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# Dynamics of a delayed discrete semi-ratio-dependent predator-prey system with Holling type IV functional response

- Hongying Lu
^{1}Email author and - Weiguo Wang
^{1}Email author

**2011**:7

https://doi.org/10.1186/1687-1847-2011-7

© Lu and Wang; licensee Springer. 2011

**Received:**16 February 2011**Accepted:**6 June 2011**Published:**6 June 2011

## Abstract

A discrete semi-ratio-dependent predator-prey system with Holling type IV functional response and time delay is investigated. It is proved the general nonautonomous system is permanent and globally attractive under some appropriate conditions. Furthermore, if the system is periodic one, some sufficient conditions are established, which guarantee the existence and global attractivity of positive periodic solutions. We show that the conditions for the permanence of the system and the global attractivity of positive periodic solutions depend on the delay, so, we call it profitless.

## Keywords

- Discrete
- Semi-ratio-dependent
- Holling type IV functional response
- Permanence
- Global attractivity

## Introduction

where *x*
_{1}(*t*), *x*
_{2}(*t*) stand for the population density of the prey and the predator at time *t*, respectively. In (1.1), it has been assumed that the prey grows logistically with growth rate *r*
_{1} (*t*) and carrying capacity *r*
_{1}(*t*)/*a*
_{11}(*t*) in the absence of predation. The predator consumes the prey according to the functional response *f* (*t*, *x*
_{1}(*t*)) and grows logistically with growth rate *r*
_{2} (*t*) and carrying capacity *x*
_{1}(*t*)/*a*
_{21}(*t*) proportional to the population size of the prey (or prey abundance). *a*
_{21} (*t*) is a measure of the food quality that the prey provides, which is converted to predator birth. For more background and biological adjustments of system (1.1), we can see [1–7] and the references cited therein.

Using Gaines and Mawhins continuation theorem of coincidence degree theory and by constructing an appropriate Lyapunov functional, they obtained a set of sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions of the system (1.2).

where *x*
_{1}(*k*), *x*
_{2}(*k*) stand for the density of the prey and the predator at *k* th generation, respectively. *m* ≠ 0 is a constant. *τ* denotes the time delay due to negative feedback of the prey population.

For convenience, throughout this article, we let *Z*, *Z*
^{+}, *R*
^{+}, and *R*
^{2} denote the sets of all integers, nonnegative integers, nonnegative real numbers, and two-dimensional Euclidian vector space, respectively, and use the notations: *f* ^{
u
} = sup_{
k∈Z+}{*f*(*k*)}, *f*
^{
l
} = inf_{
k∈Z+}{*f*(*k*)}, for any bounded sequence {*f*(*k*)}.

In this article, we always assume that for all *i*, *j* = 1, 2, (*H*
_{1}) *r*
_{
i
} (*k*), *a*
_{
ij
} (*k*) are all positive bounded sequences such that
,
; *τ* is a nonnegative integer.

*x*

_{1}(

*k*),

*x*

_{2}(

*k*)} which defines for

*Z*

^{+}and which satisfies system (1.3) for

*Z*

^{+}. Motivated by application of system (1.3) in population dynamics, we assume that solutions of system (1.3) satisfy the following initial conditions

The exponential forms of system (1.3) assure that the solution of system (1.3) with initial conditions (1.4) remains positive.

The principle aim of this article is to study the dynamic behaviors of system (1.3), such as permanence, global attractivity, existence, and global attractivity of positive periodic solutions. To the best of our knowledge, no work has been done for the discrete non-autonomous difference system (1.3). The organization of this article is as follows. In the next section, we explore the permanent property of the system (1.3). We study globally attractive property of the system (1.3) and the periodic property of system (1.3). At last, the conclusion ends with brief remarks.

## Permanence

First, we introduce a definition and some lemmas which are useful in the proof of the main results of this section.

**Definition 2.1**. System (1.3) is said to be permanent, if there are positive constants

*m*

_{ i }and

*M*

_{ i }, such that for each positive solution (

*x*

_{1}(

*k*),

*x*

_{2}(

*k*))

^{ T }of system (1.3) satisfies

Lemmas 2.1 and 2.2 are Theorem 2.1 in [19] and Lemma 2.2 in [14].

**Lemma 2.1**.

*Let*.

*For any fixed*

*k*,

*g*(

*k*,

*r*)

*is a non-decreasing function*,

*and for k ≥ k*

_{0},

*the following inequalities hold:*

*If y*(*k*
_{0}) ≤ *u*(*k*
_{0}), *then y*(*k*) ≤ *u*(*k*) *for all k* ≥ *k*
_{0}.

*where* {*a*(*k*)} *and* {*b*(*k*)} *are strictly positive sequences of real numbers defined for k* ∈ *Z*
^{+}
*and* 0 *< a*
^{
l
} ≤ *a*
^{
u
} , 0 *< b*
^{
l
} ≤ *b*
^{
u
} .

**Lemma 2.2**. *Any solution of system (2*.*1) with initial condition* *N*(0) > 0

**Theorem 2.1**. *Assume that* (*H*
_{1}) *holds*, *assume further that*

*holds*. *Then system (1*.*3) is permanent*.

*Proof*. Let

*x*(

*k*) = (

*x*

_{1}(

*k*),

*x*

_{2}(

*k*))

^{ T }be any positive solution of system (1.3) with initial conditions (1.4), from the first equation of the system (1.3), it follows that

*ε >*0 small enough, it follows from (2.7) that there exists enough large

*K*

_{1}such that for

*k*≥

*K*

_{1},

From (2.7) and (2.10) that there exists enough large *K*
_{2}
*> K*
_{1} such that for *i* = 1, 2

Consequently, combining (2.7), (2.11), (2.22) with (2.28), system (1.3) is permanent. This completes the proof of Theorem 2.1.

## Global attractivity

Now, we study the global attractivity of the positive solution of system (1.3). To do so, we first introduce a definition and prove a lemma which will be useful to our main result.

**Definition 3.1**. A positive solution (

*x*

_{1}(

*k*),

*x*

_{2}(

*k*))

^{ T }of system (1.3) is said to be globally attractive if each other solution of system (1.3) satisfies

**Lemma 3.1**.

*For any two positive solutions*(

*x*

_{1}(

*k*),

*x*

_{2}(

*k*))

^{ T }

*and*

*of system (1.3)*,

*we have*

Thus we can easily obtain (3.1) by substituting (3.3) and (3.4) into (3.2). The proof of Lemma 3.1 is completed.

Now we are in the position of stating the main result on the global attractivity of system (1.3).

**Theorem 3.1**.

*In addition to*(

*H*

_{1})

*-*(

*H*

_{2}),

*assume further that*(

*H*

_{3})

*there exist positive constants*

*λ*

_{1},

*λ*

_{2}

*such that*

*holds*,

*where ρ*,

*ϱ*,

*σ are defined by (3.23)*.

*Then for any two positive solutions*(

*x*

_{1}(

*k*),

*x*

_{2}(

*k*))

^{ T }

*and*

*of system (1.3)*,

*one has*

*Proof*. Let (*x*
_{1}(*k*), *x*
_{2}(*k*)) ^{
T
} and
be two arbitrary solutions of system (1.3). To prove Theorem 3.1, for the first equation of system (1.3), we will consider the following three steps,

*k*

_{0}such that

*m*

_{ i }≤

*x*

_{ i }(

*k*) for

*k > k*

_{0}and

*i*= 1, 2. Therefore, for all

*k*>

*k*

_{0}+

*τ*, we can obtain that

*V*(

*k*)

*>*0 and

*V*(

*k*

_{0}+

*τ*)

*<*+∞. Calculating the difference of

*V*along the solution of system (1.3), we have that for

*k*≥

*k*

_{0}+

*τ*,

This completes the proof of Theorem 3.1.

In the following section, we consider the periodic property of system (1.3).

## Existence and global attractivity of positive periodic solutions

In this section, we assume that all the coefficients of system (1.3) are positive sequences with common periodic *ω*, where *ω* is a fixed positive integer, stands for the prescribed common period of the parameters in system (1.3), then the system (1.3) is an *ω*-periodic system for this case. And so the coefficients of system (1.3) will naturally satisfy assumption (*H*
_{1}).

In order to obtain the existence of positive periodic solutions of system (1.3), we first make the following preparations that will be basic for this section.

*L*: Dom

*L*∩

*X*→

*Z*is a linear operator and

*λ*, is a parameter. Let

*P*and

*Q*denote two projectors such that

*J*: Im

*Q*→ Ker

*L*is an isomorphism of Im

*Q*onto Ker

*L*. Recall that a linear mapping

*L*: Dom

*L*∩

*X*→

*Z*with Ker

*L*=

*L*

^{-1}(0) and Im

*L*=

*L*(Dom

*L*), will be called a Fredholm mapping if the following two conditions hold:

- (i)
Ker

*L*has a finite dimension; - (ii)
Im

*L*is closed and has a finite codimension.

Recall also that the codimension of Im*L* is dimension of *Z/* Im*L*, i.e., the dimension of the cokernel coker *L* of *L*.

When *L* is a Fredholm mapping, its index is the integer Ind*L* = dim Ker*L* - codim Im*L*.

We shall say that a mapping *N* is *L*-compact on Ω if the mapping
is continuous,
is bounded and
is compact. i.e., it is continuous and
is relatively compact, where *K*
_{
P
} : Im*L* → Dom*L* ∩ Ker*P* is an inverse of the restriction *L*
_{
P
} of *L* to *DomL* ∩ *KerP*, so that *LK*
_{
P
} = *I* and *K*
_{
P
} = *I* - *P*. The following Lemma is from Gains and Mawhin [24].

**Lemma 4.1**. (Continuation Theorem)

*Let X, Z be two Banach spaces and L be a Fredholm mapping of index zero*.

*Assume that*

*is L-compact on*

*with*Ω

*open bounded in X. Furthermore assume:*

- (a)
*For each**λ*∈ (0, 1),*x*∈ ∂Ω ∩*DomL*,*Lx*≠*λNx*, - (b)
*QNx*≠ 0*for each ×*∈ ∂Ω ∩*KerL*, - (c)
*deg*{*JQNx*, Ω ∩*KerL*, 0} ≠ 0.

*Then the equation Lx* = *Nx has at least one solution lying in Dom*
.

*where* {*f*(*k*)} *is an ω-periodic sequence*.

**Lemma 4.2**.

*Let f : Z*→

*R be ω-periodic, i.e., f*(

*k*+

*ω*) =

*f*(

*k*),

*then for any fixed k*

_{1},

*k*

_{2}∈

*I*

_{ ω }

*and any k*∈

*Z*,

*one has*

*for a*= (

*a*

_{1},

*a*

_{2})

^{ T }∈

*R*

^{2},

*define*|

*a*| = max{

*a*

_{1},

*a*

_{2}}.

*Let l*

^{ ω }⊂

*l*

_{2}

*denote the subspace of all ω-periodic sequences equipped with the usual supremum norm*||·||,

*i.e*.,

*Then it follows that l*
^{
ω
} *is a finite dimensional Banach space*.

**Theorem 4.1**. *Assume that*

*holds*.

*Then periodic system (1.3) has at least one positive ω-periodic solution*.

*Proof*. Since solutions of system (1.3) remained positive for

*k*≥ 0, we let

It is easy to see that if (4.2) has one *ω*-periodic solution
, then (1.3) has one positive *ω*-periodic solution
. Therefore, to complete the proof, it is only to show that (4.2) has at least one *ω*-periodic solution.

*X*=

*Z*=

*l*

^{ ω }. Denote by

*L*:

*X*→

*X*the difference operator given by

*Lu*= {(

*Lu*)(

*k*)} with (

*Lu*)(

*k*) =

*u*(

*k*+ 1) -

*u*(

*k*), for

*u*∈

*X*and

*k*∈

*Z*, and

*N*:

*X*→

*X*as follows:

then it follows that *L* is a Fredholm mapping of index zero.

*L*)

*K*

_{ P }: Im

*L*→ Dom

*L*∩ Ker

*P*exists and is given by

*u*∈

*X*is a solution of (4.3) for a certain

*λ*∈ (0, 1). Summing on both sides of (4.3) from 0 to

*ω*- 1 with respect to

*k*, we obtain

*u*= {(

*u*

_{1}(

*k*),

*u*

_{2}(

*k*))

^{ T }} ∈

*X*. Then there exist

*ξ*

_{ i },

*η*

_{ i }∈

*I*

_{ ω },

*i*= 1, 2 such that

Clearly, *H*
_{
i
} (*i* = 1, 2) are independent of *λ*.

*u*

_{1}(

*k*),

*u*

_{2}(

*k*))

^{ T }∈

*R*

^{2}. By the similar argument of (4.7), (4.10), (4.13), and (4.17), we can derive the solutions (

*u*

_{1}(

*k*),

*u*

_{2}(

*k*))

^{ T }of (4.20) that satisfy

*H*=

*H*

_{1}+

*H*

_{2}+

*C*, here,

*C*is taken sufficiently large such that

*C*≥ |

*K*

_{1}| + |

*k*

_{1}| + |

*K*

_{2}| + |

*k*

_{2}|. Now we take Ω = {(

*u*

_{1}(

*k*),

*u*

_{2}(

*k*))

^{ T }∈

*X*: || (

*u*

_{1}(

*k*),

*u*

_{2}(

*k*))

^{ T }||

*< H*}. Now we check the conditions of Lemma 4.1.

- (a)
From (4.15) and (4.19), one can see that for each

*λ*∈ (0, 1),*u*∈ ∂Ω ∩ Dom*L*,*Lu*≠*λNu*. - (b)

*u*

_{1}(

*k*),

*u*

_{2}(

*k*))

^{ T }is the constant solution of system (4.20) with

*μ*= 1. From (4.21), we have || (

*u*

_{1},

*u*

_{2})

^{ T }||

*< H*. This contradiction implies that for each

*u*∈ ∂Ω ∩ Ker

*L*,

*QNu*≠ 0.

- (c)we will prove that condition (c) of Lemma 4.1 is satisfied. To this end, we define
*ϕ*: Dom*L*× 0[1] →*X*by

*μ*∈ 0[1] is a parameter. When (

*u*

_{1},

*u*

_{2})

^{ T }∈ ∂Ω ∩ Ker

*L*, (

*u*

_{1},

*u*

_{2})

^{ T }is a constant vector in

*R*

^{2}with || (

*u*

_{1},

*u*

_{2})

^{ T }|| =

*H*. From (4.21), we know that

*ϕ*(

*u*

_{1},

*u*

_{2},

*μ*) ≠ (0, 0)

^{ T }on ∂Ω ∩ Ker

*L*. Hence, due to homotopy invariance theorem of topology degree and taking

*J*=

*I*: Im

*Q*→ Ker

*L*, we have

Hence, is bounded. Obviously, is continuous.

Thus, the set is equicontinuous and uniformly bounded.

By applying Ascoli-Arzela theorem, one can see that
is compact. Consequently, *N* is *L*-compact.

By now we have verified all the requirements of Lemma 4.1. Hence system (4.2) has at least one *ω*-periodic solution. This ends the proof of Theorem 4.1.

By constructing similar Lyapunov function to those of Theorem 3.1, and using Theorem 4.1, we have the following Theorem 4.2.

**Theorem 4.2**. Assume that the conditions of (*H*
_{2})-(*H*
_{4}) hold. Then the positive periodic solution of periodic system (1.3) is globally attractive.

## Concluding remarks

In this article, a discrete time semi-ratio-dependent predator-prey system with Holling type IV functional response and time delay is investigated. By using comparison theorem and further developing the analytical technique of [14, 21], we prove the system (1.3) is permanent under some appropriate conditions. Further, by constructing the suitable Lyapunov function, we show that the system (1.3) is globally attractive under some appropriate conditions. If the system (1.3) is periodic one, by using the continuous theorem of coincidence degree theory and Theorem 3.1, some sufficient conditions are established, which guarantee the existence and global attractivity of positive periodic solutions of the system (1.3). We note that the time delay has an effect on the permanence and the global attractivity of periodic solution of system (1.3), but time delay has no effect on the existence of positive periodic solutions.

## Declarations

### Acknowledgements

The authors are grateful to the Associate Editor, R. L. Pouso, and referees for a number of helpful suggestions that have greatly improved our original submission. This work is supported by the National Natural Science Foundation of China (No.70901016), Excellent Talents Program of Liaoning Educational Committee (No.2008RC15), and Innovation Method Fund of China (No.2009IM010400-1-39).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.