Applied Mathematics and Computation 218 (2011) 502–513
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Dispersal permanence of a periodic predator–prey system with Holling type-IV functional response Meihua Huang a,⇑, Xuepeng Li b a b
School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, People’s Republic of China School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t In this paper, we study the permanence of a class of periodic predator–prey system with Holling type-IV functional response where the prey disperses in patchy environment with two patches, and provide a sufficient and necessary condition to guarantee permanence of the system. Finally, two examples are presented to illustrate the application of our main results. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Predator–prey Permanence Holling type-IV Dispersal
1. Introduction Many kinds of predator–prey systems have been studied, especially, permanence of the predator–prey system, for example in [1–6] and the references cited therein. But, it is difficult to get some sufficient and necessary conditions for the permanence of a time dependent predator–prey system. Already, in [6], Cui proposed the following periodic predator–prey system with Beddington–DeAngelis functional response:
x_ 1 ¼ x1 b1 ðtÞ a1 ðtÞx1
c1 ðtÞy þ DðtÞðx2 x1 Þ; eðtÞ þ bðtÞx1 þ cðtÞy
x_ 2 ¼ x2 ½b2 ðtÞ a2 ðtÞx2 þ DðtÞðx1 x2 Þ; c1 ðtÞx1 y_ ¼ y dðtÞ þ qðtÞy : eðtÞ þ bðtÞx1 þ cðtÞy He obtained a sufficient and necessary condition to guarantee the predator and prey species to be permanent. In this paper, we consider the permanence of the following periodic predator–prey system with Holling type-IV functional response:
x_ 1 ¼ x1 b1 ðtÞ a1 ðtÞx1
c1 ðtÞy þ DðtÞðx2 x1 Þ; eðtÞ þ bðtÞx1 þ x21
x_ 2 ¼ x2 ½b2 ðtÞ a2 ðtÞx2 þ DðtÞðx1 x2 Þ; c2 ðtÞx1 qðtÞy : y_ ¼ y dðtÞ þ 2 eðtÞ þ bðtÞx1 þ x1
ð1:1Þ
where x1 and x2 denote the density of prey species in patch 1 and in patch 2 respectively, and y is the density of predator species that preys on x1 ai(t), bi(t), ci(t)(i = 1, 2), d(t), e(t), b(t), q(t) and D(t) are all positive, x-periodic and continuous for t P 0. ⇑ Corresponding author. E-mail address:
[email protected] (M. Huang). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.092
M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
503
The periodic functions in (1.1) have the following biological meanings. bi(t) is the intrinsic growth rate for prey species in patch i(i = 1, 2); ai(t) represents the self-inhibition coefficient; and D(t) is the diffusion coefficient of prey species from patch i to patch j(i – j, i, j = 1, 2). The death rate of the predator population is proportional to the existing predator population and to its square with coefficients d(t) and q(t), respectively. c1(t) and c2(t) give the coefficients that relate to the conversion rate of the prey biomass (in the patch 1) into predator biomass. c1 ðtÞx1 c1 ðtÞx1 The function eðtÞþbðtÞx 2 represents the functional response of predator to the prey in patch 1. Let uðt; x1 Þ ¼ eðtÞþbðtÞx þx2 , then 1 þx1 1 1 we have @ @x1
uðt; x1 Þ P 0; 0 < x1 ðtÞ 6
@ @x1
uðt; x1 Þ < 0;
pffiffiffiffiffiffiffiffi eðtÞ;
pffiffiffiffiffiffiffiffi x1 ðtÞ > eðtÞ:
ð1:2Þ
The aim of this paper is, by further developing the analysis technique of Cui [6], to obtain a sufficient and necessary condition to guarantee the predator and prey species to be permanent of the system (1.1). The organization of this paper is as follows. In Section 2, we introduce some notations, definitions and lemmas which will be essential to our proofs. In Section 3, we state and prove the main result of this paper. In Section 4, we give two examples which together show the feasibility of our results. We discuss the biological meanings of the main result in Section 5. 2. Notations, definitions and preliminaries In this section, we introduce some definitions and notations and state some results which will be useful in subsequent sections. Let C denote the space of all bounded continuous functions f : R ! R; C 0þ the set of nonnegative f 2 C, and C+ the set of all f 2 C such that f is bounded below by a positive constant. Given f 2 C, we denote
f M ¼ sup f ðtÞ;
f L ¼ inf f ðtÞ tP0
tP0
and define the lower average AL(f) and upper average AM(f) of f by
AL ðf Þ ¼ lim inf ðt sÞ1
Z
r!1 tsPr
t
f ðsÞds
s
and
AM ðf Þ ¼ lim sup ðt sÞ1 r!1 tsPr
Z
t
f ðsÞds; s
respectively. If f 2 C is x-periodic, we define the average Ax(f) of f on the time interval [0, x] by
Ax ðf Þ ¼ x1
Z
x
f ðtÞdt:
0
Definition 2.1. The system of differential equations
x_ ¼ Fðt; xÞ;
x 2 Rnþ ;
is said to be permanent if there exists a compact set K in the interior of Rnþ ¼ fðx1 ; x2 ; . . . ; xn Þ 2 Rn : xi P 0; i ¼ 1; 2; . . . ; ng, such that all solutions starting in the interior of Rnþ ultimately enter and remain in K. Lemma 2.1 (see [7]). Let x(t) and y(t) be solution of
x_ ¼ Fðt; xÞ and
y_ ¼ Gðt; yÞ; respectively, where both systems are assumed to have the uniqueness property for initial value problems. Assume both x(t) and y(t) belong to a domain D Rn for [t0, t1], in which one of the two systems is cooperative and
Fðt; zÞ 6 Gðt; zÞ;
ðt; zÞ 2 ½t0 ; t 1 D:
If x(t0) 6 y(t0) then x(t1) 6 y(t1). If F = G and x(t0) < y(t0) then x (t1) < y(t1).To prove the permanence of the species in (1.1), we need the information on the periodic logistic models with and without dispersal. Lemma 2.2 (see [8]). The problem
x_ ¼ x½bðtÞ aðtÞx;
x 2 Cþ;
has exactly one canonical solution U if a 2 C+, b 2 C and AL(b) > 0.
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M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
For the following dispersal logistic equations:
x_ 1 ¼ x1 ½b1 ðtÞ a1 ðtÞx1 þ DðtÞðx2 x1 Þ; x_ 2 ¼ x2 ½b2 ðtÞ a2 ðtÞx2 þ DðtÞðx1 x2 Þ;
ð2:2Þ
Lemma 2.3 (see [9]). Suppose that bi(t), ai(t)(i = 1, 2) and D(t) are positive and x- periodic functions, then (2.2) has a positive and x-periodic solution x1 ðtÞ; x2 ðtÞ , which is globally asymptotically stable with respect to R2þ n fOg. 3. Main result Theorem 3.1. Suppose that
x1 ðtÞ <
pffiffiffiffiffiffiffiffi eðtÞ;
tP0
ð3:1Þ
holds, then system (1.1) is permanent if and only if
Ax dðtÞ þ
c2 ðtÞx1 ðtÞ > 0: eðtÞ þ bðtÞx1 ðtÞ þ x2 1 ðtÞ
ð3:2Þ
Here x1 ðtÞ; x2 ðtÞ is the globally and asymptotically stable x-periodic solution of (2.2) given by Lemma 2.3. From the proof of Theorem 3.1, we also have the following Corollary 3.1. Corollary 3.1. Assume that (3.1) and
Ax dðtÞ þ
c2 ðtÞx1 ðtÞ 60 eðtÞ þ bðtÞx1 ðtÞ þ x2 1 ðtÞ
ð3:3Þ
hold, where x1 ðtÞ; x2 ðtÞ is the globally and asymptotically stable x-periodic solution of (2.2) given by Lemma 2.3. Then any solution of system (1.1) with positive initial condition satisfies
lim yðtÞ ¼ 0:
t!1
To prove this theorem, we need several propositions. In the rest of this paper we denote (x1(t), x2(t), y(t)) be any solution of (1.1) with positive initial condition. Proposition 3.1. There exist positive constants Mx and My, such that
lim sup xi ðtÞ 6 M x ;
t!1
lim sup yðtÞ 6 M y ;
t!1
i ¼ 1; 2:
ð3:4Þ
Proof. Obviously, R3þ is a positively invariant set of (1.1). Given any positive solution (x1(t), x2(t), y(t)) of (1.1), we have
x_ i 6 xi ½bi ðtÞ ai ðtÞxi þ DðtÞðxj xi Þ;
i; j ¼ 1; 2;
j – i;
on the other hand, the following auxiliary equations:
u_ i ¼ ui ½bi ðtÞ ai ðtÞui þ DðtÞðuj ui Þ;
i; j ¼ 1; 2;
j – i;
has a unique globally asymptotically stable positive x-periodic solution with ui(0) = xi(0), by Lemma 2.1 we have
xi ðtÞ 6 ui ðtÞ;
ð3:5Þ
x1 ðtÞ; x2 ðtÞ
. Let (u1(t), u2(t)) be the solution of (3.5)
i ¼ 1; 2 for t P 0:
Moreover, from the global stability of x1 ðtÞ; x2 ðtÞ , for every given e(0 < e < 1), there exists T0 > 0, such that
ui ðtÞ < xi ðtÞ þ e for t > T 0 ; hence
xi ðtÞ < xi ðtÞ þ e;
i ¼ 1; 2 for t > T 0 :
In addition we have
y_ 6 yðdðtÞ þ
c2 ðtÞ x1 ðtÞ þ e qðtÞyÞ; eðtÞ
t P T 0:
By Lemmas 2.1 and 2.2, there exists T1 > T0, such that
yðtÞ < y ðtÞ þ e;
for t > T 1 ;
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M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
where y⁄(t) is the positive and globally asymptotically stable x-periodic solution of the following auxiliary logistic equation:
v_ ¼ v
c2 ðtÞðx1 ðtÞ þ eÞ qðtÞv : dðtÞ þ eðtÞ
Denote M x ¼ max06t6x xi ðtÞ þ 1 : i ¼ 1; 2 and My = max06t6x{y⁄(t) + 1}, then (3.4) holds for system (1.1).
h
Proposition 3.2. Suppose (3.2) holds, then there exists a positive constant gx such that
lim sup x1 ðtÞ P gx :
ð3:6Þ
t!1
Proof. Suppose that (3.6) is not true, then there is a sequence fzm g R3þ , such that
lim sup x1 ðt; zm Þ <
t!1
1 ; m
m ¼ 1; 2; . . . ;
ð3:7Þ
where (x1(t, zm), x2(t, zm), y(t, zm)) is the solution of (1.1) with initial values (x1(0, zm), x2(0, zm), y(0, zm)) = zm. Choose positive e sufficiently small satisfies:
c2 ðtÞe Þ < 0; eðtÞ c1 ðtÞe b1 ðtÞ > 0; eðtÞ c1 ðtÞe expðaxÞ a1 ðtÞe; b2 ðtÞ a2 ðtÞe > 0: /e ðtÞ ¼ min b1 ðtÞ eðtÞ
Ax ðdðtÞ þ
ð3:8Þ ð3:9Þ ð3:10Þ
n o 2 ðtÞ where a ¼ max06t6x dðtÞ þ ceðtÞ . By (3.7), for the given e > 0, there exists a positive integer N0, such that
lim sup x1 ðt; zm Þ <
t!1
1
ð3:11Þ ðmÞ
for m > N0. In the rest of this proof we always assume that m > N0. Above inequality implies that there exists s1
> 0, such that
x1 ðt; zm Þ < e ðmÞ
for t P s1 , and further
c ðtÞe _ zm Þ 6 yðt; zm Þ dðtÞ þ 2 yðt; qðtÞyðt; zm Þ eðtÞ ðmÞ
for t P s1 . By (3.8), any solution v(t) of the following equation:
v_ ¼ v
dðtÞ þ
c2 ðtÞe qðtÞv ; eðtÞ
with positive initial condition satisfies
lim v ðtÞ ¼ 0:
t!1
By Lemma 2.1, we have
lim yðt; zm Þ ¼ 0:
t!1
Therefore, there is a
ðmÞ sðmÞ > s1 such that 2
ðmÞ
yðt; zm Þ < e for t P s2 :
ð3:12Þ
This leads to
c1 ðtÞe x_ 1 ðt; zm Þ P x1 ðt; zm Þ b1 ðtÞ a1 ðtÞx1 ðt; zm Þ þ DðtÞðx2 ðt; zm Þ x1 ðt; zm ÞÞx_ 2 ðt; zm Þ eðtÞ ¼ x2 ðt; zm Þ½b2 ðtÞ a2 ðtÞx2 ðt; zm Þ þ DðtÞðx1 ðt; zm Þ x2 ðt; zm ÞÞ ðmÞ 2 .
for t P s
Let (u1(t), u2(t)) be any positive solution of the following auxiliary equations:
c1 ðtÞe u_ 1 ¼ u1 b1 ðtÞ a1 ðtÞu1 þ DðtÞðu2 u1 Þ; eðtÞ u_ 2 ¼ u2 ½b2 ðtÞ a2 ðtÞu2 þ DðtÞðu1 u2 Þ:
ð3:13Þ
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M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
By (3.9) and Lemma 2.3, (3.13) has a unique positive and x-periodic solution u1 ðtÞ; u2 ðtÞ , which is globally asymptotically stable. So we have
xi ðt; zm Þ >
ui ðtÞ ; 2
i ¼ 1; 2
from Lemma 2.1 for sufficiently large t > 0 and m > N0, which is a contradiction with (3.7). This completes the proof.
h
Proposition 3.3. There exists a positive constant cx such that
lim inf qx ðtÞ P cx ;
ð3:14Þ
t!1
where qx(t) = x1(t) + x2(t). Proof. For each positive e in ((3.8)–(3.10)), we choose P = lx(l is a positive integer) sufficiently large such that
M exp
Z 0
P
c2 ðtÞe dðtÞ þ qðtÞe dt < e; eðtÞ
ð3:15Þ
where M = max{Mx, My}. Let positive integer m big enough such that
gx m where
< e;
ln m > 2Pf0 ;
ð3:16Þ
c1 ðtÞMy f0 ¼ max b1 ðtÞ þ a1 ðtÞMx þ ; b2 ðtÞ þ a2 ðtÞM x : 0 6 t 6 x eðtÞ
Suppose that (3.14) is not true, then for each m in (3.16) there exists zm 2 R3þ , such that
lim inf qx ðt; zm Þ <
t!1
gx
m2
:
On the other hand, by Proposition 3.2, we have
lim sup qx ðt; zm Þ P lim sup x1 ðt; zm Þ P gx :
t!1
t!1
Hence there are two sequences {sq} and {tq} satisfying the following conditions:
0 < s1 < t 1 < s2 < t2 < < sq < tq < ; sq ! 1 as q ! 1 and
qx ðsq ; zm Þ ¼
gx m
;
qx ðtq ; zm Þ ¼
gx m2
;
gx m2
< qx ðt; zm Þ <
gx m
;
t 2 ðsq ; t q Þ:
ð3:17Þ
By Proposition 3.1, for a given integer m > 0, there is a T1 > 0, such that
xi ðt; zm Þ 6 M x ; yðt; zm Þ 6 M y ; fort P T 1
and i ¼ 1; 2:
Because of sq ? 1 as q ? 1, there is a positive integer K, such that sq > T1 as q P K, hence
c1 ðtÞMy x_ 1 ðt; zm Þ P x1 ðt; zm Þ b1 ðtÞ a1 ðtÞM x þ DðtÞðx2 ðt; zm Þ x1 ðt; zm ÞÞ eðtÞ x_ 2 ðt; zm Þ P x2 ðt; zm Þ½b2 ðtÞ a2 ðtÞM x þ DðtÞðx1 ðt; zm Þ x2 ðt; zm ÞÞ
for q P K, so
q_ x ðt; zm Þ P f0 qx ðt; zm Þ for q P K, t 2 [sq, tq]. Integrating it from sq to tq we have
qx ðtq ; zm Þ P qx ðsq ; zm Þ exp
Z
tq
ðf0 Þdt:
sq
By (3.16) and (3.17) we have
tq sq > 2P where q P K. Note that
xi ðt; zm Þ < e;
i ¼ 1; 2;
t 2 ½sq ; t q
ð3:18Þ
from (3.16) and (3.17). For the positive e satisfying (3.8, 3.9,3.10) and (3.15), we have the following two circumstances for y(t, zm):
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M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
(i) y(t, zm) P e for all t 2 [sq, sq + P]; (ii) there exists sq1 2 [sq, sq + P], such that y(sq1, zm) < e.If (i) holds, by (3.15) and (3.18) we have
e 6 yðsq þ P; zm Þ 6 yðsq ; zm Þ exp
Z
Z P c2 ðtÞe c2 ðtÞe dðtÞ þ dðtÞ þ qðtÞe dt 6 M exp qðtÞe dt < e; eðtÞ eðtÞ 0
sq þP
sq
which is a contradiction. If (ii) holds,we now claim that
t 2 ðsq1 ; t q :
yðt; zm Þ 6 e expðaxÞ; Otherwise, there exists
ð3:19Þ
sq2 2 ðsq1 ; tq such that
yðsq2 ; zm Þ > e expðaxÞ:
sq3 2 ðsq1 ; sq2 Þ such that
By the continuity of y(t, zm), there must exist
yðsq3 ; zm Þ ¼ e and
yðt; zm Þ > e for t 2 ðsq3 ; sq2 Þ: Denote P1 the nonnegative integer such that
sq2 2 ðsq3 þ P1 x; sq3 þ ðP1 þ 1Þx, by (3.8) we obtain
c2 ðtÞe qðtÞe dt eðtÞ sq3 Z sq3 þP1 x Z sq2 ! c2 ðtÞe ¼ e exp dðtÞ þ þ qðtÞe dt < e expðaxÞ: eðtÞ sq3 sq3 þP1 x
e expðaxÞ < yðsq2 ; zm Þ < yðsq3 ; zm Þ exp
Z sq2
dðtÞ þ
This contradiction establishes that (3.19) is true, particularly (3.19) holds for t 2 [sq + P, tq]. By (3.17) and (3.10), we have
gx m2
¼ qx ðt q ; zm Þ P qx ðsq þ P; zm Þ exp
Z
tq
sq þP
/e ðtÞdt >
gx m2
;
which is also a contradiction. This completes the proof. h
Proposition 3.4. There exists positive constants cxi(i = 1, 2) such that
lim inf xi ðtÞ P cxi
t!1
i ¼ 1; 2:
ð3:20Þ
Proof. (3.14) implies that there exists T2 P T1 such that
qx ðtÞ ¼ x1 ðtÞ þ x2 ðtÞ P cx for t P T 2 : Hence,
x_ 1 ¼ x1 b1 ðtÞ 2DðtÞ a1 ðtÞx1
c1 ðtÞy cM L M 2 1 My þ DðtÞqx ðtÞ P aM Þx1 þ DL cx :¼ F 1 ðx1 Þ 1 x1 þ ðb1 2D 2 eL eðtÞ þ bðtÞx1 þ x1
and
L M 2 x_ 2 P aM x2 þ DL cx :¼ F 2 ðx2 Þ 2 x2 þ b2 2D for t P T2. The algebraic equation F1(x1) = 0 gives us one positive root L
~x1 ¼
b1 2DM
cM My 1 eL
þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 cM My L b1 2DM 1eL þ 4DL aM 1 cx 2aM 1
:
Clearly, F1(x1) > 0 for every positive number x1 ð0 < x1 < ~ x1 Þ. Choose cx1 ð0 < cx1 < ~ x1 Þ; x_ 1 jx1 ¼cx1 P F 1 ðcx1 Þ > 0. If x1(T2) P cx1 then it also holds for t P T2; if x(T2) < cx1, then
x_ 1 ðT 2 Þ P inffF 1 ðx1 Þj0 6 x1 < cx1 g > 0; there must exist T3(PT2), such that x1(t) > cx1 for t P T3. Similarly, there exists cx2 > 0 and T4(PT3), such that x2(t) > cx2 for t P T4. This completes the proof. h
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M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
Proposition 3.5. Suppose that (3.2) holds, then there exists a positive constant gy such that
lim sup yðtÞ > gy :
ð3:21Þ
t!1
Proof. By (3.2), we can choose positive constant e0 < e such that
Ax ðwe0 ðtÞÞ > 0;
ð3:22Þ
where
we0 ðtÞ ¼ dðtÞ þ
c2 ðtÞ x1 ðtÞ e0 2 qðtÞe0 : eðtÞ þ bðtÞ x1 ðtÞ e0 þ x1 ðtÞ e0
bL eL Consider the following equations with parameter a 0 < a < 2c1 M :
2ac1 ðtÞ a1 ðtÞx1 þ DðtÞðx2 x1 Þ; x_ 1 ¼ x1 b1 ðtÞ eðtÞ x_ 2 ¼ x2 ½b2 ðtÞ a2 ðtÞx2 þ DðtÞðx1 x2 Þ:
1
ð3:23Þ
By Lemma 2.3, (3.23) has a unique x-periodic solution (x1a(t), x2a(t)), which is globally asymptotically stable. Let ð~ x1a ðtÞ; ~ x2a ðtÞÞ be the solution of (3.23) with initial condition ~ xia ð0Þ ¼ xi ð0Þ; i ¼ 1; 2, where x1 ðtÞ; x2 ðtÞ be the positive and x-periodic solution of (2.2), then for the given e0, there exists T5 P T4, such that
e0
j~x1a ðtÞ x1a ðtÞj <
for t P T 5 :
4
By the continuity of solution to parameter, we have ð~ x1a ðtÞ; ~ x2a ðtÞÞ ! x1 ðtÞ; x2 ðtÞ uniformly in [T5, T5 + x] as a ? 0. Hence for L L
e0 > 0, there exists positive a0 ¼ a0 ðe0 Þ < b2c1 eM such that 1
~x1a ðtÞ x ðtÞ < e0 1 4
for t 2 ½T 5 ; T 5 þ x;
0 < a < a0 :
So we have
x1a ðtÞ x ðtÞ 6 j~x1a ðtÞ x1a ðtÞj þ j~x1a ðtÞ x ðtÞj < e0 1 1 2 for t 2 [T5, T5 + x]. Since x1a(t) and x1 ðtÞ are all x-periodic, we have
x1a ðtÞ x ðtÞ < e0 1 2
for t P 0;
0 < a < a0 :
Choosing constant a1(0 < a1 < a0, 2a1 < e0), then
x1a1 ðtÞ P x1 ðtÞ
e0 2
;
t P 0:
ð3:24Þ
Suppose that conclusion (3.21) is not true, then there exists Z 2 R3þ , for the positive solution (x1(t), x2(t), y(t)) of (1.1) with initial condition (x1(0), x2(0), y(0)) = Z, we have
lim sup yðtÞ < a1 :
t!1
So there exists T6 P T5 such that
yðtÞ < 2a1 ;
for t P T 6
ð3:25Þ
and hence
2a1 c1 ðtÞ a1 ðtÞx1 þ DðtÞðx2 x1 Þ; x_ 1 P x1 b1 ðtÞ eðtÞ x_ 2 ¼ x2 ½b2 ðtÞ a2 ðtÞx2 þ DðtÞðx1 x2 Þ: Let (u1(t), u2(t)) be the solution of (3.23) with a = a1 and ui(T6) = xi(T6), i = 1, 2, by Lemma 2.1 we know that
xi ðtÞ P ui ðtÞ;
t P T 6;
i ¼ 1; 2:
By the globally asymptotical stability of ðx1a1 ðtÞ; x2a1 ðtÞÞ, for given e0, there exists T7 P T6 such that
ju1 ðtÞ x1a1 ðtÞj <
e0 2
for t P T 7 :
M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
509
So we have
x1 ðtÞ P u1 ðtÞ > x1a1 ðtÞ
e0 2
t P T7
;
and hence
x1 ðtÞ > x1 ðtÞ e0 ;
t P T7:
This implies
_ yðtÞ P we0 ðtÞyðtÞ for t P T 7 : integrating above inequality from T7 to t yields
yðtÞ P yðT 7 Þ exp
Z
t
T7
we0 ðtÞdt:
By (3.22) we know that y(t) ? 1 as t ? 1, which is a contradiction. This completes the proof. h Proposition 3.6. Under assumption (3.2), there exists a positive constant cy such that
lim inf yðtÞ > cy :
ð3:26Þ
t!1
Proof. For each positive e0 in (3.22), we choose P > 0 sufficiently large such that
Z
a
0
we0 ðtÞdt > 0
ð3:27Þ
for a P P. Further, let positive integer m big enough such that
gy mþ1
< a1 ð2a1 < e0 Þ;
lnðm þ 1Þ > 2P max fdðtÞ þ qðtÞM y g:
ð3:28Þ
06t6x
Suppose that (3.26) is not true, then for each m in (3.28) there exists zm 2 R3þ , such that
lim inf yðt; zm Þ <
t!1
gy
:
ðm þ 1Þ2
But
lim sup yðt; zm Þ > gy :
t!1
According to Proposition 3.5. Hence there are two sequences {sq} and {tq} satisfying the following conditions:
0 < s1 < t 1 < s2 < t2 < < sq < tq < ; sq ! 1 as q ! 1 and
yðsq ; zm Þ ¼
gx mþ1
;
yðt q ; zm Þ ¼
gx ðm þ 1Þ2
;
gy ðm þ 1Þ2
< yðt; zm Þ <
gy mþ1
;
t 2 ðsq ; t q Þ:
ð3:29Þ
By Proposition 3.1, for a given integer m > 0, there is a T1 > 0, such that
yðt; zm Þ 6 M y ;
for t P T 1 :
Because of sq ? 1 as q ? 1, there is a positive integer K, such that sq > T1 as q P K, hence
_ zm Þ P yðt; zm Þ½dðtÞ qðtÞMy yðt; for q P K, t 2 [sq, tq]. Integrating above inequality from sq to tq we get
yðt q ; zm Þ P yðsq ; zm Þ exp
Z
tq
dðtÞ qðtÞM y dt:
sq
Combining (3.28) and (3.29) we have
tq sq > 2P and
tq sq ! 1; where m ? 1 and q P K.
ð3:30Þ
510
M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
Further we have
yðt; zm Þ < 2a1 ;
t 2 ½sq ; tq
from (3.28). In addition, for t 2 [sq, tq], we have
2a1 c1 ðtÞ a1 ðtÞx1 ðt; zm Þ þ DðtÞðx2 ðt; zm Þ x1 ðt; zm ÞÞ; x_ 1 ðt; zm Þ P x1 ðt; zm Þ b1 ðtÞ eðtÞ x_ 2 ðt; zm Þ ¼ x2 ðt; zm Þ½b2 ðtÞ a2 ðtÞx2 ðt; zm Þ þ DðtÞðx1 ðt; zm Þ x2 ðt; zm ÞÞ:
Let (u1(t), u2(t)) be the solution of (3.23) with a = a1 and ui(sq) = xi(sq, zm), by Lemma 2.1 we have
xi ðt; zm Þ P ui ðtÞ;
t 2 ½sq ; tq :
Further, by Propositions 3.1, 3.4 and sq ? 1 as q ? 1, we can choose K1 > K, such that
cxi 6 xi ðsq ; zm Þ 6 Mx ; i ¼ 1; 2 holds for q P K1. For a = a1, (3.23) has a unique x-periodic solution ðx1a1 ðtÞ; x2a1 ðtÞÞ, which is globally asymptotically stable. Because (3.23) is a periodic system, the periodic solution ðx1a1 ðtÞ; x2a1 ðtÞÞ is uniformly asymptotically stable in R3þ (see [10]). By definitions 11.2 and 11.5 in [10], for the given set X and each e0, there is a corresponding T0 = T0(e0)(>P) such that if (u1(sq), u2(sq)) = (x1(sq, zm), x2(sq, zm)) 2 X for some sq > 0 then
ju1 ðtÞ x1a1 ðtÞj <
e0 2
for all t P T0 + sq. Hence
u1 ðtÞ P x1a1 ðtÞ
e0 2
t P T 0 þ sq :
;
Combining (3.24) we have
u1 ðtÞ P x1 ðtÞ e0 ;
t P T 0 þ sq :
From (3.30), there exists a positive integer N1 P N0, such that tq > sq + 2T0 > sq + 2P for m P N1 and q P K1. So we have
x1 ðt; zm Þ P x1 ðtÞ e0 ;
t 2 ½sq þ T 0 ; t q
as m P N1 and q P K1. Hence
_ zm Þ P we0 ðtÞyðt; zm Þ yðt; for t 2 [sq + T0, tq]. Integrating above inequality from sq + T0 to tq yields
yðtq ; zm Þ P yðsq þ T 0 ; zm Þ exp
Z
tq
sq þT 0
we0 ðtÞdt;
that is to say
gy 2
ðm þ 1Þ
P
gy 2
ðm þ 1Þ
exp
Z
tq sq þT 0
we0 ðtÞdt >
gy ðm þ 1Þ2
;
which is a contradiction. This completes the proof. h Combining the Propositions 3.1, 3.2, 3.33.4, 3.5, 3.6, we completes the proof of the sufficiency of Theorem 3.1. To prove the necessity of Theorem 3.1, we will show that
lim yðtÞ ¼ 0
t!1
under the following condition:
Ax ðdðtÞ þ
c2 ðtÞx1 ðtÞ Þ 6 0: eðtÞ þ bðtÞx1 ðtÞ þ x2 1 ðtÞ
In fact, by (3.3) we know that for every given e(0 < e < 1), there exists e1 > 0 and e0 > 0 such that
Ax
! c2 ðtÞ x1 ðtÞ þ e1 e dðtÞ þ 2 qðtÞe 6 Ax ðqðtÞÞ 6 e0 : 2 eðtÞ þ bðtÞ x1 ðtÞ þ e1 þ x1 ðtÞ þ e1
ð3:31Þ
M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
511
Since
x_ 1 6 x1 ½b1 ðtÞ a1 ðtÞx1 þ DðtÞðx2 x1 Þ; x_ 2 ¼ x2 ½b2 ðtÞ a2 ðtÞx2 þ DðtÞðx1 x2 Þ: we know that for the given e1 there exists T(1) > 0 such that
for t P T ð1Þ :
x1 ðtÞ 6 x1 ðtÞ þ e1
By (3.31) and (3.1) we have
Ax ðdðtÞ þ
c2 ðtÞx1 ðtÞ qðtÞeÞ 6 e0 eðtÞ þ bðtÞx1 ðtÞ þ x21 ðtÞ
ð3:32Þ
for t P T(1). Firstly, there must exist T(2) such that y(T(2)) < e. Otherwise, we have
e 6 yðtÞ 6 yðT ð1Þ Þ exp
Z T
t ð1Þ
½dðsÞ þ
c2 ðsÞx1 ðsÞ qðsÞeds ! 0 as t ! 0: eðsÞ þ bðsÞx1 ðsÞ þ x21 ðsÞ
This implies e 6 0, which is a contradiction. Let
MðeÞ ¼ max
06t6x
dðtÞ þ
c2 ðtÞx1 ðtÞ : e þ qðtÞ eðtÞ þ bðtÞx1 ðtÞ þ x21 ðtÞ
By Proposition 3.1, we know that x1(t) is bounded. So M(e) is also bounded for e 2 [0, 1]. Secondly we have
yðtÞ 6 e expðMðeÞxÞ for t P T ð2Þ : (3)
Otherwise, there exists T
(2)
>T
ð3:33Þ
such that
ð3Þ
yðT Þ > e expðMðeÞxÞ: By the continuity of y(t), there must exist T ð4Þ 2 ðT ð2Þ ; T ð3Þ Þ such that y(T(4)) = e and y(t) > e for t 2 ðT ð4Þ ; T ð3Þ . Let P1 be the nonnegative integer such that T ð3Þ 2 ðT ð4Þ þ P1 x; T ð4Þ þ ðP1 þ 1Þx, by (3.32) we have
c2 ðtÞx1 ðtÞ dt e qðtÞ eðtÞ þ bðtÞx1 ðtÞ þ x21 ðtÞ T ð4Þ ! Z T ð4Þ þP1 x Z T ð3Þ c2 ðtÞx1 ðtÞ ¼ e exp dt < e expðMðeÞxÞ: þ e dðtÞ þ qðtÞ eðtÞ þ bðtÞx1 ðtÞ þ x21 ðtÞ T ð4Þ T ð4Þ þP 1 x
e expðMðeÞxÞ < yðT ð3Þ Þ < yðT ð4Þ Þ exp
Z
T ð3Þ
dðtÞ þ
which is a contradiction. This implies (3.33) holds. Further by the arbitrariness of e we know that y(t) ? 0 as t ? 1. This completes the proof of Theorem 3.1. 4. Examples Examples 1. Consider the following predator–prey system:
t 2 þ cos y 10 þ 4ðx2 x1 Þ; x_ 1 ¼ x1 4 2x1 20 þ x1 þ x21 h i x2 x_ 2 ¼ x2 6 þ 4ðx1 x2 Þ; 2 t 2 þ cos x1 1 cos t 10 þ ð2 þ sin tÞy : y_ ¼ y 10 100 20 þ x1 þ x21
ð4:1Þ
t In this case, corresponding to system (1.1), one has a1 ðtÞ ¼ 2; b1 ðtÞ ¼ 4; a2 ðtÞ ¼ 12 ; b2 ðtÞ ¼ 6; DðtÞ ¼ 4; c1 ðtÞ ¼ c2 ðtÞ ¼ 2 þ cos ; 10 1 t eðtÞ ¼ 20; bðtÞ ¼ 1; dðtÞ ¼ 10 þ cos ; qðtÞ ¼ 2 þ sin t: One could easily see that 100
x_ 1 ¼ 4x2 2x21 ; 1 x_ 2 ¼ 4x1 þ 2x2 x22 2
ð4:2Þ
has a unique positive periodic solution x1 ðtÞ; x2 ðtÞ ¼ ð4; 8Þ, i.e. the positive periodic solution is the positive equilibrium. By simple computation, one has
Ax dðtÞ þ
c2 ðtÞx1 ðtÞ 2 1 ¼ > 0: 2 10 10 eðtÞ þ bðtÞx1 ðtÞ þ x1 ðtÞ
512
M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
Using Theorem 3.1, we know that system (4.1) is permanent. Fig. 1 shows the dynamic behavior of the system (4.1) with initial condition (x1(0), x2(0), y(0)) = (15, 10, 5), here t 2 [0, 100]. Examples 2. Consider the following predator–prey system:
t 2 þ cos y 10 þ 4ðx2 x1 Þ; x_ 1 ¼ x1 4 2x1 20 þ x1 þ x21 h i x2 þ 4ðx1 x2 Þ; x_ 2 ¼ x2 6 2 t 2 þ cos x1 3 cos t 10 y_ ¼ y ð2 þ sin tÞy : þ 10 100 20 þ x1 þ x21
Here, we only replace d(t) by
3 10
ð4:3Þ
t þ cos , all the other coefficients of system (4.3) are the same to that of system (4.1). 100
16
x1 x2
14
y
x1(t),x2(t),y(t)
12 10 8 6 4 2
0
20
40
60
80
100
time t Fig. 1. The dynamic behavior of the system (4.1).
5
y
4.5 4 3.5
y(t)
3 2.5 2 1.5 1 0.5 0
0
20
40
60 time t
Fig. 2. The dynamic behavior of the predator.
80
100
M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513
513
One could easily see that in this case
Ax dðtÞ þ
c2 ðtÞx1 ðtÞ 2 3 < 0: ¼ eðtÞ þ bðtÞx1 ðtÞ þ x 21 ðtÞ 10 10
Hence, corresponding to Corollary 3.1, we know that any positive solution of system (4.3) satisfies limt?1y(t) = 0. Fig. 2 shows the dynamic behavior of the predator in system (4.3) with initial condition (x1(0), x2(0), y(0)) = (15, 10, 5), here t 2 [0, 100]. 5. Discussion In this paper, a model which describes a periodic predator–prey system with Holling type-IV functional response is proposed. By Lemma 2.3, system (1.1) without predator y(t) has a unique positive periodic solution which is globally asymptotically stable. Theorem 3.1 says that system (1.1) with y(t) is permanent under (3.2) if the prey dispersal system (2.2) has such a globally asymptotically stable positive x-periodic solution. Otherwise, if (3.2) is not true then the predator goes to extinct by Corollary 3.1. c2 ðtÞx1 ðtÞ In (3.2), the term eðtÞþbðtÞx ðtÞþx2 ðtÞ describes the growth of the predator by foraging the prey in patch 1, of which quantity is 1 1 specified as x1 ðtÞ. Note that x1 ðtÞ; x2 ðtÞ is a globally asymptotically stable periodic solution in prey dispersal system (2.2) and the predator is confined only in patch 1. Hence condition (3.2) implies that the growth by foraging minus the death for predator is positive on the average. If it is negative, the extinction is inevitable for the predator. Since we have assumed condition (3.1) holds for the functional response of predator to the prey in the patch 1, it is natural to discuss that under the assumption (3.1) not always holds, whether one could obtain the sufficient and necessary condition which ensure the permanence of the system (1.1) or not. The study of this problem is our future work. References [1] J. Cui, L. Chen, Permanent and extinction in logistic and Lotka–Volterra systems with diffusion, J. Math. Anal. Appl. 258 (2001) 512–535. [2] Z. Teng, Uniform persistence of the periodic predator–prey Lotka–Volterra systems, Appl. Anal. 72 (1999) 339–352. [3] R. Xu, L.S. Chen, Persistence and global stability for three-species ratio-dependent predator–prey system with time delays, J. Systems Sci. Math. Sci. 21 (2) (2001) 204–212. [4] W. Wang, Z. Ma, Harmless delays for uniform persistence, J. Math. Anal. Appl. 158 (1991) 256–268. [5] H.I. Freedman, S. Ruan, M. Tang, Uniform persistence near a closed positively invariant set, J. Dyn. Differ. Equa. 6 (1994) 583–600. [6] J. Cui, Dispersal permanence of a periodic predator-prey system with Beddington–DeAngelis functional response, Nonlinear Anal. 64 (2006) 440–456. [7] H.L. Smith, Cooperative systems of differential equation with concave nonlinearities, Nonlinear Anal. 10 (1986) 1037–1052. [8] X.Q. Zhao, The qualitative analysis of N-species Lotka–Volterra periodic competition systems, Math. Comp. Model 15 (1991) 3–8. [9] R. Mahbuba, L.S. Chen, On the nonautonomous Lotka–Volterra competition system with diffusion, Differ. Equa. Dyn. Syst. 2 (1994) 243–253. [10] T. Yoshizawa, Stability by Liapunov’s Second Method, Mathematical Society of Japan, Tokyo, 1966. publication no. 9.