Existence of positive periodic solutions for a generalized predator prey model with diffusion feedback controls

(1)

R E S E A R C H

Open Access

Existence of positive periodic solutions for a

generalized predator-prey model with

diffusion feedback controls

Chaoxiong Du

1*

_{and Yusen Wu}

2

*_{Correspondence:}

[email protected]; [email protected]

1_{Department of Mathematics,}

Shaoyang University, Shaoyang, Hunan 422000, P.R. China Full list of author information is available at the end of the article

Abstract

By employing the continuation theorem of coincidence degree theory, we derive a suﬃcient condition for the existence and attractivity of at least a positive periodic solution of the generalized predator-prey model with exploited term.

MSC: 92D25; 34C25; 34K15

Keywords: prey-predator model; diﬀusion; positive periodic solution; coincidence degree

1 Introduction

In recent years, the existence of positive periodic solutions of the prey-predator model has been widely studied [–]. The qualitative analysis of predator-prey systems is an interest-ing mathematical problem and has attracted a great attention of many mathematicians and biologists [, ]. Recently, Xu and Chen [] investigated the two-species ratio-dependent predator-prey diﬀerent model with time delay. Since a realistic model requires the inclu-sion of the eﬀect of changing environment, recently, Shihua and Feng [] have considered the following model:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x_(t) =x(t)(a(t) –a(t)x(t) –_m₍a_t₎_x_(t₍)_tx₎₊(_xt_)₍_t₎)

+D(t)(x(t) –x(t)) –h(t),

x_(t) =x(t)(a(t) –a(t)x(t)) +D(t)(x(t) –x(t)) –h(t),

x_(t) =x(t)(–a(t) –a(t)x(t) +_m₍_t₎a_x_₍(_tt_–)xτ)+(t–xτ()t–τ)) –h(t),

(.)

whereDi(t) (i= , ),ai(t) (i= , , ),a(t),a(t),a(t),a(t) andm(t) are strictly

posi-tive continuousw-periodic functions.

In the paper, we will study the following model:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x_(t) =x(t)(g(t,x(t)) –_m₍a_t₎_x_(t₍)_tx₎₊(_xt_)₍_t₎)

+D(t)(x(t) –x(t)) –h(t),

x_(t) =x(t)(g(t,x(t)) +D(t)(x(t) –x(t)) –h(t),

x_(t) =x(t)(–a(t) –a(t)x(t) +_m₍_t₎a_x_₍(_tt_–)xτ)+(t–xτ()t–τ)) –h(t),

(.)

(2)

whereDi(t) (i= , ),a(t),a(t),m(t) are the same as in model (.). Some assumptions

on the above functions ongi(t,x) (i= , ) will be given in next section.

Our aim in this paper is to establish a suﬃcient condition for the existence and attrac-tivity of at least a positivew-periodic solution of model (.).

2 Main result

To obtain the existence of positive periodic solutions of system (.), we summarize some concepts and results from [] that will be basic for this section.

LetX,Zbe Banach spaces, letL:DomL⊂X→Ybe a linear mapping, and letN:X→ Zbe a continuous mapping. The mappingLwill be called a Fredholm mapping of index zero ifdim KerL=codim ImL< +∞andImLis closed inZ. IfLis a Fredholm mapping of index zero, there exist continuous projectorsP:X→ZandQ:Z→Zsuch thatImP= KerLandImL=KerQ=Im(I–Q). It follows thatL/DomL∩KerP: (I–P)X→ImLis invertible. We denote the inverse of that map byKp. Ifis an open-bounded subset ofX, the mappingNwill be calledL-compact onifQN() is bounded andKp(I–Q)N:→

Xis compact. SinceImQis isomorphic toKerL, there exists an isomorphismJ:ImQ→ KerL.

In the proof of our existence theorem, we will use the continuation theorem of Gaines and Mawhin [].

Lemma .[] Let L be a Fredholm mapping of index zero and let N be L-compact on. Suppose the following:

(i) for eachλ∈(, ),every solutionxofLx=λNxis such thatx∈∂; (ii) QNx= for eachx∈∂∩KerL;

(iii) deg{JQN,∩KerL, } = .

TheLx=Nxhas at least one solution inDomL∩. For convenience, we introduce the notations

f =  w

w



f(t)dt, fL= min

t∈[,w]

f(t), fM= max

t∈[,w]

f(t),

wheref is a continuousw-periodic function.

Our main result on the global existence of a positive periodic solution of system (.) is stated in the following theorem.

Theorem . Assume that

(H) there exists a constantAsuch that for∀x∈R,t∈R,whenx≥A,

g

t,ex ≤;

(H) there exists a constantBsuch that for∀x∈R,whenx≥B,

g

t,ex ≤;

(H) there exists a constantC(C<A)such that for∀x∈R,t∈R,whenx≤C,

exg

t,ex ≥

a

m M

(3)

(H) there exists a constantD(D<B)such that for∀x∈R,t∈R,whenx≤D,

exg

t,ex ≥hM_;

(H) aMed>hlml.

Then system(.)has at least one positive w-periodic solution.

Proof Consider the system ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u(t) =g(t,eu(t)) – a(t)e

u_(t) m(t)eu(t)+eu(t)

+D(t)(eu(t)–u(t)– ) –h(t)e–u(t),

u_(t) =g(t,eu(t)) +D(t)(eu(t)–u(t)– ) –h(t)e–u(t),

u_(t) = –a(t) –a(t)eu(t)+ a(t)e

u_(t–τ)

m(t)eu(t)+eu(t–τ) –h(t)e

–u(t)_.

(.)

Letxi(t) =eui(t),i= , , , then system (.) changes into system (.). Hence it is easy to see

that system (.) has aw-periodic solution (u∗_(t),u∗_(t),u∗_(t))T_{, then (e}u∗(t)_,_eu∗(t)_,_eu∗(t)₎T

is a positivew-periodic solution of system (.). Therefore, for (.) to have at least one positivew-periodic solution, it is suﬃcient that (.) has at least onew-periodic solution. In order to apply Lemma . to system (.), we take

X=Z=u(t) =u(t),u(t),u(t)

T

∈CR,R ,u(t+w) =u(t)

and

u=u(t),u(t),u(t)

T

=



i=

max

t∈[,w]

ui(t)

for anyu∈Z(orZ). ThenXandZare Banach spaces with the norm · . Let

Nu= ⎡ ⎢ ⎢ ⎣

g(t,eu(t)) – a(t)e

u_(t)

m(t)eu(t)+eu(t)+D(t)(e

u(t)–u(t)_{– ) –}_h

(t)e–u(t)

g(t,eu(t)) +D(t)(eu(t)–u(t)– ) –h(t)e–u(t)

–a(t) –a(t)eu(t)+ a(t)e

u_(t–τ)

m(t)eu(t)+eu(t–τ) –h(t)e

–u(t)

⎤ ⎥ ⎥

⎦, u∈X,

Lu=u=du(t)

dt , pu=

 w

w



u(t)dt, u∈X;

Qz=  w

w



z(t)dt, z∈Z.

Then it follows that

KerL=R, ImL=

z∈Z: w



z(t)dt= 

is closed inZ,

dim KerL=  =codim ImL,

andP,Qare continuous projectors such that

(4)

Therefore,Lis a Fredholm mapping of index zero. Furthermore, the generalized inverse

Obviously,QN and Kp(I–Q)N are continuous. It is not diﬃcult to show thatKp(I– Q)N() is compact for any open bounded⊂Xby using the Arzela-Ascoli theorem. Moreover,QN() is clearly bounded. Thus,NisL-compact onwith any open bounded set⊂X.

(5)

It is clear that

u_i(ξi) = , ui(ηi) = , i= , , .

From this and system (.), we obtain

g

ξ,eu(ξ) –

a(ξ)eu(ξ)

m(ξ)eu(ξ)+eu(ξ)

+D(ξ)

eu(ξ)–u(ξ)_{– } _–_h

(ξ)e–u(ξ)= , (.)

g

ξ,eu(ξ) +D(ξ)

eu(ξ)–u(ξ)_{– } _–_h

(ξ)e–u(ξ)= , (.)

–a(ξ) –a(ξ)eu(ξ)+

a(ξ)eu(ξ–τ)

m(ξ)eu(ξ–τ)+e

u_(ξ–τ) –h(ξ)e

–u(ξ)_{= ,} _(.)

g

η,eu(η) –

a(η)eu(η)

m(η)eu(η)+eu(η)

+D(η)

eu(η)–u(η)_{– }

–h(η)e–u(η)= , (.)

g

η,eu(η) +D(η)

eu(η)–u(η)_{– } _–_h

(η)e–u(η)= , (.)

–a(η) –a(η)eu(η)+

a(η)eu(η–τ)

m(η)eu(η–τ)+e

u(η–τ) –h(η)e

–u(η)_{= .} _(.)

There are two cases to consider for (.) and (.). Case . Assume thatu(ξ)≥u(ξ), thenu(ξ)≥u(ξ).

From this and (.), we have

g

ξ,eu(ξ) =

a(ξ)eu(ξ)

m(ξ)eu(ξ)+eu(ξ)

–D(ξ)

eu(ξ)–u(ξ)_{– } ₊_h

(ξ)e–u(ξ)> ,

which, together with condition (H) in Theorem ., gives

u(ξ) <A. (.)

Thus

u(ξ)≤u(ξ) <A. (.)

Case . Assume thatu(ξ)≤u(ξ), thenu(ξ) <u(ξ).

From this and (.), we have

g

ξ,eu(ξ) = –D(ξ)

eu(ξ)–u(ξ)_{– } ₊_h

(ξ)e–u(ξ)> ,

which, together with condition (H) in Theorem ., gives

u(ξ) <B. (.)

Thus

(6)

From Case  and Case , we obtain

u(ξ) <max{A,B} def

=d, (.)

u(ξ) <max{A,B}=d. (.)

From (.), we get

aeu(ξ)≤a(ξ)eu(ξ)≤

a(ξ)eu(ξ–τ)

m(ξ)eu(ξ–τ)+e

u(ξ–τ) <a M

.

Thus

u(ξ)≤ln

aM

al



def

=d. (.)

There are two cases to consider for (.) and (.).

Case . Assume thatu(η)≤u(η), thenu(η) <u(η). From this and (.), we have

g

η,eu(η) =

a(η)eu(η)

m(η)eu(η)+eu(η)

–D(η)

eu(η)–u(η)_{– } ₊_h

(η)e–u(η)

< a(η)e

u(η)

m(η)eu(η)+eu(η)

+h(η)e–u(η)<

a

m M

+hM_ e–u(η)_,

which, together with condition (H) in Theorem ., gives

u(η) >C. (.)

Hence

u(η) >u(η) >C. (.)

Case . Assume thatu(η)≥u(η), thenu(η)≥u(η). From this and (.), we have

g

η,eu(η) = –D(η)

eu(η)–u(η)_{– } ₊_h

(η)e–u(η)<hMe–u(η),

which, together with condition (H) in Theorem ., gives

u(η) >D. (.)

Hence

u(η) >u(η) >D. (.)

From Case  and Case , we have

u(η) >min{C,D}def=ρ, (.)

(7)

From Theorem .(H), we get

u(η) >ln

hl_ aM

–al

. (.)

From (.)-(.), we obtain, for∀t∈R, _u__(t)_≤max|d|,|ρ|

def

= R,

_u_(t)≤max|d|,|ρ|

def

=R

and

u(t)≤max

|a|,|ρ|

def

=R.

Clearly,Ri (i= , , ) are independent ofλ. DenoteM=



i=Ri+R, hereR is taken

suﬃciently large such that each solution (α∗,β∗,γ∗)Tof the system

g

t,eα – ae

γ

m(t)eγ+eα

+D

eβ–α–  –h¯e–α= ,

g

t,eβ +D

eα–β–  –h¯e–β= , (.)

–a–aer+

aeα

m(t)eγ+eα

–h¯e–γ = 

satisﬁes(α∗,β∗,γ∗)T=|α∗|+|β∗|+|γ∗|<M, provided that system (.) has a solution or a number of solutions, and that

max|d|,|ρ|

+max|d|,|ρ|

+max|d|,|ρ|

<M,

whereti∈(,w) will appear inQNubelow.

Now we take={u= (u(t),u(t),u(t))T∈X:u<M}. This satisﬁes condition (i) of

Lemma .. Whenu∈∂∩KerL=∂∩R_,_u_{is a constant vector in}_R_with

i=|ui|=M.

If system (.) has one or more solutions, then

QNu= ⎡ ⎢ ⎣

g(t,eu) – ae

u

m(t)eu+eu +D(e

u–u_{– ) –}_h¯

e–u

g(t,eu) +D(eu–u– ) –h¯e–u

–a–aeu+ ae

u

m(t)eu₊_eu–h¯e–u

⎤ ⎥

⎦= (, , )T,

whereti∈(,w) are one constant.

If system (.) does not have a solution, then naturally

QNu= (, , )T.

This shows that condition (ii) of Lemma . is satisfied finally. We will prove that condi-tion (iii) of Lemma . is satisfied. We only prove that whenu∈∂∩KerL=∂∩R_,

deg{JQNu,∂∩KerL, (, , )T} = . Whenu∈∂∩KerL=∂∩R,uis a constant vec-tor inR_with

(8)

Step . We prove

it follows the arguments of (.)-(.) that

|ui|<Ri, i= , , .

According to topological degree theory, we have

(9)

Step . We prove

deg

g

t,eu ,g

t,eu , –aeu+

aeu

m(t)eu+eu

T

,∩KerL, (, , )T

=deg

a–aeu,g

t,eu , –aeu+

aeu

T

,∩KerL, (, , )T

,

wherea,aare two chosen positive constants such that

C<ln a a

<A.

To this end, we deﬁne the mappingφ:DomL×[, ]→Xby

φ(u,u,u,μ) =μ

⎡ ⎢ ⎣

a–aeu

g(t,eu)

–aeu+ ae

u_

m(t)eu+eu

⎤ ⎥ ⎦

+ ( –μ)

⎡ ⎢ ⎣

g(t,eu)

g(t,eu)

u m(t)eu₊_eu

⎤ ⎥ ⎦

= ⎡ ⎢ ⎣

μ(a–aeu) + ( –μ)g(t,eu)

g(t,eu)

u m(t)eu+eu

⎤ ⎥ ⎦,

whereμ∈[, ] is a parameter. We will prove that whenu∈∂∩KerL,φ(u,u,u,μ)=

(, , )T_{. When}_u_∈_∂_∩_Ker_L₌_∂_∩_R_,_u_{is a constant vector in}_R_with

i=|ui|=M.

Now we consider two possible cases:

(i) u≥A; (ii) u<A.

(i) When u≥A, from condition (iii) in Theorem ., we have g(t,eu)≤.

More-over,a–aeu≤a–aeA< , thusμ(a–aeu) + ( –μ)g(t,eu) < . Therefore,

φ(u,u,u,μ)= (, , )T.

(ii) Whenu<A, ifu≤C, from condition (H) in Theorem ., we haveg(t,eu) > .

However,a–aeu≥a–aeC> . Therefore,φ(u,u,u,μ)= (, , )T. Ifu>C, we

also consider two possible cases: (a)u≥B; (b)u<B. (a) Whenu≥B, from condition

(H) in Theorem ., we have

g

t,eu < .

Thereforeφ(u,u,u,μ)= (, , )T. (b) Whenu<B, ifu≤D, then from condition

(H) in Theorem ., we obtaing(t,eu) > . Consequently,φ(u,u,u,μ)= (, , )T.

Ifu>D, we can claim whenu∈∂∩KerL=∂∩R,φ(u,u,u,μ)= (, , )T.

Oth-erwise, from

–aeu+

aeu

(10)

we have

eu_<a

a

and

eu_>–ae

ρ₊_(–a

eρ)+  –am(t)aeρ

am(t)

> ,

i.e.,

u<lna–lna,

u>ln

–aeρ+

(–aeρ)+  –am(t)aeρ

am(t)

.

Thus

|u|<max

|d|,|ρ|

, |u|<max

|d|,|ρ|

and

|u|<max

|d|,|ρ|

.

Therefore



i=

|ui|<max

|d|,|ρ|

+max|d|,|ρ|

+max|d|,|ρ|

<M,

which contradicts the fact that_i_=|ui|=M. Based on the above discussion, for anyu∈ ∂∩KerL, we haveφ(u,u,u,μ)= (, , )T. According to topological degree theory,

we obtain

deg

g

t,eu ,g

t,eu , –aeu+

aeu

T

,∩KerL, (, , )T

=degφ(u,u,u, )T,∩KerL, (, , )T

=degφ(u,u,u, )T,∩KerL, (, , )T

=deg

a–aeu,g

t,eu , –aeu+

aeu

T

,∩KerL, (, , )T

.

Step . We prove

deg

t,eu , –aeu+

aeu

T

,∩KerL, (, , )T

=deg

a–aeu,a–aeu, –aeu+

aeu

T

,∩KerL, (, , )T

(11)

To this end, we deﬁne the mappingφ:DomL×[, ]→Xby

φ(u,u,u,μ) =μ

⎡ ⎢ ⎣

a–aeu

a–aeu

⎤ ⎥ ⎦

+ ( –μ)

⎡ ⎢ ⎣

a–aeu

g(t,eu)

⎤ ⎥ ⎦

= ⎡ ⎢ ⎣

a–aeu

μ(a–aeu) + ( –μ)g(t,eu)

u_

m(t)eu+eu

⎤ ⎥ ⎦,

whereμ∈[, ] is a parameter anda,aare two chosen positive constants such thatD<

ln a

a <B. We will prove that whenu∈∂∩KerL,φ(u,u,u,μ)= (, , )

T_{. If it is not}

true, then the constant vectorusatisﬁesφ(u,u,u,μ)= (, , )Twith



i=|ui|=M.

Thus we have ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

a–aeu= , (.)

μ(a–aeu) + ( –μ)g(t,eu) = , (.)

u_

m(t)eu+eu = . (.)

(.) implies

C<u=ln

a

a

<A. (.)

We claim thatu<B; otherwise, ifu≥B, then from condition (H) in Theorem ., we

have

( –u)g

t,eu < .

Consequently,

μ

a–aeu + ( –μ)g

t,eu < ,

which contradicts (.). We also claim thatu>D. Ifu≤D, theng(t,eu) > . However,

a–aeu>a–aeD> .

Thus

u

a–aeu + ( –μ)g

t,eu > ,

which contradicts (.). (.) gives

–aeu+

aeu

(12)

that is,

u<lna–lna,

u>ln

–aeρ+

(–aeρ)+  –am(t)aeρ

am(t)

.

Thus

|u|<max

|d|,|ρ|

, |u|<max

|d|,|ρ|

and

|u|<max

|d|,|ρ|

.

Therefore



i=

|ui|<max

|d|,|ρ|

+max|d|,|ρ|

+max|d|,|ρ|

<M,

which leads to a contradiction. Therefore, by means of topological degree theory, we have

deg

t,eu , –aeu+

aeu

,∩KerL, (, , )T

=degφ(u,u,u, )T,∩KerL, (, , )T

=degφ(u,u,u, )T,∩KerL, (, , )T

=deg

a–aeu,a–aeu, –aeu+

aeu

,∩KerL, (, , )T

.

From the proof of the three steps above, we obtain

degJQNu,∩KerL, (, , )T

=deg

a–aeu,a–aeu, –aeu+

aeu

,∩KerL, (, , )T

.

Because of condition (H) in Theorem ., the system of algebraic equations

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

a–ax= ,

a–ay= ,

–aI+_m₍a_t_₎x_z₊_x= 

has a unique solution (x∗,y∗,z∗)T_{which satisﬁes}

x∗= a a

> , y∗= a a

> , z∗=ax

∗₊_(a

x∗)+ am(t)ax∗

am(t)

(13)

Thus

deg

a–aeu,a–aeu, –aeu+

aeu

,∩KerL, (, , )T

=sign

–ax∗  

 –ay∗ 

· · ·  –az∗– am(t)x

∗_z∗

[m(t)z∗+x∗]

= –.

Therefore, from (.), we have

degJQNu,∩KerL, (, , )T= –.

This completes the proof of Theorem ..

3 An example Consider the system

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x_=x(t)(a(t) –a(t)x(t) –_m₍a_t₎_x_(t₍)_tx₎₊(_xt_)₍_t₎) +D(t)(x(t) –x(t)),

x_=x(t)(a(t) –a(t)x(t)) +D(t)(x(t) –x(t)),

x_=x(t)(–a(t) +_m₍_t₎a_x_₍(_tt_–)_Tx₎₊(t–_x_T₍)_t_–_T₎),

(.)

whereτ>  is a positive constant, all the parameters are positive continuousw-periodic functions with periodicw> .

In Theorem .,g(t,ex) =a(t) –a(t)ex,g(t,ex) =a(t) –a(t)ex. It is easily shown

that ifx≥ln(am

al_) =A, theng(t,e

x₎_≤_{ and if}_x_≥_ln(aM

al_) =B, theng(t,e

x₎_≤_{. We also}

can show if

x≤lna

M

 – (am) M

al



=C,

g

t,ex ≥

a

m M

and ifx≤lnaM

al_ =D, theng(t,e x_{) > .}

(H) am > (am) M_,

(H) a>a.

By Theorem ., we have the following theorem.

Theorem . If(H)and(H)hold,the system(.)has at least one positive w-periodic

solution.Consider the system

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x_=x(t)(a(t) –a(t)x(t) –_ma₍_t(₎t_x)x(t)x(t)

(t)+x(t)) +D(t)(x(t) –x(t)),

x_=x(t)(a(t) –a(t)x(t)) +D(t)(x(t) –x(t)),

x_=x(t)(–a(t) +

a(t)x(t–τ) m(t)x

(t–τ)+x(t–τ)),

(14)

where z> is a positive constant,all the parameters are positive continuous w-periodic functions with periodic w> .

In Theorem .,g(t,ex) =a(t) –a(t)ex,g(t,ex) =a(t) –a(t)ex. It is easily shown that

ifx≥lnaM

al_ =A, theng(t,e

x₎_≤_{ and if}_x_≥_lnaM

al_ =B, theng(t,e

x₎_≤_{. We also can show}

ifx≤lna M

–(

a_

√m) M

al_ =C,g(t,e

x₎_≥₍ a

√m)

m_{and if}_x_≤_lnaM

al_=D, theng(t,e x₎_≥_.

(H_) aM

 > (a√m) M_,

(H_) a>a.

By Theorem ., we have the following theorem.

Theorem . If(H_)and(H_)hold,system(.)has at least one positive w-periodic solu-tion.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, CD and YW, contributed to each part of this study equally and read and approved the ﬁnal version of the manuscript.

Author details

1_{Department of Mathematics, Shaoyang University, Shaoyang, Hunan 422000, P.R. China.}2_{School of Mathematics and}

Statistics, Henan University of Science and Technology, Luoyang, Henan 471003, P.R. China.

Acknowledgements

This research is partially supported by the National Natural Science Foundation of China (11071222, 11101126), the Hunan provincial Natural Science Foundation of China (12JJ3008) and the Research Fund of Hunan provincial Education Department (11B113).

Received: 27 November 2012 Accepted: 15 February 2013 Published: 8 March 2013

References

1. Li, YK: Positive periodic solution of neutral predator-prey system. Appl. Math. Mech.20(5), 545-550 (1999) (in Chinese) 2. Zhang, ZQ, Wang, ZC: The existence of a periodic solution for a generalized prey-predator system with delay. Math.

Proc. Camb. Philos. Soc.137, 475-487 (2004)

3. Zhang, ZQ, Zeng, XW: Periodic solution of a nonautonomous stage-structured single species model with diﬀusion. Q. Appl. Math.63(2), 277-289 (2005)

4. Jose, F, Santiago, V: An approximation for prey-predator models with time delay. Physica D110(3-4), 313-322 (1997) 5. Xiao, YN, Chen, LS: Modeling and analysis of predator-prey model with disease in the prey. Math. Biosci.171(1), 59-82

(2001)

6. Xu, R, Chen, LS: Persistence and stability for two-species ratio-dependent predator-prey system with time delay in a two-patch environment. Comput. Math. Appl.40, 577-588 (2000)

7. Shihua, C, Feng, W: Positive periodic solution of two-species ratio-dependent predator-prey system with time delay in two-patch environment. Appl. Math. Comput.150, 737-748 (2004)

8. Gaines, RE, Mawhin, JL: Coincidence Degree and Non-Linear Diﬀerential Equations. Springer, Berlin (1977)

doi:10.1186/1687-1847-2013-52