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Positive periodic solutions for a neutral delay ratio-dependent predator–prey model with a Holling type II functional response
Nonlinear Analysis: Real World Applications 12 (2011) 3252–3260
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Positive periodic solutions for a neutral delay ratio-dependent predator–prey model with a Holling type II functional response✩ Guirong Liu ∗ , Jurang Yan School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China
article
abstract
info
Article history: Received 1 November 2010 Accepted 24 May 2011
By using a continuation theorem based on coincidence degree theory, some new and interesting sufficient conditions are obtained for the existence of positive periodic solutions for a neutral delay ratio-dependent predator–prey model with the Holling type II functional response:
Keywords: Predator–prey model Ratio-dependent Periodic solution Neutral Coincidence degree
c (t )x(t )y(t ) , x′ (t ) = x(t )[a(t ) − b(t )x(t − σ (t )) − h(t )x′ (t − σ (t ))] − my ( t ) + x( t ) [ ] f ( t ) x ( t − τ ( t )) ′ . y (t ) = y(t ) −d(t ) + my(t − τ (t )) + x(t − τ (t )) An example is presented to illustrate the feasibility of our main results. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The dynamic relationship between predator and prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. The traditional predator–prey models have been studied extensively (see, for example, [1–5] and references cited therein). Recently, there has been growing explicit biological and physiological evidence [6–8] that in many situations, especially when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator–prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as saying that the per capita predator growth rate should be a function of the ratio of prey to predator abundance, and so should be the so-called ratio-dependent functional response. This is strongly supported by numerous field and laboratory experiments and observations. On the basis of the Michaelis–Menten or Holling type II function, Arditi and Ginzburg [9] proposed a ratio-dependent function of the form c
P
x
y
= m+
x y
= x y
cx my + x
✩ This work was supported by the Natural Science Foundation of China (No. 11001157), Tianyuan Mathematics Fund of China (No. 10826080) and the Youth Science Foundation of Shanxi Province (No. 2009021001-1, No. 2010021001-1). ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (G. Liu), [email protected] (J. Yan).
1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.05.024
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and the following ratio-dependent predator–prey model:
cxy ′ x = x(a − bx) − my + x , (1.1) fx y′ = y −d + . my + x Here x(t ) and y(t ) represent the densities of the prey and the predator at time t, respectively. a/b is the carrying capacity, d > 0 is the death rate of the predator, and a, c , m and f /c are positive constants that stand for the intrinsic growth rate of the prey, capture rate, half-saturation constant and conversion rate of the predator, respectively. Subsequently, many authors [8–10] have observed that system (1.1) exhibits much richer, more complicated, and more reasonable or acceptable dynamics. Beretta and Kuang [10] introduced a single discrete time delay into the predator equation in the foregoing model, i.e.,
cxy ′ x = x(a − bx) − my + x , fx(t − τ ) ′ y = y − d + . my(t − τ ) + x(t − τ )
(1.2)
The ratio-dependent predator–prey models with or without time delays have been studied by many researchers recently and a very rich dynamics has been observed (see, for example, [11–20] and references cited therein). In view of periodicity of the actual environment, Fan and Wang [14] established verifiable criteria for the global existence of positive periodic solutions of a more general delayed ratio-dependent predator–prey model with periodic coefficients of the form
[ ] ∫ t c (t )x(t )y(t ) ′ , x ( t ) = x ( t ) a ( t ) − b ( t ) k ( t − s ) x ( s ) ds − my(t ) + x(t ) −∞ [ ] f (t )x(t − τ (t )) y′ (t ) = y(t ) −d(t ) + . my(t − τ (t )) + x(t − τ (t ))
(1.3)
In 1991, Kuang [21] studied the local stability and oscillation of the following neutral delay Gause-type predator–prey system:
[ ] ′ x′ (t ) = rx(t ) 1 − x(t − τ ) + ρ x (t − τ ) − y(t )p(x(t )), K
(1.4)
y (t ) = y(t )[−α + β p(x(t − σ ))]. ′
In this paper, motivated by the above work, we will consider the following neutral delay ratio-dependent predator–prey model with a Holling type II functional response:
c (t )x(t )y(t ) , x′ (t ) = x(t )[a(t ) − b(t )x(t − σ (t )) − h(t )x′ (t − σ (t ))] − my(t ) + x(t ) ] [ f (t )x(t − τ (t )) ′ . y (t ) = y(t ) −d(t ) + my(t − τ (t )) + x(t − τ (t ))
(1.5)
As pointed out by Kuang [22], it would be of interest to study the existence of periodic solutions for periodic systems with time delay. The periodic solutions play the same role as is played by the equilibria in autonomous systems. In addition, in view of the fact that many predator–prey systems display sustained fluctuations, it is thus desirable to construct predator–prey models capable of producing periodic solutions. To our knowledge, no such work has been done on the global existence of positive periodic solutions of (1.5). Our aim in this paper is, using the coincidence degree theory developed by Gaines and Mawhin [23], to derive a set of easily verifiable sufficient conditions for the existence of positive periodic solutions of system (1.5). For convenience, we will use the notation
|r |0 = max {|r (t )|}, t ∈[0,ω]
r¯ =
1
ω
ω
∫ 0
r (t )dt ,
rˆ =
1
ω
ω
∫
|r (t )|dt ,
0
where r (t ) is a continuous ω-periodic function. In this paper, we always make the following assumptions for system (1.5).
(H1 ) m > 0 is a constant; a, σ , τ ∈ C (R, R), b, c , d, f , h ∈ C (R, [0, +∞)) are ω-periodic functions. In addition, a¯ > 0, d¯ > 0, b¯ > 0. h(t ) (H2 ) h ∈ C 1 (R, R), σ ∈ C 2 (R, R); σ ′ (t ) < 1 and g ′ (t ) < b(t ) for any t ∈ [0, ω], where g (t ) = 1−σ ′ (t ) . ′ 1−σ (t ) 2a¯ B (H3 ) |h|0 e < 1, where B = ln 2a¯ maxt ∈[0,ω] b(t )−g ′ (t ) + |g |0 maxt ∈[0,ω] b(t )−g ′ (t ) + (ˆa + a¯ )ω. (H4 ) c¯ < ma¯ . (H5 ) d¯ < f¯ .
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2. The existence of a positive periodic solution In this section, we shall study the existence of at least one positive periodic solution of system (1.5). The method to be used in this paper involves the applications of the continuation theorem of the coincidence degree. For the readers’ convenience, we introduce a few concepts and results concerning the coincidence degree as follows. Let X , Z be real Banach spaces, L : Dom L ⊂ X → Z be a linear mapping, and N : X → Z be a continuous mapping. The mapping L is said to be a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and Im L is closed in Z . If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X → X and Q : Z → Z such that Im P = Ker L, Ker Q = Im L = Im(I − Q ). It follows that the restriction LP of L to Dom L ∩ Ker P : (I − P )X → Im L is invertible. Denote the inverse of LP by KP . The mapping N is said to be L-compact on Ω if Ω is an open bounded subset of X , 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. Lemma 2.1 (Continuation Theorem [23, p. 40]). Let Ω ⊂ X be an open bounded set, L be a Fredholm mapping of index zero and N be L-compact on Ω . Suppose that: (i) for each λ ∈ (0, 1), x ∈ ∂ Ω ∩ Dom L, Lx ̸= λNx; (ii) for each x ∈ ∂ Ω ∩ Ker L, QNx ̸= 0; (iii) deg{JQN , Ω ∩ Ker L, 0} ̸= 0. Then Lx = Nx has at least one solution in Ω ∩ Dom L. We are now in a position to state and prove our main result. Theorem 2.1. Assume that (H1 )–(H5 ) hold. Then system (1.5) has at least one ω-periodic solution with strictly positive components. Proof. Consider the following system:
u1 ′ (t ) = a(t ) − b(t )eu1 (t −σ (t )) − h(t )eu1 (t −σ (t )) u1 ′ (t − σ (t )) − u2 ′ (t ) = −d(t ) +
f (t )e
u1 (t −τ (t ))
meu2 (t −τ (t )) + eu1
c (t )eu2 (t ) meu2 (t ) + eu1 (t )
, (2.1)
, (t −τ (t ))
where all functions are defined as ones in system (1.5). It is easy to see that if system (2.1) has one ω-periodic solution ∗ ∗ (u∗1 (t ), u∗2 (t ))T , then (x∗ (t ), y∗ (t ))T = (eu1 (t ) , eu2 (t ) )T is a positive ω-periodic solution of system (1.5). Therefore, to complete the proof it suffices to show that system (2.1) has at least one ω-periodic solution. Take X = u = (u1 (t ), u2 (t ))T ∈ C 1 (R, R2 ) : ui (t + ω) = ui (t ), t ∈ R, i = 1, 2
and Z = u = (u1 (t ), u2 (t ))T ∈ C (R, R2 ) : ui (t + ω) = ui (t ), t ∈ R, i = 1, 2
and define
|u|∞ = max {|u1 (t )| + |u2 (t )|}, t ∈[0,ω]
‖u‖ = |u|∞ + |u′ |∞ .
Then X and Z are Banach spaces when they are endowed with the norms ‖ · ‖ and | · |∞ , respectively. Let L : X → Z and N : X → Z be L(u1 (t ), u2 (t ))T = (u1 ′ (t ), u2 ′ (t ))T and
c (t )eu2 (t ) u1 (t −σ (t )) u1 (t −σ (t )) ′ ] − h(t )e u1 (t − σ (t )) − a(t ) − b(t )e u (t ) meu2 (t ) + eu1 (t ) . N 1 = u1 (t −τ (t )) u2 ( t ) f (t )e −d(t ) + u ( t −τ ( t )) u ( t −τ ( t )) me 2 +e 1 [
With this notation, system (2.1) can be written in the form Lu = Nu,
u ∈ X.
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ω Obviously, Ker L = R2 , Im L = (u1 (t ), u2 (t ))T ∈ Z : 0 ui (t )dt = 0, i = 1, 2 is closed in Z , and dim Ker L = codim Im L = 2. Therefore L is a Fredholm mapping of index zero. Now define two projectors P : X → X and Q : Z → Z as [ ] [ ] [ ] u1 ( t ) u (t ) u¯ ∈X P 1 = ¯1 , u2 u2 ( t ) u2 (t ) and u1 (t ) u¯ Q = ¯1 , u2 u2 (t )
[
[ ]
]
[
u1 ( t ) ∈ Z. u2 ( t )
]
Then P and Q are continuous projectors such that Im P = Ker L,
Ker Q = Im L = Im(I − Q ).
Furthermore, the generalized inverse (to L) KP : Im L → Ker P ∩ Dom L has the form KP (u) =
t
∫
u(s)ds − 0
ω
∫
1
ω
t
∫
u(s)dsdt .
0
0
Then QN : X → Z and KP (I − Q )N : X → X read
1
∫
ω
[
a(t ) − (b(t ) − g ′ (t ))eu1 (t −σ (t )) −
c (t )eu2 (t )
]
ω 0 meu2 (t ) + eu1 (t ) (QN )u = [ ] ∫ 1 ω f (t )eu1 (t −τ (t )) −d(t ) + dt u ( 2 ω 0 me t −τ (t )) + eu1 (t −τ (t ))
dt
and
∫ t [ 0 (KP (I − Q )N )u = ∫ t [
a(s) − (b(s) − g ′ (s))eu1 (s−σ (s)) −
−d(s) +
0
1
∫
ω
f (s)e
u1 (s−τ (s))
c (s)eu2 (s) meu2 (s)
eu1 (s)
ds − g (t )eu1 (t −σ (t )) + g (0)eu1 (−σ (0))
ds
a(s) − (b(s) − g ′ (s))eu1 (s−σ (s)) −
]
meu2 (s−τ (s)) + eu1 (s−τ (s))
∫ t[
+
]
c (s)eu2 (s)
]
dsdt
ω 0 0 meu2 (s) + eu1 (s) ∫ ω 1 u1 (t −σ (t )) u1 (−σ (0)) − − g (t )e − g (0)e dt ∫ ∫ω [0 ] u1 (s−τ (s)) 1 ω t f (s)e −d(s) + dsdt ω 0 0 meu2 (s−τ (s)) + eu1 (s−τ (s)) ∫ ω [ ] 1 c (s)eu2 (s) t ′ u1 (s−σ (s)) − a(s) − (b(s) − g (s))e − ds ω 2 0 meu2 (s) + eu1 (s) . − ] [ ∫ ω u1 (s−τ (s)) t 1 f (s)e − ds −d(s) + u ( s −τ ( s )) u ( s −τ ( s )) ω 2 0 me 2 +e 1 It is obvious that QN and KP (I − Q )N are continuous by the Lebesgue theorem, and using the Arzela–Ascoli theorem it is not difficult to show that QN (Ω ) is bounded, and KP (I − Q )N (Ω ) is compact for any open bounded set Ω ⊂ X . Hence N is L-compact on Ω for any open bounded set Ω ⊂ X . In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset Ω ⊂ X . Corresponding to the operator equation Lu = λNu, λ ∈ (0, 1), we have
[ ′ u ( t ) = λ a(t ) − b(t )eu1 (t −σ (t )) − h(t )eu1 (t −σ (t )) u1 ′ (t − σ (t )) − 1 [ u2 ′ (t ) = λ −d(t ) +
f (t )eu1 (t −τ (t )) meu2 (t −τ (t )) + eu1 (t −τ (t ))
]
c (t )eu2 (t )
]
meu2 (t ) + eu1 (t )
, (2.2)
.
Suppose that (u1 (t ), u2 (t ))T ∈ X is a solution of (2.2) for a certain λ ∈ (0, 1). Integrating (2.2) over the interval [0, ω] leads to ω
∫ 0
[
a(t ) − b(t )eu1 (t −σ (t )) − h(t )eu1 (t −σ (t )) u1 ′ (t − σ (t )) −
c (t )eu2 (t ) meu2 (t ) + eu1 (t )
] dt = 0
(2.3)
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and ω
∫
[
f (t )eu1 (t −τ (t ))
−d(t ) +
]
dt = 0.
meu2 (t −τ (t )) + eu1 (t −τ (t ))
0
(2.4)
Note that ω
∫
h(t )eu1 (t −σ (t )) u1 ′ (t − σ (t ))dt =
ω
∫
h( t ) 1−
0
0
σ ′ (t )
′
eu1 (t −σ (t )) dt =
ω eu1 (t −σ (t )) 0
∫
ω
′
g (t ) eu1 (t −σ (t )) dt
0
g (t )e = g (t ) − 0 ∫ ω g ′ (t )eu1 (t −σ (t )) dt , =−
ω
∫
′
u1 (t −σ (t ))
dt
0
which, together with (2.3), yields ω
∫
[
u1 (t −σ (t ))
(b(t ) − g (t ))e ′
+
0
c (t )eu2 (t )
]
meu2 (t ) + eu1 (t )
dt = a¯ ω.
(2.5)
From (2.4), we have ω
∫
f (t )eu1 (t −τ (t )) meu2 (t −τ (t )) + eu1 (t −τ (t ))
0
dt = d¯ ω.
(2.6)
From (2.2), (2.5) and (H2 ), one can find that ω
∫ 0
∫ ω d c (t )eu2 (t ) ′ u1 (t −σ (t )) u1 (t −σ (t )) dt − dt u1 (t ) + λg (t )e a(t ) − (b(t ) − g (t ))e dt = λ u2 (t ) + eu1 (t ) me 0 ] ∫ ω ∫ ω[ c (t )eu2 (t ) ′ u1 (t −σ (t )) |a(t )| dt + ≤ dt (b(t ) − g (t ))e + meu2 (t ) + eu1 (t ) 0 0 = (ˆa + a¯ )ω.
(2.7)
Let t = φ(p) be the inverse function of p = t − σ (t ). It is easy to see that b(φ(p)), g ′ (φ(p)) and σ ′ (φ(p)) are all ω-periodic functions. Further, it follows from (2.5) and (H2 ) that a¯ ω ≥
ω
∫
(b(t ) − g ′ (t ))eu1 (t −σ (t )) dt
0
ω−σ (ω)
∫ =
−σ (0) ∫ ω
(b(φ(p)) − g ′ (φ(p)))eu1 (p)
b(φ(p)) − g ′ (φ(p))
=
1−
0
σ ′ (φ(p))
eu1 (p) dp =
1 1 − σ ′ (φ(p)) ω
∫ 0
dp
b(φ(t )) − g ′ (φ(t )) 1 − σ ′ (φ(t ))
eu1 (t ) dt ,
which yields that 2a¯ ω ≥
∫
ω
[
b(φ(t )) − g ′ (φ(t )) 1 − σ ′ (φ(t ))
0
]
eu1 (t ) + (b(t ) − g ′ (t ))eu1 (t −σ (t )) dt .
According to the mean value theorem of differential calculus, we see that there exists ξ ∈ [0, ω] such that b(φ(ξ )) − g ′ (φ(ξ )) 1 − σ ′ (φ(ξ ))
eu1 (ξ ) + (b(ξ ) − g ′ (ξ ))eu1 (ξ −σ (ξ )) ≤ 2a¯ .
This, together with (H2 ), yields
[
u1 (ξ ) ≤ ln 2a¯ max
t ∈[0,ω]
1 − σ ′ (t )
]
b(t ) − g ′ (t )
and eu1 (ξ −σ (ξ )) ≤ max
t ∈[0,ω]
2a¯ b(t ) − g ′ (t )
,
G. Liu, J. Yan / Nonlinear Analysis: Real World Applications 12 (2011) 3252–3260
3257
which, together with (2.7), imply that, for any t ∈ [0, ω], u1 (t ) + λg (t )e
ω
d u1 (t −σ (t )) ≤ u1 (ξ ) + λg (ξ )e + dt u1 (t ) + λg (t )e dt 0 [ ] 1 − σ ′ (t ) 2a¯ ≤ ln 2a¯ max + |g |0 max + (ˆa + a¯ )ω = B. t ∈[0,ω] b(t ) − g ′ (t ) t ∈[0,ω] b(t ) − g ′ (t )
u1 (t −σ (t ))
∫
u1 (ξ −σ (ξ ))
As λg (t )eu1 (t −σ (t )) ≥ 0, one can find that u1 (t ) ≤ B,
t ∈ [0, ω].
(2.8)
In view of (2.2) and (2.8), for any t ∈ [0, ω], we obtain
c (t )eu2 (t ) u1 (t −σ (t )) u1 (t −σ (t )) ′ |u1 (t )| = λ a(t ) − b(t )e − h(t )e u1 (t − σ (t )) − meu2 (t ) + eu1 (t ) ′
≤ |a|0 + |b|0 eB + |h|0 eB |u1 ′ |0 +
1 m
|c |0 .
This, together with (H3 ), implies that
|u1 |0 ≤
[ ] 1 B |a|0 + |b|0 e + |c |0 =: D1 .
1
′
1 − |h|0 eB
(2.9)
m
Note that (u1 (t ), u2 (t ))T ∈ X . Then there exist ξi , ηi ∈ [0, ω] such that ui (ξi ) = min {ui (t )},
ui (ηi ) = max {ui (t )},
t ∈[0,ω]
i = 1, 2.
t ∈[0,ω]
(2.10)
Suppose that q(t ) = b(t ) − g ′ (t ), t ∈ [0, ω]. Hence, it follows from (2.5) that ω
∫
a¯ ω − q¯ eu1 (η1 ) ω ≤ a¯ ω −
(b(t ) − g ′ (t ))eu1 (t −σ (t )) dt
0
ω
∫ =
c (t )eu2 (t ) meu2 (t )
0
+
eu1 (t )
dt ≤
1 m
c¯ ω,
which, together with (H4 ), yields u1 (η1 ) ≥ ln
ma¯ − c¯
mq¯
.
Further, it follows from (2.9) that, for any t ∈ [0, ω], u1 (t ) ≥ u1 (η1 ) −
ω
∫
|u1 ′ (t )|dt ≥ ln
0
ma¯ − c¯
mq¯
− D1 ω =: D2 .
(2.11)
From (2.8) and (2.11), we have
|u1 |0 ≤ max{|B|, |D2 |} =: D3 .
(2.12)
From (2.6) and (2.10), one can find that d¯ ω ≤
ω
∫
f (t )eu1 (t −τ (t )) meu2 (ξ2 ) + eu1
0
dt ≤ (t −τ (t ))
eD 3 meu2 (ξ2 ) + eD3
f¯ ω
and d¯ ω ≥
ω
∫ 0
f (t )eu1 (t −τ (t )) meu2 (η2 ) + eu1
dt ≥ (t −τ (t ))
e− D 3 meu2 (η2 ) + e−D3
f¯ ω.
From (H1 ) and (H5 ), we have u2 (ξ2 ) ≤ D3 + ln
¯
f − d¯ md¯
and u2 (η2 ) ≥ −D3 + ln
¯
f − d¯ md¯
.
In addition, from (2.2) and (2.6), we have ω
∫ 0
|u2 ′ (t )|dt = λ
ω
∫ 0
f (t )eu1 (t −τ (t )) −d(t ) + dt ≤ (dˆ + d¯ )ω. u ( t −τ ( t )) u ( t −τ ( t )) 2 1 me +e
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Further, for any t ∈ [0, ω], one can find that u2 (t ) ≤ u2 (ξ2 ) +
ω
∫
|u2 ′ (t )|dt ≤ D3 + ln
¯
f − d¯
md¯
0
+ (dˆ + d¯ )ω =: D4
and u2 (t ) ≥ u2 (η2 ) −
ω
∫
|u2 ′ (t )|dt ≥ −D3 + ln
¯
f − d¯
md¯
0
− (dˆ + d¯ )ω =: D5 ,
which yield
|u2 |0 ≤ max{|D4 |, |D5 |} =: D6 .
(2.13)
From (2.2), we have
|u2 ′ |0 ≤ |d|0 + |f |0 =: D7 .
(2.14)
From (2.9), (2.12)–(2.14), we have
‖u‖ = |u|∞ + |u′ |∞ ≤ D1 + D3 + D6 + D7 .
(2.15)
In addition, it follows from (H4 ) and (H5 ) that the system of algebraic equations eu2
¯ u1 − c¯ a¯ − be
meu2 + eu1 u1 e − d¯ + f¯ =0 u me 2 + eu1
= 0, (2.16)
have a unique solution (u∗1 , u∗2 )T ∈ R2 . Moreover, a single calculation yields u∗1 = ln
a¯ b¯
−
c¯ (f¯ − d¯ )
mb¯ f¯
,
u∗2 = u∗1 + ln
¯
f − d¯
md¯
.
Set D = D1 + D3 + D6 + D7 + D0 , where D0 is taken sufficiently large that the unique solution (u∗1 , u∗2 )T of (2.16) satisfies
∗ ∗ T (u , u ) = |u∗ | + |u∗ | < D0 . 1
2
1
2
Clearly, D is independent of λ. We now take
Ω = (u1 (t ), u2 (t ))T ∈ X : (u1 (t ), u2 (t ))T < D . This satisfies condition (i) in Lemma 2.1. When (u1 (t ), u2 (t ))T ∈ ∂ Ω ∩ Ker L = ∂ Ω ∩ R2 , (u1 (t ), u2 (t ))T is a constant vector in R2 with |u1 | + |u2 | = D. Thus, we have
[ ]
u QN 1 u2
¯ u1 − c¯ a¯ − be
=
− d¯ + f¯
eu 2
[ ]
0 meu2 + eu1 ̸= 0 . eu1
meu2 + eu1
This proves that condition (ii) in Lemma 2.1 is satisfied. Taking J = I : Im Q → Ker L, (u1 , u2 )T → (u1 , u2 )T , a direct calculation shows that
∗
∗
∗
eu1 eu2
¯ u1 2 −be + c¯ u∗ u∗ 2 + e 1 me deg{JQN , Ω ∩ Ker L, 0} = sgn det ∗ ∗ meu1 eu2 f¯ ∗ ∗ 2 meu2 + eu1 ∗ 2u∗ u e 1e 2 = sgn mb¯ f¯ ̸= 0. 2 ∗ ∗ meu2 + eu1
∗
∗
eu1 eu2
∗ ∗ 2 meu2 + eu1 ∗ ∗ meu1 eu2 −f¯ ∗ 2 u∗ u me 2 + e 1 −¯c
We have now proved that Ω satisfies all the requirements in Lemma 2.1. Hence, (2.1) has at least one ω-periodic solution. Accordingly, system (1.5) has at least one ω-periodic solution with strictly positive components. The proof of Theorem 2.1 is complete.
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Remark 2.1. It is easy to see that (H5 ) is also the necessary condition for the existence of positive ω-periodic solutions of system (1.5). Remark 2.2. From the proof of Theorem 2.1, we see that Theorem 2.1 is also valid if h(t ) ≡ 0. Consequently, we can obtain the following corollary. Corollary 2.1. Assume that (H1 ), (H4 ) and (H5 ) hold. Then the following delayed ratio-dependent predator–prey model with a Holling type II functional response:
c (t )x(t )y(t ) ′ , x (t ) = x(t )[a(t ) − b(t )x(t − σ (t ))] − my(t ) + x(t ) ] [ f (t )x(t − τ (t )) y′ (t ) = y(t ) −d(t ) + . my(t − τ (t )) + x(t − τ (t )) has at least one ω-periodic solution with strictly positive components. Next consider the following neutral ratio-dependent predator–prey model with state-dependent delays:
c (t )x(t )y(t ) , x′ (t ) = x(t )[a(t ) − b(t )x(t − σ (t , x(t ), y(t ))) − h(t )x′ (t − σ (t , x(t ), y(t )))] − my(t ) + x(t ) ] [ f (t )x(t − τ (t , x(t ), y(t ))) ′ . y (t ) = y(t ) −d(t ) + my(t − τ (t , x(t ), y(t ))) + x(t − τ (t , x(t ), y(t )))
(2.17)
where σ (t , x, y) and τ (t , x, y) are continuous functions and ω-periodic functions with respect to t. Theorem 2.2. Suppose that (H1 )–(H5 ) hold. Then system (2.17) has at least one ω-periodic solution with strictly positive components. Proof. The proof is similar to the proof of Theorem 2.1 and hence is omitted here.
In order to illustrate the feasibility of our results, we give the following example.
− 13 cos(20π t ), b(t ) = 10 + sin(20π t ), c (t ) = 1 − 12 sin(20π t ), 20 d(t ) = 2 + cos(20π t ), f (t ) = 3 − sin(20π t ), h(t ) = 1 + 21 sin(20π t ), σ (t ) = 1 + 211π cos(20π t ). A straightforward Example 2.1. In system (1.5), suppose that m = 4, a(t ) =
1 2
calculation shows that g (t ) ≡ 1,
aˆ = a¯ =
1 2
,
c¯ = 1,
d¯ = 2,
f¯ = 3,
ω=
1 10
,
|g |0 = 1,
|h|0 =
41 21
and
B = ln
41 231
+
19 90
.
Further, |h|0 eB ≈ 0.4280. Hence, σ ′ (t ) < 1, g ′ (t ) < b(t ) for any t ∈ [0, ω] and
|h|0 eB < 1,
c¯ < ma¯ ,
d¯ < f¯ .
Consequently, all the conditions in Theorem 2.1 hold. Using Theorem 2.1, we know that system (1.5) has at least one 1 -periodic solution with strictly positive components. 10 References [1] Y. Tao, X. Wang, X. Song, Effect of prey refuge on a harvested predator–prey model with generalized functional response, Commun. Nonlinear Sci. Numer. Simul. 16 (2) (2011) 1052–1059. [2] J.L. Bravo, M. Fernández, M. Gámez, B. Granados, A. Tineo, Existence of a polycycle in non-Lipschitz Gause-type predator–prey models, J. Math. Anal. Appl. 373 (2) (2011) 512–520. [3] L. Zhang, C. Zhang, Rich dynamic of a stage-structured prey–predator model with cannibalism and periodic attacking rate, Commun. Nonlinear Sci. Numer. Simul. 15 (12) (2010) 4029–4040. [4] X. Ding, J. Jiang, Positive periodic solutions in delayed Gause-type predator–prey systems, J. Math. Anal. Appl. 339 (2008) 1220–1230. [5] Z. Zhang, Z. Hou, L. Wang, Multiplicity of positive periodic solutions to a generalized delayed predator–prey system with stocking, Nonlinear Anal. 68 (2008) 2608–2622. [6] S.B. Hsu, T.W. Hwang, Y. Kuang, Global analysis of the Michaelis–Menten type ratio-dependent predator–prey system, J. Math. Biol. 42 (2001) 489–506. [7] C. Jost, O. Arino, R. Arditi, About deterministic extinction in ratio-dependent predator–prey models, Bull. Math. Biol. 61 (1999) 19–32. [8] Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predator–prey system, J. Math. Biol. 36 (1998) 389–406. [9] R. Arditi, L.R. Ginzburg, Coupling in predator–prey dynamics: ratio-dependence, J. Theor. Biol. 139 (1989) 311–326. [10] E. Beretta, Y. Kuang, Global analysis in some delayed ratio-dependent predator–prey systems, Nonlinear Anal. 32 (1998) 381–408. [11] M.U. Akhmet, M. Beklioglu, T. Ergenc, V.I. Tkachenko, An impulsive ratio-dependent predator–prey system with diffusion, Nonlinear Anal. RWA 7 (2006) 1255–1267.
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[12] B. Dai, N. Zhang, J. Zou, Permanence for the Michaelis–Menten type discrete three-species ratio-dependent food chain model with delay, J. Math. Anal. Appl. 324 (2006) 728–738. [13] D.L. DeAngelis, J.N. Holland, Emergence of ratio-dependent and predator-dependent functional responses for pollination mutualism and seed parasitism, Ecol. Model. 191 (2006) 551–556. [14] M. Fan, K. Wang, Periodicity in a delayed ratio-dependent predator–prey system, J. Math. Anal. Appl. 262 (2001) 179–190. [15] Y.H. Fan, W.T. Li, Global asymptotic stability of a ratio-dependent predator–prey system with diffusion, J. Comput. Appl. Math. 188 (2006) 205–227. [16] X. Liu, G. Li, G. Luo, Positive periodic solution for a two-species ratio-dependent predator–prey system with time delay and impulse, J. Math. Anal. Appl. 325 (2007) 715–723. [17] K. Ryu, I. Ahn, Positive solutions for ratio-dependent predator–prey interaction systems, J. Differential Equations 218 (2005) 117–135. [18] T. Saha, M. Bandyopadhyay, Dynamical analysis of a delayed ratio-dependent prey–predator model within fluctuating environment, Appl. Math. Comput. 196 (2008) 458–478. [19] L.L. Wang, W.T. Li, Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator–prey model with Holling type functional response, J. Comput. Appl. Math. 162 (2004) 341–357. [20] M. Wang, Stationary patterns for a prey–predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D 196 (2004) 172–192. [21] Y. Kuang, On neutral delay logistic Gause-type predator–prey systems, Dyn. Stab. Syst. 6 (1991) 173–189. [22] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [23] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Positive periodic solutions for a neutral delay ratio-dependent predator–prey model with a Holling type II functional response✩ Guirong Liu ∗ , Jurang Yan School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China
article
abstract
info
Article history: Received 1 November 2010 Accepted 24 May 2011
By using a continuation theorem based on coincidence degree theory, some new and interesting sufficient conditions are obtained for the existence of positive periodic solutions for a neutral delay ratio-dependent predator–prey model with the Holling type II functional response:
Keywords: Predator–prey model Ratio-dependent Periodic solution Neutral Coincidence degree
c (t )x(t )y(t ) , x′ (t ) = x(t )[a(t ) − b(t )x(t − σ (t )) − h(t )x′ (t − σ (t ))] − my ( t ) + x( t ) [ ] f ( t ) x ( t − τ ( t )) ′ . y (t ) = y(t ) −d(t ) + my(t − τ (t )) + x(t − τ (t )) An example is presented to illustrate the feasibility of our main results. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The dynamic relationship between predator and prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. The traditional predator–prey models have been studied extensively (see, for example, [1–5] and references cited therein). Recently, there has been growing explicit biological and physiological evidence [6–8] that in many situations, especially when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator–prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as saying that the per capita predator growth rate should be a function of the ratio of prey to predator abundance, and so should be the so-called ratio-dependent functional response. This is strongly supported by numerous field and laboratory experiments and observations. On the basis of the Michaelis–Menten or Holling type II function, Arditi and Ginzburg [9] proposed a ratio-dependent function of the form c
P
x
y
= m+
x y
= x y
cx my + x
✩ This work was supported by the Natural Science Foundation of China (No. 11001157), Tianyuan Mathematics Fund of China (No. 10826080) and the Youth Science Foundation of Shanxi Province (No. 2009021001-1, No. 2010021001-1). ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (G. Liu), [email protected] (J. Yan).
1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.05.024
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and the following ratio-dependent predator–prey model:
cxy ′ x = x(a − bx) − my + x , (1.1) fx y′ = y −d + . my + x Here x(t ) and y(t ) represent the densities of the prey and the predator at time t, respectively. a/b is the carrying capacity, d > 0 is the death rate of the predator, and a, c , m and f /c are positive constants that stand for the intrinsic growth rate of the prey, capture rate, half-saturation constant and conversion rate of the predator, respectively. Subsequently, many authors [8–10] have observed that system (1.1) exhibits much richer, more complicated, and more reasonable or acceptable dynamics. Beretta and Kuang [10] introduced a single discrete time delay into the predator equation in the foregoing model, i.e.,
cxy ′ x = x(a − bx) − my + x , fx(t − τ ) ′ y = y − d + . my(t − τ ) + x(t − τ )
(1.2)
The ratio-dependent predator–prey models with or without time delays have been studied by many researchers recently and a very rich dynamics has been observed (see, for example, [11–20] and references cited therein). In view of periodicity of the actual environment, Fan and Wang [14] established verifiable criteria for the global existence of positive periodic solutions of a more general delayed ratio-dependent predator–prey model with periodic coefficients of the form
[ ] ∫ t c (t )x(t )y(t ) ′ , x ( t ) = x ( t ) a ( t ) − b ( t ) k ( t − s ) x ( s ) ds − my(t ) + x(t ) −∞ [ ] f (t )x(t − τ (t )) y′ (t ) = y(t ) −d(t ) + . my(t − τ (t )) + x(t − τ (t ))
(1.3)
In 1991, Kuang [21] studied the local stability and oscillation of the following neutral delay Gause-type predator–prey system:
[ ] ′ x′ (t ) = rx(t ) 1 − x(t − τ ) + ρ x (t − τ ) − y(t )p(x(t )), K
(1.4)
y (t ) = y(t )[−α + β p(x(t − σ ))]. ′
In this paper, motivated by the above work, we will consider the following neutral delay ratio-dependent predator–prey model with a Holling type II functional response:
c (t )x(t )y(t ) , x′ (t ) = x(t )[a(t ) − b(t )x(t − σ (t )) − h(t )x′ (t − σ (t ))] − my(t ) + x(t ) ] [ f (t )x(t − τ (t )) ′ . y (t ) = y(t ) −d(t ) + my(t − τ (t )) + x(t − τ (t ))
(1.5)
As pointed out by Kuang [22], it would be of interest to study the existence of periodic solutions for periodic systems with time delay. The periodic solutions play the same role as is played by the equilibria in autonomous systems. In addition, in view of the fact that many predator–prey systems display sustained fluctuations, it is thus desirable to construct predator–prey models capable of producing periodic solutions. To our knowledge, no such work has been done on the global existence of positive periodic solutions of (1.5). Our aim in this paper is, using the coincidence degree theory developed by Gaines and Mawhin [23], to derive a set of easily verifiable sufficient conditions for the existence of positive periodic solutions of system (1.5). For convenience, we will use the notation
|r |0 = max {|r (t )|}, t ∈[0,ω]
r¯ =
1
ω
ω
∫ 0
r (t )dt ,
rˆ =
1
ω
ω
∫
|r (t )|dt ,
0
where r (t ) is a continuous ω-periodic function. In this paper, we always make the following assumptions for system (1.5).
(H1 ) m > 0 is a constant; a, σ , τ ∈ C (R, R), b, c , d, f , h ∈ C (R, [0, +∞)) are ω-periodic functions. In addition, a¯ > 0, d¯ > 0, b¯ > 0. h(t ) (H2 ) h ∈ C 1 (R, R), σ ∈ C 2 (R, R); σ ′ (t ) < 1 and g ′ (t ) < b(t ) for any t ∈ [0, ω], where g (t ) = 1−σ ′ (t ) . ′ 1−σ (t ) 2a¯ B (H3 ) |h|0 e < 1, where B = ln 2a¯ maxt ∈[0,ω] b(t )−g ′ (t ) + |g |0 maxt ∈[0,ω] b(t )−g ′ (t ) + (ˆa + a¯ )ω. (H4 ) c¯ < ma¯ . (H5 ) d¯ < f¯ .
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2. The existence of a positive periodic solution In this section, we shall study the existence of at least one positive periodic solution of system (1.5). The method to be used in this paper involves the applications of the continuation theorem of the coincidence degree. For the readers’ convenience, we introduce a few concepts and results concerning the coincidence degree as follows. Let X , Z be real Banach spaces, L : Dom L ⊂ X → Z be a linear mapping, and N : X → Z be a continuous mapping. The mapping L is said to be a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and Im L is closed in Z . If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X → X and Q : Z → Z such that Im P = Ker L, Ker Q = Im L = Im(I − Q ). It follows that the restriction LP of L to Dom L ∩ Ker P : (I − P )X → Im L is invertible. Denote the inverse of LP by KP . The mapping N is said to be L-compact on Ω if Ω is an open bounded subset of X , 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. Lemma 2.1 (Continuation Theorem [23, p. 40]). Let Ω ⊂ X be an open bounded set, L be a Fredholm mapping of index zero and N be L-compact on Ω . Suppose that: (i) for each λ ∈ (0, 1), x ∈ ∂ Ω ∩ Dom L, Lx ̸= λNx; (ii) for each x ∈ ∂ Ω ∩ Ker L, QNx ̸= 0; (iii) deg{JQN , Ω ∩ Ker L, 0} ̸= 0. Then Lx = Nx has at least one solution in Ω ∩ Dom L. We are now in a position to state and prove our main result. Theorem 2.1. Assume that (H1 )–(H5 ) hold. Then system (1.5) has at least one ω-periodic solution with strictly positive components. Proof. Consider the following system:
u1 ′ (t ) = a(t ) − b(t )eu1 (t −σ (t )) − h(t )eu1 (t −σ (t )) u1 ′ (t − σ (t )) − u2 ′ (t ) = −d(t ) +
f (t )e
u1 (t −τ (t ))
meu2 (t −τ (t )) + eu1
c (t )eu2 (t ) meu2 (t ) + eu1 (t )
, (2.1)
, (t −τ (t ))
where all functions are defined as ones in system (1.5). It is easy to see that if system (2.1) has one ω-periodic solution ∗ ∗ (u∗1 (t ), u∗2 (t ))T , then (x∗ (t ), y∗ (t ))T = (eu1 (t ) , eu2 (t ) )T is a positive ω-periodic solution of system (1.5). Therefore, to complete the proof it suffices to show that system (2.1) has at least one ω-periodic solution. Take X = u = (u1 (t ), u2 (t ))T ∈ C 1 (R, R2 ) : ui (t + ω) = ui (t ), t ∈ R, i = 1, 2
and Z = u = (u1 (t ), u2 (t ))T ∈ C (R, R2 ) : ui (t + ω) = ui (t ), t ∈ R, i = 1, 2
and define
|u|∞ = max {|u1 (t )| + |u2 (t )|}, t ∈[0,ω]
‖u‖ = |u|∞ + |u′ |∞ .
Then X and Z are Banach spaces when they are endowed with the norms ‖ · ‖ and | · |∞ , respectively. Let L : X → Z and N : X → Z be L(u1 (t ), u2 (t ))T = (u1 ′ (t ), u2 ′ (t ))T and
c (t )eu2 (t ) u1 (t −σ (t )) u1 (t −σ (t )) ′ ] − h(t )e u1 (t − σ (t )) − a(t ) − b(t )e u (t ) meu2 (t ) + eu1 (t ) . N 1 = u1 (t −τ (t )) u2 ( t ) f (t )e −d(t ) + u ( t −τ ( t )) u ( t −τ ( t )) me 2 +e 1 [
With this notation, system (2.1) can be written in the form Lu = Nu,
u ∈ X.
G. Liu, J. Yan / Nonlinear Analysis: Real World Applications 12 (2011) 3252–3260
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ω Obviously, Ker L = R2 , Im L = (u1 (t ), u2 (t ))T ∈ Z : 0 ui (t )dt = 0, i = 1, 2 is closed in Z , and dim Ker L = codim Im L = 2. Therefore L is a Fredholm mapping of index zero. Now define two projectors P : X → X and Q : Z → Z as [ ] [ ] [ ] u1 ( t ) u (t ) u¯ ∈X P 1 = ¯1 , u2 u2 ( t ) u2 (t ) and u1 (t ) u¯ Q = ¯1 , u2 u2 (t )
[
[ ]
]
[
u1 ( t ) ∈ Z. u2 ( t )
]
Then P and Q are continuous projectors such that Im P = Ker L,
Ker Q = Im L = Im(I − Q ).
Furthermore, the generalized inverse (to L) KP : Im L → Ker P ∩ Dom L has the form KP (u) =
t
∫
u(s)ds − 0
ω
∫
1
ω
t
∫
u(s)dsdt .
0
0
Then QN : X → Z and KP (I − Q )N : X → X read
1
∫
ω
[
a(t ) − (b(t ) − g ′ (t ))eu1 (t −σ (t )) −
c (t )eu2 (t )
]
ω 0 meu2 (t ) + eu1 (t ) (QN )u = [ ] ∫ 1 ω f (t )eu1 (t −τ (t )) −d(t ) + dt u ( 2 ω 0 me t −τ (t )) + eu1 (t −τ (t ))
dt
and
∫ t [ 0 (KP (I − Q )N )u = ∫ t [
a(s) − (b(s) − g ′ (s))eu1 (s−σ (s)) −
−d(s) +
0
1
∫
ω
f (s)e
u1 (s−τ (s))
c (s)eu2 (s) meu2 (s)
eu1 (s)
ds − g (t )eu1 (t −σ (t )) + g (0)eu1 (−σ (0))
ds
a(s) − (b(s) − g ′ (s))eu1 (s−σ (s)) −
]
meu2 (s−τ (s)) + eu1 (s−τ (s))
∫ t[
+
]
c (s)eu2 (s)
]
dsdt
ω 0 0 meu2 (s) + eu1 (s) ∫ ω 1 u1 (t −σ (t )) u1 (−σ (0)) − − g (t )e − g (0)e dt ∫ ∫ω [0 ] u1 (s−τ (s)) 1 ω t f (s)e −d(s) + dsdt ω 0 0 meu2 (s−τ (s)) + eu1 (s−τ (s)) ∫ ω [ ] 1 c (s)eu2 (s) t ′ u1 (s−σ (s)) − a(s) − (b(s) − g (s))e − ds ω 2 0 meu2 (s) + eu1 (s) . − ] [ ∫ ω u1 (s−τ (s)) t 1 f (s)e − ds −d(s) + u ( s −τ ( s )) u ( s −τ ( s )) ω 2 0 me 2 +e 1 It is obvious that QN and KP (I − Q )N are continuous by the Lebesgue theorem, and using the Arzela–Ascoli theorem it is not difficult to show that QN (Ω ) is bounded, and KP (I − Q )N (Ω ) is compact for any open bounded set Ω ⊂ X . Hence N is L-compact on Ω for any open bounded set Ω ⊂ X . In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset Ω ⊂ X . Corresponding to the operator equation Lu = λNu, λ ∈ (0, 1), we have
[ ′ u ( t ) = λ a(t ) − b(t )eu1 (t −σ (t )) − h(t )eu1 (t −σ (t )) u1 ′ (t − σ (t )) − 1 [ u2 ′ (t ) = λ −d(t ) +
f (t )eu1 (t −τ (t )) meu2 (t −τ (t )) + eu1 (t −τ (t ))
]
c (t )eu2 (t )
]
meu2 (t ) + eu1 (t )
, (2.2)
.
Suppose that (u1 (t ), u2 (t ))T ∈ X is a solution of (2.2) for a certain λ ∈ (0, 1). Integrating (2.2) over the interval [0, ω] leads to ω
∫ 0
[
a(t ) − b(t )eu1 (t −σ (t )) − h(t )eu1 (t −σ (t )) u1 ′ (t − σ (t )) −
c (t )eu2 (t ) meu2 (t ) + eu1 (t )
] dt = 0
(2.3)
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G. Liu, J. Yan / Nonlinear Analysis: Real World Applications 12 (2011) 3252–3260
and ω
∫
[
f (t )eu1 (t −τ (t ))
−d(t ) +
]
dt = 0.
meu2 (t −τ (t )) + eu1 (t −τ (t ))
0
(2.4)
Note that ω
∫
h(t )eu1 (t −σ (t )) u1 ′ (t − σ (t ))dt =
ω
∫
h( t ) 1−
0
0
σ ′ (t )
′
eu1 (t −σ (t )) dt =
ω eu1 (t −σ (t )) 0
∫
ω
′
g (t ) eu1 (t −σ (t )) dt
0
g (t )e = g (t ) − 0 ∫ ω g ′ (t )eu1 (t −σ (t )) dt , =−
ω
∫
′
u1 (t −σ (t ))
dt
0
which, together with (2.3), yields ω
∫
[
u1 (t −σ (t ))
(b(t ) − g (t ))e ′
+
0
c (t )eu2 (t )
]
meu2 (t ) + eu1 (t )
dt = a¯ ω.
(2.5)
From (2.4), we have ω
∫
f (t )eu1 (t −τ (t )) meu2 (t −τ (t )) + eu1 (t −τ (t ))
0
dt = d¯ ω.
(2.6)
From (2.2), (2.5) and (H2 ), one can find that ω
∫ 0
∫ ω d c (t )eu2 (t ) ′ u1 (t −σ (t )) u1 (t −σ (t )) dt − dt u1 (t ) + λg (t )e a(t ) − (b(t ) − g (t ))e dt = λ u2 (t ) + eu1 (t ) me 0 ] ∫ ω ∫ ω[ c (t )eu2 (t ) ′ u1 (t −σ (t )) |a(t )| dt + ≤ dt (b(t ) − g (t ))e + meu2 (t ) + eu1 (t ) 0 0 = (ˆa + a¯ )ω.
(2.7)
Let t = φ(p) be the inverse function of p = t − σ (t ). It is easy to see that b(φ(p)), g ′ (φ(p)) and σ ′ (φ(p)) are all ω-periodic functions. Further, it follows from (2.5) and (H2 ) that a¯ ω ≥
ω
∫
(b(t ) − g ′ (t ))eu1 (t −σ (t )) dt
0
ω−σ (ω)
∫ =
−σ (0) ∫ ω
(b(φ(p)) − g ′ (φ(p)))eu1 (p)
b(φ(p)) − g ′ (φ(p))
=
1−
0
σ ′ (φ(p))
eu1 (p) dp =
1 1 − σ ′ (φ(p)) ω
∫ 0
dp
b(φ(t )) − g ′ (φ(t )) 1 − σ ′ (φ(t ))
eu1 (t ) dt ,
which yields that 2a¯ ω ≥
∫
ω
[
b(φ(t )) − g ′ (φ(t )) 1 − σ ′ (φ(t ))
0
]
eu1 (t ) + (b(t ) − g ′ (t ))eu1 (t −σ (t )) dt .
According to the mean value theorem of differential calculus, we see that there exists ξ ∈ [0, ω] such that b(φ(ξ )) − g ′ (φ(ξ )) 1 − σ ′ (φ(ξ ))
eu1 (ξ ) + (b(ξ ) − g ′ (ξ ))eu1 (ξ −σ (ξ )) ≤ 2a¯ .
This, together with (H2 ), yields
[
u1 (ξ ) ≤ ln 2a¯ max
t ∈[0,ω]
1 − σ ′ (t )
]
b(t ) − g ′ (t )
and eu1 (ξ −σ (ξ )) ≤ max
t ∈[0,ω]
2a¯ b(t ) − g ′ (t )
,
G. Liu, J. Yan / Nonlinear Analysis: Real World Applications 12 (2011) 3252–3260
3257
which, together with (2.7), imply that, for any t ∈ [0, ω], u1 (t ) + λg (t )e
ω
d u1 (t −σ (t )) ≤ u1 (ξ ) + λg (ξ )e + dt u1 (t ) + λg (t )e dt 0 [ ] 1 − σ ′ (t ) 2a¯ ≤ ln 2a¯ max + |g |0 max + (ˆa + a¯ )ω = B. t ∈[0,ω] b(t ) − g ′ (t ) t ∈[0,ω] b(t ) − g ′ (t )
u1 (t −σ (t ))
∫
u1 (ξ −σ (ξ ))
As λg (t )eu1 (t −σ (t )) ≥ 0, one can find that u1 (t ) ≤ B,
t ∈ [0, ω].
(2.8)
In view of (2.2) and (2.8), for any t ∈ [0, ω], we obtain
c (t )eu2 (t ) u1 (t −σ (t )) u1 (t −σ (t )) ′ |u1 (t )| = λ a(t ) − b(t )e − h(t )e u1 (t − σ (t )) − meu2 (t ) + eu1 (t ) ′
≤ |a|0 + |b|0 eB + |h|0 eB |u1 ′ |0 +
1 m
|c |0 .
This, together with (H3 ), implies that
|u1 |0 ≤
[ ] 1 B |a|0 + |b|0 e + |c |0 =: D1 .
1
′
1 − |h|0 eB
(2.9)
m
Note that (u1 (t ), u2 (t ))T ∈ X . Then there exist ξi , ηi ∈ [0, ω] such that ui (ξi ) = min {ui (t )},
ui (ηi ) = max {ui (t )},
t ∈[0,ω]
i = 1, 2.
t ∈[0,ω]
(2.10)
Suppose that q(t ) = b(t ) − g ′ (t ), t ∈ [0, ω]. Hence, it follows from (2.5) that ω
∫
a¯ ω − q¯ eu1 (η1 ) ω ≤ a¯ ω −
(b(t ) − g ′ (t ))eu1 (t −σ (t )) dt
0
ω
∫ =
c (t )eu2 (t ) meu2 (t )
0
+
eu1 (t )
dt ≤
1 m
c¯ ω,
which, together with (H4 ), yields u1 (η1 ) ≥ ln
ma¯ − c¯
mq¯
.
Further, it follows from (2.9) that, for any t ∈ [0, ω], u1 (t ) ≥ u1 (η1 ) −
ω
∫
|u1 ′ (t )|dt ≥ ln
0
ma¯ − c¯
mq¯
− D1 ω =: D2 .
(2.11)
From (2.8) and (2.11), we have
|u1 |0 ≤ max{|B|, |D2 |} =: D3 .
(2.12)
From (2.6) and (2.10), one can find that d¯ ω ≤
ω
∫
f (t )eu1 (t −τ (t )) meu2 (ξ2 ) + eu1
0
dt ≤ (t −τ (t ))
eD 3 meu2 (ξ2 ) + eD3
f¯ ω
and d¯ ω ≥
ω
∫ 0
f (t )eu1 (t −τ (t )) meu2 (η2 ) + eu1
dt ≥ (t −τ (t ))
e− D 3 meu2 (η2 ) + e−D3
f¯ ω.
From (H1 ) and (H5 ), we have u2 (ξ2 ) ≤ D3 + ln
¯
f − d¯ md¯
and u2 (η2 ) ≥ −D3 + ln
¯
f − d¯ md¯
.
In addition, from (2.2) and (2.6), we have ω
∫ 0
|u2 ′ (t )|dt = λ
ω
∫ 0
f (t )eu1 (t −τ (t )) −d(t ) + dt ≤ (dˆ + d¯ )ω. u ( t −τ ( t )) u ( t −τ ( t )) 2 1 me +e
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G. Liu, J. Yan / Nonlinear Analysis: Real World Applications 12 (2011) 3252–3260
Further, for any t ∈ [0, ω], one can find that u2 (t ) ≤ u2 (ξ2 ) +
ω
∫
|u2 ′ (t )|dt ≤ D3 + ln
¯
f − d¯
md¯
0
+ (dˆ + d¯ )ω =: D4
and u2 (t ) ≥ u2 (η2 ) −
ω
∫
|u2 ′ (t )|dt ≥ −D3 + ln
¯
f − d¯
md¯
0
− (dˆ + d¯ )ω =: D5 ,
which yield
|u2 |0 ≤ max{|D4 |, |D5 |} =: D6 .
(2.13)
From (2.2), we have
|u2 ′ |0 ≤ |d|0 + |f |0 =: D7 .
(2.14)
From (2.9), (2.12)–(2.14), we have
‖u‖ = |u|∞ + |u′ |∞ ≤ D1 + D3 + D6 + D7 .
(2.15)
In addition, it follows from (H4 ) and (H5 ) that the system of algebraic equations eu2
¯ u1 − c¯ a¯ − be
meu2 + eu1 u1 e − d¯ + f¯ =0 u me 2 + eu1
= 0, (2.16)
have a unique solution (u∗1 , u∗2 )T ∈ R2 . Moreover, a single calculation yields u∗1 = ln
a¯ b¯
−
c¯ (f¯ − d¯ )
mb¯ f¯
,
u∗2 = u∗1 + ln
¯
f − d¯
md¯
.
Set D = D1 + D3 + D6 + D7 + D0 , where D0 is taken sufficiently large that the unique solution (u∗1 , u∗2 )T of (2.16) satisfies
∗ ∗ T (u , u ) = |u∗ | + |u∗ | < D0 . 1
2
1
2
Clearly, D is independent of λ. We now take
Ω = (u1 (t ), u2 (t ))T ∈ X : (u1 (t ), u2 (t ))T < D . This satisfies condition (i) in Lemma 2.1. When (u1 (t ), u2 (t ))T ∈ ∂ Ω ∩ Ker L = ∂ Ω ∩ R2 , (u1 (t ), u2 (t ))T is a constant vector in R2 with |u1 | + |u2 | = D. Thus, we have
[ ]
u QN 1 u2
¯ u1 − c¯ a¯ − be
=
− d¯ + f¯
eu 2
[ ]
0 meu2 + eu1 ̸= 0 . eu1
meu2 + eu1
This proves that condition (ii) in Lemma 2.1 is satisfied. Taking J = I : Im Q → Ker L, (u1 , u2 )T → (u1 , u2 )T , a direct calculation shows that
∗
∗
∗
eu1 eu2
¯ u1 2 −be + c¯ u∗ u∗ 2 + e 1 me deg{JQN , Ω ∩ Ker L, 0} = sgn det ∗ ∗ meu1 eu2 f¯ ∗ ∗ 2 meu2 + eu1 ∗ 2u∗ u e 1e 2 = sgn mb¯ f¯ ̸= 0. 2 ∗ ∗ meu2 + eu1
∗
∗
eu1 eu2
∗ ∗ 2 meu2 + eu1 ∗ ∗ meu1 eu2 −f¯ ∗ 2 u∗ u me 2 + e 1 −¯c
We have now proved that Ω satisfies all the requirements in Lemma 2.1. Hence, (2.1) has at least one ω-periodic solution. Accordingly, system (1.5) has at least one ω-periodic solution with strictly positive components. The proof of Theorem 2.1 is complete.
G. Liu, J. Yan / Nonlinear Analysis: Real World Applications 12 (2011) 3252–3260
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Remark 2.1. It is easy to see that (H5 ) is also the necessary condition for the existence of positive ω-periodic solutions of system (1.5). Remark 2.2. From the proof of Theorem 2.1, we see that Theorem 2.1 is also valid if h(t ) ≡ 0. Consequently, we can obtain the following corollary. Corollary 2.1. Assume that (H1 ), (H4 ) and (H5 ) hold. Then the following delayed ratio-dependent predator–prey model with a Holling type II functional response:
c (t )x(t )y(t ) ′ , x (t ) = x(t )[a(t ) − b(t )x(t − σ (t ))] − my(t ) + x(t ) ] [ f (t )x(t − τ (t )) y′ (t ) = y(t ) −d(t ) + . my(t − τ (t )) + x(t − τ (t )) has at least one ω-periodic solution with strictly positive components. Next consider the following neutral ratio-dependent predator–prey model with state-dependent delays:
c (t )x(t )y(t ) , x′ (t ) = x(t )[a(t ) − b(t )x(t − σ (t , x(t ), y(t ))) − h(t )x′ (t − σ (t , x(t ), y(t )))] − my(t ) + x(t ) ] [ f (t )x(t − τ (t , x(t ), y(t ))) ′ . y (t ) = y(t ) −d(t ) + my(t − τ (t , x(t ), y(t ))) + x(t − τ (t , x(t ), y(t )))
(2.17)
where σ (t , x, y) and τ (t , x, y) are continuous functions and ω-periodic functions with respect to t. Theorem 2.2. Suppose that (H1 )–(H5 ) hold. Then system (2.17) has at least one ω-periodic solution with strictly positive components. Proof. The proof is similar to the proof of Theorem 2.1 and hence is omitted here.
In order to illustrate the feasibility of our results, we give the following example.
− 13 cos(20π t ), b(t ) = 10 + sin(20π t ), c (t ) = 1 − 12 sin(20π t ), 20 d(t ) = 2 + cos(20π t ), f (t ) = 3 − sin(20π t ), h(t ) = 1 + 21 sin(20π t ), σ (t ) = 1 + 211π cos(20π t ). A straightforward Example 2.1. In system (1.5), suppose that m = 4, a(t ) =
1 2
calculation shows that g (t ) ≡ 1,
aˆ = a¯ =
1 2
,
c¯ = 1,
d¯ = 2,
f¯ = 3,
ω=
1 10
,
|g |0 = 1,
|h|0 =
41 21
and
B = ln
41 231
+
19 90
.
Further, |h|0 eB ≈ 0.4280. Hence, σ ′ (t ) < 1, g ′ (t ) < b(t ) for any t ∈ [0, ω] and
|h|0 eB < 1,
c¯ < ma¯ ,
d¯ < f¯ .
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