- Email: [email protected]
Positive periodic solutions in a non-selective harvesting predator–prey model with multiple delays
J. Math. Anal. Appl. 395 (2012) 298–306
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Positive periodic solutions in a non-selective harvesting predator–prey model with multiple delays✩ Guodong Zhang a,b , Yi Shen a,∗ , Boshan Chen b a
Department of Control Science and Engineering and the Key Laboratory of Education Ministry of Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan 430074, China b
College of Mathematics and statistics, Hubei Normal University, Huangshi 435002, China
article
abstract
info
Article history: Received 10 October 2011 Available online 26 May 2012 Submitted by M. Iannelli
This paper is concerned with a predator–prey model with Hassell–Varley type functional response, non-selective harvesting and multiple delays. Some new sufficient conditions are obtained for the existence of positive periodic solutions by applying the coincidence degree theorem. In addition, the results are delay-dependent, which is different from the previous work. Finally, simulations illustrate the effectiveness of our results. © 2012 Elsevier Inc. All rights reserved.
Keywords: Predator–prey model Periodic solution Harvesting Time delay Coincidence degree theorem
1. Introduction Lotka [1] and Volterra [2] introduced the first predator–prey model in 1925 and 1926, respectively. After that many more complicated but realistic predator–prey models have been formulated by ecologists and mathematicians. In general, a predator–prey system may have the form
dx(t ) x =r 1− x − yφ(x), dt
k
dy(t ) = y(−d + µφ(x))
(1.1)
dt
where φ(x) is the functional response function, which reflects the capture ability of the predator to prey. For more biological meaning, the reader may consult Freedman [3] and Murry [4]. Recently, there has been a growing explicit biological and physiological evidence [5–7] that in many situations, especially when predators have to share or compete for food, a more suitable general predator–prey theory should be based on the socalled ratio-dependent theory. This is strongly supported by numerous field and laboratory experiments and observations [8,9]. For this reason, Kuang and Beretta [5], Jost et al. [6], Hsu et al. [7] and Maiti et al. [10] investigated the following
✩ This work is supported by the National Science Foundation of China under Grant Nos. 60974021, the Key Program of National Natural Science Foundation of China under Grant Nos. 61134012 and also supported by China Postdoctoral Science Foundation funded project under Grant Nos. 2012M511615. ∗ Corresponding author. E-mail addresses: [email protected] (G. Zhang), [email protected] (Y. Shen), [email protected] (B. Chen).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.05.045
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
299
Michaelis–Menten type ratio-dependent predator–prey system:
dx(t ) axy x x− = r 1 − , dt k my + x dy(t ) bx = y −d + , dt my +x x(0) > 0, y(0) > 0
(1.2)
and gave a nice systematic work on the global qualitative analysis of this system. In many earlier studies, it has been shown that harvesting has a strong impact on dynamic evolution of a population, e.g., see [11–17]. So the study of the population dynamics with harvesting is becoming a very important subject in mathematical bio-economics. The predator–prey food chain model with harvesting is generally described as
x x dx(t ) = r 1 − x − y φ − h1 , dt k y dy(t ) x = y −d + µφ − h2 dt
(1.3)
y
where h1 and h2 represent harvesting terms. In fact, as was pointed by Kuang [18] that any model of species dynamics without delays is an approximation at best. And human needs are not invariable for a long time. In detail, human needs increase as biological resources become abundant, while human needs decrease as biological resource is exiguous. So in the literature [13], the authors focus on varying harvesting rate in system (1.3) with time delay as follows:
dx(t ) a1 xy = r1 x − b1 x2 − − c1 x, dt x + k1 dy(t ) a2 y(t − τ ) = y r2 − − c2 y dt x(t − τ ) + k2
(1.4)
where c1 and c2 are the harvesting coefficients of prey and predator respectively. They show that the harvest effort on both prey and predator influences the stability of the system. Motivated by the above results, and the literature [11,19–23] and considering that the delay may occur in the preys and predators, in this paper, we discuss a non-autonomous predator–prey model with Hassell–Varley (See [7,19,22]) type functional response and two delays in the system as follows:
dx(t ) = x(t )(r1 (t ) − b(t )x(t − τ1 (t )) − dt dy (t )
dt
a1 (t )y(t )
) − c1 (t )x(t ),
myγ (t ) + x(t ) a2 (t )x(t − τ2 (t )) = y(t )(−r2 (t ) + ) − c2 (t )y(t ) myγ (t ) + x(t − τ2 (t ))
(1.5)
where a1 , a2 , b, c1 , c2 , r1 , r2 , τ1 , τ2 are continuously nonnegative periodic functions with period ω; m ≥ 0, γ ∈ (0, 1], is called the Hassell–Varley constant. For θ ∈ [−τ¯ , 0], we use the notation xt (θ ) = x(t + θ ), then the initial conditions of system (1.5) take the form x0 (θ ) = φ(θ ), x(0) = φ > 0, y(0) > 0, ∀ θ ∈ [−τ¯ , 0], where τ¯ , supt ∈[0,ω] {τ1 (t ), τ2 (t )}, φ ∈ C ([−τ¯ , 0], R) with the norm ∥z ∥I = supt ∈[−τ¯ ,0] ∥z (t )∥. The organization of this paper is as follows. In Section 2, by applying the coincidence degree theorem, we obtain the new sufficient conditions for the existence of positive periodic solutions for system (1.5). It is interesting that the results are delay-dependent, which is different from the previous works that are delay-independent, and our model is more reasonable and general. In Section 3, numerical simulations are performed to illustrate the effectiveness of our results. Finally, this paper ends by a conclusion. 2. Existence of periodic solutions In order to obtain the existence of positive periodic solutions for system (1.5), we first make the following preparations. Let X and Y be two Banach spaces, L : Dom L ∩ X → Y be a linear mapping and N : X → Y be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if there exist continuous projections P : X → X and Q : Y → Y such that Im P = Ker L and Im L = Ker Q = Im(I − Q ). It follows that the mapping L|dom L ∩ Ker P : (I − P )X → Im L has an inverse mapping, denoted by Kp . For an open bounded subset of X , the mapping N is called L-compact on Ω if QN (Ω ) is bounded and Kp (I − Q )N : Ω → X is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L. Here we state the Gaines–Mawhin Lemma from [24], which is a main tool in the proofs of our theorems. Lemma 2.1 (Continuation Theorem (see [24, p. 40])). Let L be a Fredholm mapping of index zero. Assume that N : Ω → Z is L-compact on Ω with Ω open bounded in X . Furthermore, assume: (a) For each λ ∈ (0, 1), every solution x of Lx ̸= λNx is such that x ∈ ∂ Ω ∩ Dom L. (b) QNx ̸= 0 for each x ∈ ∂ Ω ∩ Ker L and the Brouwer degree, deg (JNQ , Ω ∩ Ker L, 0) ̸= 0. Then, the operator equation Lx = Nx has at least one solution in Dom L ∩ Ω .
300
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306 dτ (t )
Lemma 2.2 (See [25]). If τ ∈ C 1 (R, R) with τ (t + ω) ≡ τ (t ) and dt < 1 for t ∈ [0, ω], then function µ(t ) = t − τ (t ) has a unique inverse µ−1 (t ) satisfying µ ∈ C (R, R) with µ−1 (s + ω) ≡ µ−1 (s) + ω for s ∈ [0, ω]. In what follows, for convenience, we shall use the notation f¯ =
1
ω
ω
f (t )dt ,
f M = max f (t ), t ∈[0,ω]
0
f L = min f (t ), t ∈[0,ω]
where f is a periodic continuous function with period ω. We are now in a position to state our result on the existence of periodic solution of system (1.5). Theorem 2.1. For system (1.5), assume that: (i) τ1′ (t ) < 1 and τ2′ (t ) < 1, for t ∈ R, (ii) a¯ 2 > r¯2 + c¯2 , m(¯r1 − c¯1 ) > a¯ 1 , (iii) The following algebraic equation set
¯ − Γ = (x, y) : r¯1 − c¯1 − bx
a¯ 1 y myγ + x
= 0, −¯r2 − c¯2 +
a¯ 2 x myγ + x
=0
(2.1)
has a finite number of real-valued positive solution. Then system (1.5) has at least one positive periodic solution. Proof. For practical biological meaning, we only focus on the positive periodic solutions to system (1.5). Now, we consider the following system:
u′1 (t ) = r1 (t ) − c1 (t ) − b(t )eu1 (t −τ1 (t )) − u′2 (t ) = −(r2 (t ) + c2 (t )) +
a2 (t )e
a1 (t )eu2 (t )
meγ u2 (t ) u1 (t −τ2 (t ))
+ eu1 (t )
, (2.2)
meγ u2 (t ) + eu1 (t −τ2 (t ))
where all functions are defined as ones in system (1.5). It is easy to see that if system (2.2) has one ω-periodic solution (u∗1 (t ), u∗2 (t ))T , then (x∗ (t ), y∗ (t ))T = (exp{u∗1 (t )}, exp{u∗2 (t )})T is a positive ω-periodic solution of system (1.5). Therefore, to complete the proof it suffices to show that system (2.2) has one ω-periodic solution. Take X = Y = {u(t ) = (u1 (t ), u2 (t ))T ∈ C (R, R2 )T : ui (t + ω) = ui (t ), i = 1, 2} and
∥u∥ = max (|u1 (t )| + |u2 (t )|) for u ∈ X t ∈[0,ω]
then X and Y are Banach spaces with the norm ∥ · ∥. Define operators L, P and Q in the following form, respectively, L : Dom L ∩ X → Y ,
Lu(t ) = (u′1 (t ), u′2 (t ))T ,
P (u) = u(0), Q (u) = u¯ ,
where Dom L = {(u1 (t ), u2 (t ))T ∈ C (R, R2 )T } and N : X → Y ,
a1 (t )eu2 (t ) u1 (t −τ1 (t )) − r1 (t ) − c1 (t ) − b(t )e u1 (t ) meγ u2 (t ) + eu1 (t ) N = u1 (t −τ2 (t )) u2 (t ) a2 (t )e −(r2 (t ) + c2 (t )) + γ u ( t ) u ( t −τ ( t )) 2 me 2 + e 1
then Ker L = R2 ,
Im L =
ω
(u1 (t ), u2 (t ))T ∈ Y :
u(t )dt = 0
is closed in Y ,
0
dim Ker L = codim Im L = 2 and P , Q are continuous projectors such that Im P = Ker L,
Ker Q = Im L = Im(I − Q ),
which means that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp : Im L → Im P ∩ Dom L is given by the following form: Kp (u) =
t
u(s)ds − 0
1
ω
ω
0
t
u(s)dsdt . 0
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
301
Therefore, we have
ω
1
ω
r1 (t ) − c1 (t ) − b(t )eu1 (t −τ1 (t )) −
0
QNu =
1
a1 (t )eu2 (t )
ω
ω
−(r2 (t ) + c2 (t )) +
0
a2 (t )e
meγ u2 (t ) + eu1 (t )
u1 (t −τ2 (t ))
dt
,
meγ u2 (t ) + eu1 (t −τ2 (t ))
dt
and
t Kp (I − Q )Nu =
0
r1 (s) − c1 (s) − b(s)eu1 (s−τ1 (s)) −
a1 (s)eu2 (s)
meγ u2 (s) + eu1 (s) a2 (s)eu1 (s−τ2 (s)) ds γ me u2 (s) + eu1 (s−τ2 (s))
t −(r2 (s) + c2 (s)) + 0
ds
− A − B(t ), where
1
ω
ω
0
A=
1
t
r1 (s) − c1 (s) − b(s)e
0ω t
ω
0
0
u1 (s−τ1 (s))
−
a1 (s)eu2 (s)
dsdt
meγ u2 (s) + eu1 (s) a2 (s)eu1 (s−τ2 (s)) −(r2 (s) + c2 (s)) + dsdt meγ u2 (s) + eu1 (s−τ2 (s))
and
ω 1 a1 (s)eu2 (s) u1 (s−τ1 (s)) − r ( s ) − c ( s ) − b ( s ) e − ds 1 1 ω 2 0 meγ u2 (s) + eu1 (s) . B(t ) = ω 1 a2 (s)eu1 (s−τ2 (s)) t − −(r2 (s) + c2 (s)) + ds γ u ( s ) u ( s −τ ( s )) 2 ω 2 0 me 2 + e 1
t
Clearly, QN and Kp (I − Q )N are continuous. By using the Arzela–Ascoli theorem, it is not difficult to prove that Kp (I − Q )N (Ω ) is compact for any open bounded set Ω ∈ X . Moreover, QN (Ω ) is bounded. Therefore, N is L-compact on Ω . In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset Ω . Corresponding to the operator equation Lu = λNu, for λ ∈ (0, 1), we have
a1 (t )eu2 (t ) ′ u (t −τ (t )) , u1 (t ) = λ r1 (t ) − c1 (t ) − b(t )e 1 1 − meγ u2 (t ) + eu1 (t ) a2 (t )eu1 (t −τ2 (t )) u′2 (t ) = λ −(r2 (t ) + c2 (t )) + . γ me u2 (t ) + eu1 (t −τ2 (t ))
(2.3)
Integration of both sides of the first equation in Eqs. (2.3) gives
ω(¯r1 − c¯1 ) =
ω
a1 (t )eu2 (t )
b(t )eu1 (t −τ1 (t )) +
0
meγ u2 (t ) + eu1 (t )
dt .
(2.4)
In view of Lemma 2.2, conditions (i) and (ii), we obtain ω
b(t )e
u1 (t −τ1 (t ))
dt =
0
0
ω
b(µ−1 (t ))eu1 (t ) 1 − τ1′ (µ−1 (t ))
dt ,
which together with (2.4) gives
ω(¯r1 − c¯1 ) =
ω
0
b(µ−1 (t ))eu1 (t ) 1 − τ1′ (µ−1 (t ))
ω
dt + 0
a1 (t )eu2 (t ) meγ u2 (t ) + eu1 (t )
dt ,
(2.5)
which yields ω
0
eu1 (t ) dt ≤
ω(¯r1 − c¯1 ) , ωN1 , ΛL1
b(µ−1 (t ))
where Λ1 = 1−τ ′ (µ−1 (t )) . 1
(2.6)
302
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
Multiplying both sides of the second equation of Eqs. (2.3) by exp{γ u2 (t )} and integrating them from 0 to ω, we have ω
(r2 (t ) + c2 (t ))eγ u2 (t ) dt =
ω
0
< =
a2 (t )eu1 (t −τ2 (t ))+γ u2 (t ) meγ u2 (t ) + eu1 (t −τ2 (t ))
0 aM 2
ω
m
dt
eu1 (t −τ2 (t )) dt ,
0
aM 2
ω
m
eu1 ( t ) 1 − τ2′ (µ−1 (t ))
0
dt <
M aM 2 Λ2
m
ω
eu1 (t ) dt ,
0
where Λ2 = 1−τ ′ (µ1 −1 (t )) . And together with (2.6) yields 2 ω
e
γ u2 (t )
dt <
0
M aM 2 Λ2
m(r2L + c2L )
ω
eu1 (t ) dt ≤
0
M ω(¯r1 − c¯1 )aM 2 Λ2 , ω N2 . mΛL1 (r2L + c2L )
(2.7)
If u2 (t ) ≥ 0, then eγ u2 (t ) ≥ 1 and (2.6) gives N2 ≥
1
ω
ω
eγ u2 (t ) dt ≥ 1,
0
which implies that there must be constant ξ1 ∈ [0, ω] such that u2 (ξ1 ) ≤
ln N2
γ
,
(2.8)
which together with (2.6) and (2.7) yields N1 ≥
1
ω
e
ω
u1 ( t )
dt >
0
m(r2L + c2L )
ω
M ω aM 2 Λ2
eγ u2 (t ) dt ≥
m(r2L + c2L )
0
M aM 2 Λ2
,
which yields that there is a constant ζ1 ∈ [0, ω] such that
m(r2L + c2L ) . u1 (ζ1 ) ≤ max |ln N1 |, ln M aM 2 Λ2
(2.9)
On the other hand, if u2 (t ) < 0, (2.5) yields
ω(¯r1 − c¯1 ) < ΛM 1
ω
eu1 (t ) dt +
0
1
ω
m
a1 (t )e(1−γ )u2 (t ) dt ,
(2.10)
0
and 0 < e(1−γ )u2 (t ) ≤ 1, it follows from (2.10) that
ω(¯r1 − c¯1 ) <
ΛM 1
ω
eu1 (t ) dt +
0
ω¯a1 m
,
together with condition (ii), we have ω
eu1 (t ) dt >
0
ω [m(¯r1 − c¯1 ) − a¯ 1 ] , ωK1 > 0, mΛM 1
(2.11)
and by (2.6), we get N1 ≥
1
ω
ω
eu1 (t ) dt > K1 ,
0
which yields that there exists constant ζ2 ∈ [0, ω] such that u1 (ζ2 ) ≤ max{|ln N1 |, |ln K1 |}.
(2.12)
By (2.9) and (2.12), we know that there exists constant ζ ∈ [0, ω] such that
m(r2L + c2L ) , H1 . u1 (ζ ) ≤ max |ln N1 |, |ln K1 |, ln aM Λ M
2
2
(2.13)
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
303
Fig. 1. A positive periodic solution of System 3.1 with τ1 (t ) = 0.15, τ2 (t ) = 0.2.
Meanwhile, from condition (ii), and the second equation of (2.3) and (2.11) we obtain 0 < ω[¯a2 − (¯r2 + c¯2 )] =
ω
0
ma2 (t )eγ u2 (t ) meγ u2 (t )
+ eu1 (t −τ2 (t ))
dt
ω 1 ( 0 e2γ u2 (t ) dt ) 2 ≤ dt ≤ ω 1 eu1 (t −τ2 (t )) 0 ( 0 e2u1 (t −τ2 (t )) dt ) 2 ω ω 1 1 ( 0 e2γ u2 (t ) dt ) 2 ( 0 e2γ u2 (t ) dt ) 2 maM 2 = maM ≤ ω 2 ω 1 1 1 ( 0 Λ2 e2u1 (t ) dt ) 2 (ΛL2 ) 2 ( 0 e2u1 (t ) dt ) 2 ω 12 maM 2 2γ u2 (t ) ≤ e dt , 1 0 K1 (ωΛL2 ) 2 maM 2
ω
eγ u2 (t )
maM 2
thus, we have 1
ω
ω
e
2γ u2 (t )
dt ≥
ΛL2
ωK1 (¯a2 − (¯r2 + c¯2 )) maM 2
0
2
, K22 ,
(2.14)
which yields that there is a constant ξ2 ∈ [0, ω] such that u2 (ξ2 ) ≥
ln K2
γ
.
(2.15)
Thus, by (2.8) and (2.15), there must be a constant ξ ∈ [0, ω] such that
ln N2 ln K2 , H2 . u2 (ξ ) ≤ max , γ γ
(2.16)
Now, by (2.3), (2.13) and (2.16), we obtain
|u1 (t )| ≤ |u1 (ζ )| +
1 2
0
ω
|u′1 (t )|dt ≤ H1 + ω(¯r1 − c¯1 ) , D1
(2.17)
304
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
Fig. 2. A positive periodic solution of System 3.2 with τ1 (t ) = 0.18 − 0.15 cos 2t , τ2 (t ) = 0.25 + 0.12 sin t.
and
|u2 (t )| ≤ |u2 (ξ )| +
ω
1 2
|u′2 (t )|dt ≤ H2 + ω(¯r2 + c¯2 ) , D2 .
(2.18)
0
From condition (ii), we denote by (u∗1i (t ), u∗2i (t )), i = 1, 2, . . . , k, all the real-valued solutions of the following algebraic equation set
¯ u1 (t ) − r¯1 − c¯1 − be −¯r2 − c¯2 +
a¯ 1 eu2 (t ) meγ u2 (t ) u1 ( t )
+ eu1 (t )
a¯ 2 e
meγ u2 (t )
+ eu1 (t )
= 0,
= 0,
then, we take
D0 , exp
max
1≤i≤k,t ∈[0,ω]
{u∗1i (t ), u∗2i (t )} .
(2.19)
And now, together with (2.17)–(2.19), we can set D = D1 + D2 + D0 , and take Ω = {u = (u1 (t ), u2 (t ))T ∈ X : ∥u∥ < D}. It is clear that Ω verifies the requirement in Lemma 2.1. If u ∈ ∂ Ω ∩ Ker L = ∂ Ω ∩ R2 , then u is a constant vector in R2 with ∥u∥ = D satisfying
a¯ 1 eu2 (t ) u1 (t ) ¯ − r¯1 − c¯1 − be meγ u2 (t ) + eu1 (t ) ̸= 0. QNu = u1 (t ) a¯ 2 e −¯r2 − c¯2 + γ u ( t ) u ( t ) me 2 + e 1
By simple computing, we can get the Jacobian matrix as follows:
a¯ 1 eu2 +u1 ¯ u1 −be + (meγ u2 + eu1 )2 J¯(u1 , u2 ) = ma¯ 2 eγ u2 +u1 (meγ u2 + eu1 )2
ma¯ 1 (γ − 1)e(1+γ )u2 − a¯ 1 eu2 +u1
(meγ u2 + eu1 )2 mγ a¯ 2 eγ u2 +u1 − (meγ u2 + eu1 )2
.
(2.20)
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
305
Fig. 3. No positive periodic solution of System 3.3 when τ1 (t ) = 0.4 + 0.3 cos 4t , τ2 (t ) = 0.2.
By (2.20), we can easily know that det J¯(u1 , u2 ) > 0, ∀(u1 , u2 ) ∈ R2 , which yields deg{JQN , Ω ∩ Ker L, 0} =
sgn detJ¯(u∗1i , u∗2i ) = k > 0.
(u∗1i ,u∗2i )∈QN −1 (0)
By now we have proved that Ω verifies all the requirements in Lemma 2.1. Then, we get that Eqs. (2.2) have at least one periodic solution (u∗1 (t ), u∗2 (t ))T with period ω in Dom L ∩ Ω , which implies that Eqs. (1.5) have at least one positive periodic
solution (eu1 (t ) , eu2 (t ) ) with period ω. This completes the proof of Theorem 2.1. ∗
∗
Corollary 2.1. Suppose that (ii) and (iii) hold, and τ1 (t ), τ2 (t ) are both nonnegative constants, then, system (1.5) has at least one positive periodic solution. 3. Applications Now, we perform some numerical simulations to illustrate our analysis by using MATLAB(7.0) programming. System 3.1. In system (1.5), we choose r1 (t ) = 8 + 0.2 sin t , b(t ) = 2 + 0.1 cos t , a1 (t ) = 6 + 0.2 sin t , m = 1, γ = r2 (t ) = 3 + 0.3 cos t , a2 (t ) = 5 + 0.2 sin t , c1 (t ) = 1 + 0.1 sin t , c2 (t ) = 1 − 0.2 cos t , τ1 (t ) = 0.15, τ2 (t ) = 0.2.
1 2
,
It is obvious that the conditions (i) and (ii) hold, Moreover, the algebraic equation set Γ has only one positive solution , 1225 ), which together with Corollary 2.1 yields that System 3.1 has at least one positive periodic solution. (x∗ , y∗ ) = ( 70 23 2116 Take the initial value by (x(0), y(0)) = (2.3, 0.85), Fig. 1 clearly shows the dynamic behaviors of the solution (x(t ), y(t )), which is a positive periodic solution of System 3.1. System 3.2. In this system, we replace τ1 (t ) = 0.15, τ2 (t ) = 0.2 in System 3.1 by τ1 (t ) = 0.18 − 0.15 cos 2t , τ2 (t ) = 0.25 + 0.12 sin t, respectively. Obviously, the conditions (i)–(iii) hold, choosing the same initial value as in System 3.1, from Fig. 2 we can get the details of its dynamic behaviors of the positive periodic solution (x(t ), y(t )).
306
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
System 3.3. In this system, we only replace τ1 (t ) = 0.15 in System 3.1 by τ1 (t ) = 0.4 + 0.3 cos 4t, then, τ1′ (t ) = −1.2 sin 4t, which does not satisfy the condition (i), but the conditions (ii) and (iii) hold; choosing the same initial value as in System 3.1, one can easily see the extinction of the predator in Fig. 3. Thus, there is no positive periodic solution of the system. 4. Conclusion In this paper, we made an effort toward analyzing the existence of periodic solutions of the non-autonomous predator–prey system with Hassell–Varley type functional response, non-selective harvesting and multiple delays. From the simulation of systems 2 and 3, one can find that the conditions (i) are sharp. So we can conclude that the positive periodic solutions for system (1.5) are delay-dependent, which is different from the previous works that are delay-independent, and our model is more reasonable and general. In addition to delayed biological systems, we believe that our derived analysis can provide valuable design insights and allow an in-depth understanding of key design implications of biological systems. Acknowledgments The authors gratefully acknowledge anonymous referees’ comments and patient work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
A. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. V. Volterra, Variazioni e fluttuazioni del numero di individui in specie animali conviventi, Mem. Accd. Lincei. 2 (1926) 31–113. H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel-Dekker, NewYork, 1980. J.D. Murry, Mathematical Biology, Springer, Berlin, NewYork, 1989. Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predator–prey system, J. Math. Biol. 36 (1998) 389–406. C. Jost, O. Arino, R. Arditi, About deterministic extinction in ratio-dependent predator–prey models, Bull. Math. Biol. 61 (1999) 19–32. S.-B. Hsu, T.-W. Hwang, Y. Kuang, Global dynamics of a predator–prey model with Hassell–Varley type functional response, J. Math. Biol. 10 (2008) 1–15. I. Hanski, The functional response of predator: worries bout scale, TREE 6 (1991) 141–142. R. Arditi, N. Perrin, H. Saiah, Functional response and heterogeneities: an experiment test with cladocerans, OIKOS 60 (1991) 69–75. A. Maiti, M. Jana, G. Samanta, Deterministic and stochastic analysis of a ratio-dependent predator–prey system with delay, Nonlinear Anal. Model. Control 12 (2007) 383–398. T.K. Kar, U.K. Pahari, Non-selective harvesting in prey-predator models with delay, Commun. Nonlinear Sci. Numer. Simul. 11 (2006) 499–509. G.D. Zhang, B.S. Chen, L.L. Zhu, Yi Shen, Hopf bifurcation for a differential-algebraic biological economic system with time delay, Appl. Math. Comput. 218 (2012) 7717–7726. T.K. Kar, A. Ghorai, Dynamic behaviour of a delayed predator–prey model with harvesting, Appl. Math. Comput. 217 (2011) 9085–9104. G. Dai, M. Tang, Coexistence region and global dynamics of a harvested predator–prey system, SIAM J. Appl. Math. 58 (1) (1998) 193–210. D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator–prey system with constant rate harvesting, SIAM Appl. Math. 65 (2005) 737–753. G.D. Zhang, L.L. Zhu, B.S. Chen, Hopf bifurcation in a delayed differential-algebraic biological economic system, Nonlinear Analysis RWA 12 (2011) 1708–1719. Y.H. Fan, L.L. Wang, Multiplicity of periodic solutions for a delayed ratio-dependent predator–prey model with Holling type III functional response and harvesting terms, J. Math. Anal. Appl. 365 (2010) 525–540. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. M. Hassell, G. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature 223 (1969) 1133–1136. M. Fazly, M. Hesaaraki, Periodic solutions for predator–prey systems with Beddington–DeAngelis functional response on time scales, Nonlinear Analysis RWA 9 (2008) 1224–1235. C.J. Zhao, On a periodic predator–prey system with time delays, J. Math. Anal. Appl. 331 (2007) 978–985. K. Wang, Periodic solutions to a delayed predator–prey model with Hassell–Varley type functional response, Nonlinear Anal. RWA 12 (2011) 137–145. G.R. Liu, J.R. Yan, Existence of positive periodic solutions for neutral delay Gause-type predator–prey system, Appl. Math. Model. 35 (2011) 5741–5750. R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977. S. Lu, On the existence of positive periodic solutions to a Lotka–Volterra cooperative population model with multiple delays, Nonlinear Anal. 68 (2008) 1746–1753.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Positive periodic solutions in a non-selective harvesting predator–prey model with multiple delays✩ Guodong Zhang a,b , Yi Shen a,∗ , Boshan Chen b a
Department of Control Science and Engineering and the Key Laboratory of Education Ministry of Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan 430074, China b
College of Mathematics and statistics, Hubei Normal University, Huangshi 435002, China
article
abstract
info
Article history: Received 10 October 2011 Available online 26 May 2012 Submitted by M. Iannelli
This paper is concerned with a predator–prey model with Hassell–Varley type functional response, non-selective harvesting and multiple delays. Some new sufficient conditions are obtained for the existence of positive periodic solutions by applying the coincidence degree theorem. In addition, the results are delay-dependent, which is different from the previous work. Finally, simulations illustrate the effectiveness of our results. © 2012 Elsevier Inc. All rights reserved.
Keywords: Predator–prey model Periodic solution Harvesting Time delay Coincidence degree theorem
1. Introduction Lotka [1] and Volterra [2] introduced the first predator–prey model in 1925 and 1926, respectively. After that many more complicated but realistic predator–prey models have been formulated by ecologists and mathematicians. In general, a predator–prey system may have the form
dx(t ) x =r 1− x − yφ(x), dt
k
dy(t ) = y(−d + µφ(x))
(1.1)
dt
where φ(x) is the functional response function, which reflects the capture ability of the predator to prey. For more biological meaning, the reader may consult Freedman [3] and Murry [4]. Recently, there has been a growing explicit biological and physiological evidence [5–7] that in many situations, especially when predators have to share or compete for food, a more suitable general predator–prey theory should be based on the socalled ratio-dependent theory. This is strongly supported by numerous field and laboratory experiments and observations [8,9]. For this reason, Kuang and Beretta [5], Jost et al. [6], Hsu et al. [7] and Maiti et al. [10] investigated the following
✩ This work is supported by the National Science Foundation of China under Grant Nos. 60974021, the Key Program of National Natural Science Foundation of China under Grant Nos. 61134012 and also supported by China Postdoctoral Science Foundation funded project under Grant Nos. 2012M511615. ∗ Corresponding author. E-mail addresses: [email protected] (G. Zhang), [email protected] (Y. Shen), [email protected] (B. Chen).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.05.045
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
299
Michaelis–Menten type ratio-dependent predator–prey system:
dx(t ) axy x x− = r 1 − , dt k my + x dy(t ) bx = y −d + , dt my +x x(0) > 0, y(0) > 0
(1.2)
and gave a nice systematic work on the global qualitative analysis of this system. In many earlier studies, it has been shown that harvesting has a strong impact on dynamic evolution of a population, e.g., see [11–17]. So the study of the population dynamics with harvesting is becoming a very important subject in mathematical bio-economics. The predator–prey food chain model with harvesting is generally described as
x x dx(t ) = r 1 − x − y φ − h1 , dt k y dy(t ) x = y −d + µφ − h2 dt
(1.3)
y
where h1 and h2 represent harvesting terms. In fact, as was pointed by Kuang [18] that any model of species dynamics without delays is an approximation at best. And human needs are not invariable for a long time. In detail, human needs increase as biological resources become abundant, while human needs decrease as biological resource is exiguous. So in the literature [13], the authors focus on varying harvesting rate in system (1.3) with time delay as follows:
dx(t ) a1 xy = r1 x − b1 x2 − − c1 x, dt x + k1 dy(t ) a2 y(t − τ ) = y r2 − − c2 y dt x(t − τ ) + k2
(1.4)
where c1 and c2 are the harvesting coefficients of prey and predator respectively. They show that the harvest effort on both prey and predator influences the stability of the system. Motivated by the above results, and the literature [11,19–23] and considering that the delay may occur in the preys and predators, in this paper, we discuss a non-autonomous predator–prey model with Hassell–Varley (See [7,19,22]) type functional response and two delays in the system as follows:
dx(t ) = x(t )(r1 (t ) − b(t )x(t − τ1 (t )) − dt dy (t )
dt
a1 (t )y(t )
) − c1 (t )x(t ),
myγ (t ) + x(t ) a2 (t )x(t − τ2 (t )) = y(t )(−r2 (t ) + ) − c2 (t )y(t ) myγ (t ) + x(t − τ2 (t ))
(1.5)
where a1 , a2 , b, c1 , c2 , r1 , r2 , τ1 , τ2 are continuously nonnegative periodic functions with period ω; m ≥ 0, γ ∈ (0, 1], is called the Hassell–Varley constant. For θ ∈ [−τ¯ , 0], we use the notation xt (θ ) = x(t + θ ), then the initial conditions of system (1.5) take the form x0 (θ ) = φ(θ ), x(0) = φ > 0, y(0) > 0, ∀ θ ∈ [−τ¯ , 0], where τ¯ , supt ∈[0,ω] {τ1 (t ), τ2 (t )}, φ ∈ C ([−τ¯ , 0], R) with the norm ∥z ∥I = supt ∈[−τ¯ ,0] ∥z (t )∥. The organization of this paper is as follows. In Section 2, by applying the coincidence degree theorem, we obtain the new sufficient conditions for the existence of positive periodic solutions for system (1.5). It is interesting that the results are delay-dependent, which is different from the previous works that are delay-independent, and our model is more reasonable and general. In Section 3, numerical simulations are performed to illustrate the effectiveness of our results. Finally, this paper ends by a conclusion. 2. Existence of periodic solutions In order to obtain the existence of positive periodic solutions for system (1.5), we first make the following preparations. Let X and Y be two Banach spaces, L : Dom L ∩ X → Y be a linear mapping and N : X → Y be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if there exist continuous projections P : X → X and Q : Y → Y such that Im P = Ker L and Im L = Ker Q = Im(I − Q ). It follows that the mapping L|dom L ∩ Ker P : (I − P )X → Im L has an inverse mapping, denoted by Kp . For an open bounded subset of X , the mapping N is called L-compact on Ω if QN (Ω ) is bounded and Kp (I − Q )N : Ω → X is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L. Here we state the Gaines–Mawhin Lemma from [24], which is a main tool in the proofs of our theorems. Lemma 2.1 (Continuation Theorem (see [24, p. 40])). Let L be a Fredholm mapping of index zero. Assume that N : Ω → Z is L-compact on Ω with Ω open bounded in X . Furthermore, assume: (a) For each λ ∈ (0, 1), every solution x of Lx ̸= λNx is such that x ∈ ∂ Ω ∩ Dom L. (b) QNx ̸= 0 for each x ∈ ∂ Ω ∩ Ker L and the Brouwer degree, deg (JNQ , Ω ∩ Ker L, 0) ̸= 0. Then, the operator equation Lx = Nx has at least one solution in Dom L ∩ Ω .
300
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306 dτ (t )
Lemma 2.2 (See [25]). If τ ∈ C 1 (R, R) with τ (t + ω) ≡ τ (t ) and dt < 1 for t ∈ [0, ω], then function µ(t ) = t − τ (t ) has a unique inverse µ−1 (t ) satisfying µ ∈ C (R, R) with µ−1 (s + ω) ≡ µ−1 (s) + ω for s ∈ [0, ω]. In what follows, for convenience, we shall use the notation f¯ =
1
ω
ω
f (t )dt ,
f M = max f (t ), t ∈[0,ω]
0
f L = min f (t ), t ∈[0,ω]
where f is a periodic continuous function with period ω. We are now in a position to state our result on the existence of periodic solution of system (1.5). Theorem 2.1. For system (1.5), assume that: (i) τ1′ (t ) < 1 and τ2′ (t ) < 1, for t ∈ R, (ii) a¯ 2 > r¯2 + c¯2 , m(¯r1 − c¯1 ) > a¯ 1 , (iii) The following algebraic equation set
¯ − Γ = (x, y) : r¯1 − c¯1 − bx
a¯ 1 y myγ + x
= 0, −¯r2 − c¯2 +
a¯ 2 x myγ + x
=0
(2.1)
has a finite number of real-valued positive solution. Then system (1.5) has at least one positive periodic solution. Proof. For practical biological meaning, we only focus on the positive periodic solutions to system (1.5). Now, we consider the following system:
u′1 (t ) = r1 (t ) − c1 (t ) − b(t )eu1 (t −τ1 (t )) − u′2 (t ) = −(r2 (t ) + c2 (t )) +
a2 (t )e
a1 (t )eu2 (t )
meγ u2 (t ) u1 (t −τ2 (t ))
+ eu1 (t )
, (2.2)
meγ u2 (t ) + eu1 (t −τ2 (t ))
where all functions are defined as ones in system (1.5). It is easy to see that if system (2.2) has one ω-periodic solution (u∗1 (t ), u∗2 (t ))T , then (x∗ (t ), y∗ (t ))T = (exp{u∗1 (t )}, exp{u∗2 (t )})T is a positive ω-periodic solution of system (1.5). Therefore, to complete the proof it suffices to show that system (2.2) has one ω-periodic solution. Take X = Y = {u(t ) = (u1 (t ), u2 (t ))T ∈ C (R, R2 )T : ui (t + ω) = ui (t ), i = 1, 2} and
∥u∥ = max (|u1 (t )| + |u2 (t )|) for u ∈ X t ∈[0,ω]
then X and Y are Banach spaces with the norm ∥ · ∥. Define operators L, P and Q in the following form, respectively, L : Dom L ∩ X → Y ,
Lu(t ) = (u′1 (t ), u′2 (t ))T ,
P (u) = u(0), Q (u) = u¯ ,
where Dom L = {(u1 (t ), u2 (t ))T ∈ C (R, R2 )T } and N : X → Y ,
a1 (t )eu2 (t ) u1 (t −τ1 (t )) − r1 (t ) − c1 (t ) − b(t )e u1 (t ) meγ u2 (t ) + eu1 (t ) N = u1 (t −τ2 (t )) u2 (t ) a2 (t )e −(r2 (t ) + c2 (t )) + γ u ( t ) u ( t −τ ( t )) 2 me 2 + e 1
then Ker L = R2 ,
Im L =
ω
(u1 (t ), u2 (t ))T ∈ Y :
u(t )dt = 0
is closed in Y ,
0
dim Ker L = codim Im L = 2 and P , Q are continuous projectors such that Im P = Ker L,
Ker Q = Im L = Im(I − Q ),
which means that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp : Im L → Im P ∩ Dom L is given by the following form: Kp (u) =
t
u(s)ds − 0
1
ω
ω
0
t
u(s)dsdt . 0
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
301
Therefore, we have
ω
1
ω
r1 (t ) − c1 (t ) − b(t )eu1 (t −τ1 (t )) −
0
QNu =
1
a1 (t )eu2 (t )
ω
ω
−(r2 (t ) + c2 (t )) +
0
a2 (t )e
meγ u2 (t ) + eu1 (t )
u1 (t −τ2 (t ))
dt
,
meγ u2 (t ) + eu1 (t −τ2 (t ))
dt
and
t Kp (I − Q )Nu =
0
r1 (s) − c1 (s) − b(s)eu1 (s−τ1 (s)) −
a1 (s)eu2 (s)
meγ u2 (s) + eu1 (s) a2 (s)eu1 (s−τ2 (s)) ds γ me u2 (s) + eu1 (s−τ2 (s))
t −(r2 (s) + c2 (s)) + 0
ds
− A − B(t ), where
1
ω
ω
0
A=
1
t
r1 (s) − c1 (s) − b(s)e
0ω t
ω
0
0
u1 (s−τ1 (s))
−
a1 (s)eu2 (s)
dsdt
meγ u2 (s) + eu1 (s) a2 (s)eu1 (s−τ2 (s)) −(r2 (s) + c2 (s)) + dsdt meγ u2 (s) + eu1 (s−τ2 (s))
and
ω 1 a1 (s)eu2 (s) u1 (s−τ1 (s)) − r ( s ) − c ( s ) − b ( s ) e − ds 1 1 ω 2 0 meγ u2 (s) + eu1 (s) . B(t ) = ω 1 a2 (s)eu1 (s−τ2 (s)) t − −(r2 (s) + c2 (s)) + ds γ u ( s ) u ( s −τ ( s )) 2 ω 2 0 me 2 + e 1
t
Clearly, QN and Kp (I − Q )N are continuous. By using the Arzela–Ascoli theorem, it is not difficult to prove that Kp (I − Q )N (Ω ) is compact for any open bounded set Ω ∈ X . Moreover, QN (Ω ) is bounded. Therefore, N is L-compact on Ω . In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset Ω . Corresponding to the operator equation Lu = λNu, for λ ∈ (0, 1), we have
a1 (t )eu2 (t ) ′ u (t −τ (t )) , u1 (t ) = λ r1 (t ) − c1 (t ) − b(t )e 1 1 − meγ u2 (t ) + eu1 (t ) a2 (t )eu1 (t −τ2 (t )) u′2 (t ) = λ −(r2 (t ) + c2 (t )) + . γ me u2 (t ) + eu1 (t −τ2 (t ))
(2.3)
Integration of both sides of the first equation in Eqs. (2.3) gives
ω(¯r1 − c¯1 ) =
ω
a1 (t )eu2 (t )
b(t )eu1 (t −τ1 (t )) +
0
meγ u2 (t ) + eu1 (t )
dt .
(2.4)
In view of Lemma 2.2, conditions (i) and (ii), we obtain ω
b(t )e
u1 (t −τ1 (t ))
dt =
0
0
ω
b(µ−1 (t ))eu1 (t ) 1 − τ1′ (µ−1 (t ))
dt ,
which together with (2.4) gives
ω(¯r1 − c¯1 ) =
ω
0
b(µ−1 (t ))eu1 (t ) 1 − τ1′ (µ−1 (t ))
ω
dt + 0
a1 (t )eu2 (t ) meγ u2 (t ) + eu1 (t )
dt ,
(2.5)
which yields ω
0
eu1 (t ) dt ≤
ω(¯r1 − c¯1 ) , ωN1 , ΛL1
b(µ−1 (t ))
where Λ1 = 1−τ ′ (µ−1 (t )) . 1
(2.6)
302
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
Multiplying both sides of the second equation of Eqs. (2.3) by exp{γ u2 (t )} and integrating them from 0 to ω, we have ω
(r2 (t ) + c2 (t ))eγ u2 (t ) dt =
ω
0
< =
a2 (t )eu1 (t −τ2 (t ))+γ u2 (t ) meγ u2 (t ) + eu1 (t −τ2 (t ))
0 aM 2
ω
m
dt
eu1 (t −τ2 (t )) dt ,
0
aM 2
ω
m
eu1 ( t ) 1 − τ2′ (µ−1 (t ))
0
dt <
M aM 2 Λ2
m
ω
eu1 (t ) dt ,
0
where Λ2 = 1−τ ′ (µ1 −1 (t )) . And together with (2.6) yields 2 ω
e
γ u2 (t )
dt <
0
M aM 2 Λ2
m(r2L + c2L )
ω
eu1 (t ) dt ≤
0
M ω(¯r1 − c¯1 )aM 2 Λ2 , ω N2 . mΛL1 (r2L + c2L )
(2.7)
If u2 (t ) ≥ 0, then eγ u2 (t ) ≥ 1 and (2.6) gives N2 ≥
1
ω
ω
eγ u2 (t ) dt ≥ 1,
0
which implies that there must be constant ξ1 ∈ [0, ω] such that u2 (ξ1 ) ≤
ln N2
γ
,
(2.8)
which together with (2.6) and (2.7) yields N1 ≥
1
ω
e
ω
u1 ( t )
dt >
0
m(r2L + c2L )
ω
M ω aM 2 Λ2
eγ u2 (t ) dt ≥
m(r2L + c2L )
0
M aM 2 Λ2
,
which yields that there is a constant ζ1 ∈ [0, ω] such that
m(r2L + c2L ) . u1 (ζ1 ) ≤ max |ln N1 |, ln M aM 2 Λ2
(2.9)
On the other hand, if u2 (t ) < 0, (2.5) yields
ω(¯r1 − c¯1 ) < ΛM 1
ω
eu1 (t ) dt +
0
1
ω
m
a1 (t )e(1−γ )u2 (t ) dt ,
(2.10)
0
and 0 < e(1−γ )u2 (t ) ≤ 1, it follows from (2.10) that
ω(¯r1 − c¯1 ) <
ΛM 1
ω
eu1 (t ) dt +
0
ω¯a1 m
,
together with condition (ii), we have ω
eu1 (t ) dt >
0
ω [m(¯r1 − c¯1 ) − a¯ 1 ] , ωK1 > 0, mΛM 1
(2.11)
and by (2.6), we get N1 ≥
1
ω
ω
eu1 (t ) dt > K1 ,
0
which yields that there exists constant ζ2 ∈ [0, ω] such that u1 (ζ2 ) ≤ max{|ln N1 |, |ln K1 |}.
(2.12)
By (2.9) and (2.12), we know that there exists constant ζ ∈ [0, ω] such that
m(r2L + c2L ) , H1 . u1 (ζ ) ≤ max |ln N1 |, |ln K1 |, ln aM Λ M
2
2
(2.13)
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
303
Fig. 1. A positive periodic solution of System 3.1 with τ1 (t ) = 0.15, τ2 (t ) = 0.2.
Meanwhile, from condition (ii), and the second equation of (2.3) and (2.11) we obtain 0 < ω[¯a2 − (¯r2 + c¯2 )] =
ω
0
ma2 (t )eγ u2 (t ) meγ u2 (t )
+ eu1 (t −τ2 (t ))
dt
ω 1 ( 0 e2γ u2 (t ) dt ) 2 ≤ dt ≤ ω 1 eu1 (t −τ2 (t )) 0 ( 0 e2u1 (t −τ2 (t )) dt ) 2 ω ω 1 1 ( 0 e2γ u2 (t ) dt ) 2 ( 0 e2γ u2 (t ) dt ) 2 maM 2 = maM ≤ ω 2 ω 1 1 1 ( 0 Λ2 e2u1 (t ) dt ) 2 (ΛL2 ) 2 ( 0 e2u1 (t ) dt ) 2 ω 12 maM 2 2γ u2 (t ) ≤ e dt , 1 0 K1 (ωΛL2 ) 2 maM 2
ω
eγ u2 (t )
maM 2
thus, we have 1
ω
ω
e
2γ u2 (t )
dt ≥
ΛL2
ωK1 (¯a2 − (¯r2 + c¯2 )) maM 2
0
2
, K22 ,
(2.14)
which yields that there is a constant ξ2 ∈ [0, ω] such that u2 (ξ2 ) ≥
ln K2
γ
.
(2.15)
Thus, by (2.8) and (2.15), there must be a constant ξ ∈ [0, ω] such that
ln N2 ln K2 , H2 . u2 (ξ ) ≤ max , γ γ
(2.16)
Now, by (2.3), (2.13) and (2.16), we obtain
|u1 (t )| ≤ |u1 (ζ )| +
1 2
0
ω
|u′1 (t )|dt ≤ H1 + ω(¯r1 − c¯1 ) , D1
(2.17)
304
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
Fig. 2. A positive periodic solution of System 3.2 with τ1 (t ) = 0.18 − 0.15 cos 2t , τ2 (t ) = 0.25 + 0.12 sin t.
and
|u2 (t )| ≤ |u2 (ξ )| +
ω
1 2
|u′2 (t )|dt ≤ H2 + ω(¯r2 + c¯2 ) , D2 .
(2.18)
0
From condition (ii), we denote by (u∗1i (t ), u∗2i (t )), i = 1, 2, . . . , k, all the real-valued solutions of the following algebraic equation set
¯ u1 (t ) − r¯1 − c¯1 − be −¯r2 − c¯2 +
a¯ 1 eu2 (t ) meγ u2 (t ) u1 ( t )
+ eu1 (t )
a¯ 2 e
meγ u2 (t )
+ eu1 (t )
= 0,
= 0,
then, we take
D0 , exp
max
1≤i≤k,t ∈[0,ω]
{u∗1i (t ), u∗2i (t )} .
(2.19)
And now, together with (2.17)–(2.19), we can set D = D1 + D2 + D0 , and take Ω = {u = (u1 (t ), u2 (t ))T ∈ X : ∥u∥ < D}. It is clear that Ω verifies the requirement in Lemma 2.1. If u ∈ ∂ Ω ∩ Ker L = ∂ Ω ∩ R2 , then u is a constant vector in R2 with ∥u∥ = D satisfying
a¯ 1 eu2 (t ) u1 (t ) ¯ − r¯1 − c¯1 − be meγ u2 (t ) + eu1 (t ) ̸= 0. QNu = u1 (t ) a¯ 2 e −¯r2 − c¯2 + γ u ( t ) u ( t ) me 2 + e 1
By simple computing, we can get the Jacobian matrix as follows:
a¯ 1 eu2 +u1 ¯ u1 −be + (meγ u2 + eu1 )2 J¯(u1 , u2 ) = ma¯ 2 eγ u2 +u1 (meγ u2 + eu1 )2
ma¯ 1 (γ − 1)e(1+γ )u2 − a¯ 1 eu2 +u1
(meγ u2 + eu1 )2 mγ a¯ 2 eγ u2 +u1 − (meγ u2 + eu1 )2
.
(2.20)
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
305
Fig. 3. No positive periodic solution of System 3.3 when τ1 (t ) = 0.4 + 0.3 cos 4t , τ2 (t ) = 0.2.
By (2.20), we can easily know that det J¯(u1 , u2 ) > 0, ∀(u1 , u2 ) ∈ R2 , which yields deg{JQN , Ω ∩ Ker L, 0} =
sgn detJ¯(u∗1i , u∗2i ) = k > 0.
(u∗1i ,u∗2i )∈QN −1 (0)
By now we have proved that Ω verifies all the requirements in Lemma 2.1. Then, we get that Eqs. (2.2) have at least one periodic solution (u∗1 (t ), u∗2 (t ))T with period ω in Dom L ∩ Ω , which implies that Eqs. (1.5) have at least one positive periodic
solution (eu1 (t ) , eu2 (t ) ) with period ω. This completes the proof of Theorem 2.1. ∗
∗
Corollary 2.1. Suppose that (ii) and (iii) hold, and τ1 (t ), τ2 (t ) are both nonnegative constants, then, system (1.5) has at least one positive periodic solution. 3. Applications Now, we perform some numerical simulations to illustrate our analysis by using MATLAB(7.0) programming. System 3.1. In system (1.5), we choose r1 (t ) = 8 + 0.2 sin t , b(t ) = 2 + 0.1 cos t , a1 (t ) = 6 + 0.2 sin t , m = 1, γ = r2 (t ) = 3 + 0.3 cos t , a2 (t ) = 5 + 0.2 sin t , c1 (t ) = 1 + 0.1 sin t , c2 (t ) = 1 − 0.2 cos t , τ1 (t ) = 0.15, τ2 (t ) = 0.2.
1 2
,
It is obvious that the conditions (i) and (ii) hold, Moreover, the algebraic equation set Γ has only one positive solution , 1225 ), which together with Corollary 2.1 yields that System 3.1 has at least one positive periodic solution. (x∗ , y∗ ) = ( 70 23 2116 Take the initial value by (x(0), y(0)) = (2.3, 0.85), Fig. 1 clearly shows the dynamic behaviors of the solution (x(t ), y(t )), which is a positive periodic solution of System 3.1. System 3.2. In this system, we replace τ1 (t ) = 0.15, τ2 (t ) = 0.2 in System 3.1 by τ1 (t ) = 0.18 − 0.15 cos 2t , τ2 (t ) = 0.25 + 0.12 sin t, respectively. Obviously, the conditions (i)–(iii) hold, choosing the same initial value as in System 3.1, from Fig. 2 we can get the details of its dynamic behaviors of the positive periodic solution (x(t ), y(t )).
306
G. Zhang et al. / J. Math. Anal. Appl. 395 (2012) 298–306
System 3.3. In this system, we only replace τ1 (t ) = 0.15 in System 3.1 by τ1 (t ) = 0.4 + 0.3 cos 4t, then, τ1′ (t ) = −1.2 sin 4t, which does not satisfy the condition (i), but the conditions (ii) and (iii) hold; choosing the same initial value as in System 3.1, one can easily see the extinction of the predator in Fig. 3. Thus, there is no positive periodic solution of the system. 4. Conclusion In this paper, we made an effort toward analyzing the existence of periodic solutions of the non-autonomous predator–prey system with Hassell–Varley type functional response, non-selective harvesting and multiple delays. From the simulation of systems 2 and 3, one can find that the conditions (i) are sharp. So we can conclude that the positive periodic solutions for system (1.5) are delay-dependent, which is different from the previous works that are delay-independent, and our model is more reasonable and general. In addition to delayed biological systems, we believe that our derived analysis can provide valuable design insights and allow an in-depth understanding of key design implications of biological systems. Acknowledgments The authors gratefully acknowledge anonymous referees’ comments and patient work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
A. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. V. Volterra, Variazioni e fluttuazioni del numero di individui in specie animali conviventi, Mem. Accd. Lincei. 2 (1926) 31–113. H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel-Dekker, NewYork, 1980. J.D. Murry, Mathematical Biology, Springer, Berlin, NewYork, 1989. Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predator–prey system, J. Math. Biol. 36 (1998) 389–406. C. Jost, O. Arino, R. Arditi, About deterministic extinction in ratio-dependent predator–prey models, Bull. Math. Biol. 61 (1999) 19–32. S.-B. Hsu, T.-W. Hwang, Y. Kuang, Global dynamics of a predator–prey model with Hassell–Varley type functional response, J. Math. Biol. 10 (2008) 1–15. I. Hanski, The functional response of predator: worries bout scale, TREE 6 (1991) 141–142. R. Arditi, N. Perrin, H. Saiah, Functional response and heterogeneities: an experiment test with cladocerans, OIKOS 60 (1991) 69–75. A. Maiti, M. Jana, G. Samanta, Deterministic and stochastic analysis of a ratio-dependent predator–prey system with delay, Nonlinear Anal. Model. Control 12 (2007) 383–398. T.K. Kar, U.K. Pahari, Non-selective harvesting in prey-predator models with delay, Commun. Nonlinear Sci. Numer. Simul. 11 (2006) 499–509. G.D. Zhang, B.S. Chen, L.L. Zhu, Yi Shen, Hopf bifurcation for a differential-algebraic biological economic system with time delay, Appl. Math. Comput. 218 (2012) 7717–7726. T.K. Kar, A. Ghorai, Dynamic behaviour of a delayed predator–prey model with harvesting, Appl. Math. Comput. 217 (2011) 9085–9104. G. Dai, M. Tang, Coexistence region and global dynamics of a harvested predator–prey system, SIAM J. Appl. Math. 58 (1) (1998) 193–210. D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator–prey system with constant rate harvesting, SIAM Appl. Math. 65 (2005) 737–753. G.D. Zhang, L.L. Zhu, B.S. Chen, Hopf bifurcation in a delayed differential-algebraic biological economic system, Nonlinear Analysis RWA 12 (2011) 1708–1719. Y.H. Fan, L.L. Wang, Multiplicity of periodic solutions for a delayed ratio-dependent predator–prey model with Holling type III functional response and harvesting terms, J. Math. Anal. Appl. 365 (2010) 525–540. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. M. Hassell, G. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature 223 (1969) 1133–1136. M. Fazly, M. Hesaaraki, Periodic solutions for predator–prey systems with Beddington–DeAngelis functional response on time scales, Nonlinear Analysis RWA 9 (2008) 1224–1235. C.J. Zhao, On a periodic predator–prey system with time delays, J. Math. Anal. Appl. 331 (2007) 978–985. K. Wang, Periodic solutions to a delayed predator–prey model with Hassell–Varley type functional response, Nonlinear Anal. RWA 12 (2011) 137–145. G.R. Liu, J.R. Yan, Existence of positive periodic solutions for neutral delay Gause-type predator–prey system, Appl. Math. Model. 35 (2011) 5741–5750. R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977. S. Lu, On the existence of positive periodic solutions to a Lotka–Volterra cooperative population model with multiple delays, Nonlinear Anal. 68 (2008) 1746–1753.