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Dynamics of species in a model with two predators and one prey

Ta Viet Tona,∗, Nguyen Trong Hieub

aDepartment of Applied Physics, Graduate School of Engineering, Osaka University, Suita Osaka 565-0871, Japan

bFaculty of Mathematics, Mechanics and Informatics, Hanoi National University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:

Received 2 May 2010

Accepted 27 April 2011

Communicated by Ravi Agarwal

MSC:

34C27

34D05

Keywords:

Predator–prey system

Beddington–DeAngelis functional response

Permanence

Extinction

Periodic solution

Asymptotic stability

Liapunov function

a b s t r a c t

In this paper, we study a predator–prey model which has one prey and two predators with Beddington–DeAngelis functional responses Firstly, we establish a set of sufficient conditions for the permanence and extinction of species Secondly, the periodicity of positive solutions is studied Thirdly, by using Liapunov functions and the continuation theorem in coincidence degree theory, we show the global asymptotic stability of such solutions Finally, we give some numerical examples to illustrate the behavior of the model

© 2011 Elsevier Ltd All rights reserved

1 Introduction

The dynamical relationship between predators and prey has been studied by several authors for a long time In those researches, to represent the average number of prey killed per individual predator per unit of time, a functional, called the functional response, was introduced The functional response can depend on only the prey’s density or both the prey’s and the predator’s densities However, some biologists have argued that in many situations, especially when predators have to search for food, the functional response should depend on both the prey’s and the predator’s densities [1–6] One

of the most popular functional responses is the fractional one as in the following prey–predator model It is called the Beddington–DeAngelis functional response:







x′1=x1(a1−b1x1) − c1x1x2

x′2= −a2x2+ c2x1x2

In this model, x i(t)represents the population density of species X i at time t(i≥1); X1is the prey and X2is the predator

At time t,a1(t)is the intrinsic growth rate of X1and a i(t)is the death rate of X2;b1(t)measures the inhibiting effect of

the environment on X1 This model was originally proposed by Beddington [7] and DeAngelis et al [8] independently Since the appearance of these two investigations, there have been many other ones for analogous systems with diffusion in a

∗Corresponding author Tel.: +81 6 6879 4249.

E-mail addresses:taviet.ton@ap.eng.osaka-u.ac.jp (T.V Ton), hieungt@vnu.edu.vn (N.T Hieu).

0362-546X/$ – see front matter © 2011 Elsevier Ltd All rights reserved.

constant environment [9–14] However, a constant environment is rarely the case in real life Most natural environments are physically highly variable, i.e., the coefficients in those models should depend on time [15–18] In order to continue studying such models, in this paper, we consider a predator–prey model of one prey and two predators with Beddington–DeAngelis functional responses:















x′1=x1[a1(t) −b1(t)x1]− c2(t)x1x2

α(t) + β(t)x1+ γ (t)x2−

c3(t)x1x3

α(t) + β(t)x1+ γ (t)x3,

x′2=x2

[

α(t) + β(t)x1+ γ (t)x2−b2(t)x3

] ,

x′3=x3

[

α(t) + β(t)x1+ γ (t)x3−b3(t)x2

]

(1.1)

Here x i(t)represents the population density of species X i at time t(i≥1),X1is the prey and X2,X3are the predators Two predators share one prey and it is assumed that there are two types of competition between the two predators The first type

is direct interference where individuals of each predator species act with aggression against individuals of the other predator

species In our model, this type of competition is described by the coefficients b2(t)and b3(t) The second type of competition

is interference competition that occurs during hunting because predators spend time interacting with each other rather than seeking prey Here we assume that there is no competition of that type between individuals of the two different predator species Therefore, the Beddington–DeAngelis functional responses are of the form d i(t)x1

α(t)+β(t)x1 + γ (t)x i (i=2,3) We use the same coefficientsα, β, γin the functional responses of both predators, since it is assumed that both predators take the same time to handle a prey once they encounter it and that individuals of each predator species interfere with each other when hunting by exactly the same amount in both species This assumption is somewhat restrictive from the biological viewpoint, but it could be removed without greatly changing the analysis of system(1.1)

Throughout this paper, it is assumed that the functions a i(t),b ij(t),c i(t),d i(t), α(t), β(t), γ (t) (1 ≤ i,j ≤ 3)are continuous on R and bounded above and below by some positive constants

This article is organized as follows Section2provides some definitions and notation In Section3, we state some results

on invariant sets, and the permanence and extinction of system(1.1) Then, the asymptotic stability of solution is proved

by using a Liapunov function In Section4, we continue using other Liapunov functions and the continuation theorem

in coincidence degree theory to show the existence and global stability of a positive periodic solution The final section illustrates the behavior of system(1.1)by some computational results and gives our conclusion

2 Definitions and notation

In this section we introduce some basic definitions and facts which will be used throughout this paper Let R3

+ = { (x1,x2,x3) ∈R3|x i >0(i≥1)} Denote by x(t) = (x1(t),x2(t),x3(t))the solution of system(1.1)with initial condition

solutions x(t)with positive initial values, i.e., x0∈R3+ Let g(t)be a continuous function; for brevity, instead of writing g(t)

we write g If g is bounded on R, we denote

g u=sup

t∈ R

g(t), g l=inf

t∈Rg(t), andgˆ = 1

ω

0 g(t)dt, if g is a periodic function with periodω The global existence and uniqueness of solution of system

(1.1)are guaranteed by the properties of the map defined by the right hand side of system(1.1)[19] We have the following lemma

Lemma 2.1 Both the non-negative and positive cones of R3are positively invariant for(1.1).

Proof The solution x(t)of(1.1)with initial value x0satisfies















x1=x01exp

t0

[

a1−b1x1− c2x2

c3x3

]

du

 ,

x2=x02exp

t0

[

]

du

 ,

x3=x03exp

t0

[

]

du



The conclusion follows immediately for all t∈ [t0, ∞) The proof is complete 

Definition 2.2 System(1.1)is said to be permanent if there exist some positiveδj(j=1,2)such that

δ1≤lim inf

t→∞ x i(t) ≤lim sup

t→∞

for all solutions of(1.1)

Definition 2.3 A set A ⊂ R3+is called an ultimately bounded region of system(1.1)if for any solution x(t)of(1.1)with

positive initial values, there exists T1>0 such that x(t) ∈A for all t ≥t0+T1

Definition 2.4 A bounded non-negative solution x∗(t)of(1.1)is said to be globally asymptotically stable (or globally

attractive) if any other solution x(t)of(1.1)with positive initial values satisfies limt→∞∑3

i= 1|x i(t) −x∗

i(t)| =0

Remark 2.5 It is easy to see that if a solution of(1.1)is globally asymptotically stable, then so are all solutions In this case, system(1.1)is also said to be globally asymptotically stable

3 The model with general coefficients

Letϵ ≥0 be sufficiently small Put

Mϵ

1 = a u1

b l

1

i = d u i Mϵ

1−a l iαl

a lγl ,

mϵ

1= a l1γl−c2u−c3u

b u1γl − ϵ,

mϵ

i = d l i mϵ

1− (a u i +b u i Mϵ

j)(βu mϵ

1+ αu) (a u i +b u i Mϵ

j)γu (i,j≥2,i̸=j),

(3.1)

then Mϵ

i >mϵ

i (i≥1) We will show that max{m0

i,0} (i≥1)are the lower bounds for the limiting bounds of species X ias

time t tends to infinity This is obvious when m0i ≤0 Therefore, it is assumed that m0i >0

Hypothesis 3.1 m0i >0(i≥1)

Theorem 3.2 Under Hypothesis 3.1 , for any sufficiently smallϵ > 0 such that mϵ

i > 0(i ≥ 1), a set Γϵ defined by

Γϵ = { (x1,x2,x3) ∈R3|mϵ

i <x i<Mϵ

i (i≥1)}is positively invariant with respect to system(1.1).

Proof Throughout this proof, we use the fact that the solution to the equation

X′(t) =A(t,X)X(t)[B−X(t)] (B̸=0)

is given by

X(t) = BX

0exp



t

t0BA(s,X(s))ds



X0exp



t

t0BA(s,X(s))ds





,

where t0≥0 and X0=X(t0) Consider the solution of system(1.1)with an initial value x0∈Γϵ FromLemma 2.1and from the first equation of(1.1), we have

x′1(t) ≤ x1(t)[a1(t) −b1(t)x1(t)]

≤ x1(t)[a u−b l1x1(t)]

= b l1x1(t)(M10−x1).

Using the comparison theorem gives

x1(t) ≤ x0M0exp{a u(t−t0)}

x0[exp{a u(t−t0)} −1] +M0

1exp{a u(t−t0)}

x0[exp{a u(t−t0)} −1] +Mϵ

1

< Mϵ

It follows from the third equation of(1.1)and from(3.2)that

x′2 ≤ −a l2x2+ d u2x1x2

αl+ βl x1+ γl x2

1x2

= x2 (d u2Mϵ

1−a l2αl) −a l2γl x2

αl+ γl x2

αl+ γl x2x2(Mϵ

2−x2).

Putting

C2(t) = a l2γl

and using the comparison theorem again yields

x2(t) ≤ M

ϵ

2x0exp



Mϵ 2

t

t0C2(s)ds



x0exp



Mϵ 2

t

t0C2(s)ds





2

<Mϵ

Similarly, x3(t) <Mϵ

3for every t≥t0 Now, by the first equation of(1.1), it implies that

x′1(t) ≥x1



a l1−c2u+c u3



=b u x1(m0−x1).

Since x01>mϵ

1, by the comparison theorem, we obtain

x1(t) ≥ x0m0exp{b u m0(t−t0)}

x0[exp{b u m0(t−t0)} −1] +m0 >mϵ

1 for all t≥t0.

Similarly, for i,j≥2(i̸=j),

x′i = −a i x i+ d i x1x i

α + βx1+ γx i −b i x i x j

≥ − (a u i +b u i Mϵ

j)x i+ d l i mϵ

1x i

1+ γu x i

=



d l mϵ

1− (a u

i +b u

i Mϵ

j)(βu mϵ

1+ αu)x i− (a u

i +b u

i Mϵ

j)γu x2

i

1+ γu x i

i +b u

i Mϵ

j)γu

1+ γu x i x i(mϵ

i −x i),

from which follows that x i(t) >mϵ

i for all t≥t0 We complete the proof 

In the next theorem, the permanence of system(1.1)is shown A treatment called practical persistence to prove the permanence of models and its application to various types of models can be seen in [20,9,21]

Theorem 3.3 Under Hypothesis 3.1 , for any sufficiently smallϵ >0 such that mϵ

i >0,

mϵ

i ≤lim inf

t→∞ x i(t) ≤lim sup

t→∞

x i(t) ≤Mϵ

i (i≥1).

Consequently, system(1.1)is permanent.

Proof According to the proof ofTheorem 3.2we have

x1(t) ≤ x0M0exp{a u(t−t0)}

x01[exp{a u1(t−t0)} −1] +M10.

Thus, lim supt→∞x1(t) ≤M0, i.e., there exists t1≥t0such that x1(t) <Mϵ

1for all t ≥t1 By the same arguments as made for(3.4), it follows that

x2(t) ≤ M

ϵ

2x12exp



Mϵ 2

t

t1C2(s)ds



x1exp



Mϵ 2

t

t1C2(s)ds





2

from which it is implied that 0<x2(t) ≤max{Mϵ

2,x12}for all t≥t1, where x12=x2(t1) Then from(3.3), inft≥t1C2(s) >0

By using(3.5), we have lim supt→∞x2(t) ≤Mϵ

2 Similarly, lim supt→∞x3(t) ≤Mϵ

3and lim inft→∞x i(t) ≥mϵ

i (i≥1) The permanence follows fromDefinition 2.2 The proof is complete 

Theorem 3.4 Let i∈ {2,3} If M i0<0 then lim t→∞x i(t) =0, i.e., the ith predator goes to extinction.

Proof It follows from M0

i <0 that Mϵ

i <0 with a sufficiently smallϵ Similarly to the proof ofTheorem 3.2we have

x′i(t) ≤ αl a+lγ γl l

x i x i(Mϵ

Thus, there exists C≥0 such that limt→∞x i(t) =C and C ≤x i(t) ≤x0

i for all t≥t0 If C>0 then from(3.6)there exists

contradicts x i(t) >0 for all t≥t0 Hence, limt→∞x i(t) =0 

In order to consider the global asymptotic stability of system(1.1), we need the following result called Barbalat’s lemma

Lemma 3.5 (See [ 22 ]) Let h be a real number and f be a non-negative function defined on[h, +∞)such that f is integrable and uniformly continuous on[h, +∞) Then lim t→∞f(t) =0.

Theorem 3.6 Suppose that Hypothesis 3.1 holds and letϵ > 0 be sufficiently small such that mϵ

i > 0(i ≥ 1) Let x∗be a solution of system(1.1)satisfying

lim sup

t→∞



2)

u2(mϵ

1,Mϵ

βc3Mϵ

3)

u3(mϵ

1,Mϵ

3)



<0,

lim sup

t→∞



1)

u i(Mϵ

1,mϵ

i) −

d iγmϵ 1

u i(mϵ

1,Mϵ

i)



<0,

(3.7)

where u i(a,b) = (α + βx∗1+ γx∗i)(α + βa+ γb) (i,j≥2,i̸=j) Then x∗is globally asymptotically stable.

Proof Let x be the other solution of(1.1) FromTheorem 3.3,Γϵis an ultimately bounded region of(1.1) Then there exists

T1>0 such that x,x∗∈Γϵfor all t≥t0+T1 Consider a Liapunov function defined by V(t) = ∑3

i= 1|ln x i−ln x∗

i| ,t ≥t0

A direct calculation of the right derivative D+V(t)of V(t)along the solution of(1.1)gives

D+V(t) =

3

−

i= 1 sgn(x i−x∗i)



x′

i

x i −

x∗

i

′

x∗

i



= sgn(x1−x∗1)



−c2



x2

x∗2

α + βx∗1+ γx∗2





x3

x∗3

1+ γx∗ 3



−b1(x1−x∗1)



+d2sgn(x2−x∗2)

[

x1

x∗ 1

b2

d2(x3−x∗3)

]

+d3sgn(x3−x∗3)

[

x1

x∗ 1

b3

d3(x2−x∗2)

]

≤ −b1|x1−x∗1| −c2sgn(x1−x∗1) α(x2−x∗2) + β(x∗1x2−x1x∗2)

u2(x1,x2)

−c3sgn(x1−x∗1) α(x3−x∗

1x3−x1x∗

3)

u3(x1,x3) +d2sgn(x2−x∗2) α(x1−x∗

1) + γ (x1x∗

2−x2x∗

1)

u2(x1,x2) +b2|x3−x∗3| +d3sgn(x3−x∗3) α(x1−x∗1) + γ (x1x∗3−x3x∗1)

u3(x1,x3) +b3|x2−x∗2|

It follows from x,x∗∈Γϵfor t≥t0+T1and x1x∗

i −x∗

1x i =x1(x∗

i −x i) +x i(x1−x∗

1) (i=2,3)that

D+V(t) ≤ −b1|x1−x∗1| −c2sgn(x1−x∗1) (α + βx1)(x2−x∗

2) − βx2(x1−x∗

1)

u2(x1,x2)

−c3sgn(x1−x∗1) (α + βx1)(x3−x∗3) − βx3(x1−x∗1)

u3(x1,x3) +d2sgn(x2−x∗2) (α + γx2)(x1−x∗1) − γx1(x2−x∗2)

u (x ,x ) +b2|x3−x∗3|

+d3sgn(x3−x∗3) (α + γx3)(x1−x∗1) − γx1(x3−x∗3)

u2(x1,x3) +b3|x2−x∗2|

≤

[

2)

u2(mϵ

1,Mϵ

βc3Mϵ

3)

u3(mϵ

1,Mϵ

3)

]

|x1−x∗1|

+ [

1)

u2(Mϵ

1,mϵ

2) −

d2γmϵ 1

u2(mϵ

1,Mϵ

2)

]

|x2−x∗2|

+ [

1)

u3(Mϵ

1,mϵ

3) −

d3γmϵ 1

u3(mϵ

1,Mϵ

3)

]

Combining(3.7)and(3.8)gives the existence of a positive numberµ >0 and of T2≥t0+T1such that

3

−

i= 1

Integrating both sides of(3.9)from T2to t yields

T2

−

i= 1

|x i−x∗i|



ThenT t

2



i= 1|x i−x∗

i|



i= 1|x i−x∗

i| ∈L1([T2, ∞))

On the other hand, it follows from x,x∗ ∈ Γϵ for all t ≥ t0+T1and from the equations of(1.1)that the derivatives

of x i(t),x∗i(t)(i ≥ 1)are bounded on[T2, ∞) As a consequence∑3

i= 1|x i−x∗i|is uniformly continuous on[T2, ∞) By

Lemma 3.5we have limt→∞

i= 1|x i−x∗

i| =0, which completes the proof 

4 The model with periodic coefficients

In this section, we assume that the coefficients in system(1.1)areω-periodic in t and bounded above and below by some

positive constants We study the existence and stability of a periodic solution of this system To do this, we will employ

an alternative approach to establish some criteria in terms of the average of the related functions over an interval of the common period That is continuation theorem in coincidence degree theory, which has been successfully used to establish criteria for the existence of positive periodic solutions of some mathematical models of predator–prey type; we refer the reader to [23–26] To this end, we shall summarize in the following a few concepts and results from [27] that will be basic for this section

Let X and Y be two Banach spaces, let L:Dom L⊂X→Y be a linear mapping, and let N:X→Y be a continuous mapping.

The mapping L will be called a Fredholm mapping of index zero if the following conditions hold:

(i) Im L is closed;

(ii) dim Ker L=codim Im L< ∞

If L is a Fredholm mapping of index zero and there exist continuous projections P:X → X and Q:Y → Y such that

Im P=Ker L,Im L=Ker Q =Im(I−Q), it follows that

L p=L|Dom L∩Ker P: (I−P)X→Im L

is invertible We denote by K pthe inverse of that map IfΩis an open bounded subset of X, the mapping N will be called

L-compact onΩ¯ if the mapping QN : ¯Ω → Y is continuous and bounded, and K p(I−Q)N : ¯Ω → X is compact, i.e.,

it is continuous and K p(I−Q)N( ¯Ω)is relatively compact Since Im Q is isomorphic to Ker L, there exists an isomorphism

J:Im Q →Ker L The following continuation theorem is from [27]

Lemma 4.1 (Continuation Theorem) Let X and Y be two Banach spaces and L a Fredholm mapping of index zero Assume that

N: ¯Ω →Y is L-compact onΩ¯ withΩis open and bounded in X Furthermore, assume that

(a) for eachλ ∈ (0,1),x∈ ∂Ω∩Dom L, Lx̸= λNx;

(b) for each x∈ ∂Ω∩Ker L,QNx̸=0;

(c) deg{QNx,Ω∩Ker L,0} ̸=0;

then the operator equation Lx=Nx has at least one solution in Dom L∩ ¯Ω.

We now put

L11=lnˆa1

ˆ

b1, H11=lnaˆ1

ˆ

b1

+2aˆ1ω,

L12=ln



ˆ



c2+c3

γ

L i1=lnaˆ1(ˆd i− ˆa iβl)exp{2aˆ1ω} − ˆa i bˆ1αl

ˆ

a i bˆ1γl ,

H i1=2



d i

β



L i2=lnd iexp{H12} − (αu+ βuexp{H12} )(ˆa i+ ˆb iexp{H j1} )

H i2=L i2−2





d i

β



The convention here is that ln x= −∞if x≤0 In the next theorem, a sufficient condition for existence of anω-periodic solution of(1.1)is presented

Theorem 4.2 If L i2> −∞ (i≥1)then system(1.1)has at least one positiveω-periodic solution.

Proof Put x i(t) =exp{u i(t)} (i≥1), then system(1.1)becomes











u′1=a1−b1exp{u1} − c2exp{u2}

c3exp{u3}

u′2= −a2+ d2exp{u1}

u′3= −a3+ d3exp{u1}

(4.1)

Let X=Y= {u= (u1,u2,u3)T ∈C1(R,R3) |u i(t) =u i(t+ ω) (i≥1)}with‖u‖ = ∑3

i= 1maxt∈[ 0 ,T]|u i(t)|,u∈X Then

X,Y are both Banach spaces with the above norm‖ · ‖ Let

N

u1

u2

u3



=

N1(t)

N2(t)

N3(t)



=









a1−b1exp{u1} − c2exp{u2}

c3exp{u3}







 ,

L

u1

u2

u3



=





u′1

u′2

u′3



u1

u2

u3



u1

u2

u3



=











1 ω

0

u1(t)dt

1 ω

0

u2(t)dt

1 ω

0

u3(t)dt









 ,

u1

u2

u3



Then Ker L= {u∈X|u= (h1,h2,h3)T ∈R3} ,Im L= 

u∈Y| ω

0 u i(t)dt=0(i≥1), and dim Ker L=3=codim Im L Since Im L is closed in Y,L is a Fredholm mapping of index zero It is easy to show that P,Q are continuous projections such

that Im P=Ker L,Im L=Ker Q =Im(I−Q) Furthermore, the generalized inverse (to L) K P:Im L→Dom L∩Ker P exists

and is given by

K P

u1

u2

u3



=









0

u1(s)ds− 1

T

0

0

u1(s)dsdt

0

u2(s)ds− 1

T

0

0

u2(s)dsdt

u3(s)ds− 1

T

u3(s)dsdt









Obviously, QN and K P(I − Q)N are continuous It is easy to see that N is L-compact onΩ¯ with any open bounded setΩ⊂X.

Now we will find an appropriate open, bounded subsetΩfor application of the continuation theorem Corresponding to

the operator equation Lu= λNu, λ ∈ (0,1), we have

u′1= λ



a1−b1exp{u1} − c2exp{u2}

c3exp{u3}

 ,

u′2= λ





u′3= λ





Suppose that(u1,u2,u3) ∈X is an arbitrary solution of system(4.2)for a certainλ ∈ (0,1) Integrating both sides of(4.2)

over the interval[0, ω], we obtain

ˆ

a1ω = ∫ ω

0



b1exp{u1} + c2exp{u2}

c3exp{u3}



dt,

ˆ

a iω +

0

b iexp{u j}dt =

0

d iexp{u1}dt

≤

0

d i





d i

β



It follows from(4.2)and(4.3)that for i,j≥2(i̸=j),

0

|u1(t)′|dt ≤ λ



0

a1dt+

0

b1exp{u1}dt+

0

c2exp{u2}

+

0

c3exp{u3}



dt

< 2aˆ1ω,

0

|u i(t)′

|dt<2





d i

β

 ω.

Since u∈X, there existξi, ηi ∈ [0, ω]such that

u i(ξi) = min

t∈[ 0 ,ω]u i(t), u i(ηi) = max

From the first equation of(4.3)and(4.4), we obtainaˆ1ω ≥ 0ωb1exp{u1(ξ1)}dt = ˆb1ωexp{u1(ξ1)}, from which follows

u1(ξ1) <L11 Hence

u1(t) ≤u1(ξ1) +

0

|u′1(t)|dt<L11+2ˆa1ω =H11 for all t≥0.

On the other hand, from the first equation of(4.3)and(4.4), we also have

ˆ

a1ω ≤

0

b1exp{u1(η1)}dt+

0

c2(t) +c3(t)

=



ˆ

b1exp{u1(η1)} + 



c2+c3

γ

ω.

Then for any t≥0,

u1(t) ≥ u1(η1) −

0

|u′1(t)|dt

 ˆ



c2+c3

γ

−2aˆ1ω

From the arguments above, we have H12 ≤u1(t) ≤H11for all t ∈ [0, ω] It then follows from the second equation of(4.3)

and(4.4)that

ˆ

a iω ≤ ∫ ω

0

d iexp{u1}dt

≤

0

d iexp{H11}dt

αl+ βlexp{H11} + γlexp{u i(ξi)}

αl+ βlexp{H11} + γlexp{u i(ξi)} ,

from which it is implied that

u i(ξi) ≤ ln (ˆd i− ˆa iβl)exp{H11} − ˆa iαl

ˆ

a iγl

=lnaˆ1(ˆd i− ˆa iβl)exp{2aˆ1ω} − ˆa i bˆ1αl

ˆ

a i bˆ1γl , and then

u i(t) ≤ u i(ξi) +

0

|u′i(t)|dt

≤lnaˆ1(ˆd i− ˆa iβl)exp{2ˆa1ω} − ˆa i bˆ1αl

ˆ

a i bˆ1γl +2



d i

β

 ω

Similarly, for i,j≥2(i̸=j)and t≥0, we have

ˆ

a iω =

0

[

diexp{u1}

]

dt

≥

0

[

diexp{H12}

αu+ βuexp{H12} + γuexp{u i(ηi)} −b iexp{H j1}

]

dt

=



ˆ

d iexp{H12}

αu+ βuexp{H12} + γuexp{u i(ηi)} − ˆb iexp{H j1}

 ω, from which it is implied that

u i(ηi) ≥lnd iexp{H12} − (αu+ βuexp{H12} )(ˆa i+ ˆb iexp{H j1} )

and

u i(t) ≥ u i(ηi) −

0

|u′i(t)|dt

≥lnd iexp{H12} − (αu+ βuexp{H12} )(ˆa i+ ˆb iexp{H j1} )





d i

β

 ω

Put B i =max{|H i1| , |H i2|} (i≥1), then maxt∈[ 0 ,ω]|u i| ≤B i Thus, for any solution u∈X of(4.2), we have‖u‖ ≤ ∑3

i= 1B i,

and, clearly, B i(i ≥ 1)are independent ofλ Taking B = ∑4

i= 1B i where B4is taken sufficiently large such that B4 ≥

i= 1

j= 1|L ij|and lettingΩ = {u ∈ X| ‖u‖ < B}, thenΩ satisfies the condition (a) ofLemma 4.1 To compute the Brouwer degree, let us consider the homotopy

where G:R3→R3,

G(u) =









ˆ

a1− ˆb1exp{u1}

−ˆa2− ˆb2exp{u3} +1

ω

0

d2(t)exp{u1}dt

−ˆa3− ˆb3exp{u2} +1

ω

0

d3(t)exp{u1}dt









We have

Hµ(u) =











ˆ

a1− ˆb1exp{u1} − 1

ω

0

µc3exp{u3}

]

dt

−ˆa2− ˆb2exp{u3} + 1

ω

0

d2exp{u1}dt

−ˆa3− ˆb3exp{u2} + 1

ω

0

d3exp{u1}dt











By carrying out similar arguments to those above, one can easily show that any solution u∗of the equation Hµ(u) =0∈R3

i ≤L i2(i≥1) Thus, 0̸∈Hµ(∂Ω∩Ker L)forµ ∈ [0,1], and then QN(∂Ω∩Ker L) ̸=0.

Note that the isomorphism J can be the identity mapping I; since Im P=Ker L, by the invariance property of homotopy, we

have

deg(JQN,Ω∩Ker L,0) =deg(QN,Ω∩Ker L,0)









 det









∂f2(u1,u2)

∂u1

∂f2(u1,u2)

∂f3(u1,u3)

∂f3(u1,u3)

∂u3



















 ˆ

b1exp{u1}

 ∂f2(u1,u2)

∂u2

∂f3(u1,u3)

ˆ

b3exp{u2+u3}



where deg(·, ·, ·)is the Brouwer degree [28] and

f2(u1,u2) = ω1

0

d2exp{u1}dt

f3(u1,u3) = ω1

0

d3exp{u1}dt

It is easy to see that the functions f i(u1,u i)are decreasing in u i∈R(i≥2) Then

∂f2(u1,u2)

∂u2

∂f3(u1,u3)

Combining(4.5)and(4.6)gives deg(JQN,Ω∩Ker L,0) = −1̸=0 By now we have proved thatΩverifies all requirements

ofLemma 4.1, then Lu = Nu has at least one solution in Dom L∩ ¯Ω, i.e.,(4.1)has at least oneω-periodic solution u∗in

Dom L∩ ¯Ω Set x∗

i =exp{u∗

i} (i≥1), then x∗is anω-periodic solution of system(1.1)with strictly positive components We complete the proof 

Corollary 4.3 If the ω-periodic solution x∗ in Theorem 4.2 satisfies the assumptions in Theorem 3.6 , then x∗ is globally asymptotically stable.

Proof The proof of this corollary is derived directly fromTheorems 3.6and4.2 

5 Numerical examples and conclusion

In this section, we present some numerical examples As a first example, we consider the case a1=4.7+sin(πt),b1=

2.4−cos(2.7t),c2=10.1+2.2 sin(1.4t),c3=9.3+cos(2.8t),a2= (1.1−cos(2πt))/2.5,b2=1.4+sin(0.8t),d2=9.9−

0.4 sin(0.6t),a3= (2.2−cos(1.7t))/6,b3= (2.3+1.3 sin(3.2t))/2,d3=8.5+sin(0.9πt), α = (1.2−cos(2t))/4, β =