Positive Periodic Solutions for Neutral Delay Ratio Dependent Predator Prey Model with Holling Tanner Functional Response

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Volume 2011, Article ID 376862,14pages doi:10.1155/2011/376862

Research Article

Positive Periodic Solutions for Neutral

Delay Ratio-Dependent Predator-Prey Model with

Holling-Tanner Functional Response

Guirong Liu,

1

_{Sanhu Wang,}

2

_{and Jurang Yan}

1

1_{School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China}

2_{Department of Computer, Luliang University, Luliang, Shanxi 033000, China}

Correspondence should be addressed to Guirong Liu,[email protected]

Received 31 December 2010; Revised 25 April 2011; Accepted 29 May 2011

Academic Editor: Robert Redfield

Copyrightq2011 Guirong Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By using a continuation theorem based on coincidence degree theory, we establish some easily verifiable criteria for the existence of positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling-Tanner functional responsext xtrt−atxt−σt− btxt−σt−ctxtyt/htyt xt,yt ytdt−ftyt−τt/xt−τt.

1. Introduction

The dynamic relationship between the predator and the prey has long been and will continue to be one of the dominant themes in population dynamics due to its universal existence and importance in nature1. In order to precisely describe the real ecological interactions between species such as mite and spider mite, lynx and hare, and sparrow and sparrow hawk, described by Tanner2and Wollkind et al.3, May4developed the Holling-Tanner prey-predator model

xt rxt

1− xt K

−mxtyt

xt q ,

yt yt

s

1−hyt xt

.

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In system1.1,xtandytstand for prey and predator density at timet.r,K,m,q,s,h are positive constants that stand for prey intrinsic growth rate, carrying capacity, capturing rate, half-capturing saturation constant, predator intrinsic growth rate, and conversion rate of prey into predators biomass, respectively.

Nowadays attention have been paid by many authors to Holling-Tanner predator-prey modelsee5–7.

Recently, there is a growing explicit biological and physiological evidence8–10that in many situations, especially when predators have to search for foodand, 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 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 observations11,12. Generally, a ratio-dependent Holling-Tanner predator-prey model takes the form of

xt rxt

1−xt K

− mxtyt

qyt xt,

yt yt

s

1−hyt xt

.

1.2

Liang and Pan13obtained results for the global stability of the positive equilibrium of1.2. However, time delays of one type or another have been incorporated into biological models by many researchers; we refer to the monographs of Cushing14, Gopalsamy15, Kuang16, and MacDonald17for general delayed biological systems. Time delay due to gestation is a common example, because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, more realistic models of population interactions should take into account the eﬀect of time delays. Recently, Saha and Chakrabarti18considered the following delayed ratio-dependent Holling-Tanner predator-prey model:

xt rxt

1−xt−τ K

− mxtyt

qyt xt,

yt yt

s

1−hyt xt

.

1.3

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Kuang21studied the local stability and oscillation of the following neutral delay Gause-type predator-prey system:

xt rxt

1−xt−τ ρx _t₋_τ

K

−ytpxt,

yt yt−αβpxt−σ.

1.4

In this paper, motivated by the above work, we will consider the following neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response:

xt xtrt−atxt−σt−btxt−σt− ctxtyt htyt xt,

yt yt

dt−ftyt−τt xt−τt

.

1.5

As pointed out by Freedman and Wu22and Kuang16, 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 played by the equilibria of 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 of1.5.

For convenience, we will use the following notations:

_p

0_tmax_∈_0,ω pt, p

1 ω

ω

0

ptdt, p 1 ω

ω

0

_p_t_dt, _1.6

whereptis a continuousω-periodic function.

In this paper, we always make the following assumptions for system1.5.

H1rt, at, bt, ct, dt, ft, ht, τt, and σt are continuous ω-periodic

functions. In addition,r > 0,d > 0, and at > 0,ct > 0,ft > 0,ht > 0 for any t∈0, ω;

H2b∈C1R,0,∞,σ∈C2R, R,σt<1, andgt>0, where

gt at−qt, qt bt

1−σt, t∈R. 1.7

H3eBmax{|b|0,|q|0}<1, where

Bln

2rmax

t∈0,ω

1−σt gt

q₀max

t∈0,ω

2r gt

rrω. 1.8

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Our aim in this paper is, by using the coincidence degree theory developed by Gaines and Mawhin23, to derive a set of easily verifiable suﬃcient conditions for the existence of positive periodic solutions of system1.5.

2. Existence of Positive Periodic Solution

In this section, we will 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 coincidence degree. For the readers’ convenience, we introduce a few concepts and results about the coincidence degree as follows.

LetX, Zbe real Banach spaces,L: DomL⊂X → Za linear mapping, andN:X → Z a continuous mapping.

The mapping L is said to be a Fredholm mapping of index zero if dim KerL co dim ImL <∞and ImLis closed inZ.

IfLis a Fredholm mapping of index zero, then there exist continuous projectorsP : X → XandQ:Z → Z, such that ImP KerL, KerQ ImL ImI−Q. It follows that the restrictionLP ofLto DomL∩KerP :I−PX → ImLis invertible. Denote the inverse

ofLP byKP.

The mappingNis said to beL-compact onΩifΩis an open bounded subset ofX, QNΩis bounded, andKPI−QN:Ω → Xis compact.

Since ImQis isomorphic to KerL, there exists an isomorphismJ: ImQ → KerL.

Lemma 2.1Continuation Theorem23, page 40. LetΩ ⊂ X be an open bounded set,Lbe a

Fredholm mapping of index zero, andNL-compact onΩ. Suppose

ifor eachλ∈0,1,x∈∂Ω∩DomL, Lx /λNx;

iifor eachx∈∂Ω∩KerL,QNx /0;

iiideg{JQN,Ω∩KerL,0}/0.

ThenLxNxhas at least one solution inΩ∩DomL.

We are now in a position to state and prove our main result.

Theorem 2.2. Assume that (H1)–(H4) hold. Then system1.5has at least oneω-periodic solution

with strictly positive components.

Proof. Consider the following system:

u₁t rt−ateu1t−σt−bteu1t−σtu₁t−σt− cte

u2t

hteu2teu1t,

u₂t dt−fte

u2t−τt

eu1t−τt,

2.1

where all functions are defined as ones in system1.5. It is easy to see that if system2.1 has oneω-periodic solutionu∗₁t, u∗₂tT, then x∗t, y∗tT eu₁∗t_{, e}u∗₂tT _{is a positive}

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Take

Xu u1t, u2tT ∈C1

R, R2:uitω uit, t∈R, i1,2

,

Zu u1t, u2tT ∈C

R, R2:uitω uit, t∈R, i1,2

2.2

and denote

|u|_∞ max

t∈0,ω{|u1t||u2t|}, u |u|∞u

∞. 2.3

ThenX andZ are Banach spaces when they are endowed with the norms · and | · |∞, respectively. LetL:X → ZandN:X → Zbe

Lu1t, u2tT

u₁t, u₂tT,

N

u1t

u2t

⎡ ⎢ ⎢ ⎢ ⎣

rt−ateu1t−σt−bteu1t−σtu₁t−σt− cte

u2t

hteu2teu1t

dt−fteu2t−τt eu1t−τt

⎤ ⎥ ⎥ ⎥

⎦.

2.4

With these notations, system2.1can be written in the form

LuNu, u∈X. 2.5

Obviously, KerL R2_{, Im}_L _{_u

1t, u2tT ∈Z :

ω

0 uitdt 0, i 1,2}is closed inZ,

and dim KerL co dim ImL 2. Therefore,Lis a Fredholm mapping of index zero. Now define two projectorsP :X → XandQ:Z → Zas

Pu1t, u2tT u1, u2T, u1t, u2tT ∈X,

Qu1t, u2tT u1, u2T, u1t, u2tT ∈Z.

2.6

ThenP andQare continuous projectors such that ImP KerL, KerQ ImL ImI−Q. Furthermore, the generalized inversetoLKP : ImL → KerP∩DomLhas the form

KPu

_t

0

usds− 1 ω

_ω

0

_t

0

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Note that

_ω

0

bteu1t−σtu₁t−σtdt

_ω

0

bt 1−σt

eu1t−σtdt

ω

0

qteu1t−σtdt

qteu1t−σtω

0 −

ω

0

qteu1t−σtdt

−

ω

0

qteu1t−σtdt.

2.8

ThenQN :X → ZandKPI−QN:X → Xread

QNu

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 ω _ω 0

rt−gteu1t−σt− cteu2t hteu2teu1t

dt 1 ω _ω 0

dt ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦,

KPI−QNu

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ _t 0

rs−gseu1s−σs− cseu2s hseu2seu1s

ds−qteu1t−σtq0eu1−σ0

_t

0

ds−fseu2s−τs eu1s−τs

ds ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ − ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 ω _ω 0 _t 0

rs−gseu1s−σs− cseu2s hseu2seu1s

dsdt−1 ω

_ω

0

qteu1t−σt−q0eu1−σ0dt

1 ω ω 0 t 0

ds−fse

u2s−τs

eu1s−τs

dsdt ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ − ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ t ω − 1 2 ω 0

rs−gseu1s−σs− cse

u2s

hseu2seu1s

ds t ω − 1 2 ω 0

ds−fse

u2s−τs

eu1s−τs

ds ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦. 2.9

It is obvious thatQNandKPI−QNare continuous by the Lebesgue theorem, and using the

Arzela-Ascoli theorem it is not diﬃcult to show thatQNΩis bounded andKPI−QNΩ

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In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subsetΩ⊂X.

Corresponding to the operator equationLuλNu,λ∈0,1, we have

u₁t λ

u2t

hteu2teu1t

,

u₂t λ

.

2.10

Suppose thatu1t, u2tT ∈Xis a solution of2.10for a certainλ∈0,1. Integrating2.10

over the interval0, ωleads to

ω

0

u2t

hteu2teu1t

dt0, 2.11

ω

0

dt−fte

u2t−τt

eu1t−τt

dt0. 2.12

It follows from2.8and2.11that

ω

0

gteu1t−σt cte

u2t

hteu2teu1t

dtrω. 2.13

From2.12, we have

_ω

0

fteu2t−τt

eu1t−τtdtdω. 2.14

FromH2,2.10, and2.13, one can find

ω

0

_dtdu1t λqteu1t−σt

dtλ

ω

0

rt−gteu1t−σt−

cteu2t hteu2teu1t

dt

≤

_ω

0

|rt|dt

_ω

0

gteu1t−σt cteu2t hteu2teu1t

dt

rrω.

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Lettψvbe the inverse function ofvt−σt. It is easy to see thatgψvandσψv are allω-periodic functions. Further, it follows from2.13,H1, andH2that

rω≥

_ω

0

gteu1t−σtdt

_ω₋_σω

−σ0 g

ψveu1v 1

1−σψvdv

ω

0

gψv 1−σψve

u1v_dvω

0

gψt 1−σψte

u1t_dt,

2.16

which yields that

2rω≥

_ω

0

gψt 1−σψte

u1t_g_t_eu1t−σt

dt. 2.17

According to the mean value theorem of diﬀerential calculus, we see that there exists ξ ∈

0, ωsuch that

gψξ 1−σψξe

u1ξ_g_ξ_eu1ξ−σξ_≤_2r. _2.18

This, together withH2, yields

u1ξ≤ln

2rmax

t∈0,ω

1−σt gt

, eu1ξ−σξ≤ max

t∈0,ω

2r gt

, 2.19

which, together with2.15andH3, imply that, for anyt∈0, ω,

u1t λqteu1t−σt≤u1ξ λqξeu1ξ−σξ

ω

0

_dtdu1t λqteu1t−σt

dt

≤ln

2rmax

t∈0,ω

1−σt gt

q₀max

t∈0,ω

2r gt

rrωB.

2.20

Asλqteu1t−σt ≥0, one can find that

u1t≤B, t∈0, ω. 2.21

Sinceu1t, u2tT ∈X, there existξi, ηi∈0, ω i1,2such that

uiξi min

t∈0,ω{uit}, ui

ηi

max

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From2.13andH4, we obtain

rω≤

_ω

0

gteu1t−σt ct ht

dt≤kωeu1η1gω, 2.23

which, together withH4, implies that

u1

η1

≥lnr−k

g . 2.24

In view of2.10,2.13, and2.21, we obtain

_ω

0

u₁tdtλ

_ω

0

rt−ateu1t−σt−bteu1t−σtu1t−σt−

dt ≤ ω 0

|rt|dt

ω

0

dt ω 0

ateu1t−σtdt

ω

0

bteu1t−σtu₁t−σtdt

≤rrω|a|₀eB_ω_eB

_ω

0

btu₁t−σtdt.

2.25

In addition,

_ω

0

btu₁t−σtdt

_ω₋_σω

−σ0

b

ψvu₁v 1 1−σψv

dv,

≤q₀

_ω₋_σω

−σ0

u₁vdvq₀

_ω

0

u₁vdvq₀

_ω

0

u₁tdt,

2.26

which, together withH3, implies that

ω

0

_u

1tdt≤

ω 1−q₀eB

rr|a|₀eB. 2.27

It follows from2.24and2.27that, for anyt∈0, ω,

u1t≥u1 η1 − ω 0 _u

1tdt≥ln

r−k g −

ω 1−q₀eB

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which, together with2.21, implies that

|u1|₀ ≤max |B|,β1:β2. 2.29

From2.10, we obtain

u₁₀ ≤ |r|₀|a|₀eB|b|₀eBu₁₀|k|₀. 2.30

It follows fromH3that

u₁₀≤ 1

1− |b|₀eB

|r|₀|a|₀eB_|_k_|

0

:β3. 2.31

In view of2.14, we obtain

dω≥

_ω

0

fteu2ξ2

eB dtfω

eu2ξ2 eB ,

dω≤

_ω

0

fteu2η2

eβ1 dtfω

eu2η2 eβ1 .

2.32

Further,

u2ξ2≤Bln

d

f, u2

η2

≥β1ln

d

f. 2.33

It follows that from2.10and2.14, we obtain

ω

0

_u

2tdt≤

ω

0

|dt|dt

ω

0

fte

u2t−τt

eu1t−τtdt

ddω. 2.34

From2.33and2.34, one can find that, for anyt∈0, ω,

u2t≥u2

η2

−

ω

0

_u

2tdt≥β1lnd

f −

ddω:β4,

u2t≤u2ξ2

ω

0

_u

2tdt≤Bln

d f

ddω:β5,

2.35

which imply that

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In view of2.10, we have

u₂₀ ≤ |d|₀f₀e

β5

eβ1 :β7. 2.37

From2.29,2.31,2.36, and2.37, we obtain

u |u|_∞u_∞≤β2β3β6β7. 2.38

FromH4, the algebraic equations

r−geu1− 1 ω

ω

0

cteu2

hteu2_eu1dt0,

d−feu2−u1 0

2.39

have a unique solutionu∗₁, u∗₂T ∈R2_{, where}

u∗₁ln

1 g r−

1 ω

ω

0

dct

dht fdt

!

, u∗₂u∗₁lnd

f. 2.40

Setββ2β3β6β7β0, whereβ0is taken suﬃciently large such that the unique solution

of2.39satisfies u∗₁, u∗₂T |u∗₁||u∗₂|< β0. Clearly,βis independent ofλ.

We now take

Ω u1t, u2tT ∈X :"""u1t, u2tT"""< β

. 2.41

This satisfies condition i in Lemma 2.1. When u1t, u2tT ∈ ∂Ω ∩KerL ∂Ω∩R2,

u1t, u2tTis a constant vector inR2with|u1||u2|β. Thus, we have

QN

u1

u2

⎡

⎣r−geu1−ω1

_ω

0

cteu2

hteu2_eu1dt

d−feu2−u1

⎤

⎦_/

0 0

. 2.42

This proves that conditioniiinLemma 2.1is satisfied.

TakingJ I: ImQ → KerL,u1, u2T → u1, u2T, a direct calculation shows that

deg{JQN,Ω∩KerL,0}

sgndet

⎡ ⎢

⎣−ge

u∗₁ 1

ωe

u∗₁_eu∗₂

ω

0

ct

hteu∗2eu∗12

dt− 1 ωe

u∗₁_eu∗₂

ω

0

ct

hteu∗2eu∗12

dt

feu∗2−u∗1−feu∗2−u∗1

⎤ ⎥ ⎦

sgnfgeu∗2

/

0.

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By now we have proved thatΩsatisfies all the requirements inLemma 2.1. Hence,2.1has at least oneω-periodic solution. Accordingly, system1.5has at least oneω-periodic solution with strictly positive components. The proof ofTheorem 2.2is complete.

Remark 2.3. From the proof ofTheorem 2.2, we see thatTheorem 2.2is also valid ifbt ≡0

fort∈R. Consequently, we can obtain the following corollary.

Corollary 2.4. Assume that (H1), (H4) hold, andσ ∈C2R, R,σt<1. Then the following delay

ratio-dependent predator-prey model with Holling-Tanner functional response

xt xtrt−atxt−σt− ctxtyt htyt xt,

yt yt

dt−ftyt−τt xt−τt

2.44

has at least oneω-periodic solution with strictly positive components.

Next consider the following neutral ratio-dependent predator-prey system with state-dependent delays:

xt xtrt−atxt−σt−btxt−σt− ctxtyt htyt xt,

yt yt

dt−fty

t−τ1

t, xt, yt xt−τ1

t, xt, yt

,

2.45

whereτ1t, x, yis a continuous function andω-periodic function with respect tot.

Theorem 2.5. Assume that (H1)–(H4) hold. Then system2.45has at least oneω-periodic solution

with strictly positive components.

Proof. The proof is similar to that ofTheorem 2.2and hence is omitted here.

3. Discussion

In this paper, we have discussed the combined effects of periodicity of the ecological and environmental parameters and time delays due to the negative feedback of the predator density and gestations on the dynamics of a neutral delay ratio-dependent predator-prey model. By using Gaines and Mawhin’s continuation theorem of coincidence degree theory, we have established sufficient conditions for the existence of positive periodic solutions to a neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response. ByTheorem 2.2, we see that system1.5will have at least oneω-periodic solution with strictly positive components if a the density-dependent coefficient of the prey is sufficiently large, the neutral coefficientbis sufficiently small, and c/h < r, wherec,h, r stand for capturing rate, half-capturing saturation coefficient, and prey intrinsic growth rate, respectively.

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of positive periodic solutions to system1.5. However,σ the time delay due to gestation plays the important role in determining the existence of positive periodic solutions of1.5.

From the results in this paper, we can find that the neutral term eﬀects are quite significant.

Acknowledgments

This work was supported by the Natural Science Foundation of China no. 11001157, Tianyuan Mathematics Fund of Chinano. 10826080and the Youth Science Foundation of Shanxi Provinceno. 2009021001-1, no. 2010021001-1.

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