这篇作文的题目是关于课外体育活动。利用跑步锻炼这个题材，作者阐述了自己对人生现实的认知，充满了积极向上的期待。
I'M GOING RUNNING TODAY. I am not concerned about my calorie consumption for the day, nor am I anxious to get in shape for the winter season. I just want to go running。
I used to dislike running. "If you don't win this game, you're all running five miles tomorrow," the field hockey coach used to warn, during those last days of October when the average temperature seemed to be decreasing exponentially. And so, occasionally, my grief-stricken team would run numerous miserable laps around the fields. At the end of these excursions, our faces and limbs would be numb, and we would all have developed those notorious flu-like symptoms; but the running made us better in the long run, I suppose. Nevertheless, I counted down the days until the end of the field hockey season, vowing never to put on a pair of running shoes again. Then I surprised myself by signing up for outdoor track in the second half of sophomore year. I was foolish to have believed that I could ever escape this insidious and magnetic addiction。
Anyone would have thought that I'd be off the team in a few days, but the last week of January caught me splashing through puddles of melted ice, and February winds nearly blew me off the track. I looked forward to practices this time around, to the claps and the persistent cheers of my fellow trackies. I was feeling a "runner's high" spurred by the endorphins released by exercise. But to attribute my affinity for running solely to chemistry diminishes the personal importance that running has for me。
I like running—in the cool shade of the towering oak trees, and in the warm sunlight spilling over the horizon, and in the drops of rain falling gently from the clouds. Certain things become clear to me when I'm running—only while running did I realize that "hippopotami" is possibly the funniest word in the English language, and only while running did I realize that the travel section of The York Times does not necessarily provide an accurate depiction of the entire world. Running lends me precious moments to contemplate my life: while running I find time to dream about changing the world, to think about recent death of a classmate, or to wonder about the secret to college admission
Running is the awareness of hurdles between me and the finish line; running is the desire to overcome them. Running is putting up with aches and pains, relishing the knowledge that, in the end, I will have built strength and endurance. Running is the instant clarity of vision with which I can see my future just one hundred yards in the distance; it is the understanding that these crucial steps will determine victory or defeat。
Running is not the most important thing in the world to me, but it is what fulfills me when time permits. And right now, before the sun goes down, I like to take advantage of the road that lies ahead。
解析：
要完全理解这篇作文，有必要提到据说是比尔·盖茨送给年轻人的十一条忠告：
1. 生活是不公平的，你要去学会适应它；
2. 这世界不会在意你的自尊，这世界指望你在自我感觉良好之前先要有所成就；
3. 高中刚毕业后你不会一下就拿到年薪六万美金的职位，你也不会很快成为拥有车载电话的公司副总，这些都要你自己挣得；
4. 如果你认为你的老师严厉，等你有了老板后再比较，老板可不是终身的；
5. 翻烤汉堡包并不有损你的尊严。你的长辈们用另一个词来描述这份工作，他们称之为机遇；
6. 如果你搞砸了，那不是你父母的错，不要只会发牢骚，要学会吸取教训；
7. 在你出生之前，你的父母并非像现在这样乏味。他们变成今天这个样子是因为这些年来他们一直在为你付账单，给你洗衣服，听你大谈你是如何的酷。在你大谈拯救雨林以免遭受你父母辈的寄生虫的危害时，先把你自己衣橱里的跳蚤除去；
8. 你的学校也许已经不再分优等生和劣等生，但生活却仍在划分；有些学校已经废除了不及格并给你想要多少就多少的机会让你得到正确的答案。但在现实生活中，却完全不同；
9. 生活不分学期，你并没有暑假可以休息，也没有几个人乐于帮你发现自我。你得用你自己的时间去发现；
10. 电视并不是真实的生活，在现实生活中，人们得离开咖啡屋去干自己的工作；
11. 善待那些看似怪异的人，很有可能有一天你会不得不为他们打工。
美国的大学教育是普通教育，培养有一技之长、对社会有用并且能适应社会的人。现实社会，不可避免会有很多不公平的地方，要成功，需要有顽强的心理素质。名牌大学对学生未来的发展期望很高，对学生承受压力、正视挫折的能力非常看重。很多大学的命题作文直接或间接地考察学生面对人生逆境的表现；而一个聪明的学生也会利用机会展示自己面对挑战的勇气和进取心。
在这篇作文里，作者开始就提到了自己早年在曲棍球队的经历：一个粗暴有虐待倾向的教练和惩罚性的长跑。尽管心里很不乐意，作者并没有放弃，反而以一种适应的态度去对待并最终迷上了这项运动。径赛队同伴的鼓励，让我们看到了作者珍惜友爱和社会的温情；作者的感悟，让我们既看到了作者走向社会的心理准备，又充满了积极的人生向往。一般的作文要求500单词左右，这篇文章共503单词，在有限的空间，包含了磨难，毅力，关怀，理解，憧憬。全文词汇优雅丰富，修辞巧妙，用了很多排比句，画面感非常强，感染力也非常强。在具体写作技巧上，有二点值得一提：
1. 使用了不少科学词汇，如指数般(exponentially)，内啡肽(endorphin)，爱好(affinity)，这些词汇的应用显然有利于叩击麻省理工学院的大门。
2. 巧妙甚至狡猾地使用了幽默。幽默是个双刃剑，往往容易弄巧成拙，一般人在作文里会尽量避免。然而，作者却大胆地调侃道：跑步时，会去猜想大学招生的秘密 --这简直是在向正在阅读此作文的招生人员叫阵！但是，说这句话的时候，招生的人应该已经为其经历和毅力所触动，而且前面谈到河马单词，已经把作文的节奏调得轻松，这句话会让招生人员会心一笑，拉近了彼此的距离。而随后梦幻般的紧凑道白，为这篇作文留下了非常美妙的收尾。
Lauren 是个可男可女的名字，但从第一段谈论控制体重保持身材就可看出是个女孩。的确，这篇文章透着一股女孩气，精灵机警，如同金庸小说里的某位人物。
本文建立了数学模型用以研究具有种群内控制的时滞两种群竞争生态系统,并讨论了成熟时滞以及捕获努力对种群动力学的影响。研究了三个非负平衡点以及唯一正平衡点存在的条件。分析了系统解的正定性与有界性。通过分析对应的特征方程,研究了系统在非负平衡点以及正平衡点附近的局部稳定性。利用迭代算法研究了非负平衡点的全局稳定性。通过建立适当Lypunov函数,研究了唯一正平衡点的全局稳定性。最后,通过数值仿真验证本文研究结果Dynamics Analysis in a Delayed
Two-Species
Competition Model with Harvest Eﬀort and
Nonlinear Intraspeciﬁc Regulation
LIU Chao12, WANG Xiao-Min3, Yue Wen-Quan4
1 Institute of Systems Science, Northeastern University, Shenyang 110004
2 State Key Laboratory of Integrated Automation of Process Industry, Northeastern
University, Shenyang, 110819
3 School of Mathematics and Statistics, Northeastern University at Qinhuangdao,
Qinhuangdao, 066004
4 Hebei Academy of Agricultural and Forestry Sciences, Shijiazhuang, 050000
Abstract: In this paper, a mathematical model is established to investigate interaction and
coexistence mechanism of two-species competition ecosystem with nonlinear intraspeciﬁc
regulation. The combined dynamic eﬀects of maturation delay and harvest eﬀort on
population dynamics are discussed. Conditions for existence of three nonnegative boundary
equilibria and a unique interior equilibrium are investigated. Positivity and boundedness of
solutions are analytically studied. By analyzing associated characteristic equation, local
stability of the proposed model around nonnegative boundary equilibrium and interior
equilibrium is discussed, respectively. Furthermore, global stability of the nonnegative
boundary equilibrium is investigated based on an iterative technique. By constructing an
appropriate Lyapunov functional, global stability of the unique interior equilibrium is also
discussed. Numerical simulations are carried out to show consistency with theoretical analysis.
Key words: Operations research and control theory; maturation delay, harvest eﬀort,
nonlinear intraspeciﬁc regulation, global stability analysis
0 Introduction
Competition is an interaction among competing species, in which the ﬁtness of one is
lowered by the presence of another species within ecosystem. Generally, competition is very
important in determining the characteristics of species, and there are two types of competition,
intraspeciﬁc competition and interspeciﬁc competition [1, 2].
Intraspeciﬁc competition is an interaction in population ecology, whereby members of the
same species compete for limited resources. When resources are limited, an increase in popula-
tion size reduces the quantity of resources available for each individual, reducing the per capita
ﬁtness in the population. As a result, the growth rate of a population slows as intraspeciﬁc
competition becomes more intense, making it a negatively density dependent process [1, 2]. In
the natural world, intraspeciﬁc competition phenomenon can be observed in a variety of ways.
Bird songs are often signals to other birds that they are not welcome in that area, which are
used to defend territories that contain breeding areas, shelter, and food. Many wild canine and
feline species mark their territories with scent, which tells other members of the same species
in that area that they have claimed the territory and all of the resources within it. Intraspeciﬁc
competition for mates can be quite dramatically observed through ornamental features. For
example, male peacocks display beautiful plumage to attract females. Male deer ﬁght each
other for mates with their large antlers; the larger set of antlers usually wins this competition
[3, 4, 5].
Interspeciﬁc competition refers to the competition between two or more species for some
limiting resource. This limiting resource can be food or nutrients, space, mates, nesting sites,
anything for which demand is greater than supply. When one species is a better competi-
tor, interspeciﬁc competition negatively inﬂuences the other species by reducing population
- 2 -
sizes and/or growth rates, which in turn aﬀects population dynamics of the competitor [1, 4].
Generally, interspeciﬁc competition has the potential to alter populations, communities and
the evolution of interacting species. On an individual organism level, interspeciﬁc competi-
tion can occur as interference or exploitative competition. There are some vivid interspeciﬁc
competition examples in the natural world, if a tree species in a dense forest grows taller than
surrounding tree species, it is able to absorb more of the incoming sunlight. However, less
sunlight is then available for the trees that are shaded by the taller tree. Cheetahs and lions
can also be in interspeciﬁc competition, since both species feed on the same prey, and can be
negatively impacted by the presence of the other because they will have less food [3, 4, 5].
In the 1920s, the dynamic impacts of intraspeciﬁc and interspeciﬁc competition on pop-
ulation dynamics have been discussed in [6, 7]. Under some necessary simpliﬁed assumptions
that there are not migration, and the carrying capacities and competition coeﬃcients for both
species are constants, Lotka and Volterra propose a mathematical model in [6, 7], which takes
the following form,
{
x˙1(t) = x1(t)(b1− a11x1(t) − a12x2(t)),
x˙2(t) = x2(t)(b2− a22x2(t) − a21x1(t)),
where xi(t) (i = 1, 2) represents population density of the competing ith species at time t,
respectively. bi(i = 1, 2) denotes the birth rate of the corresponding species; aij(i, j =
1, 2, i̸= j) is the corresponding linear reduction of the ith species’ rate growth by interspeciﬁc
competition, the j th species. aii(i = 1, 2) stands for the corresponding linear reduction of the
ith species’ rate growth by intraspeciﬁc competition. It should be noted that model (1) combines
the eﬀects of each species on the other and creates a theoretical prediction of interactions that
can be used to understand how diﬀerent factors aﬀect the outcomes of competitive interactions.
However, a variety of factors, which may aﬀect the outcome of competitive interactions and
dynamics of one or both populations, are not considered in the model (1). In the 1970s, Gilpin
and Ayala [8, 9, 10] conducted experiments on drosophila dynamics to test validity of ten
competition models. One of the models accounting for the experimental results is given as
follows:
{
˙1(t) = x1(t)(b1− a11xθ11(t) − a12x2(t)),
˙2(t) = x2(t)(b2− a22xθ22 (t) − a21x1(t)),
where model (2) is an extension of the Lotka-Volterra’s model, and estimations of θi(i = 1, 2)
for drosophila in the literature [8, 9, 10] suggest that θiis typically less than one.
Along with this line of research on model (2), persistence and asymptotical stability analysis
of competing species are studied in recent decades, which can be found in [11, 12, 13, 14, 15, 16]
and references therein. By constructing appropriate Lyapunov function, Goh et al. [11] inves-
tigate global stability of the proposed model with θ1≥ 1, θ2≥ 1. Zhou et al. [12] and Fan et al.
[13] discuss global stability of the proposed model with θ1≥ 1, θ2≥ 1 and θ1≤ 1, θ2≤ 1. Chen
- 3 -
et al. [15] propose a discrete Gilpin-Ayala type multispecies competition model. For general
nonautonomous case, suﬃcient conditions which ensure permanence and global stability of the
proposed model are obtained; For periodic case, suﬃcient conditions which ensure the existence
of an unique globally stable positive periodic solution are obtained. Liao et al. [16] propose
a discrete multispecies general Gilpin-Ayala competition predator prey model. By using new
diﬀerence inequality and developing the analysis technique of Chen [14], suﬃcient conditions
are established for permanence and the global stability in [16]. Dynamic eﬀect of impulsive
control and asymptotic stability analysis on Gilpin-Ayala competition model are investigated
in [17, 18, 19, 20]. Permanence and extinction of stochastic nonautonomous Gilpin-Ayala type
competition model with delays are investigated in [21, 22, 23, 24, 25]. A delayed nonautonomous
n-species Gilpin-Ayala type competition system is proposed in [21], which is more general and
more realistic then classical Lotka-Volterra type competition model. M. Vasilova [23] studies
the stochastic Gilpin-Ayala competition model with an inﬁnite delay, veriﬁes that the environ-
mental noise included in the model does not only provide a positive global solution and certain
asymptotic results regarding a large time behavior are obtained. A stochastic Gilpin-Ayala
predator-prey model with time dependent delay is studied in [24], and suﬃcient conditions for
existence of a global positive solution of the considered model are obtained. Furthermore, suf-
ﬁcient criteria for extinction of species for a special case of the considered system are given. In
[25], the nonautonomous stochastic Gilpin-Ayala competition model with time-dependent delay
is considered. Existence and uniqueness of the global positive solution and various properties of
that solution are proved in [25]. By considering the stage structure of competing species within
competition ecosystem, model (2) is extended by incorporating maturation delay of both two
competing species, which can be found in [18, 26, 27]. In the work done in [27], the stage
structured Gilpin-Ayala competition model of discrete delays are discussed as follows:
{
x˙1(t) = b1e−d1τ1x1(t − τ1) − a11x1+1 θ1(t) − a12x1(t)x2(t),
x˙2(t) = b2e−d2τ2x2(t − τ2) − a22x1+2 θ2(t) − a21x1(t)x2(t),
(3)
where τi, i = 1, 2 denotes the maturation delay of species xi(t), respectively. The term e−diτi
denotes the corresponding surviving rate of species i during its maturation duration from
immature to mature stage. Other parameters share the same interpretations mentions in model
(2). The global asymptotical stability criteria for the coexistence equilibrium as well as the
excluding equilibria are established in [27].
It is well known that many species within competition ecosystem are of agricultural and
medical utilization, which are mostly harvested and sold with the purpose of obtaining the
economic interest [4]. It motivates the introduction of commercial harvesting into mathematical
model and dynamic analysis of harvest eﬀort on population dynamics are discussed in [1, 3, 4].
Although there are much progress on Gilpin-Ayala competing model, such models are discussed
in the sense that the above work ignore commercial harvesting, which can not vividly reﬂect
complex biological phenomena from harvested competition ecosystem. Since harvesting has
a strong impact on the dynamic evolution of a population, it is necessary to investigate the
dynamic eﬀect of harvesting on Gilpin-Ayala competing population dynamics. Based on the
above analysis, model (3) is extended by incorporating harvest eﬀort on two competing species
in this paper, and the combined dynamic eﬀect of harvest eﬀort and maturation delays on
population dynamics will be discussed in this paper.
The rest section of this paper is organized as follows: a delayed two-species competition
model with harvest eﬀort and nonlinear intraspeciﬁc regulation is established in the second
section. Conditions for existence of three nonnegative boundary equilibria and a unique interior
equilibrium are analytically investigated. Positivity and boundedness of solutions of model are
also studied. In the third section, local stability of the proposed model around nonnegative
boundary equilibrium and interior equilibrium is discussed, respectively. In the fourth section,
global stability of the nonnegative boundary equilibrium is investigated based on an iterative
technique. By constructing an appropriate Lyapunov functional, global stability of the unique
interior equilibrium is also discussed. In the ﬁfth section, numerical simulations are given
to support the theoretical ﬁndings obtained in this paper. Finally, this paper ends with a
conclusion.
1 Model Formulation
By incorporating harvest eﬀort into model (3), a delayed two-species competition model
with harvest eﬀort and nonlinear intraspeciﬁc regulation is established in this section. Further-
more, conditions for existence of three nonnegative boundary equilibria and a unique interior
equilibrium are analytically investigated. Positivity and boundedness of solutions of model are
also studied. In this paper, the model is proposed based on the following hypotheses.
(H1) xi(t) (i = 1, 2) represents population density of the competing ith species at time t,
respectively. bi(i = 1, 2) denotes the birth rate of the corresponding species; aij(i, j =
1, 2, i̸= j) is the corresponding linear reduction of the ith species’ rate growth by its
competitor, the jth species. aii(i = 1, 2) is the corresponding linear reduction of the ith
species’ rate growth by the same species, the ith species.
(H2) τi, i = 1, 2 denotes the maturation delay of species i, respectively. The term e−diτi
denotes the corresponding surviving rate of species i during its maturation duration from
immature to mature stage.
(H3) E1≥ 0 and E2≥ 0 denotes harvest eﬀort on competing species x1(t) and x2(t), respec-
tively. q1E1x1(t) and q2E2x2(t) represents the catch of species x1(t) and x2(t), respec-
tively. q1and q2represents the catchability coeﬃcients of harvest eﬀort on species x1(t)
and x2(t), respectively.
http://www.paper.edu.cn
According to (H1)-(H3), a delayed two-species competition model with harvest eﬀort and
nonlinear intraspeciﬁc regulation is proposed as follows:
{
x˙1(t) = b1e−d1τ1 x1(t − τ1) − a11x1+1 θ1(t) − a12x1(t)x2(t) − q1E1x1(t),
x˙2(t) = b2e−d2τ2x2(t − τ2) − a22x1+2 θ2(t) − a21x1(t)x2(t) − q2E2x2(t).
In this paper, model (4) will be discussed with θ1≥ 1, θ2≥ 1 and θ1< 1, θ2< 1 under
the following initial conditions:
{
x1(t) = ϕ1(t) > 0, −τ1≤ t ≤ 0,
x2(t) = ϕ2(t) > 0, −τ2≤ t ≤ 0.
Theorem 1. All solutions of model (4) with initial conditions (5) are positive for allt > 0.
Proof.Firstly, we show that x1(t) > 0 for all t > 0. If x1(t) ≤ 0 for all t > 0, then it follows
from initial conditions (5) that there exists
t0= inf{t > 0|x1(t) = 0}.
According to initial conditions (5) and deﬁnition of t0, it is easy to show that x˙1(t0) < 0.
On the other hand, based on the continuity t0> 0, it can be computed by evaluating the
model (4) along solutions at time t0,
{
b1e−d1τ1ϕ1(t0− τ1), 0 ≤ t0≤ τ1,
x˙1(t0) =
b1e−d1τ1 x1(t0− τ1), t0> τ1.
It follows from initial conditions (5) that x˙1(t0) > 0, which is a contradiction. Consequent-
ly, x1(t) > 0 for all t > 0.
By using the similar arguments, it is easy to show that x2(t) > 0 for all t > 0.
Theorem 2. All solutions of model (4) with initial conditions (5) are ultimately bounded.
Proof.It follows from the ﬁrst equation of model (4) that
x˙1(t) ≤ b1e−d1τ1x1(t − τ1) − a11x1+1 θ1(t).
Considering the following auxiliary equation
z˙(t) = b1e−d1τ1z(t − τ1) − a11z1+θ1(t),
where z(t) = ϕ1(t) for t ∈ [−τ1, 0].
It is easy to show that z(t) ≥ x1(t) > 0 for all t > 0. Furthermore, it follows from Theorem
2 in [28] that z(t) is ultimately bounded, which implies there exists N1> 0 and T1> τ1
satisfying that x1(t) < N1for all t ≥ T1− τ1.
- 6 -
http://www.paper.edu.cn
By using the similar arguments, it is easy to show that there exists N2> 0 and T2> τ2
satisfying that x2(t) < N2for all t ≥ T2− τ2.
Consequently, all solutions of model (4) with initial conditions (5) are ultimately bounded.
According to model (4), nonnegative boundary equilibrium and interior equilibrium satis-
ﬁes the following equation,
{
x1(b1e−d1τ1− a11xθ11 − a12x2− q1E1) = 0,
x2(b2e−d2τ2 − a22xθ22 − a21x1− q2E2) = 0.
For the sake of simplicity, some transformations are proposed,
(7)
f1= (
b1
a11
1
) θ1 , f2= (
b2
a22
1
) θ2 , c12=
a12f1
b1
q1
, c21=
q2
a21f2
b2
,
η1= d1τ1, η2= d2τ2, p1=
and then Eqn. (7) can be rewritten as follows:
{
b1
, p2=
b2
,
b1x1(e−η1 − (xf11)θ1 −c12f1x2 − p1E1) = 0,
b2x2(e−η2− (xf22)θ2−c21f2x1 − p2E2) = 0.
(8)
By solving Eqn. (8), three nonnegative boundary equilibria and a unique interior equilib-
rium are as follows:
(i) M0(0, 0), which biologically implies that all competing species die out.
(ii) M1(˜1, 0), which biologically implies that x1species survives and x2species dies out. ˜1
takes the following form based on Eqn. (8),
˜1= f1(e−η1 − p1E1)1
θ1 , or ˜1=
f2(e−η2 − p2E2)
c21
,
which derives that M1(˜1, 0) exists provided that
e−η1
0 < E1< , or 0 < E2<
p1
e−η2
p2
.
(iii) M2(0, ˆ2), which biologically implies that x2species survives and x1species dies out. ˆ2
takes the following form based on Eqn. (8),
1
ˆ2= f2(e−η2− p2E2) θ2 , or ˆ2=
which derives that M2(0, ˆ2) exists provided that
e−η2
f1(e−η1− p1E1)
c12
e−η1
,
0 < E2<
p2
, or 0 < E1<
- 7 -
p1
.
(iv) A unique interior equilibrium M ∗(x∗1, x∗2) biologically implies that all competing species
survive and M ∗(x∗1, x∗2) exists provided that
or
{
{
1
f1c21(e−η1 − p1E1) θ1 > f2(e−η2 − p2E2),
1
f2c12(e−η2− p2E2) θ2> f1(e−η1− p1E1).
1
f1c21(e−η1− p1E1) θ1 < f2(e−η2− p2E2),
1
f2c12(e−η2 − p2E2) θ2 < f1(e−η1 − p1E1).
2 Local Stability Analysis
In this section, by analyzing associated characteristic equation, local stability of model (4)
around nonnegative boundary equilibrium and interior equilibrium is discussed, respectively.
Theorem 3. IfE1>p11andE2>p12hold, then model (4) is locally stable aroundM0.
Proof.It follows from model (4) that the characteristic equation evaluated around M0(0, 0) is
as follows,
(λ − b1e−(d1+λ)τ1+ q1E1)(λ − b2e−(d2+λ)τ2+ q2E2) = 0.
(11)
Let G1(λ) = λ and G2(λ) = b1e−(d1+λ)τ1 − q1E1, it easy to show that curve of G1(λ) and
G2(λ) must intersect at some negative value provided that E1>p11.
Similarly, let G3(λ) = b2e−(d2+λ)τ2 − q2E2, it easy to show that curve of G1(λ) and G3(λ)
must intersect at some negative value provided that E2>1
p2.
It follows from the above analysis that all eigenvalues of Eqn. (11) satisﬁes that Reλ1< 0
and Reλ2< 0. Hence, according to Routh-Hurwitz criteria [4], model (4) is locally stable
around M0provided that E1>p11and E2>p12.
Theorem 4. If 0 < E1<e−p1η1 −p11(ba2e21−fη12)θ1 holds, then model (4) is locally stable around
1
M1(˜1, 0)i.e. (f1(e−η1− p1E1) θ1 , 0).
Proof.It follows from model (4) that the characteristic equation evaluated around M1(˜1, 0) is
as follows,
(λ − b1e−(d1+λ)τ1+ a11˜θ11 (1 + θ1) + q1E1)(λ − b2e−(d2+λ)τ2+ a21˜1+ q2E2) = 0.
Let G4(λ) = λ − b1e−(d1+λ)τ1+ a11˜θ11 (1 + θ1) + q1E1. By substituting λ1= u1+ iv1into
G4(λ) = 0, where u1and v1are real number, it follows from Eqn. (7) that
G4(λ1) = u1+ iv1− b1e−(d1+u1)τ1e−iv1τ1+b1e−d1τ1+a11θ1˜θ11= 0.
According to (13), it can be obtained that ReG4(λ1) = 0.
In the following part, we will show that u1< 0. If u1≥ 0, then
ReG4(λ1) ≥ b1(1 − e−(d1+u1)τ1cos(v1τ1)) + θ1(b1e−d1τ1− q1E1) > 0,
which is a contradiction. Hence, it can be concluded that u1< 0.
Furthermore, consider the equation
λ − b2e−(d2+λ)τ2+ a21˜1+ q2E2= 0.
By substituting λ2= u2+ iv2into Eqn. (14), where u2and v2are real number, it follows
from Eqn. (7) that
|u2+ a21˜1+ q2E2+ iv2| = b2e−η2 |e−λ——2τ2 |,
If u2≥ 0, then we have,
(u2+ a21˜1+ q2E2)2+ v22 ≤ (b2e−η2)2,
and
a21˜1+ q2E2≤ b2e−η2.
It follows from the above analysis and Eqn. (8) that
−
E1≥
1 b
[e−η1 − (
2e
η1
)θ1],
p1
a21f1
which is a contradiction to the assumption of Theorem 4. Hence, it can be concluded that
u2< 0.
Based on the above analysis, all eigenvalues of Eqn.(12) satisﬁes that Reλ1< 0, Reλ2< 0.
It follows from Routh-Hurwitz criteria [4] that model (4) is locally stable around M1(f1(e−η1 −
1
−η
1
−
η2
p1E1) θ1 , 0) provided that 0 < E1<ep1−p11(b2e
θ
a21f1)1.
Theorem 5. If 0 < E2<e−p2η2 −p12(ba1e12−fη21)θ2 holds, then model (4) is locally stable around
1
M2(0, ˆ2)i.e. (0, f2(e−η2− p2E2) θ2 ).
Proof.It follows from model (4) that the characteristic equation evaluated around M2(0, ˆ2) is
as follows,
(λ − b1e−(d1+λ)τ1+ a12ˆ2+ q1E1)(λ − b2e−(d2+λ)τ2+ a22ˆθ22 (1 + θ2) + q2E2) = 0.
(15)
Let G5(λ) = λ − b2e−(d2+λ)τ2+ a22ˆθ22 (1 + θ2) + q2E2. By substituting λ3= u3+ iv3into
G5(λ) = 0, where u3and v3are real number, it follows from Eqn. (7) that
G5(λ3) = u3+ iv3− b2e−(d2+u2)τ2e−iv2τ2+b2e−d2τ2+a22θ2ˆθ22= 0.
According to (16), it can be obtained that ReG5(λ3) = 0.
- 9 -
(16)
In the following part, we will show that u3< 0. If u3≥ 0, then
ReG5(λ3) ≥ b2(1 − e−(d2+u3)τ2cos(v3τ2)) + θ2(b2e−d2τ2− q2E2) > 0,
which is a contradiction. Hence, it can be concluded that u3< 0.
Furthermore, consider the equation
λ − b1e−(d1+λ)τ1+ a12ˆ2+ q1E1= 0.
By substituting λ4= u4+ iv4into Eqn. (17), where u4and v4are real number, it follows
from Eqn. (7) that
|u4+ a12ˆ2+ q1E1+ iv4| = b1e−η1 |e−λ4τ1 |,
If u4≥ 0, then we have,
(u4+ a12ˆ2+ q1E1)2+ v42 ≤ (b1e−η1)2,
and
a12ˆ2+ q1E1≤ b1e−η1.
It follows from the above analysis and Eqn. (8) that
−
E2≥
1 b
[e−η2 − (
1e
η1
)θ2],
p2
a12f2
which is a contradiction to the assumption of Theorem 5. Hence, it can be concluded that
u4< 0.
Based on the above analysis, all eigenvalues of Eqn.(15) satisﬁes that Reλ3< 0, Reλ4< 0.
It follows from Routh-Hurwitz criteria [4] that model (4) is locally stable around M2(0, f2(e−η2 −
1
−η
2
−
η1
p2E2) θ2 ) provided that 0 < E2<ep2−p12(b1e
θ
a12f2)2.
It follows from Theorem 3 that both competing species x1and x2will face up with ex-
tinction when harvest eﬀort E1and E2crosses critical valuep11andp12, respectively. If harvest
eﬀort E1is constrained within certain range (0,e−pη1
− 1
b
−
η2
p1(2e
θ
1
a21f1)1), extinction of species x1can
be avoided while interspeciﬁc competition has negative eﬀect on survival of competing species
x2, which can be found in Theorem 4.
Similarly, extinction of species x2can be avoided while interspeciﬁc competition has neg-
ative eﬀect on survival of competing species x1provided that harvest eﬀort E2is constrained
within (0,e−η2
−
η1
p2 −p12(b1e
θ
a12f2)2), which can be found in Theorem 5.
Theorem 6. Model (4) is locally stable around the interior equilibriumM ∗(x∗1, x∗2)provided
that inequality (10) holds.
Proof.The characteristic equation of model (4) around the interior equilibrium M ∗(x∗1, x∗2) is
as follows:
[λ − b1e−(λ+d1)τ1+ a11(1 + θ1)x∗1θ1+a12x∗2+ q1E1]
×[λ − b2e−(λ+d2)τ2+ a22(1 + θ2)x∗2θ2+a21x∗1+ q2E2] − a12a21x∗1x∗2= 0.
(18)
By virtue of Eqn. (8), Eqn. (18) can be rewritten as follows,
[λ + b1e−η1(1 − e−λτ1) + a11θ1x∗1θ1][λ + b2e−η2(1 − e−λτ2) + a22θ2x∗2θ2]− a12a21x∗1x∗2= 0. (19)
Let λ5= u5+ iv5be an arbitrary solution of Eqn. (19), where u5and v5are real number.
By substituting λ5= u5+ iv5into Eqn. (19), it gives that,
(A1+ iB1)(A2+ iB2) − a12a21x∗1x∗2= 0,
where Akand Bk(k = 1, 2) are deﬁned as follows:
A1= u5− b1e−(d1+u5)τ1cos(v5τ1) + a11(1 + θ1)x∗1θ1+ a12x∗2+ q1E1,
B1= v5+ b1e−(d1+u5)τ1sin(v5τ1),
A2= u5− b2e−(d2+u5)τ2 cos(v5τ2) + a22(1 + θ2)x∗2θ2+ a21x∗1+ q2E2,
B1= v5+ b2e−(d2+u5)τ2sin(v5τ2).
Further computations show that
A1A2− B1B2= a12a21x∗1x∗2, A1B2+ A2B1= 0,
which derives that
A1A2≤ a12a21x∗1x∗2.
We claim that u5< 0. On the contrary, based on Eqn. (8), it is easy to show that
A1≥ a11θ1x∗1θ1+b1e−η1 − b1e−η1−u5τ1 cos(v5τ1)
≥ a11θ1x∗1θ1+b1e−η1− b1e−η1> 0,
and
A2≥ a22θ2x∗2θ2+b2e−η2− b2e−η2−u5τ2cos(v5τ2)
≥ a22θ2x∗2θ2+b2e−η2− b2e−η2> 0.
Hence, we have
A1A2≥ a11a22θ1θ2x∗1θ1x∗2θ2.
Based on (22) and (23), it can be derived that
a12a21(x∗1)1−θ1(x∗2)1−θ2≥ a11a22θ1θ2.
On the other hand, denote Li, i = 1, 2 as the curve about x1and x2in R2+, respectively.
{
−d
L1: x2=b1e
−
1τ1−q1E1
a12
− |
a11 a12xθ11, |
L2: x1=b2ed2aτ212−q2E2−aa2221xθ22.
By simple computation, the slope kLi of curve Li, i = 1, 2 can be obtained as follows:
kL1=dx2|L1=−a11
θ1(x1)θ1−1,kL2=
dx2
|L2=−
1
.
dx1
dx1
a22
θ
a12
a21θ2(x2)2−1
If inequality (10) holds, then it is easy to show that kL1< kL2< 0 around the interior
equilibrium M ∗(x∗1, x∗2), which derives that
a11a22θ1θ2(x∗1)θ1−1(x∗2)θ2−1> a12a21,
which contradicts to (24). Consequently, it can be concluded that u5< 0.
Since λ5= u5+ iv5is an arbitrary solution of Eqn. (19) and Reλ5< 0, it follows from
Routh-Hurwitz criteria [4] that model (4) is locally stable around M ∗(x∗1, x∗2) provided that
inequality (10) holds.
It is easy to show that model (4) is locally unstable around the nonnegative boundary
equilibrium M1(˜1, 0) and M2(0, ˆ2) provided that inequality (10) holds. Consequently, local
stability around M1(˜1, 0) and M2(0, ˆ2) can not be preserved while the interior equilibrium
M ∗ is locally stable.
3 Global Stability Analysis
In this section, global stability of the nonnegative boundary equilibrium M1and M2is
investigated based on an iterative technique, respectively. By constructing an appropriate
Lyapunov functional, global stability of the unique interior equilibrium is also discussed.
Lemma 1. [27] Considering the following diﬀerential equation,
z˙(t) = αz(t − τ1) − βz1+θ(t) − γz(t).
Ifα > γ ≥ 0, thenlimt→∞ z(t) = ( α−β γ)1θ.
Theorem 7. If the following equalities hold
0 < E1<
e−η1
p1
−1(
p1
b2e−η2
a21f1
)θ1, 0 < E2<
e−η2
p2
−1(
p2
b1e−η1
a12f2
)θ2,
(26)
thenM1(˜1, 0) is globally asymptotically stable, wherefi, ηi, pi,i = 1, 2have been deﬁned in
(8).
- 12 -
Proof. According to (26), it follows from Theorem 4 that model (4) is locally stable around
M1, and the global stability of M1will be proved in the following part.
Let
I1= lim inf x1(t), J1= lim sup x1(t),
t→∞
t→∞
I2= lim inf x2(t), J2= lim sup x2(t).
t→∞
t→∞
It follows from Theorem 1 and the ﬁrst equation of model (4) that
x˙1(t) ≤ b1e−d1τ1x1(t − τ1) − a11x1+1 θ1(t) − q1E1x1(t).
By virtue of (26), it is easy to show that b1e−d1τ1 > q1E1, and it follows from Lemma 4.1
that there exists T1> 0 and when t > T1,
−
b1ed1τ1− q1E11
x1(t) ≤ (
holds for suﬃciently small ϵ > 0.
a11
) θ1 + ϵ := V1x1
(27)
It follows from Theorem 1 and the second equation of model (4) that
x˙2(t) ≤ b2e−d2τ2x2(t − τ2) − a22x1+2 θ2(t) − q2E2x2(t).
By virtue of (26), it is easy to show that b2e−d2τ2 > q2E2, and it follows from Lemma 4.1
that there exists T2> T1and when t > T2,
−
b2ed2τ2− q2E21
x2(t) ≤ (
holds for suﬃciently small ϵ > 0.
a22
) θ2+ ϵ := V1x2
(28)
Based on (28) and the ﬁrst equation of model (4), it can be obtained that
x˙1(t) ≥ b1e−d1τ1x1(t − τ1) − a11x1+1 θ1(t) − (a12V1x2+q1E1)x1(t).
By virtue of (26), it is easy to show that b1e−d1τ1 > a12V1x2+ q1E1, and it follows from
Lemma 4.1 that there exists T3> T2and when t > T3,
−
b1ed1τ1− a12V1x2 − q1E11
x1(t) ≥ (
holds for suﬃciently small ϵ > 0.
a11
) θ1− ϵ := U1x1
(29)
Based on (27) and the second equation of model (4), it can be obtained that
x˙2(t) ≥ b2e−d2τ2x2(t − τ2) − a22x1+2 θ2(t) − (a21V1x1+q2E2)x2(t).
By virtue of (26), it is easy to show that b2e−d2τ2 > a21V1x1+ q2E2, and it follows from
Lemma 4.1 that there exists T4> T3and when t > T4,
−
b2ed2τ2 − a21V1x1 − q2E21
x2(t) ≥ (
a22
- 13 -
) θ2− ϵ := U1x2
(30)
holds for suﬃciently small ϵ > 0.
According to (30) and the ﬁrst equation of model (4), it derives that
x˙1(t) ≤ b1e−d1τ1x1(t − τ1) − a11x1+1 θ1(t) − (a12U1x2+ q1E1)x1(t).
By virtue of (26), it is easy to show that b1e−d1τ1 > a12U1x2+q1E1, and it follows from
Lemma 4.1 that there exists T5> T4and when t > T5,
−
b1ed1τ1 − a12U1x2 − q1E11
x1(t) ≤ (
holds for suﬃciently small ϵ > 0.
a11
) θ1 + ϵ := V2x1
(31)
It follows from (29) and the second equation of model (4) that
x˙2(t) ≤ b2e−d2τ2x2(t − τ2) − a22x1+2 θ2(t) − (a21U1x1+ q2E2)x2(t).
By virtue of (26), it is easy to show that b2e−d2τ2 > a21U1x1+q2E2, and it follows from
Lemma 4.1 that there exists T6> T5and when t > T6,
−
b2ed2τ2 − a21U1x1− q2E21
x2(t) ≤ (
holds for suﬃciently small ϵ > 0.
a22
) θ2 + ϵ := V2x2
(32)
Based on (32) and the ﬁrst equation of model (4), it can be obtained that
x˙1(t) ≥ b1e−d1τ1x1(t − τ1) − a11x1+1 θ1(t) − (a12V2x2+ q1E1)x1(t).
By virtue of (26), it is easy to show that b1e−d1τ1 > a12V2x2+q1E1, and it follows from
Lemma 4.1 that there exists T7> T6and when t > T7,
−
b1ed1τ1 − a12V1x2− q1E11
x1(t) ≥ (
holds for suﬃciently small ϵ > 0.
a11
) θ1 − ϵ := U2x1
(33)
Based on (31) and the second equation of model (4), it can be obtained that
x˙2(t) ≥ b2e−d2τ2 x2(t − τ2) − a22x1+2 θ2(t) − (a21V2x1+ q2E2)x2(t).
By virtue of (26), it is easy to show that b2e−d2τ2 > a21V2x1+ q2E2, and it follows from
Lemma 4.1 that there exists T8> T7and when t > T8,
−
b2ed2τ2 − a21V2x1− q2E21
x2(t) ≥ (
holds for suﬃciently small ϵ > 0.
a22
) θ2 − ϵ := U2x2
(34)
Continuing the above processes, it follows from (27)-(34) that four sequences {Vnx1}, {Vnx2},
{Unx1} and {Unx2 }, n = 1, 2, · · · can be obtained, which take the following form for n ≥ 2:
−d1τ1 −a12Unx−21−q1E1
Vnx1 = (b1e
−
a11
1
) θ1+ ϵ,
Vnx2= (b2ed2τ2 −a21Unx−11−q2E2
1
) θ2 + ϵ,
−d
a22
(35)
Unx1= (b1e
1τ1−a12Vnx2−q1E1
a11
1
) θ1 − ϵ,
−d
Unx2 = (b2e
2τ2−a21Vnx1−q2E2
a22
- 14 -
1
) θ2 − ϵ.
By virtue of (35), it can be derived that
b
−d1τ1− a12Vnx−21 − q1E1
a22(Vnx2)θ2=b2e−d2τ2 − q2E2− a21(1e
which follows that
a11
1
) θ1 ,
(36)
(
b2e−d2τ2 − q2E2− a22(Vnx2 )θ2
a21
)θ1 =
b1e−d1τ1− a12Vnx−21 − q1E1
a11
.
(37)
Since Vnx2≥ I2and {Vnx2} is monotonically decreasing based on mathematical induction,
it can be obtained that limn→∞ Vnx2 = h ≥ 0 exists.
By limiting (37) with n → ∞, it gives that
(C1− C2hθ2)θ1= C3− C4h,
where C1=b2e−d2aτ212−q2E2, C2=aa2221,C3=b1e−d1aτ111−q1E1, C4=aa1211.
According to (26) and (38), it can be computed that
−
b2ed2τ2− q2E2
(38)
C1− C2hθ2> C1− C2
a22
= 0,
which derives that C3− C4h > 0 and 0 ≤ h < h2where h2=b1e−d1aτ1−q1E1.
12
In order to show this theorem, we only need to discuss that h = 0 and the following two
cases will be considered.
Case I.θ1≥ 1, θ2≥ 1.
Firstly, denote Lk, k = 3, 4 as the curve about yj, j = 1, 2 and h in R2+, respectively.
{
L3: y1(h) = (C1− C2hθ2 )θ1,
L4: y2(h) = C3− C4h.
By simple computation, it derives that
dddy2hy11= −θ1θ2C2hθ2−−1(C1− C2hθ2)θ1−1<0,
dh2=−θ1θ2C2hθ22(C1− C2hθ2)θ1−2[(θ2− 1)(C1− C2hθ2) − (θ1− 1)θ2C2hθ2],
dy2
dh=−C4< 0,
d2y2
(39)
(40)
dh2= 0.
It follows from (40) that
d2y1
dh2=
<0, 0 ≤ h < h1,
0, h = h1,
>0, h1< h ≤ h2.
- 15 -
(41)
where h1=
C1(θ2−1)
C2[(θ2−1)+θ2(θ1−1)].
Let y(h) = y1(h) − y2(h), y1(h) and y2(h) have been deﬁned in (39). Next, four steps are
provided to show that limt→∞ x2(t) = 0.
(i) there is at most one intersection forL3andL4whenh ∈ [0, h1].
If this claim is false, then there are at least two intersection, h11and h12s.t., y(h11) = 0
and y(h12) = 0. It follows from (41) that y′′(h) < 0 when h ∈ [0, h1], which derives that
y′(h) is monotonically decreasing. Furthermore, we claim that y′(h11) ≤ 0. Otherwise, if
y′(h11) > 0, then y′(h) > 0 when h ∈ [0, h11]. It follows from y(h11) = 0 that y(0) < 0.
According to deﬁnition of y(h) and Eqn. (7), it can be obtained that y(0) = 0, which is
a contradiction.
By using similar arguments, we have y′(h12) ≤ 0, which follows that y′(h) < 0 when
h ∈ (h11, h12). On the other hand, there exists ξ1∈ (h11, h12) s.t. y′ (ξ1) = 0 based on
y(h12) = y(h12) = 0, which is a contradiction.
Hence, L3and L4have at most one intersection when h ∈ [0, h1].
(ii) there is at most one intersection for L3and L4when h ∈ [h1, h2].By using the
similar arguments in step (i), it can be shown that L3and L4have at most one intersection
when h ∈ [h1, h2].
(iii) there is no intersection forL3andL4whenh ∈ [h1, h2].
Suppose there is one intersection h21∈ (h1, h2) for L3and L4. Based on the arguments
in step (i) and (ii), we have y′(h21) ≤ 0. It follows from y′(h11) ≤ 0 and y′(h21) ≤ 0
that y′(h) ≤ 0 when h ∈ (h11, h21). On the other hand, there exists ξ2∈ (h11, h21) s.t.,
y′(ξ2) = 0, which is a contradiction.
Consequently, there is no intersection for L3and L4when h ∈ [h1, h2].
(iv) there is no intersection forL3andL4whenh ∈ (0, h1).
Otherwise, if there is one intersection h0∈ (0, h1). Since there exists no intersection for
L3and L4when h ∈ [h1, h2]. It is easy to see that h0∈ (0, h2) and the solution of Eqn.
(38) is either 0 or h0> 0.
In the following part, we will show that h̸= h0. According to the arguments for ﬁnding
sequences Unx1 and Vnx2 utilized in the above steps, we can ﬁnd two new sequences U˜nx1
and ˜nx2, such that x1≥ U˜nx1and x2≤ V˜nx2, where
{
b
−d2τ2 −a21U˜nx−11−q2E2
1
˜ x2
n= (
2e
a22
) θ2 + ϵ,
(42)
U˜nx1= (b1e−d1τ1−aa12V˜nx2−q1E1
1
) θ1 − ϵ.
11
- 16 -
By virtue of (42), it can be concluded that
(C1− C2( ˜nx2)θ2)θ1= C3− C4V˜nx−21,
where Ci(i = 1, 2, 3, 4) have been deﬁned in (38).
(43)
Similarly, it is also easy to show that limn→∞ ˜nx2= ˜h ≥ 0 exists, which follows that
(C1− C2˜θ2)θ1= C3− C4˜h.
(44)
By using the similar arguments in step (i)-(iv), we can also conclude that ˜ = 0 or ˜ = h0.
Since ˜nx2is monotonically decreasing, V˜nx2< V˜1x2=h0hold for all n ≥ 2. It follows from
the above analysis that ˜h < h0. Consequently, it can be concluded that ˜= 0.
Based on step (i)-(iv), there must be h = 0, i.e. limt→∞ x2(t) = 0. It follows from (38) that
limn→∞ Unx1 = f1(e−η1 − p1E1) θ11 = ˜x1. Since Unx1 ≤ I1≤ J1≤ ˜1. Hence, limt→∞ x1(t) = ˜x1.
Case II.θ1< 1, θ2< 1.
By using similar proof with Case I, it is easy to show limt→∞ x1(t) = ˜x1and limt→∞ x2(t) =
0.
This completes the proof of Theorem 7.
By using the similar proof, global stability analysis of the boundary equilibrium M2(0, ˆ2)
can be concluded as follows.
Theorem 8. If the following equalities hold
0 < E1<
e−η1
p1
−1(
p1
b2e−η2
a21f1
)θ1, 0 < E2<
e−η2
p2
−1(
p2
b1e−η1
a12f2
)θ2,
(45)
thenM2(0, ˆ2) is globally asymptotically stable, wherefi, ηi, pi,i = 1, 2have been deﬁned in
(8).
Theorem 9. Interior equilibrium M ∗(x∗1, x∗2) is globally stable provided that inequality (10)
holds.
Proof. Firstly, let N(z) = z − ln z − 1 for all z > 0. By simple computation, it can be obtained
that N(z) ≥ 0 for all z ≥ 0 and N(z) = 0 if and only if z = 1. Nextly, we deﬁne the Lyapunov
functional
W (t) = W1(t) + W2(t),
where W1(t) and W2(t) are deﬁned as follows:
∗
W1(t) = x∗1N (xx1(∗t)) +a12x
2
x2(t)
),
a21N (x∗
1
∫
0
2
−d
∫0
W2(t) = b1e−d1τ1x∗1
x1(t+ζ))dζ +a12b2e
2τ2x∗
x2(t+ζ)
−τ1N(
x∗
a21
2
−τ2N(
x∗
)dζ.
1
2
According to deﬁnition of W (t), it is easy to show that W (t) ≥ 0 for all t ≥ 0.
- 17 -
that
By calculating the derivative of W1(t) and W2(t) along the solution of model (4), it gives
W˙1(t) = x∗1(1−1
)[b1e−d1τ1x1(t − τ1) − a11x(1+1 θ1)(t) − a12x1(t)x2(t) − q1E1x1(t)]
+
x |
∗ 1 |
a12x∗2
a21
∗
(
x1(t)
1 −1
x∗2x2(t)
)[b2e−d2τ2x2(t − τ2) − a22x(1+2 θ2)(t)]
and
−a12x
2
a21
(
1 −1
x∗2x2(t)
∗
)[a21x1(t)x2(t) + q2E2x2(t)],
W˙2(t) = (1 −x1
x1(t)
)[b1e−d1τ1 x1(t) − a11x(1+1 θ1)(t) − a12x1(t)x2(t) − q1E1x1(t)]
x∗
+b1e−d1τ1(1 −
1
x1(t)
)[x1(t − τ1) − x1(t)]
x
+b1e−d1τ1[x1(t) − x1(t − τ1) + x∗1ln(
1(t − τ1)
x1(t)
)]
+
a12
a21
(1 −
x∗
2
x2(t)
)[b2e−d2τ2x2(t) − a22x(1+2 θ2)(t) − a21x1(t)x2(t) − q2E2x2(t)]
+
a12b2e−d2τ2
a21
a12b2e−d2τ2
(1 −
x∗
2
x2(t)
)(x2(t − τ2) − x2(t))
x
+
a21
[x2(t) − x2(t − τ2) + x∗2ln(
2(t − τ2)
x2(t)
)]
= (x1(t) − x∗1)[b1e−d1τ1− a11xθ11(t) − a12x2(t) − q1E1]
a12
+
a21
(x2(t) − x∗2)[b2e−d2τ2− a22xθ22 (t) − a21x1(t) − q2E2]
−b1e−d1τ1x∗1[x1(t − τ1) − lnx1(t − τ1) − 1]
x1(t) x1(t)
−d2τ2x∗2x2(t − τ2)
−a12b2e
a21
[
x2(t)
− lnx2(t − τ2) − 1].
x2(t)
- 18 -
Hence, it can be obtained that
W˙ (t) = (x1(t) − x∗1)[a11(x∗1)θ1+a12x∗2− a11xθ11(t) − a12x2(t)]
a12
+
a21
(x2(t) − x∗2)[a22(x∗2)θ2+ a21x∗1− a22xθ22(t) − a21x1(t)]
−
−b1e−d1τ1x∗1N (x1(t − τ1)
) −
a12b2ed2τ2x∗2
N(
x2(t − τ2)
)
x1(t)
= a11(x1(t) − x∗1)[(x∗1)θ1 − xθ11(t)] +
a21
a
12a22
a21
−
x2(t)
(x2(t) − x∗2)[(x∗2)θ2 − xθ22(t)]
−b1e−d1τ1x∗1N (x1(t − τ1)
) −
a12b2ed2τ2x∗2
N(
x2(t − τ2)
)
a12a21
x1(t)
a21
x2(t)
+(a12−
a21
)(x1(t) − x∗1)(x2(t) − x∗2)
a
= a11(x1(t) − x∗1)[(x∗1)θ1− xθ11(t)] +
12a22
a21
(x2(t) − x∗2)[(x∗2)θ2− xθ22 (t)]
−b1e−d1τ1x∗1N (x1(t − τ1)
) −
−d2τ2x∗2
a12b2e
N(
x2(t − τ2)
).
x1(t)
a21
x2(t)
By straightforward computation, it is easy to show that
x
for i = 1, 2.
[xi(t) − x∗i][(x∗i)θi− xθii(t)] ≤ 0, N (
i(t − τi)
xi(t)
) ≥ 0,
Consequently, it can be concluded that W˙ (t) ≤ 0, which follows that W (t) is bounded,
non-increasing and limt→∞ W (t) exists.
By using the similar arguments in Theorem 2 in [29], it is easy to show that
lim x1(t) = x∗1, lim x2(t) = x∗2.
t→∞
4 Numerical Simulation
t→∞
Numerical simulations are carried out to show consistency with the global stability analysis
discussed in the case of θ1≥ 1, θ2≥ 1 and θ1< 1, θ2< 1.
Numerical Simulation I.Parameter values are partially taken from [27] and [4], which are
given as follows: b1= 2, d1= 0.5, τ1= 2.8, a11= 1.5, a12= 1, q1= 1, E1= 0.2, b2= 2.5,
d2= 0.6, τ2= 3, a22= 1.5, a21= 1, q2= 1, E2= 0.1. By simple computation, it can be
−η
−η
obtained that 0 < E1<e−η1 − 1 ( b2e
2)θ1and 0 < E2<e−η2 − 1 ( b1e
1)θ2with θ1= 2
p1
p1
a21f1
p2
p2
a12f2
and θ2= 1.9. It follows from Theorem 7 that M1(0.4422, 0) is globally stable, and dynamical
responses of model (4) with θ1= 2 and θ2= 1.9 are plotted in Figure 1. Furthermore, it
can be computed that 0 < E1<e−pη1
− 1
b
−
η2
−η2
−
η1
p1(2e
θ
a21f1)1and 0 < E2<ep2−p12(b1e
θ
1
a12f2)2with
θ1= 0.45 and θ2= 0.1263. It follows from Theorem 7 that M1(0.0265, 0) is globally stable,
and dynamical responses of model (4) with θ1= 0.45 and θ2= 0.1263 are plotted in Figure 2.
- 19 -
0.7
0.6
0.5
0.4
0
0.2
0.1
0
−0.1
0
200
200
400
400
Time
600
600
800
800
1000
1000
图 1: Dynamical responses of the nonnegative boundary equilibrium M1(0.4422, 0) of model
(4) with θ1= 2 and θ2= 1.9, which indicates that M1(0.4422, 0) is globally stable.
Numerical Simulation II.Parameter values are partially taken from [27] and [4], which
are given as follows: b1= 2, d1= 0.5, τ1= 3, a11= 1.5, a12= 1, q1= 1, E1= 0.15, b2= 2.5,
d2= 0.6, τ2= 1, a22= 1.5, a21= 1, q2= 1, E2= 0.08. By simple computation, it can be
obtained that 0 < E1<e−pη11 −p11(ba221e−fη12)θ1 and 0 < E2<e−pη22 −p12(ba112e−fη21)θ2 with θ1= 2
and θ2= 1.9. It follows from Theorem 8 that M1(0, 0.9243) is globally stable, and dynamical
responses of model (4) with θ1= 2 and θ2= 1.9 are plotted in Figure 3. Furthermore, it can be
computed that 0 < E1<e−pη1
− 1
b
−η2
θ
−η
2
−
η1
p1(2e
a21f1)1and 0 < E2<ep2−p12(b1e
θ
1
a12f2)2with θ1= 0.5 and
θ2= 0.5263. It follows from Theorem 8 that M1(0, 0.7531) is globally stable, and dynamical
responses of model (4) with θ1= 0.453 and θ2= 0.5263 are plotted in Figure 4.
- 20 -
0.4
0.3
0.2
0.1
0
0
0.8
0.6
0.4
0.2
0
0
200
200
400
400
Time
600
600
800
800
1000
1000
图 2: Dynamical responses of the nonnegative boundary equilibrium M1(0.0265, 0) of model
(4) with θ1= 0.45 and θ2= 0.1263, which indicates that M1(0.0265, 0) is globally stable.
0.2
0.1
0
−0.1
0
1
0.8
0.6
0.4
0.2
0
200
200
400
400
Time
600
600
800
800
1000
1000
图 3: Dynamical responses of the nonnegative boundary equilibrium M2(0, 0.9243) of model
(4) with θ1= 2 and θ2= 1.9, which indicates that M2(0, 0.9243) is globally stable.
- 21 -
0.2
0.1
0
−0.1
0
0.8
0.7
0.6
0.5
0.4
0
200
200
400
400
Time
600
600
800
800
1000
1000
图 4: Dynamical responses of the nonnegative boundary equilibrium M2(0, 0.7531) of model
(4) with θ1= 0.5 and θ2= 0.5263, which indicates that M2(0, 0.7531) is globally stable.
Numerical Simulation III.Parameter values are partially taken from [27] and [4], which
are given as follows: b1= 2, d1= 0.5, τ1= 0.75, a11= 1.5, a12= 1, q1= 1, E1= 0.1, b2= 2.5,
d2= 0.6, τ2= 0.95, a22= 1.5, a21= 1, q2= 1, E2= 0.05. By simple computation, it can be
obtained that
1
1
f1c21(e−η1 − p1E1) θ1< f2(e−η2 − p2E2), f2c12(e−η2 − p2E2) θ2< f1(e−η1 − p1E1)
with θ1= 2 and θ2= 1.9, which implies that inequality (10) holds. It follows from Theorem 9
that M ∗(0.6247, 0.6885) is globally stable, and dynamical responses of model (4) with θ1= 2
and θ2= 1.9 are plotted in Figure 5.
Furthermore, it can be computed that
1
1
f1c21(e−η1− p1E1) θ1 < f2(e−η2− p2E2), f2c12(e−η2− p2E2) θ2 < f1(e−η1− p1E1)
with θ1= 0.23 and θ2= 0.14, which implies that inequality (10) holds. It follows from
Theorem 9 that M ∗(0.4433, 0.0308) is globally stable, and dynamical responses of model (4)
with θ1= 0.23 and θ2= 0.14 are plotted in Figure 6.
5 Conclusion
In this paper, the combined dynamic eﬀect of maturation delay and harvest eﬀort on
two-species competition ecosystem is investigated. Generally, positivity and boundedness of
- 22 -
0.65
0.6
0.55
0
0.8
0.7
0.6
0.5
0.4
0
200
200
400
400
Time
600
600
800
800
1000
1000
图 5: Dynamical responses of the unique interior equilibrium M ∗(0.6247, 0.6885) of model (4)
with θ1= 2 and θ2= 1.9, which indicates that M ∗(0.6247, 0.6885) is globally stable.
0.5
0.4
0.3
0
0.25
0.2
0.15
0.1
0.05
0
0
200
200
400
400
Time
600
600
800
800
1000
1000
图 6: Dynamical responses of the unique interior equilibrium M ∗(0.4433, 0.0308) of model (4)
with θ1= 0.23 and θ2= 0.14, which indicates that M ∗(0.4433, 0.0308) is globally stable.
- 23 -
solution biologically interpret sustainable survival of two competing species, which is studied in
Theorem 1 and Theorem 2, respectively. Some suﬃcient conditions are provided to show the
existence of three nonnegative boundary equilibria and a unique interior equilibrium. Further
attempts are carried out to discuss local stability analysis around nonnegative boundary equi-
librium and interior equilibrium. It reveals that two competing species will come to extinction
when harvest eﬀort crosses certain critical value, respectively. It is shown that increase of
the maturation delay of one species has negative eﬀect on its permanence and a suﬃciently
large maturation delay will directly lead to its extinction. Further discussions show that har-
vest eﬀort on one species may change the surviving or extinction behavior of the harvested
species and its competitor. If harvest eﬀort on competing species is constrained within certain
range, extinction of the corresponding species can be avoided while interspeciﬁc competition
has negative eﬀect on survival of its competing species. It should be noted that local stability
of boundary equilibrium can not be preserved while interior equilibrium is locally stable, which
can be found in Theorem 6. By using an iterative technique, global stability of the nonneg-
ative boundary equilibrium is investigated in the case of θ1≥ 1, θ2≥ 1 and θ1< 1, θ2< 1,
respectively. Global stability of the unique interior equilibrium is also discussed by the means
of constructing an appropriate Lyapunov functional, which shows that two competing species
coexist and sustainable development of competing population ecosystem can be achieved when
maturation delay and harvest eﬀort are constrained within certain range.
Since many species within competition ecosystem are of agricultural and medical utiliza-
tion, they are mostly harvested and sold with the purpose of obtaining the economic interest
[4]. Although there are much progress on Gilpin-Ayala competition model, such model are
discussed in the sense that previously related work ignore commercial harvesting, which can
not vividly reﬂect complex biological phenomena from harvested competition ecosystem. Since
harvesting has a strong impact on the dynamic evolution of a population, it is necessary to
investigate the dynamic eﬀect of harvesting on population dynamics. Based on above analysis,
work done in [27] is extended by incorporating harvest eﬀort on two competing species, and
combined dynamic eﬀect of harvest eﬀort and maturation delays on population dynamics are
discussed. It should be noted that global stability analysis of nonnegative boundary and interior
equilibrium are relevant to investigation of coexistence and interaction mechanism of compet-
ing species. These theoretical ﬁndings are of inspiration for administrative agency to regulate
commercial harvesting within appropriate limitations, especially for species with relatively long
maturation. It makes work done in this paper has some positive and new feature.
- 24 -
Acknowledgement
This work is supported by National Natural Science Foundation of China, grant No.
61104003, grant No. 61273008 and grant No. 61104093. Research Foundation for Doctor-
al Program of Higher Education of Education Ministry, grant No. 20110042120016. Hebei
Province Natural Science Foundation, grant No. F2011501023. Fundamental Research Fund-
s for the Central Universities, grant No. N120423009. Research Foundation for Science and
Technology Pillar Program of Northeastern University at Qinhuangdao, grant No. XNK201301.
This work is supported by State Key Laboratory of Integrated Automation of Process
Industry, Northeastern University, supported by Hong Kong Admission Scheme for Mainland
Talents and Professionals, Hong Kong Special Administrative Region.
参考文献（References）
[1] J.D. Murray, Mathematical Biology: I. An Introduction, 3rd edn., Vol. 2 [M], Springer
Verlag, London, 2002.
[2] C.R. Townsend, M. Begon, J.L. Harper,Essentials of Ecology[M], Blackwell Publishing
House, Oxford, 2008.
[3] C.W. Clark, Mathematical Bioeconomics: The Optimal management of Renewable re-
source, 2nd edn.[M], John Wiley and Sons, New York, 1990.
[4] M. Kot,Elements of Mathematical Biology[M], Cambridge University Press, Cambridge,
2001.
[5] S. Sahney, M.J. Benton, P.A. Ferry, Links between global taxonomic diversity, ecological
diversity and the expansion of vertebrates on land [J],Biology Letters, 2010, 6 (4): 544-547.
[6] A. Lotka,Elements of Physical Biology[M], Williams and Wilkins, Baltimore, Md., 1924.
[7] V. Volterra,Lecons Sur la Theorie Mathematique de la Lutte pour la Vie[M], Gauthier
Villars, Paris, 1931.
[8] M.E. Gilpin, F.J. Ayala, Global models of growth and competition [C],Proceedings of the
National Academy of Sciences, 1973, 70: 3590-3593.
[9] F.J. Ayala, M.E. Gilpin, J.G. Eherenfeld, Competition between species: Theoretical models
and experimental tests [J],Theoretical Population Biology, 1973, 4: 331-356.
[10] M.E. Gilpin, F.J. Ayala, Schoener’s model and Drosophila competition [J], Theoretical
Population Biology, 1976, 9: 12-14.
- 25 -
[11] B.S. Goh, T.T. Agnew, Stability in Gilpin and Ayala’s models of competition nedlands [J],
Journal of Mathematical Biology, 1977, 4: 275-279.
[12] K. Zhou, J. Lei, Analysis for the global stability of Gilpin Ayala competition model [J],
Journal of Sichuan University Natural Science Edition, 1987, 24: 361-366.
[13] M. Fan, K. Wang, Global periodic solutions of a generalized N-species Gilpin Ayala com-
petition model [J],Computers and Mathematics with Applications, 2000, 40: 1141-1151.
[14] Y. Chen, Z. Zhou, Stable periodic solution of a discrete periodic Lotka Volterra competition
system [J],Journal of Mathematical Analysis and Application, 2003, 277: 358-366.
[15] F.D. Chen, L.P. Wu, Z. Li, Permanence and global attractivity of the discrete Gilpin
Ayala type population model [J],Computers and Mathematics with Applications, 2007, 53:
1214-1227.
[16] X.Y. Liao, S.F. Zhou, Y.M. Chen, On permanence and global stability in a general Gilpin
Ayala competition predator prey discrete system [J],Applied Mathematics and Computa-
tion, 2007, 190: 500-509.
[17] S. Zhang, D. Tan, L.S. Chen, the periodic N-species Gilpin Ayala competition system with
impulsive eﬀect [J],Chaos, Solitons Fractals, 2005, 26: 507-517.
[18] S. Liu, E. Bereta, Competitive systems with stage structure of distributed delay type [J],
Journal of Mathematical Analysis and Application, 2006, 323: 331-343.
[19] M.X. He, Z. Li, F.D. Chen, Permanence, extinction and global attractivity of the peri-
odic Gilpin Ayala competition system with impulses [J],Nonlinear Analysis Real World
Application, 2010, 11: 2675-2685.
[20] T. Zhang, Y. Li, Positive periodic solutions for a generalized impulsive N-species Gilpin
Ayala competition system with continuously distributed delays on time scales [J],Inter-
national Journal of Biomathematics, 2011, 4(1): 23-34.
[21] F.D. Chen, Some new results on the permanence and extinction of nonautonomous Gilpin
Ayala type competition model with delays [J],Nonlinear Analysis: Real World Applica-
tions, 2006, 7(5): 1205-1222.
[22] B.S. Lian, S.G. Hu, Asymptotic behavior of the stochastic Gilpin Ayala competition models
[J],Journal of Mathematical Analysis and Application, 2008, 339: 419-428.
[23] M. Vasilova, M. Jovanovic, Stochastic Gilpin Ayala competition model with inﬁnite delay
[J],Applied Mathematics and Computation, 2011, 217(10): 4944-4959.
[24] M. Vasilova, Asymptotic behavior of a stochastic Gilpin Ayala predator-prey system with
time-dependent delay [J],Mathematical and Computer Modelling, 2013, 57(3-4): 764-781.
[25] M. Jovanovic, M. Vasilova, Dynamics of non-autonomous stochastic Gilpin Ayala compe-
tition model with time-varying delays [J], Applied Mathematics and Computation 2013,
219(12): 6946-6964.
[26] J.F.M. Al-omari, S.A. Gourley, Stability and traveling fronts in Lotka Volterra competition
models with stage structure [J],SIAM Journal on Applied Mathematics, 2003, 63: 2063-
2086.
[27] S.Q. Liu, X. Xie, J. Tang, Competing population model with nonlinear intraspeciﬁc reg-
ulation and maturation delays [J],International Journal of Biomathematics, 2012, 5(3):
Article No. 1260007.
[28] W.G. Alello, H.I. Freedman, J. Wu, Analysis of a model representing stage structured
populations growth with state dependent time delay [J],SIAM Journal on Applied Math-
ematics, 1992, 3: 855-869.
[29] G. Huang, Y. Takeuchi, W.B. Ma, Lyapunov functionals for delay diﬀerential equations
model of viral infections [J],SIAM Journal on Applied Mathematics, 2010, 70: 2693-2708.
Describe a Challenging Situation - Client Admitted to Wharton Business School
The car engine started roaring breaking the tranquility of the quiet suburb. The beams of the headlight pierced through the evening darkness. The Honda Civic picked up speed from zero to 60 miles per hour within few seconds. As I drove in the expressway, I occasionally starred at the star filled night sky and thought “Is this my destiny?” I raced the car full speed on the interstate highway. But the events of the godforsaken day kept rolling in my mind.
Months back, I was asked to join our Regional Head Quarter at London. I was elated to receive such a big promotion. With superior monetary benefits and enhanced job responsibilities, I was enjoying my time in London with my colleagues and friends. Unaware of the gathering storm at our New York Global Headquarter, I even booked tickets for my first big music concert of Metallica on 15th September 2008 and could barely wait for the day to come.
On 15th Sep, 2008, I reached office with a cheerful mood. The hustle and bustle in the corridor was bigger than usual. I sensed uneasiness in the ambience. “What happened?” I asked someone at the entrance. He looked at me with a blank face and asked “You don’t know? Lehman has filed for bankruptcy”. The news came as a jolt. We all were mindful of the worsening financial crisis, but nobody really expected an investment bank as big as Lehman Brothers to fail. I entered the office building and saw complete mayhem. With now no job in picture for anybody, people were panicking about prior financial commitments and responsibilities. Some were packing their belongings while many were calling friends and head hunters for job openings. My entire planned career, which looked in the best shape less than a year ago, seemed to turn upside down. On the giant screen on the wall, I saw news reporter screaming with their breaking news on CNN about the largest bankruptcy filing the corporate history. At the desk, I sat alongside my boss watching him drafting his resume. Amidst the on-going chaos, I kept my nerve, but things were to become worse. In the evening, I came back home. I thought that there was no better way to cheer myself up on that godforsaken day than with the Metallica concert! But within minutes, I got a call from the society office. I was asked to vacate the company apartment as the agency wasn’t sure whom to bill afterwards. I got almost choked. Without altering a single word, I packed my stuffs and left. From losing a dream job to being homeless, all in a span of few hours, I experienced the roller coaster ride of my life.
I stopped the car in the parking and checked in Hotel Plaza Inn. I took a shower. I was feeling refreshed. At the end of the day, I thought the best thing to do was just to stand quietly in the hotel balcony and enjoy the breeze. I looked at the London skyline and reminded myself that every cloud has a silver lining. I decided to stay back in London for some more time to face my destiny.
Instead of flying back to India or brooding about the situation, I continued going to office every day to network with senior associates. I realized the importance of a backup plan even when everything sails smooth as life is full of chance events. During the extra hours every day, which suddenly fell in my lap with no job, I explored around the city of London and knocked at every door for a suitable opening. It was the journey from a reckless, imprudent undergraduate with a trading job to a cautiously optimistic professional in a short span. I have continued to adapt myself to the changing world around and make the best out of every situation.
更多essay范文：请登录www.lxws.net/blog
Background
I raised this question because of an argument I am having with a question from user697473 here. The title of his question is "Formal definiton of random assignment." In the post he makes a claim that one can get unbiased estimates of the difference of treatment effect even if one group that is represented in the population is not sampled. At least that is the way I interpret the claim. He points to a paper coauthored by Don Rubin on causal modeling here. Both he and StasK argue through examples that this claim is true. This issue was also raised in another question that he posted here. In that post titled "Random assignment: why bother?" gung expresses skeptism.
In the first question my answer included a statement to the effect that without sampling all possible assignments of a potential confounding factor you cannot construct an estimate of the difference in treatment effect without assigning samples to each stratum for the confounding factor.
The Issue
I am confused about the claim. As I interpret the claim I am sure it is false. The claim is supported by two examples one given by user697473 and the other by StasK. The examples are confusing to me and in neither case is a proof or demonstration given to show that the claim is true.
I could be misinterpreting the claim but based on my interpetation it is false. To illustrate my point let's look at a famous example from the design of experiments book of the Godfather of randomization Sir Ronald Aylmer Fisher: "The Lady Tasting Tea".
In case you don't know the example I will describe it and provide a modification to clarify the example. The lady claims that she has the ability to determine just by tasting a cup of tea whether the tea was poured first or the milk was poured first. Fisher poses an experiment to test her claim by providing some cups with tea poured first and others with the milk poured first. If the Lady has the expertise that she claims she should be able to correctly identify the types better than a person who guesses at random. So the experiment is design to see if her probability of correct classification is better than 0.5. So he randomly provides the Lady with cups from both groups (in the example there were 6 cups 3 with milk poured first and 3 with tea poured first.
Now let me modify the experiment slightly. Suppose I have three brands of tea and for my population these brands are served at a tea parties equally. The question I want to ask is when given a cup of tea at a tea party can the Lady predict whether milk was poured first or the tea was poured first. I want to test whether her prediction accuracy is greater than 0.5 in the setting of tea parties. So I will apply Fisher's experiment to estimate this probability when the Lady is at one of these tea parties.
Note: I do not care whether or not she can differentiate the brands.
Suppose that in the population of tea parties the Lady's ability is:
Brand A Can predict tea first with probability 0.7 and can predict milk first with probability 0.7.
Brand B Can predict tea first with probability 0.75 and can predict milk first with probability 0.75.
Brand C Can predict tea first with probability 0.5 and can predict milk first with probability 0.6.
So she does no better than chance with Brand C but can do better than chance with Brands A and B.
For the experiment I only provide Brands A and B for her to taste. All I want to know is her ability to predict correctly in the normal tea party situation.
My Questions
Can I get an unbiased estimates of her prediction capabilities with this experiment?
If the answer to (1) is no, is user697473's claim false or have I misinterpreted it?
Randomization is used to avoid bias that is produced by confounding variables. This is important when trying to draw inference from a sample to a population. In my example the Brands are confounders. The treatment is tasting the cups. In the population she would get Brand A, Brand B and Brand C each 1/3rd of the time. If I know this and I sample Brands with unequal probabilities I claim that I can get an unbiased estimate of her prediction capabilities by taking a specific weighted average of the estimates of the proportions if I use a stratified random sample with strata totals n1, n2 and n3all greater than 0. But I cannot if n3 = 0. Furthermore, there is not other estimate I can calculate from a sample where allocation to Brand C equal 0 that will be an unbiased estimate.
Personal Statement
I choose to apply for the English Teaching department in your esteemed university not only for my great interest and burning enthusiasm in this field, but also because I am really determined to be involved in the related study and work in the coming future. I have decided that I will contribute all of my life in English Teaching. My career goal is to become an outstanding English teacher stepping into the new period in order to apply the more scientific teaching methods and ways into practice. That is just where my interest lies.
Meanwhile, I must acknowledge that my choice of TESOL never comes from my hot mind, but from the continuous learning and practices. So, it is actually a determination rather than a mere choice. Taking advantage of my excellent score in English learning, I have been involved in the social work as family teacher as well as attending some professional trainingto broaden my horizon since my freshman year. Theformer experiences enabled me to find that it is really exciting to be devoted to English Teaching field and such kind of excitation used to roll up in me every time, which made me believe that it is just the enterprise that I will be devoted to with my whole life.
Cherishing this determination, I began to make insistent efforts to achieve my dream by participating more teaching activities to accumulate the related experiences. It still occurs to me that once in an activity, I met with a student with very poor foundation in English. However, during my contact with him, I found that he was actually gifted in language learning and his English is poor just because he dislikes the boring “signals” without any “nutrition”. Taking his interest into full account, I brought out one of my speeches before and he turned out to be deeply attracted by the words of “Wine is soaking, memories are passing. Wound is recovering, just for never forgetting.” He told me that it is the first time for him to realize that English can also express the sorrows so incisively and vividly. In this way, I successful lightened his interest in English, according to which I designed a special teaching scheme for him with the Tagore poems instead of traditional texts or grammar. Besides, I even advised him to read the English edition of “The Book of Songs” to cultivate his interest and feelings to the new language. Meanwhile, I also found some original edition of “English and American Literature”, telling him to translate them into his interested literary form. Benefiting from my special teaching scheme on him, his interest in English was being heated, which witnessed his great improvement in English reading ability. This successful experience presented me with much more confidence for my future work.
The following teaching experiences after that enabled me to realize that few people are learning English as a kind of language, but as a kind of science like mathematics. So the sentence patterns are just like the mathematical formulas that are tormenting their memories since they can only put all the sentences into “frameworks”. In this way, they will easily get puzzled once the expression is changed because they are fettered in the grammar or the stupid “sentence patterns” with little understanding on the flexibility of a language. Thus, during my teaching, I have been trying to tell all of my students that grammar is just a way to help us to express our ideas in a scientific and acceptable way and they can actually write any thing in their own manner instead of thinking too much of the grammar requirements. However, I realized that my teaching way is not enough to pull up the root of the problem though I actually help a few of the students indeed because under Chinese boring education, English can be only considered as a subject instead of a language, which is just the problem existing in Chinese education system.
Therefore, I hope to find a more suitable English teaching methodfor Chinese students, which can pull them out of the fixed clues by telling them that language is not just a tool, but also anexpression and communication that carries full emotions of human beings. Generally, I have every reason to believe that cherishing this kind of attitude and enthusiasm, I must be able to make more achievements after the study in your gorgeous learning institute, which will largely improve my personal professional quality and abilities and promote me to achieve my career goal in the coming future.