# Multiple Positive Periodic Solutions for a Gilpin-Ayala Competition Predator-Prey System with Harvesting Terms

Authors: Yongzhi Liao; Yongkun Li; Xiaoyan Dou
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

By applying Mawhin’s continuation theorem of coincidence degree theory, we study the existence of multiple positive periodic solutions for a Gilpin-Ayala competition predator-prey system with harvesting terms and obtain some sufficient conditions for the existence of multiple positive periodic solutions for the system under consideration. The result of this paper is completely new. An example is employed to illustrate our result.


---

## Body

## 1. Introduction

In 1973, Ayala et al. [1] conducted experiments on fruit fly dynamics to test the validity of ten models of competition. One of the models accounting best for the experimental results is given by
(1.1)y1′=r1y1(1-(y1K1)θ1-a12y2K2),y2′=r2y2(1-(y2K2)θ2-a21y1K1).In order to fit data in their experiments and to yield significantly more accurate results, Gilpin and Ayala [2] claimed that a slightly more complicated model was needed and proposed the following competition model:
(1.2)xi′(t)=rixi(1-(xiKi)θi-∑j=1,j≠in‍aijxjKj),i=1,2,…,n,
where xi is the population density of the ith species, ri is the intrinsic exponential growth rate of the ith species, Ki is the environmental carrying capacity of species i in the absence of competition, θi provides a nonlinear measure of interspecific interference, and αij provides a measure of interspecific interference.During the past decade, many generalizations and modifications of systems (1.1) and (1.2) have been proposed and studied; see, for example, [3–10].Virtually all biological systems exist in environments which vary with time, frequently in a periodic way. Ecosystem effects and environmental variability are very important factors, and mathematical models cannot ignore, for example, year-to-year changes in weather, habitat destruction and exploitation, the expanding food surplus, and other factors that affect the population growth.Since biological and environmental parameters are naturally subjected to fluctuation in time, the effects of a periodically varying environment are considered as important selective forces on systems in a fluctuating environment. Therefore, on the one hand, models should take into account the seasonality of the periodically changing environment. Also, the exploitation of biological resources and the harvest of population species are commonly practiced in fishery, forestry, and wildlife management; the study of population dynamics with harvesting is an important subject in mathematical bioeconomics, which is related to the optimal management of renewable resources and some other issues including control issues to regulate populations (see [11–16]).Motivated by above, in this paper, we will investigate the following two species Gilpin-Ayala competition predator-prey system with harvesting terms:(1.3)y1′(t)=r1(t)y1(t){1-(y1(t)k1(t))θ1-a12(t)y2(t)k2(t)}-h1(t),y2′(t)=r2(t)y2(t){1-(y2(t)k2(t))θ2-a21(t)y1(t)k1(t)}-h2(t),
where ri(t)>0, ki(t)>0, hi(t)>0,i=1,2, a12(t) and a21(t)∈C([0,+∞),(0,+∞)) are ω-periodic functions, θi,i=1,2 are positive constants, and y1 and y2 represent the number of individuals in the prey and predator population.A very basic and important problem in the study of a population growth model with a periodic environment is the global existence and stability of a positive periodic, which plays a similar role as a globally stable equilibrium does in an autonomous model; also, on the existence of positive periodic solutions to system (1.2), few results are found in literatures. This motivates us to investigate the existence of a positive periodic or multiple positive periodic solutions for system (1.2). In fact, it is more likely for some biological species to take on multiple periodic change regulations and have multiple local stable periodic phenomena. Therefore, it is essential for us to investigate the existence of multiple positive periodic solutions for population models. Our main purpose of this paper is, by using Mawhin's continuation theorem of coincidence degree theory [17], to establish the existence of four positive periodic solutions for system (1.3). Our method used in this paper can be used to study the multiple existence of positive periodic solutions for n-species Gilpin-Ayala competition predator-prey system with harvesting terms.The organization of this paper is as follows. In Section2, we make some preparations. In Section 3, by using Mawhin's continuation theorem of coincidence degree theory, we establish sufficient conditions for the existence of multiple positive periodic solutions to system (1.3). An illustrative example is given in Section 3.


## 2. Preliminaries

For the readers' convenience, we first summarize a few concepts from [17].Let𝕏 and ℤ Banach spaces. Let L:DomL⊂𝕏→ℤ be a linear mapping and N:𝕏→ℤ be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if ImL is a closed subspace of ℤ and
(2.1)dim⁡Ker⁡L=codimIm⁡L<∞.
If L is a Fredholm mapping of index zero, then there exist continuous projectors P:𝕏→ℤ and Q:ℤ→ℤ such that ImP=KerL and ImL=KerQ=Im(I-Q). It follows that
(2.2)L|Dom⁡L∩Ker⁡P:(I-P)X⟶Im⁡L
is invertible and its inverse is denoted by KP. If Ω is a bounded open subset of 𝕏, the mapping N is called L-compact on 𝕏, if QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→𝕏 is compact. Because ImQ is isomorphic to KerL, there exists an isomorphism J:ImQ→KerL.In the proof of our existence result, we need the following continuation theorem cited from [17].Lemma 2.1 (see [17]).
LetL be a Fredholm mapping of index zero, and let N be L-compact on 𝕏. Suppose (a)
for eachλ∈(0,1), x∈∂Ω, Lx≠λNx;(b)
for eachx∈∂Ω, QNx≠0;(c)
the Brownerdeg{JQN,Ω∩KerL,0}≠0.Then Lx=Nx has at least one solution in Ω¯∩DomL.By making the substitutionsy1(t)=ex1(t)andy2(t)=ex2(t), then system (1.2) is reformulated as
(2.3)x1′(t)=r1(t){1-(ex1(t)k1(t))θ1-a12(t)ex2(t)k2(t)}-h1(t)e-x1(t),x2′(t)=r2(t){1-(ex2(t)k2(t))θ2-a21(t)ex1(t)k1(t)}-h2(t)e-x2(t).It is obvious that periodic solutions of (2.3) are positive periodic solutions of (1.3).For the sake of convenience, we introduce notations as follows:(2.4)fL=max⁡t∈[0,ω]f(t),fl=min⁡t∈[0,ω]f(t),
where f is a continuous ω-periodic function.We also introduce four assumptions and eight positive numbers as follows.Assumption (H1) (1-a12(k2L/k2))l>2(h1/r1)L/(k1θ1)l[(h1/r1)l]1-θ1.Assumption (H2) (1-a21(k1L/k1))l>2(h2/r2)L/(k2θ2)l[(h2/r2)l]1-θ2.Assumption (H3) (1-a12(k2L/k2))l>2(h1/r1)L((k1L)θ1-1/(k1l)θ1).Assumption (H4) (1-a21(k1L/k1))l>2(h2/r2)L((k2L)θ1-2/(k2l)θ2). 									(2.5)u±=(1-a12(k2L/k2))l±[(1-a12(k2L/k2))l]2-4((h1/r1)L/(k1θ1)l[(h1/r1)l]1-θ1)2/{(k1θ1)l[(h1/r1)l]1-θ1},l±=(1-a21(k1L/k1))l±[(1-a21(k1L/k1))l]2-4((h2/r2)L/(k2θ2)l[(h2/r2)l]1-θ2)2/{(k2θ2)l[(h2/r2)l]1-θ2},u¯±=(1-a12(k2L/k2))l±[(1-a12(k2L/k2))l]2-4(h1/r1)L((k1L)θ1-1/(k1l)θ1)2(k1L)θ1-1/(k1l)θ1,l¯±=(1-a21(k1L/k1))l±[(1-a21(k1L/k1))l]2-4(h2/r2)L((k2L)θ2-1/(k2l)θ2)2(k2L)θ2-1/(k2l)θ2.


## 3. Main Result

Our main result of this paper is as follows.Theorem 3.1.
Assume that one of the following conditions holds.(i)
If0<θi<1,i=1,2, then (H1)-(H2).(ii)
Ifθi≥1, i=1,2, then (H3)-(H4).(iii)
Ifθ1≥1 and 0<θ2<1, then (H2)-(H3).(iv)
Ifθ2≥1 and 0<θ1<1, then (H1)-(H4).
Then system (1.3) has at least four positive periodic solutions.Proof.
Let(3.1)X=Z={z=(x1,x2)T∈C(R,Rn):z(t+ω)=z(t)},
and define
(3.2)‖z‖=max⁡t∈[0,ω]|x1(t)|+max⁡t∈[0,ω]|x2(t)|,z∈XorZ.
Equipped with the above norm ∥·∥,X and Z are Banach spaces. Let
(3.3)N:X⟶Z,Nz(t)=(F1(z(t))F2(z(t))),
where
(3.4)F1(z(t))=r1(t){1-(ex1(t)k1(t))θ1-a12(t)ex2(t)k2(t)}-h1(t)e-x1(t),F2(z(t))=r2(t){1-(ex2(t)k2(t))θ2-a21(t)ex1(t)k1(t)}-h2(t)e-x2(t),
and Lu=z˙=dz(t)/dt. We put Pz=(1/ω)∫0ω‍z(t)dt,z∈X;Qz=(1/ω)∫0ω‍z(t)dt,z∈Z. Thus it follows that KerL=R2,ImL={z∈Z:∫0ω‍z(t)dt=0} is closed in Z,dimKerL=2=codimImL, and P,Q are continuous projectors such that
(3.5)Im⁡P=Ker⁡L,Ker⁡Q=Im⁡L=Im⁡(I-Q).
Hence, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) KP:ImL→KerP∩DomL is given by
(3.6)KP(z)=∫0t‍z(s)ds-1ω∫0ω‍∫0s‍z(s)ds.
Then
(3.7)QNz=(1ω∫0ω‍F1(z(s))ds1ω∫0ω‍F2(z(s))ds),KP(I-Q)N(z)=(∫0t‍F1(z(s))ds-1ω∫0ω‍∫0t‍F1(z(s))dsdt+(12-tω)∫0ω‍F1(z(s))ds∫0t‍F2(z(s))ds-1ω∫0ω‍∫0t‍F2(z(s))dsdt+(12-tω)∫0ω‍F2(z(s))ds).
Obviously,QN and KP(I-Q)N are continuous. It is not difficult to show that KP(I-Q)N(Ω¯) is compact for any open bounded set Ω⊂X by using the Arzela-Ascoli theorem. Moreover, QN(Ω¯) is clearly bounded. Thus, N is L-compact on Ω¯ with any open bounded set Ω⊂X.
Now, we are in the position of searching for an appropriate open, bounded subsetΩ for the application of Lemma 2.1. Corresponding to the operator equation
(3.8)Lx=λNx,λ∈(0,1),
we obtain
(3.9)x1′(t)=λr1(t){1-(ex1(t)k1(t))θ1-a12(t)ex2(t)k2(t)}-λh1(t)e-x1(t),x2′(t)=λr2(t){1-(ex2(t)k2(t))θ2-a21(t)ex1(t)k1(t)}-λh2(t)e-x2(t).
Assume that x∈𝕏 is a solution of system (3.9) for some λ∈(0,1). Let ξi, ηi∈[0,ω] such that
(3.10)xi(ξi)=max⁡t∈[0,ω]xi(t),xi(ηi)=min⁡t∈[0,ω]xi(t),i=1,2.
It is clear that xi′(ξi)=0 and xi′(ηi)=0, i=1,2. From this and system (3.9), we have
(3.11)r1(ξ1){1-(ex1(ξ1)k1(ξ1))θ1-a12(ξ1)ex2(ξ1)k2(ξ1)}-h1(ξ1)e-x1(ξ1)=0,r2(ξ2){1-(ex2(ξ2)k2(ξ2))θ2-a21(ξ2)ex1(ξ2)k1(ξ2)}-h2(ξ2)e-x2(ξ2)=0,(3.12)r1(η1){1-(ex1(η1)k1(η1))θ1-a12(η1)ex2(η1)k2(η1)}-h1(η1)e-x1(η1)=0,r2(η2){1-(ex2(η2)k2(η2))θ2-a21(η2)ex1(η2)k1(η2)}-h2(η2)e-x2(η2)=0.
The first equation of (3.11) implies
(3.13)1-(ex1(ξ1)k1(ξ1))θ1=h1ξ1r1(ξ1)e-x1(ξ1)+a12(ξ1)ex2(ξ1)k2(ξ1)>0;
that is,
(3.14)x1(ξ1)<ln⁡k1L.
Similarly from the second equation of (3.11), we have
(3.15)x2(ξ2)<ln⁡k2L.
And, the first equation of (3.12) implies
(3.16)h1(η1)r1(η1)e-x1(η1)=1-(ex1(η1)k1(η1))θ1-a12(η1)ex2(η1)k2(η1)<1;
that is,
(3.17)x1(η1)>ln⁡(h1r1)l.
Similarly from the second equation of (3.12), we have
(3.18)x2(η2)>ln⁡(h2r2)l.
Case i (for 0<θi<1,i=1,2). In view of (3.15), we have
(3.19)ex2(ξ1)<ex2(ξ2)<k2L.
Then, the first equation of (3.11) can be reduced to
(3.20)(ex1(ξ1)k1(ξ1))θ1+a12(ξ1)k2Lk2(ξ1)+h1(ξ1)r1(ξ1)e-x1(ξ1)-1>0.
Multiplying inequality (3.20) by ex1(ξ1) gives
(3.21)ex1(ξ1)(θ1-1)[k1(ξ1)]θ1e2x1(ξ1)-[1-a12(ξ1)k2Lk2(ξ1)]ex1(ξ1)+h1(ξ1)r1(ξ1)>0.
Because
(3.22)ex1(ξ1)(θ1-1)=1ex1(ξ1)(1-θ1)<1[(h1/r1)l]1-θ1,
then we have
(3.23)1(k1θ1)l[(h1/r1)l]1-θ1e2x1(ξ1)-(1-a12k2Lk2)lex1(ξ1)+(h1r1)L>0,
which implies
(3.24)x1(ξ1)>ln⁡u+orx1(ξ1)<ln⁡u-.
Similarly from the first equation of (3.12), we have
(3.25)x1(η1)>ln⁡u+orx1(η1)<ln⁡u-.
From the second equations of (3.11) and (3.12), by a parallel argument to (3.24) and (3.25), we obtain
(3.26)x2(ξ2)>ln⁡l+orx2(ξ2)<ln⁡l-,x2(η2)>ln⁡l+orx2(η2)<ln⁡l-.
From (3.14), (3.17), (3.24), and (3.25), we obtain that, for all t∈ℝ,
(3.27)ln⁡(h1r1)l<x1(t)<ln⁡u-orln⁡u+<x1(t)<ln⁡k1L.
Similarly, from (3.15), (3.18), and (3.26), we obtain that, for all t∈ℝ,
(3.28)ln⁡(h2r2)l<x2(t)<ln⁡l-orln⁡l+<x2(t)<ln⁡k2L.
Obviously, lnu±,lnl±,lnk1L,lnk2L,ln(h1/r1)l, and ln(h2/r2)l are independent of λ. Now let
(3.29)Ω1={x=(x1,x2)T∈X:ln⁡(h1r1)l<x1(t)<ln⁡u-,ln⁡(h2r2)l<x2(t)<ln⁡l-},Ω2={x=(x1,x2)T∈X:ln⁡(h1r1)l<x1(t)<ln⁡u-,ln⁡l+<x2(t)<ln⁡k2L},Ω3={x=(x1,x2)T∈X:ln⁡u+<x1(t)<ln⁡k1L,ln⁡(h2r2)l<x2(t)<ln⁡l-},Ω4={x=(x1,x2)T∈X:ln⁡u+<x1(t)<ln⁡k1L,ln⁡l+<x2(t)<ln⁡k2L}.
Then Ωi,i=1,2,3,4, are bounded open subsets of 𝕏,Ωi∩Ωj=∅,i≠j,i,j=1,2,3,4. Thus Ωi,i=1,2,3,4, satisfy the requirement (a) in Lemma 2.1.
Case ii (for θi≥1, i=1,2). From (3.20), it follows that
(3.30)ex1(ξ1)(θ1-1)[k1(ξ1)]θ1e2x1(ξ1)-[1-a12(ξ1)k2Lk2(ξ1)]ex1(ξ1)+h1(ξ1)r1(ξ1)>0.
Because
(3.31)ex1(ξ1)(θ1-1)<(k1L)θ1-1,
then we have
(3.32)(k1L)θ1-1(k1l)θ1e2x1(ξ1)-(1-a12k2Lk2)lex1(ξ1)+(h1r1)L>0,
which implies
(3.33)x1(ξ1)>ln⁡u¯+orx1(ξ1)<ln⁡u¯-.
Similarly from the first equation of (3.12), we have
(3.34)x1(η1)>ln⁡u¯+orx1(η1)<ln⁡u¯-.
From the second equations of (3.11) and (3.12), by a parallel argument to (3.33) and (3.34), we obtain
(3.35)x2(ξ2)>ln⁡l¯+orx2(ξ2)<ln⁡l¯-,x2(η2)>ln⁡l¯+orx2(η2)<ln⁡l¯-.
From (3.14), (3.17), (3.33), and (3.34), we obtain that, for all t∈ℝ,
(3.36)ln⁡(h1r1)l<x1(t)<ln⁡u¯-orln⁡u¯+<x1(t)<ln⁡k1L.
Similarly, from (3.15), (3.18), and (3.35) we obtain that, for all t∈ℝ,
(3.37)ln⁡(h2r2)l<x2(t)<ln⁡l¯-orln⁡l¯+<x2(t)<ln⁡k2L.
Obviously, lnu¯±,lnl¯±,lnk1L,lnk2L,ln(h1/r1)l, and ln(h2/r2)l are independent of λ. Now let
(3.38)Ω1={x=(x1,x2)T∈X:ln⁡(h1r1)l<x1(t)<ln⁡u¯-,ln⁡(h2r2)l<x2(t)<ln⁡l¯-},Ω2={x=(x1,x2)T∈X:ln⁡(h1r1)l<x1(t)<ln⁡u¯-,ln⁡l¯+<x2(t)<ln⁡k2L},Ω3={x=(x1,x2)T∈X:ln⁡u¯+<x1(t)<ln⁡k1L,ln⁡(h2r2)l<x2(t)<ln⁡l¯-},Ω4={x=(x1,x2)T∈X:ln⁡u¯+<x1(t)<ln⁡k1L,ln⁡l¯+<x2(t)<ln⁡k2L}.
Then Ωi,i=1,2,3,4, are bounded open subsets of 𝕏,Ωi∩Ωj=∅,i≠j,i,j=1,2,3,4. Thus Ωi,i=1,2,3,4, satisfy the requirement (a) in Lemma 2.1.
Case iii (for θ1≥1 and 0<θ2<1). From Case i and Case ii, we can easily obtain that
(3.39)Ω1={x=(x1,x2)T∈X:ln⁡(h1r1)l<x1(t)<ln⁡u¯-,ln⁡(h2r2)l<x2(t)<ln⁡l-},Ω2={x=(x1,x2)T∈X:ln⁡(h1r1)l<x1(t)<ln⁡u¯-,ln⁡l+<x2(t)<ln⁡k2L},Ω3={x=(x1,x2)T∈X:ln⁡u¯+<x1(t)<ln⁡k1L,ln⁡(h2r2)l<x2(t)<ln⁡l-},Ω4={x=(x1,x2)T∈X:ln⁡u¯+<x1(t)<ln⁡k1L,ln⁡l+<x2(t)<ln⁡k2L}.
Then Ωi,i=1,2,3,4, are bounded open subsets of 𝕏,Ωi∩Ωj=∅,i≠j,i,j=1,2,3,4. Thus Ωi,i=1,2,3,4, satisfy the requirement (a) in Lemma 2.1.
Case iv (for θ2≥1 and 0<θ1<1). From Case i and Case ii, we can easily obtain that
(3.40)Ω1={x=(x1,x2)T∈X:ln⁡(h1r1)l<x1(t)<ln⁡u-,ln⁡(h2r2)l<x2(t)<ln⁡l¯-},Ω2={x=(x1,x2)T∈X:ln⁡(h1r1)l<x1(t)<ln⁡u-,ln⁡l¯+<x2(t)<ln⁡k2L},Ω3={x=(x1,x2)T∈X:ln⁡u+<x1(t)<ln⁡k1L,ln⁡(h2r2)l<x2(t)<ln⁡l¯-},Ω4={x=(x1,x2)T∈X:ln⁡u+<x1(t)<ln⁡k1L,ln⁡l¯+<x2(t)<ln⁡k2L}.
Then Ωi,i=1,2,3,4, are bounded open subsets of 𝕏,Ωi∩Ωj=∅,i≠j,i,j=1,2,3,4. Thus Ωi,i=1,2,3,4, satisfy the requirement (a) in Lemma 2.1.
Now, we prove that(b) of Lemma 2.1 holds; that is, we prove that, when x∈∂Ωi∩KerL=∂Ωi∩ℝ2, QNu≠(0,0)T(i=1,2,3,4). If it is not true, then, when x=(x1,x2)T∈∂Ωi∩ℝ2, i=1,2,3,4, x=(x1,x2)T is constant vector satisfying(3.41)QNx=m((Nx1,Nx2)T)=m(r1(t){1-(ex1k1(t))θ1-a12(t)ex2k2(t)}-h1(t)e-x1r2(t){1-(ex2k2(t))θ2-a21(t)ex1k1}-h2(t)e-x2)=(0,0)T.
Thus there exists a point t0 such that
(3.42)(r1(t0){1-(ex1k1(t0))θ1-a12(t0)ex2k2(t0)}-h1(t0)e-x1r2(t0){1-(ex2k2(t0))θ2-a21(t0)ex1k1(t0)}-h2(t0)e-x2)=(0,0)T.
For0<θi<1,i=1,2, following the arguments of (3.11)–(3.26), we obtain
(3.43)ln⁡(h1r1)l<x1(t)<ln⁡u-orln⁡u+<x1(t)<ln⁡k1L,ln⁡(h2r2)l<x2(t)<ln⁡l-orln⁡l+<x2(t)<ln⁡k2L.
Hence x∈Ω1∩ℝ2 or x∈Ω2∩ℝ2 or x∈Ω3∩ℝ2 or x∈Ω4∩ℝ2. This contradicts the fact that x=(x1,x2)T∈∂Ωi∩ℝ2(i=1,2,3,4). This proves that (b) in Lemma 2.1 holds. By a similar argument as above process, for θi≥1(i=1,2) or θ1≥1 and 0<θ2<1 or θ2≥1 and 0<θ1<1, we can easily prove that (b) in Lemma 2.1 holds.
A direct computation gives(3.44)deg⁡{JQN,Ωi∩ker⁡L,0}=-1or1≠0,i=1,2,3,4.
Here J is taken as the identity mapping. So far we have proved that Ωi(i=1,2,3,4) satisfies all the conditions in Lemma 2.1. Hence system (2.3) has at least four periodic solutions in Ω¯i(i=1,2,3,4); that is, system (1.3) has at least four positive periodic solutions. This completes the proof.An Example
In system (1.2), let
(3.45)r1(t)=k1(t)=sin⁡t+2,r2(t)=k2(t)=cos⁡t+2,h1(t)=110000(sin⁡t+2),h2(t)=110000(cos⁡t+2),a12(t)=1100(cos⁡t+2),a21(t)=1100(sin⁡t+2).(i)
Ifθ1=θ2=1/2, then system (1.2) has at least four positive 2π-periodic solutions.(ii)
Ifθ1=θ2=1, then system (1.2) has at least four positive 2π-periodic solutions.(iii)
Ifθ1=1 and θ2=1/2 or θ2=1 and θ1=1/2, then system (1.2) has at least four positive 2π-periodic solutions.Proof.
By calculation, forθ1=θ2=1/2,
(3.46)(1-a12k2Lk2)l=299300>23100=2(h1/r1)L(k1θ1)l[(h1/r1)l]1-θ1,(1-a21k1Lk1)l=299300>23100=2(h2/r2)L(k2θ2)l[(h2/r2)l]1-θ2.
It is obvious that (H1)-(H2) hold. By Theorem 3.1 and system (1.2) have at least four positive almost periodic solutions. For θ1=θ2=1,
(3.47)(1-a12k2Lk2)l=299300>2100=2(h1r1)L(k1L)θ1-1(k1l)θ1,(3.48)(1-a21k1Lk1)l=299300>2100=2(h2r2)L(k2L)θ2-1(k2l)θ2.
It is obvious that (H3)-(H4) hold. By Theorem 3.1, system (1.2) has at least four positive 2π-periodic solutions. For θ1=1 and θ2=1/2 or θ2=1 and θ1=1/2, from (i) and (ii), the result follows from Theorem 3.1. This completes the proof.



---

*Source: 427264-2012-06-28.xml*