Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 845606, 24 pages
doi:10.1155/2010/845606
Research Article
Permanence and Extinction of
a Generalized Gause-Type Predator-Prey System
with Periodic Coefficients
Jinhuo Luo1, 2
1
2
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
College of Information Technology, Shanghai Ocean University, Shanghai 201306, China
Correspondence should be addressed to Jinhuo Luo, jhluo@shou.edu.cn
Received 9 August 2010; Revised 8 December 2010; Accepted 17 December 2010
Academic Editor: Elena Litsyn
Copyright q 2010 Jinhuo Luo. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the permanence and extinction of a generalized Gause-type predator-prey system with
periodic coefficients. We provide a sufficient and necessary condition to guarantee the predator
and prey species to be permanent and a sufficient condition for the existence of a periodic solution.
In addition we prove that when the predator population tends to extinction, the prey population
keeps oscillating above a positive population level.
1. Introduction
Permanence of a dynamical system has always been a hot issue in the past few decades.
The concept of permanence has been introduced and investigated by several authors, each
using his own terminology: “cooperativity” in the earlier papers of Schuster et al. 1, and
Hofbauer 2, “permanent coexistence” by Hutson and Vickers 3, “uniform persistence” in
Butler et al. 4, and “ecological stability” by Svirezhev and Logofet 5–7 for more detailed
statements of the concept see 8.
Many important results have been found in recent years 1–37. Some authors see
21, 28, 31, 33 have considered the following two species periodic Lotka-Volterra predatorprey system
ẋ1 t x1 tb1 t − a11 tx1 t − a12 tx1 tx2 t,
ẋ2 t x2 t−b2 t − a22 tx2 t a21 tx1 tx2 t,
1.1
2
Abstract and Applied Analysis
where bi t and aij t i, j 1, 2 are periodic functions on R with common period ω > 0
and aij t ≥ 0 for all t ∈ R. They have established sufficient and necessary conditions for
the existence of positive ω-periodic solutions of the system by using different methods,
respectively. Teng 31 has given sufficient and necessary conditions for the uniform
persistence of the system.
Cui 16 has considered the permanence of the following Lotka-Volterra predator-prey
model with periodic coefficients:
cty
,
ẋ x at − btx −
pt x
etx
ẏ y −dt − fty .
pt x
1.2
He provided a sufficient and necessary condition to guarantee the predator and prey species
to be permanent. In Theorems 2.2 and 2.3 he set a precondition that ft /
≡ 0. This restricts
the application of the theorems more or less, since many researchers often neglect the logistic
term in the predator equation when the population level of the predator is relatively low
and the competition between predators can be ignored, and it proved to be an unnecessary
precondition in our paper. However, the research methods in his work inspired me, and many
proofs, especially in the first half of this paper, are analogous to 16.
In this paper we consider the permanence of the following generalized Gause-type
predator-prey system,
ẋ x ft, x − gt, xy ,
1.3
ẏ y γtgt, xx − μt − hty ,
where ft ω, x ft, x, gt ω, x gt, x for all t and γ·, μ·, h· are all
periodic continuous functions with common period ω > 0; γ·, μ· are positive, and h·
is nonnegative. We emphasize that our model includes the case when ht ≡ 0.
In the absence of predators, system 1.3 becomes
ẋ xft, x,
1.4
where f is a real-valued function defined on
R20 t, x ∈ R2 : t ≥ 0, x ≥ 0 .
1.5
Abstract and Applied Analysis
3
Vance and Coddington 35 have studied system 1.4 and proved the existence of a
unique periodic solution under some assumptions. Apart from the assumption we mentioned
above that f·, x is ω-periodic, the other assumptions with a mild modification are as
follows.
A1 Function f is continuous and differentiable with respect to x on R20 , and ∂f/∂x is
continuous on R20 .
A2 There are continuous functions p and λ with px > 0 for x > 0 and λt ≥ 0 for
t ≥ 0, such that
∂f
≤ −pxλt for t ≥ 0, x ≥ 0,
∂x
∞
λsds ∞.
1.6
0
A3 There exist constants β ≥ 0 and K > 0, such that
i ft, K ≤ β for t ≥ 0,
tω
fs, Kds ≤ 0 for t ≥ 0.
t
ii
A4 There exist constants α ≥ 0 and 0 < δ ≤ K, such that
i ft, δ ≥ −α for t ≥ 0,
tω
fs, δds > 0 for t ≥ 0.
t
ii
In addition, we assume that the ω-periodic function g·, x satisfies the following.
A5 Function gt, x is continuous with respect to t, and Gt, x : xgt, x is strictly
monotonely increasing with respect to x,
A6 gt, x is nonnegative and is positive when x > 0,
A7 gt, x is bounded on R20 , that is there is a constant Mg such that gt, x ≤ Mg for
t, x ∈ R20 .
Traditionally Gt, x represents a grazing rate. Usually when the amount of prey
increases, the grazing rate increases and eventually tends to a maximal value as the prey
population tends to infinity. But here we do not emphasize that Gt, x is bounded because
its unnecessary theoretically. Assumption A5 means that there is a higher capture rate when
there is a larger amount of prey. gt, x is directly proportional to the mean possibility density
for each individual prey being captured, and A6, A7 suggest that there is a possibility, but
its not definitely for each individual prey being captured. In ecology Gt, x is called the
functional response or grazing function, for example, the Holling-type grazing function:
Rm xn
,
α xn
n ≥ 1,
1.7
where Rm , α > 0 or the generalized form
Rm txn
,
αt xn
n ≥ 1,
1.8
4
Abstract and Applied Analysis
where Rm t, αt > 0; the Ivlev grazing function
Rm 1 − e−λx ,
1.9
where Rm , λ > 0. Obviously all of these functions satisfy A5–A7.
In this paper we will establish sufficient and necessary conditions for the permanence
of system 1.3. In the next section we state our main results. These results are proved in
Section 3. Two applications are given in Section 4.
2. Main Results
Throughout this paper, we will assume that all the functions f·, x, g·, x, γ·, μ·, and
h· are continuous and periodic with common period ω > 0. For any continuous ω-periodic
function ut defined on R, we denote
ut 1
ω
ω
utdt,
0
uM max ut,
t∈0,ω
uL min ut.
t∈0,ω
2.1
In order to describe our main results, we first introduce a lemma.
Lemma 2.1 see 35. Suppose that f satisfies (A1)–(A4). Then system 1.4 possesses a unique
ω-periodic positive solution x∗ t which is globally asymptotically stable with respect to the positive
x-axis.
Theorem 2.2. Suppose that f satisfies (A1)–(A4), g satisfies (A5)–(A7). Then system 1.3 is
permanent provided that
γtgt, x∗ tx∗ t − μt > 0,
2.2
where x∗ t is the unique periodic solution of 1.4 given by Lemma 2.1.
Theorem 2.3. Suppose that f satisfies (A1)–(A4), g satisfies (A5)–(A7), and
γtgt, x∗ tx∗ t − μt ≤ 0,
2.3
then
i lim yt 0,
t→∞
ii lim inf xt > 0,
t→∞
for any solution xt, yt of system 1.3 with positive initial conditions, where x∗ t is the unique
periodic solution of 1.4 given by Lemma 2.1.
By Theorems 2.2 and 2.3, we have the following corollary.
Abstract and Applied Analysis
5
Corollary 2.4. Suppose that f satisfies (A1)–(A4), g satisfies (A5)–(A7). Then system 1.3 is
permanent if and only if 2.2 holds.
Lemma 2.5 see Theorem 15.5 in 30, 37. Consider a periodic system
ẋ Ft, x,
Ft ω, x Ft, x,
ω > 0,
2.4
where Ft, x ∈ CR × R2 , R2 satisfies a local Lipschitz condition. If all solutions of the above system
exist in the future and one of them is bounded, then there exists a periodic solution of period ω.
By Theorem 2.2 and Lemma 2.5, we have the following corollary see proof in
Section 3.
Corollary 2.6. Suppose f satisfies (A1)–(A4), g satisfies (A5)–(A7), and condition 2.2 holds. If in
addition gt, x satisfies a local Lipschitz condition with respect to x, then system 1.3 has a positive
ω-periodic solution.
3. Proof of Our Main Results
Lemma 3.1 see 31. If at, bt are ω-periodic functions, bt ≥ 0, for all t ∈ R and bt > 0,
then
u̇ uat − btu
3.1
has a unique nonnegative ω-periodic solution u∗ which is globally asymptotically stable with respect to
the positive u-axis. Moreover, if at > 0, then u∗ t > 0, for all t ∈ R and if at ≤ 0, then u∗ t 0.
Lemma 3.2. Suppose that f satisfies (A1)–(A4), then there exists an ε0 > 0, such that for any 0 <
ε ≤ ε0 , system
ẋ x ft, x − ε
3.2
possesses an ω-periodic positive solution xε∗ t which is globally asymptotically stable with respect to
the positive x-axis.
Proof. It suffices to show that if f satisfies A1–A4, then there exists an ε0 > 0 such that
for any 0 < ε ≤ ε0 , f f − ε satisfies A1–A4. Obviously f satisfies A1–A3. Taking
ω
ε0 1/2ω 0 fs, δds, its easy to see f satisfies A4.
From a comparison theorem see Theorem 1.1 37 and Lemma 2.1, we have the
following lemma.
Lemma 3.3. Suppose that f satisfies (A1)–(A4), xt is a solution of system 1.4, and x∗ t is the
periodic solution of system 1.4 given by Lemma 2.1. Let vt be a right maximal (right minimal)
6
Abstract and Applied Analysis
solution of a scalar differential equation v̇ wt, v on t0 , ∞ ⊂ 0, ∞, where wt, v is
continuous and satisfies wt, x ≤ ≥ xft, x for all t, x ∈ R20 . Then the following conclusions
hold.
i If vt0 ≤ ≥ xt0 , then vt ≤ ≥xt for t ≥ t0 .
ii For all ε > 0 there exist a τ ≥ t0 such that vt < x∗ t εvt > x∗ t − ε for t ≥ τ.
Proposition 3.4. Under assumptions (A1)–(A7), there exist Mx and My , such that
lim sup xt ≤ Mx ,
t→∞
lim sup yt ≤ My
t→∞
3.3
for all solutions xt, yt of system 1.3 with positive initial values.
Proof. Obviously, R2 {x, y : x ≥ 0, y ≥ 0} is a positively invariant set of system 1.3.
Given any solution xt, yt of 1.3 with positive initial values, from system 1.3 we have
ẋ ≤ xft, x.
3.4
v̇ vft, v,
3.5
The following equation
has a globally asymptotically stable positive ω-periodic solution v∗ t by Lemma 2.1. By
Lemma 3.3, there exists T1 > 0, such that
xt < v∗ t 1,
for t > T1 .
3.6
Let Mx max0≤t≤ω {v∗ t 1}. We have
lim sup xt ≤ Mx .
t→∞
3.7
By 1.3,
γ M ẋ ẏ γ M xft, x − γ M − γt gt, xxy − μty − hty2
≤ γ M xft, x − μty
≤ γ M xft, x − μL y
γ M x ft, x μL − μL γ M x y .
3.8
Denote wt γ M xt yt. Then we have
ẇ μL w ≤ γ M x ft, x μL .
3.9
Abstract and Applied Analysis
7
Since xt ≤ max0≤t≤ω {v∗ t 1}, for t ≥ T1 , there is a constant B > 0, such that
γ M x ft, x μL ≤ B
for t ≥ T1 ,
3.10
by the continuity of f. It follows that
ẇ μL w ≤ B,
for t ≥ T1 .
3.11
Notice that μL > 0. It’s easy to show that there is an A > 0, such that
lim sup wt ≤ A.
3.12
t→∞
By the notation w γ M x y, taking Mx A/γ M and My A, Proposition 3.4 is proved.
Proposition 3.5. Under assumption (A1)–(A7), there exists a positive constant ηx , such that
lim sup xt ≥ ηx ,
3.13
t→∞
for all solutions xt, yt of 1.3 with positive initial values.
Proof. Suppose that 3.13 is not true. Then there is a sequence {zm } ⊂ R2 , such that
lim sup xt, zm <
t→∞
1
,
m
m 1, 2, . . . ,
3.14
where xt, zm , yt, zm is the solution of 1.3 with x0, zm , y0, zm zm . By assumption
A5, we can choose sufficiently small positive numbers εx < 1 and εy < 1, such that
γtgt, εx εx − μt < 0,
3.15
φε t > 0,
3.16
α max μt γtgt, εx εx htεy .
3.17
where
φε t ft, εx − Mg εy expαω,
0≤t≤ω
By 3.14, for the given εx > 0, there exists a positive integer N0 , such that
lim sup xt, zm <
t→∞
1
< εx ,
m
m > N0 .
3.18
8
Abstract and Applied Analysis
m
For the rest of this proof, we assume that m > N0 . Equation 3.18 implies there exists τ1
such that
m
3.19
t ≥ τ1 ,
xt, zm < εx ,
> 0,
and further
ẏt, zm ≤ yt, zm γtgt, εx εx − μt − htyt, zm ,
m
for t ≥ τ1 .
3.20
By 3.15, and Lemma 3.1, any solution vt of the following equation,
v̇ v γtgt, εx εx − μt − htv
3.21
with positive initial conditions satisfies
lim vt 0.
3.22
lim yt, zm 0
3.23
t→∞
Hence,
t→∞
m
by Lemma 3.3. So there is a τ2
m
> τ1 , such that
yt, zm < εy ,
m
3.24
for t ≥ τ2 .
It follows that
ẋt, zm ≥ xt, zm ft, xt, zm − gt, εx εy ,
m
t ≥ τ2 .
3.25
By 3.16, the equation
ẋ x ft, xt, zm − gt, εx εy
3.26
has an ω-periodic positive solution x∗ t which is globally asymptotically stable. Hence,
xt, zm >
x∗ t
,
2
3.27
for sufficiently large t > 0 and m > N0 , which is a contradiction with 3.14. This completes
the proof of Proposition 3.5.
Abstract and Applied Analysis
9
Proposition 3.6. Under assumption (A1)–(A7), there exists a positive constant γx , such that
lim inf xt ≥ γx
3.28
t→∞
for all solutions xt, yt of system 1.3 with positive initial values.
Proof. Suppose that 3.28 is not true, then there exists a sequence {zm } ⊂ R2 such that
ηx
,
2m2
lim inf xt, zm <
t→∞
m 1, 2, . . . .
3.29
m 1, 2, . . . .
3.30
On the other hand, by Proposition 3.5, we have
lim sup xt, zm > ηx ,
t→∞
m
m
Hence, there are time sequences {sq } and {tq } satisfying
m
0 < s1
m
< t1
m
sq
m
< s2
−→ ∞,
m
m
< · · · < sq
< t2
m
tq
−→ ∞,
m
< tq
< ··· ,
as q −→ ∞,
ηx
ηx
m
m
x sq , zm ,
x tq , zm 2 < xt, zm <
m
m
3.31
m m
t ∈ sq , tq .
By Proposition 3.4, for a given positive integer m, there is a T m > 0, such that
xt, zm ≤ Mx ,
yt, zm ≤ My ,
m
3.32
for t ≥ T1 .
M
m
→ ∞ as q → ∞, there is a positive integer K m such that sq
Because of sq
q ≥ Km, and hence
ẋ ≥ xt, zm ft, Mx − Mg My : ζtxt, zm m
for q ≥ K m . Integrating 3.33 from sq
m
to tq
m
> T1
since
3.33
yields
tm
q
m
m
x tq , zm ≥ x sq , zm exp
ζtdt
m
3.34
sq
or
−
tm
q
m
sq
ζtdt ≥ ln m,
for q ≥ K m .
3.35
10
Abstract and Applied Analysis
If ζt ≥ 0, it leads to a contradiction. Otherwise, ζt < 0, we have
m
m
− sq
tq
as m −→ ∞, q ≥ K m ,
−→ ∞,
3.36
since ζt is bounded. By 3.15 and 3.16, there are constants P > 0 and N0 > 0, such that
ηx
< εx ,
m
P
My exp
m
tq
m
− sq
3.37
> 2P,
γtgt, εx εx − μt − htεy dt < εy ,
P
0
φε tdt > 0
3.38
0
for m ≥ N0 , q ≥ K m , and a ≥ P . Equation 3.37 implies
xt, zm < εx ,
m m
t ∈ s q , tq
3.39
for m ≥ N0 , q ≥ K m . For the positive εy satisfying 3.16 and 3.38, we have the following
two cases:
m
m
i yt, zm ≥ εy for all t ∈ sq , sq
ii there exists
m
τq1
∈
m m
sq , sq
P ;
m
P , such that yτq1 , zm < εy .
If i holds, by 3.38 and 3.39 we have
m
εy ≤ y sq P, zm
sm
q P m
γtgt, εx εx − μt − htεy dt
≤ y sq , zm exp
m
sq
≤ My exp
P
3.40
γtgt, εx εx − μt − htεy dt
0
< εy .
This is a contradiction.
If ii holds, we now claim that
yt, zm ≤ εy expαω,
m
m
m m
t ∈ τq1 , tq .
3.41
m
Otherwise, there exists τq2 ∈ τq1 , tq such that
m
y τq2 , zm > εy expαω.
3.42
Abstract and Applied Analysis
11
m
m
m
By the continuity of yt, zm , there must exist τq3 ∈ τq1 , τq2 such that
m
y τq3 , zm εy ,
yt, zm > εy ,
for t ∈
m m
τq3 , τq2
m
3.43
.
m
m
Denote P m as the nonnegative integer, such that τq2 ∈ τq3 P m ω, τq3 P m 1ω. By
3.15, we have
m
εy expαω < y τq2 , zm
<y
m
τq3 , zm
εy exp
τqm
exp
2
m
τq3
⎧ m
⎨ τq3 P m ω
⎩
m
τq3
γtgt, εx εx − μt − htεy dt
3.44
⎫
⎬
τqm
2
τq3 P m ω ⎭
m
γtgt, εx εx − μt − htεy dt
< εy expαω.
m
m
This contradiction establishes that 3.41 is true, particularly 3.41 holds for t ∈ sq P, tq .
By 3.31 and 3.38, we have
ηx
m
x
t
,
z
m
q
m2
tm
q
m
≥ x sq P, zm exp
m
sq P
τqm
m
x sq P, zm exp
m
sq P
>
ft, εx − Mg εy expαω dt
3.45
φε tdt
ηx
,
m2
which is also a contradiction. This completes the proof of Proposition 3.6.
Proposition 3.7. Suppose f satisfies (A1)–(A4), g satisfies (A5)–(A7), and 2.2 holds. Then there
exists a positive constant ηy such that
lim sup yt > ηy
t→∞
for all solutions xt, yt of 1.3 with positive initial values.
3.46
12
Abstract and Applied Analysis
Proof. By assumption A5 and 2.2 we can choose a constant ε1 > 0 such that
ψε1 t > 0,
3.47
ψε1 t γtgt, x∗ t − ε1 x∗ t − ε1 − μt − htε1 .
3.48
where
Consider the following equation with positive parameter α:
ẋ x ft, x − 2Mg α .
3.49
By Lemma 3.2, 3.49 has a unique positive ω-periodic solution xα∗ t which is globally
asymptotically stable since α < ε0 /2Mg . Let xα t be the solution of 3.49 with initial
condition xα 0 x∗ 0 in which x∗ t is the unique periodic solution of 1.4 given by
Lemma 2.1. Hence, for the above ε1 , there exists sufficiently large T2 > T1 , such that
|xα t − xα∗ t| <
ε1
,
4
for t ≥ T2 .
3.50
By the continuity of the solution in the parameter, we have xα t → x∗ t uniformly in
T2 , T2 ω as α → 0. Hence, for ε1 > 0 there exists α0 > 0, such that
|xα t − x∗ t| <
ε1
,
4
for t ∈ T2 , T2 ω,
0 < α < α0 .
3.51
So, we have
|xα∗ t − x∗ t| <
ε1
,
2
t ∈ T2 , T2 ω.
3.52
Notice that xα t and x∗ t are all ω-periodic, hence
|xα∗ t − x∗ t| <
ε1
,
2
t ≥ 0, 0 < α < α0 .
3.53
Choosing a constant α1 0 < α1 < α0 , 2α1 < ε1 , we have
xα∗ 1 t ≥ x∗ t −
ε1
,
2
t ≥ 0.
3.54
Suppose that 3.46 is not true. Then there exists z ∈ R2 , such that
lim sup yt, z < α1 ,
t→∞
3.55
Abstract and Applied Analysis
13
where xt, z, yt, z is the solution of 1.3 with x0, z, y0, z z. So, there exist T3 ≥ T2 ,
such that
t ≥ T3
yt, z < 2α1 < ε1 ,
3.56
and hence
ẋ ≥ xt, z ft, x − 2Mg α1 .
3.57
Let ut be the solution of 3.49 with α α1 and uT3 xT3 , z, then
xt, z ≥ ut,
t ≥ T3 .
3.58
By the global asymptotic stability of xα∗ 1 t, for the given ε ε1 /2, there exists T4 ≥ T3 , such
that
ut − xα∗ t < ε1 ,
1
2
t ≥ T4 .
3.59
Hence
xt, z ≥ ut > xα∗ 1 t −
ε1
,
2
t ≥ T4
3.60
and hence
xt, z > x∗ t − ε1 ,
t ≥ T4
3.61
from 3.54. This implies
ẏt, z ≥ yt, z γtgt, x∗ t − ε1 x∗ − ε1 − μt − htε1 ψε1 yt, z,
t ≥ T4 .
3.62
Integrating the above inequality from T4 to t yields
yt, z ≥ yT4 , z exp
t
ψε1 tdt.
3.63
T4
yt, z → ∞ as t → ∞ from 3.47 which is a contradiction. This completes the proof of
Proposition 3.7.
Proposition 3.8. Suppose f satisfies (A1)–(A4), g satisfies (A5)–(A7), and 2.2 holds. Then there
exists a positive constant γy , such that
lim inf yt ≥ γy
t→∞
for all solutions xt, yt of 1.3 with positive initial values.
3.64
14
Abstract and Applied Analysis
Proof. Otherwise, there exists a sequence {zm } ⊂ R2 , such that
lim inf yt, zm <
t→∞
ηy
m 12
m 1, 2, . . . .
,
3.65
However
lim sup yt, zm > ηy ,
t→∞
m 1, 2, . . . ,
3.66
m
m
from Proposition 3.7. Hence there are two time sequence {sq } and {tq } satisfying the
following conditions:
m
m
< t1
0 < s1
m
ηy
m
< t1
m
−→ ∞,
sq
m
y tq , zm m
< s2
tq
< yt, zm <
m 12
m
< · · · < sq
−→ ∞,
m
< tq
< ··· ,
as q −→ ∞,
ηy
m
y sq , zm ,
m1
m
By Proposition 3.4, for a given integer m > 0 there is a T1
yt, zm ≤ My ,
3.67
m m
t ∈ sq , tq .
> 0, such that
m
for t ≥ T1 .
m
3.68
m
Because of sq → ∞ as q → ∞, there is a positive integer K m , such that sq
q ≥ K m , and hence
ẏt, zm ≥ yt, zm −μt − htMy
m
m
m
> T1
as
3.69
m
for q ≥ K m and t ∈ sq , tq . Integrating the above inequality from sm to tq , we have
y
m
tq , zm
≥y
m
sq , zm
tm
q
exp
m
sq
−μt − htMy dt
3.70
or
tm
q
m
sq
μt htMy dt ≥ lnm 1,
q ≥ K m .
3.71
Because of the boundedness of the function μt htMy , we know that
m
tq
m
− sq
−→ ∞,
as m −→ ∞, q ≥ K m .
3.72
Abstract and Applied Analysis
15
By 3.47, there is a constant P > 0 and a positive integer N0 such that
ηy
m
m
< α1 < ε1 , tq − sq > 2P,
m1
q
ψε1 tdt > 0
3.73
3.74
0
for m ≥ N0 , q ≥ K m and a ≥ P . Further,
yt, zm < α1 < ε1 ,
m m
t ∈ sq , tq , m ≥ N0 , q ≥ K m .
3.75
In addition,
ẋt, zm ≥ xt, zm ft, xt, zm − Mg α1 .
m
3.76
m
Let ut be the solution of 3.49 with α α1 and usq xsq , zm . Then
xt, zm ≥ ut,
m m
t ∈ sq , tq
3.77
m
by Lemma 3.3. Further, by Propositions 3.4 and 3.6, we can choose K1
> K m such that
m
γx ≤ x sq , zm ≤ Mx
3.78
m
for q ≥ K1 . For α α1 , 3.49 has a unique positive ω-periodic solution xα∗ 1 t which is
globally asymptotically stable. In addition, by the periodicity of 3.49, the periodic solution
xα1 t is uniformly asymptotically stable with respect to the compact set Ω {x : γx ≤ x ≤
Mx }. Hence, for the given ε1 in Proposition 3.7, there exists T0 > P which is independent of
m and q, such that
ut ≥ xα∗ 1 t −
ε1
,
2
m
3.79
m
3.80
t ≥ T0 sq .
Thus
ut ≥ x∗ t − ε1 ,
t ≥ T0 sq
m
from 3.54. By 3.72, there exists a positive integer N1 ≥ N0 such that tq
m
sq
2P for m ≥
m
K1
and q ≥
m
K1 .
m
> sq
2T0 >
So we have
xt, zm ≥ x∗ t − ε1 ,
m
m
t ∈ sq T0 , tq
3.81
16
Abstract and Applied Analysis
m
since m ≥ N1 and q ≥ K1 . Hence,
ẏt, zm ≥ ψε1 tyt, zm ,
m
m
t ∈ sq T0 , tq
m
from 3.75 and 3.81. Integrationg the above inequality from sq
y
m
tq , zm
≥y
m
sq
T0 , zm exp
tm
q
m
sq T0
3.82
m
T0 to tq
yields
ψε1 tdt,
3.83
ηy
3.84
and hence
ηy
m 1
2
≥
tm
q
ηy
m 1
2
exp
m
sq T0
ψε1 tdt >
m 12
from 3.74. This is a contradiction. This completes the proof of Proposition 3.8. The result of
Theorem 2.2 follows from Propositions 3.4–3.8.
Proof of Theorem 2.3. Suppose x∗ t is the periodic solution of 1.4. For any solution
xt, yt of 1.3 with a positive initial value, there are two possible cases:
i for all t ≥ 0, xt > x∗ t;
ii there is a T > 0, such that xT ≤ x∗ t.
Now, in the above two cases, we prove limt → 0 yt 0, respectively.
i Suppose vt is a solution of 1.4 with v0 x0. Comparing 1.3 and 1.4 we
have vt ≥ xt ≥ x∗ t and limt → 0 vt − x∗ t 0. Hence
lim xt − x∗ t 0.
3.85
t→∞
From system 1.3, for all t0 ∈ 0, ∞, we have
xt0 ω − xt0 t0 ω
xsfs, xsds −
t0 ω
t0
∗
∗
x t0 ω − x t0 t0 ω
t0
xsgs, xy ds
t0
∗
∗
x sfs, x ds 0.
3.86
Abstract and Applied Analysis
17
So
xt0 ω − xt0 t0 ω
xfs, x − x fs, x ds −
∗
∗
t0
t0 ω
xgs, xy ds
t0
t0 ω
x − x fs, x x fs, x − fs, x∗ ds −
∗
∗
t0 ω
t0
xgs, xy ds
t0
t0 ω
∗
x − x fs, x x
∗ ∂f
t0
∂x
t0 ω
xgs, xy ds,
s, x θx − x ds −
∗
∗
t0
3.87
where 0 < θ < 1. By the boundedness of x, x∗ , and the continuity of f and ∂f/∂x, there must
exist a constant A > 0, such that
fs, x x∗ ∂f s, x∗ θx − x∗ ≤ A.
∂x
3.88
For all ε > 0, by 3.85, there exists a T 0 > 0, such that 0 < xt − x∗ t < ε, for all t ≥ T 0 . Hence
t0 ω
xgs, xy ds ≤ −xt0 ω xt0 Aωε
t0
−xt0 ω − x∗ t0 ω xt0 − x∗ t0 Aωε
3.89
< 0 ε Aωε
Aω 1ε
for t0 ≥ T 0 . Denote GL min0≤t≤ω x∗ tgt, x∗ t, ytL0 mint0 ≤t≤t0 ω yt. Then
L
t0 ω
0<G
y ds ≤
t0
t0 ω
xgs, xy ds ≤ Aω 1ε.
3.90
t0
Hence
0<
ytL0
1
≤
ω
t0 ω
ysds ≤
t0
A
1
L
G
ωGL
ε.
3.91
This implies
lim ytL0 lim
t0 → ∞
t0 → ∞
t0 ω
t0
ysds 0.
3.92
18
Abstract and Applied Analysis
On the other hand,
yt yt0 exp
t0 ω
γsxsgs, xs − μs − hsys ds.
3.93
t0
By the boundedness of x, y, g, and the continuity of γ, μ, and h, there exists a C > 0 such that
γtgt, xx − μt − htyt ≤ C,
t ≥ 0.
3.94
Hence
t0 ω
yt ln
≤C
ds Cω,
yt 0
for t ∈ 0, ω.
3.95
t0
For all t, s ∈ 0, ω, we have
yt yt
ys ln
ln
ys yt − ln yt ≤ 2Cω.
0
0
3.96
yt
≤ exp2Cω,
ys
3.97
It follows that
exp−2Cω ≤
|t − s| ≤ ω.
Considering 3.92, we have
lim yt 0.
t→∞
3.98
ii Consider the second case. We claim that once there is a T > 0 such that xT ≤ x∗ T ,
then
xt < x∗ t,
∀t > T .
3.99
Otherwise, suppose T mint>T {t : xt x∗ t}. Then
xt − x T
xt − x∗ T
x∗ t − x∗ T
ẋ T x T − 0 lim
lim
≥ lim
ẋ∗ T .
t → T −0
t → T −0
t → T −0
t − T
t − T
t − T
3.100
This is a contradiction since from 1.3 and 1.4 we know ẋt < ẋ∗ t at the same point
t, x ∈ R2 . So 3.99 holds.
Abstract and Applied Analysis
19
Now we show for all ε > 0, there is a T 1 > T ≥ 0 such that
y T 1 < ε.
3.101
Otherwise, suppose for all t ≥ T , yt ≥ ε. Then
ẋ ≤ x ft, x − gt, xε ,
t ≥ T .
3.102
Suppose vt is a solution of 1.4 with vT xT . Then by Lemma 2.1 we have vt →
x∗ t t → ∞, where x∗ t denotes the periodic solution of 1.4. Since ẋT < v̇T , there
exists a σ > 0 such that xt < vt for t ∈ T , T σ. Denote δt vt − xt. From 3.99 we
know δt > 0, for t ≥ T . We show that there is an ε1 > 0 such that limt → ∞ inf δt ≥ ε1 . In fact,
v̇ − ẋ vft, v − xft, x − gt, xε
v ft, v − ft, x ft, xv − x xgt, xε
∂f
v t, x θv − x ft, x v − x xgt, xε
∂x
3.103
where 0 < θ < 1. By the boundedness of x, v, and the continuity of f and ∂f/∂x, there exists
an L > 0, such that |v∂f/∂xt, x θv − x ft, x| < L. Hence
δ̇ > −Lδ xgt, xε.
3.104
Choosing T 0 > T , such that vt ≥ x∗ t/2 for t ≥ T 0 , and letting
∗ x t min0≤t≤ω gt, x∗ t/4 x∗ tε
ε1 min max δt, min
,
.
0≤t≤;ω
0≤t≤σ
4
4L
3.105
Then whenever δt ≤ ε1 for t ≥ T 0 , we have xt vt − δt ≥ x∗ t/2 − x∗ t/4 x∗ t/4.
By assumption A6, δ̇t > −Lε1 gt, x∗ t/4x∗ t/4 ≥ 0, and this implies limt → ∞ inf δt >
ε1 . Choosing T > T , such that vt < x∗ t ε1 /2 and δt ≥ ε1 for t ≥ T , we have
ε1 ε1
xt ≤ vt − ε1 ≤ x∗ t − ε1 x∗ t − ,
2
2
for t ≥ T .
3.106
Hence, there exists an ε0 > 0 such that
ε1 ∗
ε1 x t −
− μt ≤ −ε0 < 0
γtgt, xtxt − μt − htε ≤ γtg t, x∗ t −
2
2
3.107
20
Abstract and Applied Analysis
for t ≥ T , by 2.3 and assumption A5. So
ε ≤ yt ≤ y T exp
t
T
γsgs, xsxs − μs − hsε ds −→ 0 t −→ ∞.
3.108
This is a contradiction and it implies that 3.101 holds.
Second, we show that
yt ≤ ε expMεω,
for t ≥ T 1 ,
3.109
where
Mε max γtgt, x∗ tx∗ t μt htε .
0≤t≤ω
3.110
Otherwise, there exists T 2 > T 1 , such that
y T 2 > ε expMεω.
3.111
By the continuity of yt, there must exist T 3 ∈ T 1 , T 2 such that yT 3 ε and yt > ε
for t ∈ T 3 , T 2 . Let P1 be the nonnegative integer, such that T 2 ∈ T 3 P1 ω, T 3 P1 1ω, and by 2.3, A6 and 3.99, we have
ε expMεω < y T 2
T 2
< y T 3 exp
γsgs, xsxs − μs − hsε ds
T 3
"
! T 3 P ω T 2
1
ε exp
T 3
T 3 P
1ω
"
! T 3 P ω T 2
1
< ε exp
T 3
≤ ε exp
≤ ε exp
! T 2
T 3P1 ω
! T 2
T 3P1 ω
T 3 P
γsgs, xx − μs − hsε ds
γsgs, x∗ x∗ − μs − hsε ds
1ω
"
∗ ∗
γsgs, x x − μs − hsε ds
"
∗ ∗
γsgs, x x μs hsε ds
< ε expMεω,
which is a contradiction. This completes the proof of conclusion i of Theorem 2.3.
3.112
Abstract and Applied Analysis
21
Now we prove the second conclusion. Since i holds and gt, x is bounded, there is
a t1 > 0 and an ε > 0, such that gt, xy < ε ≤ ε0 for t ≥ t1 . Then we have
ẋ ≥ x ft, x − ε ,
for t ≥ t1 .
3.113
The following auxiliary equation,
v̇ v ft, v − ε
3.114
has a globally asymptotically stable positive ω-periodic solution xε∗ t by Lemma 3.2. So by
Lemma 3.3, we have
lim inf xt ≥ xε∗ t.
3.115
t→∞
This completes the proof of conclusion ii of Theorem 2.3.
Proof of Corollary 2.6. We claim that once gt, x satisfies a local Lipschitz condition with
respect to x, then the function
F t, x, y x ft, x − gt, xy
y γtgt, xx − μt − hty
3.116
satisfies a local Lipschitz condition with respect to x and y. Considering assumptions A1–
A7, this is clearly the case. So the uniqueness of solutions of system 1.3 is guaranteed,
and by Lemma 2.5 we know that there exsits an ω-periodic solution Z∗ t x∗ t, y∗ t.
From the proof of Theorem 15.5 in 37, the initial value of the periodic solution is the fix
point of the mapping T : z0 → zω; 0, z0 which is the limit point of a subsequence of the
sequence {T n z0 }∞
1 , where z0 x0 , y0 is the initial value of a bounded solution of system
1.3. Taking any positive x0 and y0 , by Theorem 2.2 we know the solution started from this
point is bounded and the limit of any subsequence of the sequence {T n z0 }∞
1 is positive. So
∗
∗
the periodic solution x t, y t is positive.
4. Examples
Example 4.1. Suppose ft, x 1 − 2 cos tx, gt, x 2xy/2 sin t x2 , γt 0.5,
μt 1/10 − 1/20 sin t, ht 1. The corresponding system is
ẋ x 1 − 2 cos tx −
2xy
,
2 sin t x2
x2
1
1
ẏ y
−
sin t .
−
2 sin t x2 10 20
4.1
Abstract and Applied Analysis
0.65
0.65
0.6
0.6
0.55
0.55
0.5
0.5
Predator population
Prey (blue line) and predator (dash line)
22
0.45
0.4
0.35
0.3
0.45
0.4
0.35
0.3
0.25
0.25
0.2
0.2
0
20
40
60
80
Start
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65
100
Prey population
Time
0.8
0.7
0.7
0.6
0.6
Start
0.5
0.5
Predator
Prey (blue line) and predator (red dash line)
a
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
20
40
60
80
100
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Prey
Time
b
Figure 1: a Simulation results of Example 4.1 with initial values x0 0.3, y0 0.5. b Simulation
results of Example 4.2 with initial values x0 0.3, y0 0.6.
We know that the periodic solution of system 3.1 is
ω
1 − exp − 0 asds
x t ω
ω
.
bt − s exp − 0 at − τdτ ds
0
∗
4.2
Abstract and Applied Analysis
23
Using this formula we can compute the periodic solution of 4.1 in absence of predators
x∗ t 2
≈
γtgt, x∗ tx∗ t
4e2π
coste2π
1 − e−2π
sinte2π − 4 − cost − sinte−2π
2
,
4 cost sint
1
− μt 2π
2π 0
1
1
−
sin t dt ≈ 0.05313 > 0.
−
2 sin t x∗ 2 10 20
4.3
x∗ 2
System 4.1 is permanent.
Example 4.2. Taking ft, x, gt, x and γt the same as in Example 4.1, and μt 1/5 −
1/10 sin t, we can compute
γtgt, x∗ tx∗ t
1
− μt 2π
2π 0
1
1
sin t dt ≈ −0.04687 < 0.
− −
2 sin t x∗ 2 5 10
4.4
x∗ 2
In this circumstance the predator population tends to extinction and the prey
population keeps oscillating.
Simulation results of the two examples are shown in Figure 1.
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