotically stable and mod(pi) C mod(Fi). Furthermore,if (1.4.1) is a w-periodic system in t,then (1.4.1) has a positive w-periodic solution.
Theorem 1.5.2 Assume system (1.2.1) satisfies (H1) - (H3) and (H7) - (Hg),then equation (1.2.1) has an almost periodic solution p(t) which is uniformly asymptotically stable. Furthermore,if (1.2.1) is a w-periodic system in t.,then (1.2.1) has a w-periodic solution.
In Chapter two,consider the following model
where xi(t),yi(t) represent the immature and mature predator population densities in the patch 1
在本论文中，分别讨论了两类带有扩散和时滞的阶段结构捕食被捕食模型。在第一章中，讨论了在两个斑块环境下，每个斑块都含有捕食者的幼年和成年种群，且在其中一个斑块上存在另一个被捕食者种群；在第二章中，讨论了仅在一个斑块上含有捕食者的两阶段结构，而成年的捕食者可以在两个斑块之间扩散，且在没有幼年种群的斑块上，存在捕食者的明显的食饵。出于季节的影响，进而考虑了模型的周期和概周期的情况。
在本文中，首先对于这两类模型，分别考虑各种群的一致持久性；通过对独立子系统的讨论得到了，当系统是周期系统时，周期正解的存在唯一和全局吸引性，当系统是概周期系统时，正概周期解的存在性和一致渐近稳定的条件，并对上述两种情况分别给出了说明其可行性的数学例子。
在以上的证明中，考虑了系统的独立子系统，使问题更加简单，条件更加简捷，并且使用了比较原理、Liapunov泛函方法、Razumikhin函数法。
在第一章中，讨论如下模型：其中，x_i(t)，y_i(t)分别表示t时刻捕食者在斑块i(i=1，2)中的幼年和成年种群的密度；z(t)表示在斑块1上被y_1(t)所捕食的种群；成年捕食者种群可以扩散，D_i(t)(i=1，2)表示它们的扩散率。
以某种生活在两个岛屿上的海鸟为例来说明本模型的生物意义，其成年海鸟的飞行能力强，可以在两个岛屿间飞行，而幼年海鸟不具有这种能力；其中一个岛屿有充足的鱼类资源，便成了它们扩散的最充分动力。而另外的岛屿，因为鱼类资源较贫乏，而忽略不计。
初始条件为
φ∈C~+，φ_i(s)＞0，i=2，5，s∈[-τ，0]；φ_i(0)＞0，i=1，3，4． (1．2．2)
对任何定义在[0，+∞)上的有界连续函数ω(t)，令。
为保证初值的连续性，设
x_i(0)=integral from -τ to 0(α_i(s)y_i(s)e~(-integral from (?) to 0 (γ_i(θ)dθ)ds)，i=1，2．(1．2．3)
主要结果为
定理l．3．l如果系统（1．2．l）满足（l．2．3），则系统（．2．l）的解最终有界．
定理l．3．2如果（HI），人）和（H3）成立，其中
（H）t一6M25 ＞ 0；
（H）ue－”’”＞ DI一Rin3；
（H。）aze””＞DZ．
则系统（二．2＊）是一致持久的．
考虑（1．2．及）的独立子系统
f．，．、。。、一I 卞1（a）da．．、。。、、，、．＿．．、．。、。、、。＿．．、
191厂J—ofu一川e“’一gin一丁）一厂l卜）忻u］十UI厂八VZ卜）一山U＊十川【）91厂厂uJ，
ILl41＿＿141＿I4＼＿I4＼＿2I41Dll＼。111＿111
IZ厂I二o3u厂厂I一h u乃“uI一们n91卜乃卜I，
【．I．、I．、一0 7】（日）d刃，、。，、D，、。＿，。、．
【pZV1＝＊2厂一川e“’一 叨u 门一厂2卜川引U十UZ卜八DI厂I’92u川，
【WI LS），ZIS），pZ【S））”I一ZIS），…3IS），…5［S）），S EI一了，UI．
（二．4．二）
定理 1．4二若（HI）～抑）成立，其中，
（兄）ZPgu。＋旦l一DZ一0一RM一了e－工”＞0；
（w2内m。＋丛一页一欧－2”＞0；
（H6）73—RMS＞0．
则。－周期系统（1．4J的每个正解是在 I。tR＋全局吸弓l的·
定理二人2若（HI）～（＆）成立，则。周期系统（二人．1）存在唯一全局吸引的正。周期解．
定理二＊S若w卜（仇）成立，则。周期系统（且J．互）存在唯一全局吸引的正。周期解．
为方便起见，定义系统（二．2川的右端函数为F（t，if，yi。l，xZ，yZ）．
定理二．5．1假设系统门．4二）满足（HI）～收）和（H7）～（Hg），其中，
（＆）2凡。。十凡一马一0一呐R—q了e－y＞0；
（W2内。＋q一马一昨e工”＞0；
（m）＞。h一Rm。ma—qsq－2”＞0，（q＞l为常数）
则系统门．4l）存在一个一致渐近稳定的正概周期僻p（t），且满足 ined…l）C lod（FI）．进而，若系统
（二A．二）是。周期系统，则（二．4．1）存在一个正的。周期解．
定理二石．2假设系统（1．2．二）满足出卜抑）和（H7）一（Hg），则方程（二．2．1）存在一个一致渐近
稳定的概周期解p（t）．如果（1．2＊）是关于t的。周期系统，则（1．2＊）也相应的存在。周期解．
In the study of the population dynamics,the population's persistence is a very interesting and important problem. In order to make the models more practical and accurate,more and more realistic factors have been considered,such as stage-structure (see [1-10]),diffusion (see [10-15,27-30]),time delay(see [14-18,31-40]),but the model with all the factors seemed rarely to be considered.
Considering all these factors,the paper discusses two types of predator-prey models which are two-patch,two-stage and two-population dynamics. The model in Chapter 1 describes that each of the two patches has the immature and mature predator populations,and there is another prey population only in one patch;The model in Chapter 2 describes that there are immature and mature predator populations only in one patch,and the mature predator can disperse between the two patches,and there is a prey population on the patch which has no immature population. Because of the season's influence,it is reasonable to assume all the coefficients are periodic and almost periodic.
Each chapter of the paper reads as follows. Firstly,the uniform persistence of the system is proved;Secondly,by using the independent subsystem method,the more brief sufficient conditions for the existence and global attractivity of the periodic solution and a mathematical example is obtained to show its practicality. Finally,the conditions of the existence of the positive almost periodic solution which is uniformly asymptotically stable are derived by the Razumikhin function method and a mathematical
example is obtained.
In the proof,by the subsystem which makes the discussion more convenient,we get more brief sufficient conditions. We also use the comparison theorem,Liapunov functional method and Razumikhin function method.
In Chapter one,consider the following model
where represent the immature and mature predator populations densities in the patch i at time t respectively. z(t) represents the prey population density in the patch 1 at time t,and yi(t) in patch 1 is its predator at time t. The mature population yt(t) can disperse between the two patches at time t,Di(t) is the diffusion coefficient in the patch i at time t,where,t = 1,2.
The models have their right background,for example,some kind of sea birds which live in two different islands,the adult birds can fly between the two islands,so the sufficient fish supply is one of most important driving forces. The initial conditions are as follows,
For a bounded continuous function w(t) defined on
For the continuity of the initial conditions,set
The main results are as follows,
Theorem 1.3.1 If system(1.2.1) satisfies (1.2.3),then the solution of (1.2.1) is ultimately bounded.
Theorem 1.3.2 If (H1),(H2)and (H3) hold,where
then (1.2.1) is uniformly persistent.
Let's consider the subsystem of (1.2.1)
Theorem 1.4.1 If (H1) - (H6) hold,where
then every positive solution of w-periodic system (1.4.1) is globally attractive in IntR .
Theorem 1.4.2 If (H1) - (H6) hold,then u;-periodic system (1.4.1) has a unique positive w-periodic solution which is globally attractive.
Theorem 1.4.3 If (H1) - (H6) hold,then w-periodic system (1.2.1) has a unique positive w-periodic solution which is globally attractive.
For convenience,define the right part of (1.2.1) as F(t,xi,yi,z,xy,3 2)-
Theorem 1.5.1 Assume system (1.4.1) satisfies (H1) - (H3) and (H7) - (H9),where,
then equation (1.4.1) has an almost periodic solution p(t) which is uniformly asymptotically stable and mod(pi) C mod(Fi). Furthermore,if (1.4.1) is a w-periodic system in t,then (1.4.1) has a positive w-periodic solution.
Theorem 1.5.2 Assume system (1.2.1) satisfies (H1) - (H3) and (H7) - (Hg),then equation (1.2.1) has an almost periodic solution p(t) which is uniformly asymptotically stable. Furthermore,if (1.2.1) is a w-periodic system in t.,then (1.2.1) has a w-periodic solution.
In Chapter two,consider the following model
where xi(t),yi(t) represent the immature and mature predator population densities in the patch 1
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