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Journal of Mathematical Analysis andApplications
journal homepage: www.elsevier.com/locate/jmaa
The vector–host epidemic model with multiple strains in apatchy environment
Zhipeng Qiu a, Qingkai Kong a, Xuezhi Li b,∗, Maia Martcheva c
a Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, PR Chinab Department of Mathematics, Xinyang Normal University, Xinyang 464000, PR Chinac Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA
a r t i c l e i n f o
Article history:Received 11 October 2012Available online 26 March 2013Submitted by Jeff Morgan
Spatial heterogeneity plays an important role in the distribution and persistence ofinfectious diseases. In this article, a vector–host epidemic model is proposed to explorethe effect of spatial heterogeneity on the evolution of vector-borne diseases. The model isa Ross–MacDonald type model with multiple competing strains on a number of patchesconnected by host migration. The multi-patch basic reproduction numbers R
j0, j =
1, 2, . . . , l are respectively derived for the model with l strains on n discrete patches.Analytical results show that if Rj
0 < 1, then strain j cannot invade the patchy environmentand dies out. The invasion reproduction numbers R
ji, i, j = 1, 2, i = j are also derived for
themodelwith two strains on n discrete patches. It is shown that the invasion reproductionnumbers R
ji, i, j = 1, 2, i = j provide threshold conditions that determine the competitive
outcomes for the two strains. Under the condition that both invasion reproductionnumbersare larger than one, the coexistence of two competing strains is rigorously proved.However, the two competing strains cannot coexist for the corresponding model withno host migration. This implies that host migration can lead to the coexistence of twocompeting strains and enhancement of pathogen genetic diversity. Global dynamics isdetermined for themodelwith two competing strains on twopatches. The results are basedon the theory of type-K monotone dynamical systems.
Twocharacteristics that aremaindrivers behind thedistribution andpersistence of infectious diseases, are hostmigrationand pathogen variability. Host migration allows the pathogen to invade new areas, and maintains the disease in areaswhere it would disappear if the area were isolated. Pathogen variability allows the pathogen to persist despite buildinghost immunity, wide-spread treatment and vaccination. These two heterogeneities of transmission have been exploredseparately in multiple studies. In this article we study them together to gain insight on the impact of spatial heterogeneityon pathogen genetic diversity.
On the one hand, understanding the transmission mechanisms for diseases with multiple strains or serotypes is criticalfor predicting the persistence and evolution of diseases. Mathematical models have provided a powerful tool to broadenour knowledge into the mechanisms [23,16] that lead to coexistence or competitive exclusion of multiple strains. The
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 13
competitive exclusion principle is a classic result in this field, which states that no two species can indefinitely occupythe same ecological niche [21]. Using a multi-strain ODE model, Bremermann and Thieme [6] proved that the principle ofcompetitive exclusion is valid with the strain with the highest reproduction number persisting, while all remaining strainsare being eliminated. Castillo-Chavez et al. [8,9] formulated a simple two-sex epidemiological model that considers thecompetitive interactions of two strains. They showed that coexistence of two competing strains is not possible except inspecial and unrealistic circumstances.
However, it is a commonphenomenon thatmultiple strains coexist in nature. For instance, dengue fever has four differentserotypes, often coexisting in the same geographical region [23]. The competitive exclusion principle leads to the conclusionthat persistent coexistence may only occur if some heterogeneity in the ecological niche is present. Identifying the factorsthat allowmultiple strains to coexist is an important topic in theoretical biology that has beenoccupying significant attentionin the last 20 years. Recent studies have shown thatmechanisms, such as superinfection [16,20], co-infection [24,25], partialcross-immunity [7], density dependent host mortality [1], different modes of transmission [13], can lead to coexistence ofstrains. In this paper, we will show that another mechanism, spatial heterogeneity, can also generate the coexistence ofmultiple competing strains in the same geographical domain.
On the other hand, spatial heterogeneities are believed to play an important role in the distribution and dynamicsof infectious diseases [22]. Spatial heterogeneity can be incorporated in epidemic models as a continuous characteristic,in the form of epidemic models with diffusion, or as a discrete characteristic where migration of individuals betweendiscrete geographical regions is considered. The discrete geographical regions can be cities, towns, states, countries orother appropriate community divisions. In recent years, several studies have focused on the transmission dynamics ofinfectious diseases in patchy environments by using deterministic meta-population epidemic models [2]. Castillo-Chavezand Yakubu [11] discussed a two-patch SIS epidemicmodel with dispersion governed by discrete equations.Wang and Zhao[36] andWang andMulone [35] proposed an epidemic model with population dispersal to describe the dynamics of diseasespread between two patches and n patches. Wang and Zhao [37] formulated a time-delayed epidemic model to describethe dynamics of disease spread among patches; an age structure is incorporated in order to simulate the phenomenon thatsome diseases only occur in the adult population; sufficient conditions are established for global extinction and uniformpersistence of the disease. Arino and van den Driessche [3] developed a multi-city epidemic model to analyze the spatialspread of infectious diseases. Dhirasakdanon, Thieme and Driessche [14] established sharp persistence results for multi-citymodels. All the previous articles consider directly transmitted diseases.
Vector-borne diseases, such as West Nile virus (WNV) and malaria, have reemerged after being nearly eliminated inthe 1950s and 1960s [18]. Migration patterns of the hosts, birds and humans, is one of the important reasons that cause theworldwide spread of the vector-borne diseases.Wonham et al. [38] have suggested that theWNVmodel should be extendedbiologically to consider bird migration. Rappole et al. [28] have provided some factors supporting the hypothesis that themigrant bird is an introductory host for the spread of WNV. Owen et al. [26] have demonstrated that migrating passerinebirds are potential dispersal vehicles for WNV. These studies show that the importance of migration on the distribution andmaintenance of infectious diseases can hardly be underestimated. Few articles have considered the effect of host migrationamong multiple patches on the dynamics of vector-borne diseases. Auger et al. [4] formulate a Ross–MacDonald model onn patches to describe the transmission dynamics of malaria. In a recent study Cosner et al. [12] consider the impact of bothshort term host movement and long-term host migration on the dynamics of vector-borne diseases. The models in [4,12]discuss only vector–host diseases represented by a single strain. In this paper, based on the model in [4] we formulate theRoss–MacDonald type model with multiple competing strains on n patches. Competitive exclusion of the strains is the onlyoutcome on a single patch. Themain question thatwe address is whether spatial heterogeneity can generate the coexistenceof multiple competing strains in a common heterogeneous geographical area.
The remaining parts of this paper are organized as follows. In the next section we formulate the Ross–MacDonald modelwith multiple competing strains on n patches. In Section 3, we derive the reproduction numbers and investigate the localstability of the model. In Section 4, we consider the threshold dynamics of a two-strain multi-patch version of the model.Section 5 is devoted to the global analysis of the two-strain two-patch version of the model. The paper ends with a briefdiscussion of the results in Section 6.
2. Model description
In this section we formulate a Ross–MacDonald type model to describe the transmission dynamics of a vector bornedisease. Vector and host populations occupy n discrete patches linked by host migration. The model also incorporates lcompeting strains.
To introduce the model let Ni(t) denote the total host population in the i-th patch which is partitioned into l + 1distinct epidemiological subclasses: susceptible and infected with strain j, j = 1, 2, . . . , l. The size of the susceptible hostpopulation on patch i is denoted by Si(t). The size of the infected host population with strain j on patch i is denoted byH j
i (t), j = 1, 2, . . . , l. Let Ti(t) denote the total vector population in the i-th patch. The vector population is also dividedinto susceptible and infected with strain j, j = 1, 2, . . . , l subclasses. The size of the susceptible vector population thatoccupies patch i is denoted byMi(t). The size of the infected with strain j vector population that occupies patch i is denotedby V j
i (t), j = 1, 2, . . . , l. We assume that the system which describes the spread of a vector borne disease with l strains in
14 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
the i-th patch is governed by the following differential equations:
dSi(t)dt
= νiNi − bi
l
j=1
αjV li
SiNi
+
lj=1
γji H
ji − νiSi,
dH ji (t)dt
= biαjVjiSiNi
− γji H
ji − νiH
ji ,
dMi(t)dt
= Λi − bi
l
j=1
βjHji
Mi
Ni− µiMi,
dV ji (t)dt
= biβjMiH j
i
Ni− µiV
ji ,
Ni = Si + Hi
j = 1, 2, . . . , l. (2.1)
Here, bi is the per capita biting rate of vectors on hosts in the i-th patch;αj, βj are the disease transmission probabilities frominfected vectors with strain j to uninfected hosts and from infected hosts with strain j to uninfected vectors, respectively;νi is the birth and the death rate of the hosts; µi is the natural death rate of the vectors; γ j
i is the recovery rate of infectedhosts with strain j in the i-th patch, andΛi is the recruitment rate of the uninfected vectors (by birth) in the i-th patch.
When the n patches are connected, we assume that only hosts can migrate among the patches since vectors are usuallyarthropods who typically move only small distances during their lifetime. Let mji ≥ 0 denote the per capita rate thatsusceptible and infected hosts of patch i leave for patch j, where i = j. Then the dynamics of the hosts and the vectorswith migration is governed by the following model:
dSi(t)dt
= νiNi − bi
l
j=1
αjVji
SiNi
+
lj=1
γji H
ji +
nk=1,k=i
mikSk −
nk=1,k=i
mkiSi − νiSi,
dH ji (t)dt
= biαjVjiSiNi
− γji H
ji +
nk=1,k=i
mikHji −
nk=1,k=i
mkiHji − νiH
ji ,
dMi(t)dt
= Λi − bi
l
j=1
βjHji
Mi
Ni− µiMi,
dV ji (t)dt
= biβjMiH j
i
Ni− µiV
ji ,
Ni = Si + Hi
(2.2)
where i = 1, 2, . . . , n, j = 1, 2, . . . , l.Adding the first l + 1 equations of the system (2.1) gives
dNi(t)dt
= 0,
and it follows that the total population size Ni(t) = N0i is a constant. Similarly, adding the last l+ 1 equations of the system
(2.1) gives
dTi(t)dt
= Λi − µiTi.
The asymptotic equilibrium values for the Ti are Ti(t) →Λiµi
:= Wi as t −→ +∞.By adding the first l + 1 equations of the system (2.2), we have
dNi(t)dt
=
nj=1,j=i
mijNj −
nj=1,j=i
mjiNi.
This system can be rewritten as
dN(t)dt
= MN(t), (2.3)
whereN is the column vector (N1,N2, . . . ,Nn)T and the superscript T denotes transpose. ThemovementmatrixM is defined
byM(i, j) = mij for i = j and
M(i, i) = −
nj=1,j=i
mji.
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 15
We assume that thematrixM is irreducible, that is, the graph of the patches is strongly connected through themovement ofhosts. If that is not the case, it follows from [4] that the system (2.2) can be divided into somedecoupled subsystems. From [4]it follows that any trajectory of the system (2.3) remains in the affine hyperplane orthogonal to the vector (1, 1, . . . , 1)Tand containing the initial condition N(0). In the affine hyperplane the system (2.3) has a positive equilibrium denoted byN = (N1, N2, . . . , Nn)
T . Moreover, the positive equilibrium N is globally asymptotically stable on the affine hyperplane.Noting that the total host and vector populations for systems (2.1) and (2.2) tend to the asymptotic states as t → +∞,
in this paper we always assume that the system (2.1) and the system (2.2) have reached the asymptotic states. Thus system(2.1) is equivalent to the following system
dH ji (t)dt
= biαjVji
N0i −
lj=1
H ji
N0i
− (γji + νi)H
ji ,
dV ji (t)dt
= biβj
Wi −
lj=1
V ji
H j
i
N0i
− µiVji ,
(2.4)
and system (2.2) can be reduced to the system as followsdH j
i (t)dt
= biαjVji
Ni −l
j=1H j
i
Ni− γ
ji H
ji +
nk=1,k=i
mikHjk −
nk=1,k=i
mkiHji − νiH
ji ,
dV ji (t)dt
= biβj
Wi −
lj=1
V ji
H j
i
Ni− µiV
ji ,
(2.5)
where i = 1, 2, . . . , n, j = 1, 2, . . . , l.In the remainder of this article we will analyze the dynamics of the systems (2.4) and (2.5) instead of (2.1) and (2.2),
respectively, and we will further investigate how spatial heterogeneity affects the dynamics and outbreaks of the vectorborne diseases with multiple strains on multiple patches.
3. The reproduction numbers and the local stability
One of the important critical threshold quantities in epidemiological modeling studies is the reproduction number. Epi-demiologically, this quantity is defined as the average number of secondary cases (infections) produced by a typical infectedindividual during the entire period of infection when this infectious individual is introduced into a completely susceptiblepopulation [15]. Mathematically, the reproduction number serves as a threshold quantity that often determines the persis-tence or eradication of the disease [1,5,19]. Generally, if the basic reproduction number is less than one, the disease cannotestablish itself in the population. If the reproduction number is greater than one the disease will be endemic. In this sectionwe derive the reproduction numbers for strain j, j = 1, 2, . . . , l, and thenwe investigate the local stabilities of the boundaryequilibria using these reproduction numbers.
We begin by introducing certain notations that will be used throughout this paper. Let
R2ln+
:= (I1, I2, . . . , I l) : H ji ≥ 0, V j
i ≥ 0, i = 1, 2, . . . , n, j = 1, 2, . . . , l.
We define a subsetΩ of R2ln+
by
Ω =
(I1, I2, . . . , I l) ∈ R2ln
+:
lj=1
H ji ≤ Ni,
lj=1
V ji ≤ Wi, i = 1, 2, . . . , n
,
where
I j = (H j1,H
j2, . . . ,H
jn, V
j1, V
j2, . . . , V
jn). (3.1)
Let ϕ(I1, I2, . . . , I l) denote the solution flow generated by (2.5). It is not difficult to see that the flow is positively invariantinΩ . For two vectors x = (x1, x2, . . . , x2n), z = (z1, z2, . . . , z2n) ∈ R2n we define an order between them as follows:
x ≤ z if xi ≤ zi, i = 1, 2, . . . , 2n.We can easily derive the reproduction number of system (2.4) which gives the isolated reproduction number for strain j
in patch i:
Rji =
b2i αjβjWi
(γji + νi)µiN0
i
. (3.2)
By using a similar argument as in the proof of Theorems 3.2 and 4.2 of [8], we can obtain the following theorem.
16 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
Theorem 3.1. For a given i ∈ 1, 2, . . . , n, the system (2.4) has the following:(1) if Rj
i < 1 for all 1 ≤ j ≤ l, then the disease for all strains will eventually die out, i.e., the disease-free equilibrium of thesystem (2.4) is globally asymptotically stable;(2) if Rj
i > 1 for some 1 ≤ j ≤ l and assume that there exists j∗ ∈ 1, 2, . . . , l such that Rj∗i > Rj
i for all j = 1, 2, . . . , l, j = j∗,then
limt→+∞
H j∗i (t) =
b2i αj∗βj∗
WiN0i
− µi(γj∗i + νi)
N0
i
biβj∗
γ
j∗i + νi + biαj∗
WiN0i
, limt→+∞
V j∗i (t) =
b2i αj∗βj∗
WiN0i
− µi(γj∗i + νi)
N0
i
biαj∗(biβj∗ + µi),
and
limt→+∞
H ji (t) = 0, lim
t→+∞V ji (t) = 0
for all j = 1, 2, . . . , l, j = j∗.
The proof of Theorem 3.1 is provided in Appendix A.Theorem 3.1 implies that if the system (2.5) has no host migration among patches then no more than one strain will
persist in the population of patch i, namely the strain with the largest reproduction number in patch i. All strains which havelower basic reproductive rates die out in patch i. In what follows we will prove that coexistence of two competing strains ina common area is possible if the system incorporates host migration among patches. This suggests that host migration, thatis spatial heterogeneity, is one of the mechanisms which can lead to the coexistence of multiple competing strains.
We now derive the basic reproduction numbers for system (2.5). Let c ∈ 1, 2, . . . , l and
then Γ c is invariant for system (2.5). The system (2.5) in Γ c isdHc
i (t)dt
= biαcV ciNi − Hc
i
Ni− γ c
i Hci +
nk=1,k=i
mikHck −
nk=1,k=i
mkiHci − νiHc
i ,
dV ci (t)dt
= biβc(Wi − V ci )
Hci
Ni− µiV c
i , i = 1, 2, . . . , n.(3.4)
It is clear that Ec0(I
c= 0) is the disease-free equilibrium (DFE) of the subsystem (3.4). Noting that the model has 2n
infected populations, namely Hci and V c
i , i = 1, 2, . . . , n, it follows that, in the notation of [34], the matrices F c and V c forthe new infection terms and the remaining transfer terms respectively, are given by
F c=
0 F c
12F c
21 0
, V c
=
V c11 00 V c
22
,
where
F c12 =
b1αc 0 · · · 00 b2αc · · · 0...
.... . .
...0 0 · · · bnαc
, F c21 =
b1βcW1
N10 · · · 0
0 b2βcW2
N2· · · 0
......
. . ....
0 0 · · · bnβcWn
Nn
,
V c11 =
γ c1 −m12 · · · −m1n
−m21 γ c2 · · · −m2n
......
. . ....
−mn1 −mn2 · · · γ cn
, V c22 =
µ1 0 · · · 00 µ2 · · · 0...
.... . .
...0 0 · · · µn
,and γ c
i = γ ci + νi +
nk=1,k=i mki.
Results in [34] imply that the basic reproduction number of the subsystem (3.4) is given byRc
0 := ρ(F c(V c)−1)
=
ρ
diag
b1βc
W1
N1, b2βc
W2
N2, . . . , bnβc
Wn
Nn
(V c
11)−1diag
b1αc
µ1,b2αc
µ2, . . . ,
bnαc
µn
12
(3.5)
where ρ(M) represents the spectral radius of the matrixM .
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 17
Following Smith [29] one can establish that the subsystem (3.4) is strongly concave. Results in [29] also imply that eitherthe origin of system (2.5) is globally asymptotically stable in Γ c defined in (3.3), or system (2.5) has a unique equilibriumEIc (Ic = Ic > 0, I j = 0, j = c) such that it is globally asymptotically stable in Γ c
\ O, where Ic, I j are defined in (3.1). Thisconclusion is based on the observation that EIc is linearly stable in Γ c , that is
Ac11 =
η1 m12 · · · m1n αcQ H1 0 · · · 0
m21 η2 · · · m2n 0 αcQ H2 · · · 0
......
. . ....
......
. . ....
mn1 mn2 · · · ηn 0 0 · · · αcQ Hn
βcQ V1 0 · · · 0 Q β
1 0 · · · 00 βcQ V
2 · · · 0 0 Q β
2 · · · 0...
.... . .
......
.... . .
...
0 0 · · · βcQ Vn 0 0 · · · Q β
n
, (3.6)
is a stable matrix, where
Q Vi =
bi(Wi − V ci )
Ni, Q H
i =bi(Ni − Hc
i )
Ni, Q β
i = −µi −biβc Hc
i
Ni,
ηi = −biαc V c
i
Ni− γ c
i −
nk=1,k=i
mki − νi, i = 1, 2, . . . , n.
Simple algebraic calculations imply that system (3.4) has an equilibrium if and only if Rc0 > 1. Thus we have the following.
Theorem 3.2. If Rc0 ≤ 1, then the disease-free equilibrium (DFE) E0 of the system (2.5) is globally asymptotically stable in Γ c .
If Rc0 > 1, then system (2.5) has a unique equilibrium EIc (Ic = Ic > 0, I j = 0, j = c) which is globally asymptotically stable in
Γc \ O.
Now we are able to state the main result in this section.
Theorem 3.3. (1) If Rj0 ≤ 1 for all 1 ≤ j ≤ l, then the DFE E0 of the system (2.5) is globally asymptotically stable inΩ .
(2) If there exists c ∈ 1, 2, . . . , l such that Rc0 > 1 and R
j0 ≤ 1 for 1 ≤ j ≤ l, j = c, then the boundary equilibrium
EIc (Ic = Ic > 0, I j = 0, j = c) is globally asymptotically stable inΩ \ (I1, I2, . . . , I l) : Ic = 0.
Proof. For a given j ∈ 1, 2, . . . , l, it follows from the system (2.5) thatdH j
i (t)dt
≤ biαjVji − γjH
ji +
nk=1,k=i
mikHjk −
nk=1,k=i
mkiHji − νiH
ji ,
dV ji (t)dt
≤ biβjWi
NiH j
i − µiVji , i = 1, 2, . . . , n.
(3.7)
Let us consider the following differential equationsdH j
i (t)dt
= biαjVji − γjH
ji +
nk=1,k=i
mikHjk −
nk=1,k=i
mkiHji − νiH
ji ,
dV ji (t)dt
= biβjWi
NiH j
i − µiVji , i = 1, 2, . . . , n.
(3.8)
Since the system (3.8) is a linear system, the global stability of the origin of the system (3.8) is determined by the stabilityof the matrix J j = F j
− V j. If Rj0 ≤ 1, Theorem 2 in [34] implies that the matrix J j is stable. Then we have limt→+∞ H j
i (t) =
0, limt→+∞ V ji (t) = 0 for all 1 ≤ i ≤ n. By the comparison principle it then follows that H j
i (t) → 0, V ji (t) → 0 as t → +∞
for all 1 ≤ i ≤ n.If R
j0 ≤ 1 for all 1 ≤ j ≤ l, then we have limt→+∞ H j
i (t) = 0, limt→+∞ V ji (t) = 0 for all 1 ≤ i ≤ n, 1 ≤ j ≤ l. We
can easily see that E0 is locally asymptotically stable inΩ . This fact implies that the disease-free equilibrium E0 is globallyasymptotically stable inΩ if R
j0 < 1 for all 1 ≤ j ≤ l.
If there exists c ∈ 1, 2, . . . , n such that Rc0 > 1 and R
j0 ≤ 1 for 1 ≤ j ≤ n, j = c , then we have that limt→+∞ H j
i (t)= 0, limt→+∞ V j
i (t) = 0 for all 1 ≤ i ≤ n, 1 ≤ j ≤ l, j = c . Furthermore, if (I1(0), I2(0), . . . , I l(0)) ∈ Ω \ (I1, I2, . . . , I l) :
Ic = 0, by using the comparison principle, we can easily prove that there exists ς > 0 such that
Hci (t) > ς, V c
i (t) > ς (3.9)
18 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
for t sufficiently large and all 1 ≤ i ≤ n. In this case the limiting system of system (2.5) is subsystem (3.4). Since Rc0 > 1,
Theorem3.2 implies that EIc (Ic > 0, I j = 0, j = c) is globally asymptotically stable inΓ c\O. Denote the flows generated by
system (2.5) byΨ (t, X). Since for any X ∈ Ω \(I1, I2, . . . , I l) : Ic = 0, the orbit Ψ (t, X) : t > 0 is precompact,ωΨ (X)thelimit set of X , exists. Let ωP
Ψ be the projection of ωΨ (X) onto Γ c . Then (3.9) implies that ωPΨ ∈ Γ c
\ 0. By Theorem 2.3in [10] we can conclude that the equilibrium EIc (Ic > 0, I j = 0, j = c) is a global attractor inΩ \ (I1, I2, . . . , I l) : Ic = 0.This completes the proof of Theorem 3.3.
4. Coexistence of two strains on n patches
In this section we consider the case of two strains on n patches. When system (2.5) has only two strains, strain 1 andstrain 2, then it can be rewritten as
dH1i (t)dt
= biα1V 1iNi − H1
i − H2i
Ni− γ 1
i H1i +
nk=1,k=i
mikH1k −
nk=1,k=i
mkiH1i − νiH1
i ,
dV 1i (t)dt
= biβ1(Wi − V 1i − V 2
i )H1
i
Ni− µiV 1
i ,
dH2i (t)dt
= biα2V 2iNi − H1
i − H2i
Ni− γ 2
i H2i +
nk=1,k=i
mikH2k −
nk=1,k=i
mkiH2i − νiH2
i ,
dV 2i (t)dt
= biβ2(Wi − V 1i − V 2
i )H2
i
Ni− µiV 2
i ,
i = 1, 2, . . . , n. (4.1)
In the case when at least one of the reproduction numbers is smaller than one, that is, either R10 ≤ 1 or R2
0 ≤ 1,Theorem 3.3 gives the global behavior of system (4.1). Therefore, we only need to consider the case when both R1
0 > 1and R2
0 > 1. When R10 > 1,R2
0 > 1, the system (4.1) has the disease-free equilibrium E0(0, 0), which is unstable, as wellas the two boundary equilibria EI1(I
1, 0), EI2(0, I2), where I j = (H j
1, Hj2, . . . , H
jn, V
j1, V
j2, . . . , V
jn), j = 1, 2. In what follows
we investigate the local stability of the boundary equilibria EI1(I1, 0), EI2(0, I
2). To this effect we define two importantquantities R2
1 ,R12 as follows
R21 = (ρ(M 2
1 ))12 , R1
2 = (ρ(M 12 ))
12 ,
where
M 21 =
diag
b1β2
W1 − V 11
N1, b2β2
W2 − V 12
N2, . . . , bnβ2
Wn − V 1n
Nn
× (V 2
11)−1diag
b1α2
N1 − H11
N1µ1, b2α2
N2 − H12
N2µ2, . . . , bnα2
Nn − H1n
Nnµn
;
M 12 =
diag
b1β1
W1 − V 21
N1, b2β1
W2 − V 22
N2, . . . , bnβ1
Wn − V 2n
Nn
× (V 1
11)−1diag
b1α1
N1 − H21
N1µ1, b2α1
N2 − H22
N2µ2, . . . , bnα1
Nn − H2n
Nnµn
and Vj11, j = 1, 2 are defined in Section 3.
Theorem 4.1. (1) If R21 > 1 (R2
1 < 1) the boundary equilibrium EI1(I1, 0) is unstable (locally stable).
(2) If R12 > 1 (R1
2 < 1) the boundary equilibrium EI2(0, I2) is unstable (locally stable).
Proof. We only prove the first point above, since the second point can be proved in a similar way.The Jacobian matrix J(EI1(I
1, 0)) at EI1(I1, 0) takes the form
J(EI1(I1, 0)) =
A111 A1
12
0 A122
,
where A111 has the same form as (3.6) and
A122 = F 1
22 − V 122,
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 19
The stability of the boundary equilibrium EI1(I1, 0) is determined by the stability of the matrices A1
11 and A122. It follows
from (3.6) that the matrix A111 is stable. We only need to investigate the stability of the matrix A1
22.Since V 1
22 has the Z pattern and it is a strictly row diagonally dominant matrix, we conclude that V 122 is a non-singular
M-matrix. It is easy to see that F 122 is non-negative, then −A1
22 = V 122 − F 1
22 has the Z pattern [34]. Thus,
s(A122) < 0 ⇐⇒ −A1
22 is a non-singular M-matrix,
where s(A122) denotes the maximum real part of all the eigenvalues of the matrix A1
22. By results in [34], we have
−A122 is a non-singularM-matrix ⇐⇒ I − F 1
22(V122)
−1 is a non-singularM-matrix.
Since F 122(V
122)
−1 is non-negative, all eigenvalues of F 122(V
122)
−1 have magnitude less than or equal to ρ(F 122(V
122)
−1). So
I − F 122(V
122)
−1 is a non-singular M-matrix ⇐⇒ ρ(F 122(V
122)
−1) < 1
⇐⇒ (ρ(M 21 ))
12 < 1.
Hence, s(A122) < 0 if and only if R2
1 < 1.We conclude that the boundary equilibrium EI1(I1, 0) is locally stable when R21 < 1.
Similarly, we have s(A122) = 0 if and only if R2
1 = 1. In addition, s(A122) > 0 if and only if R2
1 > 1. Thus, if R21 > 1 then
the boundary equilibrium EI1(I1, 0) is unstable. This concludes the proof.
Theorem 4.2. If R12 > 1 and R2
1 > 1, then there exists an ε > 0 such that for every (I1(0), I2(0)) ∈ IntR4n+
the solution(I1(t), I2(t)) of system (4.1) satisfies that
lim inft→+∞
H ji (t) ≥ ε, lim inf
t→+∞V ji (t) ≥ ε
for all i = 1, 2, . . . , n, j = 1, 2. Moreover, the system (4.1) admits at least one (component-wise) positive equilibrium.
Proof. Define
X = (I1, I2) : H ji ≥ 0, V j
i ≥ 0, i = 1, 2, . . . , n, j = 1, 2,
X0= (I1, I2) : H j
i > 0, V ji > 0, i = 1, 2, . . . , n, j = 1, 2,
∂X0= X \ X0.
To prove the theorem, it suffices to show that (4.1) is uniformly persistent with respect to (X0, ∂X0) (see [36]).First, from system (4.1), we get that both X and X0 are positively invariant. Clearly, ∂X0 is relatively closed in X and the
system (4.1) is point dissipative. If (I1(t), I2(t)) are solutions of system (4.1), we define
20 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
We can show that
M∂ = B1 ∪ B2, (4.2)
where B1 = (I1, I2) : I2 ≡ 0 and B2 = (I1, I2) : I1 ≡ 0. Let (I1(0), I2(0)) ∈ M∂ . To show that (4.2) holds, it sufficesto show that I1(t) ≡ 0 or I2(t) ≡ 0 for all t ≥ 0. We establish this result by contradiction. Suppose the result is not true.Then there exists a t1 > 0 such that, without loss of generality, H1
1 (t1) > 0,H21 (t1) > 0 and H j
i (t1) = 0, i = 2, 3, . . . , n, j =
1, 2, V ji (t1) = 0, i = 1, 2, . . . , n, j = 1, 2 (the other cases can be discussed in the same way). Since
dH ji (t)dt
≥ −
γ
ji +
nk=1,k=i
mki + νi
H j
i ,
dV ji (t)dt
≥ −µiVji
for all i = 1, 2, . . . , n, j = 1, 2, we can easily see that if there exists a t0 > 0 such that H ji (t0) > 0 or V j
i (t0) > 0 thenH j
i (t) > 0 or V ji (t) > 0 for all t > t0. Let k ∈ 2, 3, . . . , n. The irreducibility ofM implies that there exists a chain from 1 to
k, i.e., a sequence i1, i2, . . . , is ∈ 1, 2, . . . , n with i1 = 1 and is = k such that mipip+1 > 0 for 1 ≤ p ≤ s − 1. System (4.1)implies that we have
dH1i2(t)
dt
t=t0
> 0.
Then there exists a ti2(>t1) such that H1i2(t) > 0 for all t > ti2 . Similarly, there exists a tip(>tip−1), p = 3, 4, . . . , s such
that H1ip(t) > 0 for all t > tip . Since k is arbitrary, we can conclude that there exists a t1 > 0 such that H1
i (t) > 0 for alli = 1, 2, . . . , n and t > t1. From the second equation of system (4.1) we can easily see that there exists a T 1(>t1) such thatV 1i (t) > 0 for all i = 1, 2, . . . , n and t > T 1. As in the previous proof it is also easy to show that if H2
1 (0) > 0 there existsa T 2 > 0 such that H2
i (t) > 0, V 2i (t) > 0 for all i = 1, 2, . . . , n and t > T 2. Clearly, we have that H j
i (t) > 0, V ji (t) > 0,
i = 1, 2, . . . , n, j = 1, 2 for all t > maxT 1, T 2. This means that (I1(t), I2(t)) ∈ ∂X0 for all t > maxT 1, T 2
, whichcontradicts the assumption (I1(0), I2(0)) ∈ M∂ . The contradiction implies that (4.2) holds.
It is clear that there are three equilibria E0, EI1 and EI2 inM∂ . Since R21 > 1, we can choose δ1 > 0 small enough such that
r21 :=
ρ
diag
b1β2
W1 − V 11 − 2δ1
N1, b2β2
W2 − V 12 − 2δ1
N2, . . . , bnβ2
Wn − V 1n − 2δ1
Nn
× (V 211)
−1diagb1α2
N1 − H11 − 2δ1
N1µ1, b2α2
N2 − H12 − 2δ1
N2µ2, . . . , bnα2
Nn − H1n − 2δ1
Nnµn
12
> 1.
Let us consider the arbitrary positive solution (I1(t), I2(t)) of system (4.1). Now we can claim that
lim supt→+∞
maxi
H2i (t), V
2i (t) > δ1.
Suppose, for the sake of contradiction, that there is a T1 > 0 such thatH2i (t) < δ1, V 2
i (t) < δ1, i = 1, 2, . . . , n, for all t > T1.Note that
dH1i (t)dt
≤ biα1V 1iNi − H1
i
Ni− γ 1
i H1i +
nk=1,k=i
mikH1k −
nk=1,k=i
mkiH1i − νiH1
i ,
dV 1i (t)dt
≤ biβ1(Wi − V 1i )
H1i
Ni− µiV 1
i , i = 1, 2, . . . , n.
Since the equilibrium EI1(I1
= I1) of system (3.4), where c = 1, is globally asymptotically stable, by the comparison princi-ple there is a T2 > 0 such that H1
i (t) < H1i + δ1, V 1
i (t) < V 1i + δ1, i = 1, 2, . . . , n, for all t > T2. Then, for t > maxT1, T2,
we have
H1i (t) < H1
i + δ1, V 1i (t) < V 1
i + δ1, 0 < H2i (t) < δ1, 0 < V 1
i (t) < δ1
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 21
for i = 1, 2, . . . , n. Hence, from the third and fourth equations of system (4.1) we havedH2
i (t)dt
≥ biα2V 2iNi − (H1
i + 2δ1)Ni
−
γ 2i + νi +
nk=1,k=i
mki
H2
i +
nk=1,k=i
mikH2k ,
dV 2i (t)dt
≥ biβ2(Wi − (V 1i + 2δ1))
H2i
Ni− µiV 2
i , i = 1, 2, . . . , n,
(4.3)
for sufficiently large t . We consider the following auxiliary systemdH2
i (t)dt
= biα2V 2iNi − (H1
i + 2δ1)Ni
−
γ 2i + νi +
nk=1,k=i
mki
H2
i +
nk=1,k=i
mikH2k ,
dV 2i (t)dt
= biβ2(Wi − (V 1i + 2δ1))
H2i
Ni− µiV 2
i , i = 1, 2, . . . , n.
(4.4)
The coefficient matrix J21 of system (4.4) is given by
J21 =
Q γ
1 · · · m1n αc Q H1 0 · · · 0
m21 Q γ
2 · · · m2n 0 αc Q H2 · · · 0
......
. . ....
......
. . ....
mn1 mn2 · · · Q γn 0 0 · · · αc Q H
n
βc Q V1 0 · · · 0 −µ1 0 · · · 0
0 βc Q V2 · · · 0 0 −µ2 · · · 0
......
. . ....
......
. . ....
0 0 · · · βc Q Vn 0 0 · · · −µn
,
where
Q Vi =
bi(Wi − V 1i − 2δ1)
Ni, Q H
i =bi(Ni − H1
i − 2δ1)Ni
,
Q γ
j = −γ ij − µi −
nk=1,k=j
mkj, i = 1, 2, . . . , n; j = 1, 2, ·, n.
A similar discussion as for the matrices J j implies that the matrices J21 are also unstable and their principal eigenvalue iseither positive or has a positive real part λm > 0 when r21 > 1. Using the linear systems theory, we can establish that allpositive solutions of system (4.4) tend to infinity as t → ∞. Then, applying the standard comparison principle, we havethat H2
i (t) → +∞ and V 2i (t) → +∞ as t → +∞ for all i = 1, 2, . . . , n. This is a contradiction with the assumption that
H2i (t) → 0 and V 2
i (t) → 0 as t → +∞ for all i = 1, 2, . . . , n, which leads to a contradiction. The contradiction impliesthat lim supt→+∞ maxiH2
i (t), V2i (t) > δ1. Similarly, since R1
2 > 1, we can choose δ2 small enough such that
lim supt→+∞
maxi
H1i (t), V
1i (t) > δ2.
By Theorem 3.2 EI1 is a global attractor in B1 \ 0 and EI2 is a global attractor in B2 \ 0 for (4.1). By the afore-mentioned arguments, it then follows that the set E0, EI1 , EI2 is an isolated invariant set in X , and W s(E0) ∩ X0
= ∅,W s(EI1) ∩ X0
= ∅, W s(EI2) ∩ X0= ∅. Clearly, the set E0, EI1 , EI2 is acyclic in ∂X0, then Theorem 4.6 in [33] leads to the
conclusion that the system (4.1) is uniformly persistent with respect to (X0, ∂X0). Using Theorem 1.3.7 in [39], as applied tothe solution semiflow of system (4.1), we can infer that the system has a positive equilibrium. This completes the proof ofTheorem 4.2.
Theorem 3.1 implies that if the system (4.1) has no host migration among patches then no more than one strain willpersist in the population of patch i except in special and unrealistic circumstances. However, it follows from Theorem 4.2that the coexistence of two competing strains is possible if the system has host migration among dispersal patches. Thecoexistence occurs on all n patches. This indicates that the host migration, i.e., the spatial heterogeneity, can lead to thecoexistence of multiple competing strains.
22 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
5. Global behavior of two strains on two patches
Global results formulti-strain ormulti-patch systems are rare, so it is of particular interest to further study the asymptoticbehavior of the system (4.1). The high dimension of system (4.1)with general n, however, increases the difficulty in obtaininginformation on the global behavior of the system. To show the main idea and obtain results on the global behavior, whichmight be obscured by the complicated computation for the higher dimensional case, in this sectionwewill focus on the casen = 2. We shall see that the minimum dimension choice for the patchy environment enables us to do some more detailedrigorous analysis.
When n = 2, the model (4.1) becomes
dH1i (t)dt
= biα1V 1iNi − H1
i − H2i
Ni− γ 1
i H1i + mikH1
k − mkiH1i − νiH1
i ,
dV 1i (t)dt
= biβ1(Wi − V 1i − V 2
i )H1
i
Ni− µiV 1
i ,
dH2i (t)dt
= biα2V 2iNi − H1
i − H2i
Ni− γ 2
i H2i + mikH2
k − mkiH2i − νiH2
i ,
dV 2i (t)dt
= biβ2(Wi − V 1i − V 2
i )H2
i
Ni− µiV 2
i , i, k = 1, 2, i = k.
(5.1)
Straight forward computation yields that the basic reproduction number for strain j, j = 1, 2 over the whole domain canbe expressed as
Rj0 =
√22
(Rj
1)2(1 − χ j)+ (Rj
2)2(1 − ζ j)+ [((Rj
1)2(1 − χ j)− (Rj
2)2(1 − ζ j))2 + 4(Rj
1)2(Rj
2)2χ jζ j]
12
and the invasion reproduction number for strain j on patch i can be expressed as
Rji =
√22
(R
j1i)
2(1 − χ j)+ (Rj2i)
2(1 − ζ j)+
((R
j1i)
2(1 − χ j)− (Rj2i)
2(1 − ζ j))2 + 4(Rj1i)
2(Rj2i)
2χ jζ j 1
2,
where
χ j=
m21
γj1+ν1
1 +m21
γj1+ν1
+m12
γj2+ν2
; ζ j=
m12
γj2+ν2
1 +m21
γj1+ν1
+m12
γj2+ν2
;
Rjki =
b2kαjβj(Nk − H i
k)(Wk − V ik)
µk(γjk + νk)(Nk)2
, i, j, k = 1, 2, i = j.
We recall that reproduction numbers of strain j on patch i, Rji, i, j = 1, 2, are defined in (3.2). From the proof of Theorem 4.1,
and s(J j0) is the maximum real part of the eigenvalues of the matrix J j0. Since J j0 is irreducible and has non-negative off-diagonal elements, it follows form Theorem A.5 in [32] that s(J j0) is a simple eigenvalue of J j0 with a positive eigenvector.Furthermore, since the diagonal elements of −J are positive and its off-diagonal elements are non-positive, it follows fromM-matrix theory [17] that
s(J j0) < 0 ⇔
J j01 = −(γj1 + ν1 + m21) < 0,
J j02 = (γj1 + ν1 + m21)(γ
j2 + ν2 + m12)− m12m21 > 0,
J j03 = (γj1 + ν1)(γ
j2 + ν2)µ1
1 +
m21
γj1 + ν1
+m12
γj2 + ν2
((Rj
1)2(1 − χ j)− 1) < 0,
J j04 = (γj1 + ν1)(γ
j2 + ν2)µ1µ2
1 +
m21
γj1 + ν1
+m12
γj2 + ν2
× (1 − (Rj
1)2(1 − χ j)− (Rj
2)2(1 − ζ j)+ (Rj
1)2(Rj
2)2(1 − χ j
− ζ j)) > 0,
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 23
where J j0k, k = 1, 2, 3, 4 are the leading principal minors of J j0 with k rows. Consequently, a simple calculation yields that
Rj0 < 1 ⇔
(bi)2αjβjWi
µiNi(γji + νi + mki)
< 1, i, k = 1, 2, i = k,
(Rj1)
2(1 − χ j)+ (Rj2)
2(1 − ζ j)− (Rj1)
2(Rj2)
2(1 − χ j− ζ j) < 1.
Using a similar approach, we have
Rji < 1 ⇔
(bi)2αjβj(Wi − V i
i )(Ni − H ii )
µi(Ni)2(γji + νi + mki)
< 1, k = 1, 2, k = i,
(Rj1i)
2(1 − χ j)+ (Rj2i)
2(1 − ζ j)− (Rj1i)
2(Rj2i)
2(1 − χ j− ζ j) < 1.
(5.2)
It follows from Theorem 3.3 that the global behavior of system (5.1) is clear in the case when both reproduction numbersare less than one, or at most one reproduction number is greater than one, that is either R1
0 > 1 or R20 > 1. Hence, in
this section we only need to investigate the global dynamics of the model in the case when R10 > 1 and R2
0 > 1. WhenR1
0 > 1 and R20 > 1, the system (5.1) has three boundary equilibria E0, EI1(I
1, 0), EI2(0, I2), where I1 = (H1
1 , H12 , V
11 , V
12 ),
I2 = (H21 , H
22 , V
21 , V
22 ).
We begin by investigating the local dynamics of system (5.1). Any positive equilibrium must satisfy the followingalgebraic equations
Γji (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) := biαjV
jiNi − H1
i − H2i
Ni− (γ
ji + mki + νi)H
ji + mikH
jk = 0,
Θji (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) := biβj(Wi − V 1
i − V 2i )
H ji
Ni− µiV
ji = 0,
(5.3)
where i, j, k = 1, 2, i = k. From the second equation in (5.3), we haveV 1i =
biβ1WiH1i
biβ1H1i + biβ2H2
i + Niµi,
V 2i =
biβ2WiH2i
biβ1H1i + biβ2H2
i + Niµi.
(5.4)
Then, by substituting (5.4) into the first equation in (5.3), we haveFi(H1
1 ,H21 ,H
12 ,H
22 ) = H1
i ϕi(H1i ,H
2i )+ mikH1
k = 0,
Gi(H11 ,H
21 ,H
12 ,H
22 ) = H2
i ψi(H1i ,H
2i )+ mikH2
k = 0, i, k = 1, 2, i = k,(5.5)
where
ϕi(H1i ,H
2i ) =
(bi)2α1β1Wi(Ni − H1i − H2
i )
(biβ1H1i + biβ2H2
i + Niµi)Ni− (γ 1
i + νi + mki);
ψi(H1i ,H
2i ) =
(bi)2α2β2Wi(Ni − H1i − H2
i )
(biβ1H1i + biβ2H2
i + Niµi)Ni− (γ 2
i + νi + mki), i, k = 1, 2, i = k.
From (5.5) it follows that
H11 = −
1m21
H12ϕ2(H1
2 ,H22 ),
H21 = −
1m21
H22ψ2(H1
2 ,H22 ).
(5.6)
Substituting (5.6) into (5.5) yields:F(H1
2 ,H22 ) := ϕ2(H1
2 ,H22 )ϕ1
−
1m21
H12ϕ2(H1
2 ,H22 ),−
1m21
H22ψ2(H1
2 ,H22 )
− m12m21 = 0,
G(H12 ,H
22 ) := ψ2(H1
2 ,H22 )ψ1
−
1m21
H12ϕ2(H1
2 ,H22 ),−
1m21
H22ψ2(H1
2 ,H22 )
− m12m21 = 0.
(5.7)
24 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
Clearly, F(H12 , 0) = 0,G(0, H2
2 ) = 0. Eqs. (5.7) give a 2 × 2 system which only depends on H12 ,H
22 . From (5.4) and (5.6)
we can easily see that the system (5.1) has a positive equilibrium E#(H1#1 ,H
1#2 , V
1#1 , V
1#2 ,H
2#1 ,H
2#2 , V
2#1 , V
2#2 ) if and only if
(5.7) has a positive solution (H1#2 ,H
2#2 ) satisfying ϕ2(H1#
2 ,H2#2 ) < 0, ψ2(H1#
2 ,H2#2 ) < 0, i.e., (γ 1
2 + ν2 + m12)N2
b2α1W2+ 1
H1#
2 +
(γ 12 + ν2 + m12)N2β2
b2α1β1W2+ 1
H2#
2 > N2
1 −
(γ 12 + ν2 + m12)µ2N2
(b2)2α1β1W2
,
(γ 12 + ν2 + m12)β1N2
b2α2β2W2+ 1
H1#
2 +
(γ 12 + ν2 + m12)N2
b2α2W2+ 1
H2#
2 > N2
1 −
(γ 12 + ν2 + m12)µ2N2
(b2)2α2β2W2
.
After extensive algebraic calculations, we can verify that
∂F∂H1
2
(H1#
2 ,H2#2 )
> 0,∂F∂H2
2
(H1#
2 ,H2#2 )
> 0;∂G∂H1
2
(H1#
2 ,H2#2 )
> 0,∂G∂H2
2
(H1#
2 ,H2#2 )
> 0.
To obtain results on the local stability of the endemic equilibrium, we assume the following non-degeneracy assumption (H)
(H)FH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
·
GH12(H1#
2 ,H2#2 )
GH22(H1#
2 ,H2#2 )
= 1.
Then we have the following result.
Theorem 5.1. Let E#(H1#1 ,H
1#2 , V
1#1 , V
1#2 ,H
2#1 ,H
2#2 , V
2#1 , V
2#2 ) be a positive equilibrium of system (5.1) and let (H) hold.
Equilibrium E# is locally stable ifFH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
·
GH12(H1#
2 ,H2#2 )
GH22(H1#
2 ,H2#2 )
< 1, and it is unstable ifFH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
·
GH12(H1#
2 ,H2#2 )
GH22(H1#
2 ,H2#2 )
> 1.
Proof. The Jacobian matrix J(E#) at E# takes the form
J(E#) =
A11 A12A21 A22
,
where
A11 =
−
b1α1
V 1#1
N1+ γ 1
1 + ν1 + m21
m12 b1α1
N1 − H1#1 − H2#
1
N10
m21 r −
b2α1
V 1#2
N2+ γ 1
2 + m12 + ν2
0 b2α1
N2 − H1#2 − H2#
2
N2
b1β1W1 − V 1#
1 − V 2#1
N10 −
µ1 + b1β1
H1#1
N1
0
0 b2β1W2 − V 1#
2 − V 2#2
N20 −
µ2 + b2β1
H1#2
N2
;
A12 =
−b1α1V 1#1
N10 0 0
0 −b2α1V 1#2
N20 0
0 0 −b1β1H1#
1
N10
0 0 0 −b2β1H1#
2
N2
;
A21 =
−b1α2V 2#1
N10 0 0
0 −b2α2V 2#2
N20 0
0 0 −b1β2H2#
1
N10
0 0 0 −b2β2H2#
2
N2
;
A22 =
r −
b1α2
V 2#1
N1+ γ 2
1 + ν1 + m21
m12 b1α2
N1 − H1#1 − H2#
1
N10
m21 r −
b2α2
V 2#2
N2+ γ 2
2 + m12 + ν2
0 b2α2
N2 − H1#2 − H2#
2
N2
b1β2W1 − V 1#
1 − V 2#1
N10 −
µ1 + b1β2
H2#1
N1
0
0 b2β2W2 − V 1#
2 − V 2#2
N20 −
µ2 + b2β2
H2#2
N2
.
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 25
It is easy to see that if all eigenvalues of J have negative real parts then so do those of J(E#). Note that all off-diagonal elementsof J are non-negative. LetWi, i = 1, 2, . . . , 8 be the leading principal minors of J with i rows. Then straight forward algebraiccalculations give
(−1)1W1 = γ 12 + ν2 + m12 + b2α1
V 1#2
N2> 0;
(−1)2W2 =
b2α1
V 1#2
N2+ γ 1
2 + ν2 + m12
µ1 + b1β1
H1#1
N1
> 0;
(−1)3W3 =
µ1 + b1β1
H1#1
N1
µ2b2α1
V 1#2
N2+ b2β1
H1#2
N2
b2α1
V 1#2
N2+ γ 1
2 + m12 + ν2
− µ2ϕ2(H1#
2 ,H2#2 )
> 0;
(−1)4W4 =
µ2 + b2β2
H2#2
N2
µ1 + b1β1
H1#1
N1
µ2b2α1
V 1#2
N2+ b2β1
H1#2
N2
×µ2
µ2 + b2β2H2#2N2
b2α1
V 1#2
N2+ γ 1
2 + m12 + ν2
− µ2ϕ2(H1#
2 ,H2#2 )
> 0;
(−1)5W5 =
µ2 + b2β2
H2#2
N2
µ1 +
µ1
µ1 + b1β2H2#1N1
b1β1H1#
1
N1
µ2b2α1
V 1#2
N2
+ b2β1H1#
2
N2
µ2
µ2 + b2β2H2#2N2
b2α1
V 1#2
N2+ γ 1
2 + m12 + ν2
− µ2ϕ2(H1#
2 ,H2#2 )
> 0.
Furthermore, we apply tricky calculations to conclude that (−1)6W6 > 0 and (−1)7W7 > 0. The proofs for (−1)6W6 > 0and (−1)7W7 > 0 are given in Appendices B and C, respectively. Since (−1)iWi > 0, i = 1, 2, . . . , 7, it follows from thewell-knownM-matrix theory that the stability of thematrix J is determined by the sign of the determinant of J . In particular,if det(J) > 0 then the matrix J is stable, and if det(J) < 0 then the matrix J is unstable. In what follows we prove thatdet(J(E#)) = det(J) > 0 if and only if
FH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
·
GH12(H1#
2 ,H2#2 )
GH22(H1#
2 ,H2#2 )
< 1.
Moreover, we prove that det(J(E#)) = det(J) < 0 if and only if
FH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
·
GH12(H1#
2 ,H2#2 )
GH22(H1#
2 ,H2#2 )
> 1.
We begin by observing that from Eqs. (5.3), we haveΓ 1i (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Θji (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Γ 2i (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0, i, j = 1, 2.
26 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
Furthermore, we verify that
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
22 ,−Θ
21 ,Γ
11 ,−Γ
22 )
∂(H12 , V
11 , V
12 ,−V 2
2 ,−V 21 ,H
11 ,−H2
2 )
(H1#
2 ,V1#1 ,V1#
2 ,V2#1 ,V2#
2 ,H1#1 ,H2#
2 )
= W7 < 0.
The implicit function theorem then implies that there exist continuously differentiable functions H1i (H
21 ), V
ji (H
21 ),H
22 (H
21 ),
i, j = 1, 2 defined on a neighborhood∆ of H2#1 such that
(1) H1i (H
2#1 ) = H1#
i , Vji (H
2#1 ) = V j#
i ,H22 (H
2#1 ) = H2#
2 , i, j = 1, 2;(2) for H2
1 ∈ ∆, the functions H1i (H
21 ), V
ji (H
21 ),H
22 (H
21 ), i, j = 1, 2 satisfy the equations
Γ 12 (H
11 ,H
12 (H
11 ,H
21 ,H
22 ), V
11 (H
11 ,H
21 ,H
22 ), V
12 (H
11 ,H
21 ,H
22 ),
H21 ,H
22 , V
21 (H
11 ,H
21 ,H
22 ), V
22 (H
11 ,H
21 ,H
22 )) ≡ 0,
Θji (H
11 ,H
12 (H
11 ,H
21 ,H
22 ), V
11 (H
11 ,H
21 ,H
22 ), V
12 (H
11 ,H
21 ,H
22 ),
H21 ,H
22 , V
21 (H
11 ,H
21 ,H
22 ), V
22 (H
11 ,H
21 ,H
22 )) ≡ 0, i, j = 1, 2;
(3) for H21 ∈ ∆, we have
∂V 21 (H
21 )
∂H21
= −
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
22 ,−Θ
21 ,Γ
11 ,−Γ
22 )
∂(H12 ,V
11 ,V
12 ,−V2
2 ,−H21 ,H
11 ,−H2
2 )
det
∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
22 ,−Θ
21 ,Γ
11 ,−Γ
22 )
∂(H12 ,V
11 ,V
12 ,−V2
2 ,−V21 ,H
11 ,−H2
2 )
= −
D85
W7,
−∂H1
1 (H21 )
∂H21
= −
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
22 ,−Θ
21 ,Γ
11 ,−Γ
22 )
∂(H12 ,V
11 ,V
12 ,−V2
2 ,−V21 ,−H2
1 ,−H22 )
det
∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
22 ,−Θ
21 ,Γ
11 ,−Γ
22 )
∂(H12 ,V
11 ,V
12 ,−V2
2 ,−V21 ,H
11 ,−H2
2 )
=
D86
W7,
∂H22 (H
21 )
∂H21
= −
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
22 ,−Θ
21 ,Γ
11 ,−Γ
22 )
∂(H12 ,V
11 ,V
12 ,−V2
2 ,−V21 ,H
11 ,−H2
1 )
det
∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
22 ,−Θ
21 ,Γ
11 ,−Γ
22 )
∂(H12 ,V
11 ,V
12 ,−V2
2 ,−V21 ,H
11 ,−H2
2 )
= −
D87
W7,
where Dij is the complement minor of order 7 obtained by removing the i-th row and the j-th column from the matrix J .On the one hand, substituting the functions H1
i (H21 ), V
ji (H
21 ),H
22 (H
21 ), i, j = 1, 2 for the expression Γ 2
1 (H11 ,H
12 , V
11 , V
12 ,
H21 ,H
22 , V
21 , V
22 ) yields:
Υ (H21 ) := Γ 1
1 (H11 (H
21 ),H
12 (H
21 ), V
11 (H
21 ), V
12 (H
21 ),H
21 ,H
22 (H
21 ), V
21 (H
21 ), V
22 (H
21 )).
By differentiating the function Υ (H21 )with respect to H2
1 , we have
∂Υ (H21 )
∂H21
H21=H2#
1
=
b1α2
N1 − H11 − H2
1
N1
∂V 21
∂H21
− b1α2V 21
N1
∂H11
∂H12
+ m12∂H2
2
∂H21
−
b1α2
V 21
N1+ γ 2
1 + m21 + ν1
H21=H2#
1
=det(J)W7
. (5.8)
On the other hand, in Appendix C we have obtained continuously differentiable functions H11 (H
21 ,H
22 ), H
12 (H
21 ,H
22 ) from
the first equation (5.5). These functions are defined on a neighborhood∆′ of (H2#1 ,H
2#2 ) and satisfy:
H11 (H
2#1 ,H
2#2 ) = H1#
1 , H12 (H
2#1 ,H
2#2 ) = H1#
2 .
By substituting H12 = H1
2 (H21 ,H
22 ) into the second equation in (5.5), we obtain
Υ ′(H21 ,H
22 ) = H2
2ψ2(H12 (H
21 ,H
22 ),H
22 )+ m21H2
1 = 0.
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 27
From Appendix C we can see that
∂Υ ′(H21 ,H
22 )
∂H22
(H2#
1 ,H2#2 )
< 0.
The implicit function theorem then implies that there exists a continuously differentiable function H22 (H
21 ) defined on a
neighborhood ∆ of H2#1 such that
H22 (H
2#1 ) = H2#
2 ,
and
Υ ′(H21 ,H
22 (H
21 )) ≡ 0
for all H21 ∈ ∆. Moreover, from Appendix C we have
∂H22
∂H21
H21=H2#
1
= −
Υ ′
H21(H2#
1 ,H2#2 )
Υ ′
H22(H2#
1 ,H2#2 )
> 0.
Since FH12(H1#
2 ,H2#2 ) > 0 then by the implicit function theorem there exists a continuously differentiable function H1
2 (H22 )
defined on a neighborhood∆′ of H2#2 such that H1
2 (H2#2 ) = H1#
2 and F(H12 (H
22 ),H
22 ) ≡ 0 for H2
2 ∈ ∆′. Moreover, we have
∂H12
∂H22
H22=H2#
2
= −
FH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
< 0.
By substituting (5.6) and H12 = H1
2 (H22 ),H
22 = H2
2 (H21 ) into the second equation in (5.5), we obtain
Υ (H21 ) = −
1m21
H22 (H
21 )G(H
12 (H
22 (H
21 )), H
22 (H
21 )).
Differentiating the function Υ (H21 )with respect to H2
1 , we have
∂Υ (H21 )
∂H21
H21=H2#
1
= −1
m21H2#
2∂H2
2
∂H21
H21=H2#
1
GH22(H2#
1 ,H2#2 )
1 −
GH12(H2#
1 ,H2#2 )
GH22 (H
2#1 ,H2#
2 )
·
FH22(H2#
1 ,H2#2 )
FH21(H2#
1 ,H2#2 )
. (5.9)
Hence, we see that
∂Υ (H21 )
∂H21
H21=H2#
1
> 0 (5.10)
if and only if
FH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
·
GH12(H1#
2 ,H2#2 )
GH22(H1#
2 ,H2#2 )
> 1.
Moreover,
∂Υ (H21 )
∂H21
H21=H2#
1
< 0 (5.11)
if and only if
FH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
·
GH12(H1#
2 ,H2#2 )
GH22(H1#
2 ,H2#2 )
< 1.
From Eqs. (5.8), (5.10) and (5.11) we can easily see that det(J(E#)) = det(J) > 0 if and only if
FH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
·
GH12(H1#
2 ,H2#2 )
GH22(H1#
2 ,H2#2 )
< 1,
28 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
and det(J(E#)) = det(J) < 0 if and only if
FH22(H1#
2 ,H2#2 )
FH12(H1#
2 ,H2#2 )
·
GH12(H1#
2 ,H2#2 )
GH22(H1#
2 ,H2#2 )
> 1.
This completes the proof of Theorem 5.1.
We now proceed to investigate the global stability of the system (5.1). Let
Λ = (H12 ,H
22 ) ∈ R2
+: ϕ2(H1
2 ,H22 ) < 0, ψ2(H1
2 ,H22 ) < 0,
H12 + H2
2 ≤ N2,−H12ϕ2(H1
2 ,H22 )− H2
2ψ2(H12 ,H
22 ) ≤ m21N1
and
Ω = (I1, I2) ∈ R8+
: I j , (H j1,H
j2, V
j1, V
j2),H
1i + H2
i ≤ Ni, V 1i + V 2
i ≤ Wi, i = 1, 2.
The Jacobian matrix of system (5.1) at each point (I1, I2) ∈ Ω has the formA1(I1, I2) −A2(I1, I2)
−A3(I1, I2) A4(I1, I2)
,
where Ai(I1, I2), i = 1, 2, 3, 4 are all 4 × 4 matrices. One can verify that all off-diagonal entries of A1(I1, I2) and A4(I1, I2)are non-negative, and A2(I1, I2) and A3(I1, I2) are non-negative matrices. It follows from Smith [30] that the flowΦt(I1, I2)generated by (5.1) is type-K monotone in the sense that
Φt(I1, I2)≥K Φt(I1, I2) whenever (I1, I2)≥K (I1, I2) and t > 0.
Theorem 4.1.2 in [30] implies that almost all solutions of system (5.1) are convergent to the equilibria, and thus the globaldynamics of the system (5.1) is completely determined by Eqs. (5.7). Since the algebraic equations (5.7) are difficult to solveexplicitly, in what follows we only consider a special case to show the global stability. We further assume the followinghypothesis:(H′) (H1
2 ,H22 ) : F(H1
2 ,H22 ) = 0,H1
2 ≥ 0,H22 ≥ 0 ⊂ Λ and (H1
2 ,H22 ) : G(H1
2 ,H22 ) = 0,H1
2 ≥ 0,H22 ≥ 0 ⊂ Λ.
Straight forward, but tedious algebraic calculations yield that
FH12(H1
2 ,H22 ) > 0, FH2
2(H1
2 ,H22 ) > 0, GH1
2(H1
2 ,H22 ) > 0, GH2
2(H1
2 ,H22 ) > 0.
Using the implicit function theorem, from Eqs. (5.7) we can infer that there exist positive, continuously differentiablefunctions H2
2 = f (H12 ),H
12 ,= g(H2
2 ) defined on the intervals [0, H12 ] and [0, H2
2 ], respectively, such that(1) f (H1
2 ) = 0, g(H22 ) = 0;
(2) for H12 ∈ [0, H1
2 ] and H22 ∈ [0, H2
2 ], H22 = f (H1
2 ) and H12 = g(H2
2 ) satisfy F(H12 , f (H
12 )) ≡ 0 and G(g(H2
2 ),H22 ) ≡ 0;
(3) df (H12 )
dH12
= −
FH12(H1
2 ,I22 )
FH22(H1
2 ,H12 )< 0, dg(H2
2 )
dH22
= −
GH22(H1
2 ,H22 )
GH12(H1
2 ,H22 )< 0.
The last property above says that H22 = f (H1
2 ),H12 = g(H2
2 ) are both monotonically decreasing functions on the intervals[0, H1
2 ] and [0, H22 ] respectively.
Let H (H12 ) = g(f (H1
2 )),H12 ∈ [χ, H1
2 ], where
χ =
f −1(H2
2 ) if f (0) > H22 ;
0 if f (0) ≤ Im2 .
In view of the properties of f and g , H (H12 ) satisfies H ′(H1
2 ) > 0 for H12 ∈ [χ, H1
2 ]. Suppose the equation H (H12 ) = H1
2 has lpositive roots in the interval (χ, H1
2 ), which we label as H121 < H1
22 < · · · < H12l. Since each root gives a positive equilibrium
of system (5.1), then system (5.1) has l positive equilibria. The corresponding positive equilibria E∗
j (I1∗j , I
2∗j ), j = 1, 2, . . . , l
of (5.1) are given by
(I1∗j , I2∗j ) = (H1∗
1j ,H1∗2j , V
1∗1j , V
1∗2j ,H
2∗1j ,H
2∗2j , V
2∗1j , V
2∗2j ), j = 1, . . . , l, (5.12)
where
H2∗2j = f (H2∗
1j ), H1∗1j = −
1m21
H1∗2j ϕ2(H1∗
2j ,H2∗2j ), H2∗
1j = −1
m21H2∗
2j ψ2(H1∗2j ,H
2∗2j ),
V 1∗1j =
b1β1W1H1∗1j
b1β1H1∗1j + b1β2H2∗
1j + N1µ1, V 1∗
2j =b2β1W2H1∗
2j
b2β1H1∗2j + b2β2H2∗
2j + N2µ2,
V 2∗1j =
b1β2W1H2∗1j
b1β2H1∗1j + b1β2H2∗
1j + N1µ1, V 2∗
2j =b2β2W2H2∗
2j
b2β1H1∗2j + b2β2H2∗
2j + N2µ2,
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 29
respectively. Moreover, we have
(0, I2)≤K (I1∗1 , I2∗1 )≤K · · · ≤K (I1∗l , I
2∗l )≤K (I1, 0).
For convenience, let B(EI1), B(EI2), B(E∗
j ), j = 1, 2, . . . , l, denote the basins of attraction of EI1 , EI2 , E∗
j inR8+, and Cl U denote
the closure of U . Note that the flow ϕt(I1, I2) generated by the system (4.1) is type-K strongly monotone. The followingtheorem summarizes the results on existence and stability of the positive equilibria.
Theorem 5.2. Let R10 > 1, R2
0 > 1. Let (H) and (H′) hold.(1) If R2
1 > 1,R12 > 1, then system (5.1) has an odd number l of positive equilibria given by (5.12). The odd indexed positive
equilibria E∗
j , j = 1, 3, . . . , l are asymptotically stable and Cl∪j odd B(E∗
j ) = R8+. The boundary equilibria EI1 , EI2 and the even
indexed positive equilibria E∗
j , j = 2, 4, . . . , l − 1 are unstable. Moreover, if l = 1 then E∗
1 is globally asymptotically stable inR8
+\ (Γ 1
∪ Γ 2).(2) If R2
1 < 1,R12 < 1, then system (5.1) has an odd number l of positive equilibria given by (5.12). The boundary equilibria
EI1 , EI2 and the even indexed positive equilibria E∗
j , j = 2, 4, . . . , l− 1, are asymptotically stable, and Cl(∪j even B(E∗
j )∪ B(EI1)∪B(EI2)) = R8
+. The odd indexed positive equilibria E∗
j , j = 1, 3, . . . , l are unstable. Moreover, if l = 1 then there exists anunordered separatrix S containing E0 and the unique positive equilibrium E∗, and the unordered separatrix S separates the basinsof attraction of the EI1 and EI2 .(3) If R2
1 > 1,R12 < 1, then system (5.1) has an even number l of positive equilibria given by (5.12). The boundary equilibrium
EI2 and the even indexed positive equilibria E∗
j , j = 2, 4, . . . , l, are asymptotically stable, and Cl(∪j even B(E∗
j ) ∪ B(EI2)) = R8+.
The boundary equilibrium EI1 and the odd indexed positive equilibria E∗
j , j = 1, 3, . . . , l − 1 are unstable. Moreover, if l = 0,i.e., system (5.1) has no positive equilibrium, then EI2 is globally asymptotically stable in R8
+\ Γ 1.
(4) If R21 < 1,R1
2 > 1, then the system (5.1) has an even number l of positive equilibria given by (5.12). The boundaryequilibrium EI1 and the odd indexed positive equilibria E∗
j , j = 1, 3, . . . , l − 1, are asymptotically stable, and Cl(∪j odd B(E∗
j ) ∪
B(EI1)) = R8+. The boundary equilibrium EI2 and the even indexed positive equilibria E∗
j , j = 2, 4, . . . , l, are unstable. Moreover,if l = 0, i.e., system (5.1) has no positive equilibrium, then EI1 is globally asymptotically stable in R8
+\ Γ 2.
In order to prove Theorem 5.2, we need to prove the following lemmas.
Lemma 5.3. Let assumption (H′) hold. Then we have
R12 > 1(R1
2 < 1) ⇔ F(0, H22 ) < 0(F(0, H2
2 ) > 0).
Proof. From the expression of ϕ2(H12 ,H
22 ), we have
ϕ2(0, H22 ) =
(b2)2α1β1W2(N2 − H22 )
(b2β2H22 + N2µ2)N2
− (γ 12 + ν2 + m12)
=(b2)2α1β1(W2 − V 2
2 )(N2 − H22 )
µ2(N2)2− (γ 1
2 + ν2 + m12).
Thus,
ϕ2(0, H22 ) < 0 ⇔
(b2)2α1β1(W2 − V 22 )(N2 − H2
2 )
µ2(N2)2(γ12 + ν2 + m12)
< 1,
ϕ2(0, H22 ) > 0 ⇔
(b2)2α1β1(W2 − V 22 )(N2 − H2
2 )
µ2(N2)2(γ12 + ν2 + m12)
> 1.
Substituting H12 = 0,H2
2 = H22 into the expression for F(H1
2 ,H22 ) gives
F(0, H22 ) = ϕ2(0, H2
2 )ϕ1
0,−
1m21
H22ψ2(0, H2
2 )
− m12m21
=
(b2)2α1β1(W2 − V 2
2 )(N2 − H22 )
µ2(N2)2− (γ 1
2 + ν2 + m12)
×
(b1)2α1β1(W1 − V 2
1 )(N1 − H21 )
µ1(N1)2− (γ 1
1 + ν1 + m21)
− m12m21
= [(γ 12 + ν2)(γ
11 + ν1)+ m12(γ
11 + ν1)+ m21(γ
12 + ν2)]
× [1 − (Rj1i)
2(1 − χ j)− (Rj2i)
2(1 − ζ j)+ (Rj1i)
2(Rj2i)
2(1 − χ j− ζ j)].
30 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
Here, we have used the fact H21 = −
1m21
H22ψ2(0, H2
2 ). We can then conclude that
F(0, H22 ) > 0 ⇔ (R
j1i)
2(1 − χ j)+ (Rj2i)
2(1 − ζ j)− (Rj1i)
2(Rj2i)
2(1 − χ j− ζ j) < 1,
F(0, H22 ) < 0 ⇔ (R
j1i)
2(1 − χ j)+ (Rj2i)
2(1 − ζ j)− (Rj1i)
2(Rj2i)
2(1 − χ j− ζ j) > 1.
Assumption (H′) implies that ϕ2(0, H22 ) < 0. Hence, the conclusions follow immediately from (5.2).
Similarly, we can establish the following lemma whose proof is omitted.
Lemma 5.4. Let the assumption (H′) hold. Then we have
R21 > 1(R2
1 < 1) ⇔ G(H12 , 0) < 0(G(H1
2 , 0) > 0).
Now we are able to prove Theorem 5.2.Proof of Theorem 5.2. Here we only prove part (1) of the theorem. The other parts can be established similarly except thelast conclusion in part (2) which is a corollary of Theorem 1 in [31]. We consider the roots of the equation H (H1
2 ) = H12
in the interval [χ, H12 ]. If R2
1 > 1 and R12 > 1 hold, Lemmas 5.3 and 5.4 imply that F(0, H2
2 ) < 0,G(H12 , 0) < 0, i.e.,
f (0) > H22 , g(0) > H1
2 . Hence, we can easily see that χ = f −1(H22 ) > 0, g(f (χ)) = 0, g(f (H1
2 )) > H12 . The non-degeneracy
assumption (H) implies that the number l of the roots of the equationH (H12 ) = H1
2 in the interval [χ, H12 ] is odd. Let the roots
of the equation H (H12 ) = H1
2 in the interval [χ, H12 ] be H1∗
21 < H1∗22 < · · · < H1∗
2l . This means that system (5.1) has exactlyl positive equilibria given by (5.12) and three boundary equilibria E0, EI1 , EI2 . Moreover, the non-degeneracy assumption Halso implies that g ′(f (H1∗
2j ))f′(H1∗
2j ) > 1, i.e.,
FH22(H1∗
2j ,H2∗2j )
FH12(H1∗
2j ,H2∗2j )
·
GH12(H1∗
2j ,H2∗2j )
GH22(H1∗
2j ,H2#2j )
< 1
in the case when j is odd and g ′(f (H1∗2j ))f
′(H1∗2j ) < 1, that is,
FH22(H1∗
2j ,H2∗2j )
FH12(H1∗
2j ,H2∗2j )
·
GH12(H1∗
2j ,H2∗2j )
GH22(H1∗
2j ,H2#2j )
> 1
in the case when j is even. By Theorem 5.1 the odd indexed positive equilibria E∗
j are locally asymptotically stable and theeven indexed positive equilibria E∗
j are unstable. Note that the flowϕt(I1, I2) generated by (5.1) is type-K stronglymonotone.Since EI2 <K E∗
1 <K · · ·<K E∗
l <K EI1 and E∗
1 , E∗
l are both asymptotically stable, it follows from Theorem 2.2.2 in [30] thatEI2 , EI1 are both unstable. By Theorem 4.1.2 in [30] we obtain that almost all solutions of system (5.1) are convergent to theequilibria and Cl∪j odd B(E∗
j ) = R8+. This completes the proof of Theorem 5.2.
6. Discussion
In this article we study the effect of spatial heterogeneity on the transmission dynamics of vector-borne diseases withmultiple strains and on multiple patches. Based on the Ross–MacDonald multi-patch model analyzed by Auger et al. [4], weformulate an extensionmulti-patchmulti-strainmodel. Vector-borne diseases, such asmalaria, often display heterogeneityof transmission in different locations. High transmission areas neighbor low transmission areas and those are connected byhost migration. Vector-borne disease have reemerged as a major public health threat in the last 30–40 years. The reasonsfor the re-emergence are complex but they involve the evolution of the pathogens to more resilient drug-resistant strains,often persisting in different isolated regions [27]. This suggests that studying the evolution of pathogens in a spatial context(on multiple patches) is an important topic of particular interest. We believe our model here is the first one that studies theimpact of spatial heterogeneities on the evolution of pathogens.
We focus on investigating the dynamics of the multi-patch multi-strain Ross–MacDonald type model. We define themulti-patch basic reproduction numbers R
j0 for each strain. Theorem 3.3 shows that if the reproduction number for strain
j is less than one then strain j cannot invade the patchy environment and dies out over the entire domain. Theorem 3.3 alsoimplies that if the multi-patch basic reproduction numbers for all strains are less than one the disease free equilibrium isglobally asymptotically stable and the disease is eliminated from the host and the vector populations.When themulti-patchbasic reproduction numbers for all strains are greater than one, each strain can invade into the population when alone, andthus all strains compete for the same resource, the susceptible individuals.
In order to obtain further theoretical results, we systematically analyze the multi-patch multi-strain model on n discretepatches but restricting the number of strains to two. By analyzing the local stability of the single-strain equilibria, we derivethe invasion reproduction numbers R
ji , i, j = 1, 2, i = j for strain j. Applying the theory of uniform persistence of dynamical
systems the uniform persistence of two competing strains on the entire domain is rigorously proved in Theorem 4.2 underthe condition that both invasion reproduction numbers are larger than one. However, the results of Theorem 3.1 show that ifthe system has no host migration no more than one strain will persist in the population on a single patch, namely the strain
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 31
with the largest reproduction number on that patch. Multiple strains may persist, each on a separate patch, but essentially adivide and conquer strategy is adopted. When the patches are linked through migration, the divide and conquer strategy isnot an option and all strains whose reproduction number is greater than one are competing. However, Theorems 4.2 and 3.1indicate that spatial heterogeneity can lead to the coexistence of multiple competing strains on the entire domain. Hence,spatial heterogeneity supports pathogen genetic diversity. This is the main result of this article.
Finally, we examine the global behavior of the model with two competing strains on two patches. Applying the well-known M-matrix theory and the implicit function theorem a complete classification for the local stabilities of positiveequilibria is given in Theorem 5.1. We determine that the flow Φt(I1, I2) generated by the two-strain two-patch model(5.1) is type-K monotone. By applying the theory of type-K monotone dynamical systems, we provide the global behaviorof the two-strain two-patch model in Theorem 5.2 which is completely determined by the algebraic equations (5.7). Theseresults follow from the conditions on the invasion reproduction numbers as well as the non-degeneracy assumption (H) andassumption (H′).
There are still many interesting and challenging mathematical questions which need to be studied for the system (2.5).For example, we could not present the complete classification for the dynamics of the system (2.5). The main difficultystems from the high dimension of the multi-patch multi-strain model. Additionally, the model discussed here can also beextended to incorporate the other ingredients, such as the different incidences and/or different compartmental structures.It is worth noting that the methods applied to study model (5.1) are not applicable to the other general models because themonotonicity of the model (5.1) plays an essential role in our analysis. We leave these investigations for the future.
Acknowledgments
The authors are very grateful to an anonymous referee for his valuable comments and suggestions which led to animprovement of their originalmanuscript. The research of this paper is supported by the NSF of China grants (Nos 11271190,11271314, and 10911120387), the Zijin Star Project of Excellence Plan of NJUST and Qinglan Project of Jiangsu Province. Theresearch is also in part supported by NSF (USA) grant DMS-0817789.
Appendix A. Proof of Theorem 3.1
For any 1 ≤ p, q ≤ n, we consider the system
dHpi (t)dt
= biαpVpiN0
i − (Hpi + Hq
i )
N0i
− (γpi + νi)H
pi ,
dV pi (t)dt
= biβp(Wi − (V pi + V q
i ))Hp
i
N0i
− µiVpi ,
dHqi (t)dt
= biαqVqiN0i − (Hp
i + Hqi )
N0i
− (γqi + νi)H
qi ,
dV qi (t)dt
= biβq(Wi − (V pi + V q
i ))Hq
i
N0i
− µiVqi .
If we let Rpi and Rq
i be the reproduction numbers for strains p and q, then Rpi = Rp
i and Rqi = Rq
i . Since Rji ≤ 1 for all j (which
implies Rpi < 1 and Rq
i < 1) it follows from Theorem 4.1.2 in [9] that Hpi (t) → 0,Hq
i (t) → 0 and V pi (t) → 0, V q
i (t) → 0,as t → +∞. On the other hand, from the comparison principle, it follows that Hp
i (t) ≤ Hpi (t),H
qi (t) ≤ Hq
i (t) andV pi (t) ≤ V p
i (t), Vqi (t) ≤ V q
i (t) for all t ≥ 0. Thus, the disease-free equilibrium of the system (2.4) is globally asymptoticallystable. This completes the proof of part (1).
(2) For any j = j∗, consider the system
dH j∗i (t)dt
= biαj∗ Vj∗iN0
i − (H j∗i + H j
i )
N0i
− (γj∗i + νi)H
j∗i ,
dV j∗i (t)dt
= biβj∗(Wi − (V j∗i + V j
i ))H j∗
i
N0i
− µiVj∗i ,
dH ji (t)dt
= biαjVjiN0
i − (H j∗i + H j
i )
N0i
− (γji + νi)H
ji ,
dV ji (t)dt
= biβj(Wi − (V j∗i + V j
i ))H j
i
N0i
− µiVji .
Let Rj∗i and Rj
i be the reproduction numbers for strains j∗ and j, then Rj∗i = Rj∗
i and Rji = Rj
i. Hence Rj∗i > Rj
i and it followsfrom Theorem 4.1.2 in [9] that H j
i (t) → 0V ji (t) → 0, as t → +∞. Again, from the comparison principle, it follows that
32 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
H ji (t) ≤ˆH j
i (t) and V ji (t) ≤ V j
i (t) for all t ≥ 0. Thus, limt→+∞ H ji (t) = 0, limt→+∞ V j
i (t) = 0. Since j is arbitrary, we have
limt→+∞
H j∗i (t) =
b2i αj∗βj∗
WiN0i
− µi(γj∗i + νi)
N0
i
biβj∗
γ
j∗i + νi + biαj∗
WiN0i
, limt→+∞
V j∗i (t) =
b2i αj∗βj∗
WiN0i
− µi(γj∗i + νi)
N0
i
biαj∗(biβj∗ + µi),
and
limt→+∞
H ji (t) = 0, lim
t→+∞V ji (t) = 0
for all j = 1, 2, . . . , l, j = j∗. This completes the proof of Theorem 3.1.
Appendix B. Proof of (−1)6W6 > 0
From the equations in (5.3), we have
Γ 12 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Θ11 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Θ12 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Θ21 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Θ22 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0.
It is easy to verify that
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
22 ,−Θ
21 )
∂(H12 , V
11 , V
12 ,−V 2
2 ,−V 21 )
(H1#
2 ,V1#1 ,V1#
2 ,V2#1 ,V2#
2 )
= W5 < 0.
Applying the implicit function theorem we conclude that there exist continuously differentiable functions H12 (H
11 ,H
21 ,H
22 ),
V 11 (H
11 ,H
21 ,H
22 ), V
12 (H
11 ,H
21 ,H
22 ), V
21 (H
11 ,H
21 ,H
22 ), V
22 (H
11 ,H
21 ,H
22 ) defined on a neighborhood∆ of (H1#
1 ,H2#1 ,H
2#2 ) such that
(1) H12 (H
1#1 ,H
2#1 ,H
2#2 ) = H1#
2 , V11 (H
1#1 ,H
2#1 ,H
2#2 ) = V 1#
1 , V12 (H
1#1 ,H
2#1 ,H
2#2 ) = V 1#
2 ,
V 21 (H
1#1 ,H
2#1 ,H
2#2 ) = V 2#
1 , V22 (H
1#1 ,H
2#1 ,H
2#2 ) = V 2#
2 ;
(2) For (H11 ,H
21 ,H
22 ) ∈ ∆, H1
2 (H11 ,H
21 ,H
22 ), V
11 (H
11 ,H
21 ,H
22 ), V
12 (H
11 ,H
21 ,H
22 ), V
21 (H
11 ,H
21 ,H
22 ), V
22 (H
11 ,H
21 ,H
22 ) satisfy the
equationsΓ 12 (H
11 ,H
12 (H
11 ,H
21 ,H
22 ), V
11 (H
11 ,H
21 ,H
22 ), V
12 (H
11 ,H
21 ,H
22 ),
H21 ,H
22 , V
21 (H
11 ,H
21 ,H
22 ), V
22 (H
11 ,H
21 ,H
22 )) ≡ 0,
Θji (H
11 ,H
12 (H
11 ,H
21 ,H
22 ), V
11 (H
11 ,H
21 ,H
22 ), V
12 (H
11 ,H
21 ,H
22 ),
H21 ,H
22 , V
21 (H
11 ,H
21 ,H
22 ), V
22 (H
11 ,H
21 ,H
22 )) ≡ 0, i, j = 1, 2;
(3) For (H11 ,H
21 ,H
22 ) ∈ ∆, we have
∂H12 (H
11 ,H
21 ,H
22 )
∂H11
= −
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
21 ,−Θ
22 )
∂(H11 ,V
11 ,V
12 ,−V2
1 ,−V22 )
det
∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
21 ,−Θ
22 )
∂(H12 ,V
11 ,V
12 ,−V2
1 ,−V22 )
,∂V 1
1 (H11 ,H
21 ,H
22 )
∂H11
= −
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
21 ,−Θ
22 )
∂(H12 ,H
11 ,V
12 ,−V2
1 ,−V22 )
det
∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
21 ,−Θ
22 )
∂(H12 ,V
11 ,V
12 ,−V2
1 ,−V22 )
.On the one hand, substituting the functions H1
2 (H11 ,H
21 ,H
22 ), V
11 (H
11 ,H
21 ,H
22 ), V
12 (H
11 ,H
21 ,H
22 ), V
21 (H
11 ,H
21 ,H
22 ), V
22 (H
11 ,
H21 ,H
22 ) for the expression Γ 1
1 (H11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) yields that
Υ (H11 ,H
21 ,H
22 ) := Γ 1
1 (H11 ,H
12 (H
11 ,H
21 ,H
22 ), V
11 (H
11 ,H
21 ,H
22 ),
V 12 (H
11 ,H
21 ,H
22 ),H
21 ,H
22 , V
21 (H
11 ,H
21 ,H
22 ), V
22 (H
11 ,H
21 ,H
22 )).
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 33
Differentiating the function Υ (H11 ,H
21 ,H
22 )with respect to H1
1 , we have
∂Υ (H11 ,H
21 ,H
22 )
∂H11
(H1#
1 ,H2#1 ,H2#
2 )
=
m21
∂H12
∂H11
+ b1α1N1 − H1
1 − H21
N1
∂V 11
∂H11
− (b1α1V 11 + γ 1
1 + m21 + ν1)
(H1#
1 ,H2#1 ,H2#
2 )
=W6
W5. (B.1)
On the other hand, the first equation in (5.5) implies that we have
∂F2
∂H12
(H1#
1 ,H1#2 ,H2#
1 ,H2#2 )
= ϕ2(H1#2 ,H
2#2 )+ H1#
2∂ϕ2
∂H12
(H1#
2 ,H2#2 )
.
Since ϕ2(H1#2 ,H
2#2 ) < 0 and
∂ϕ2
∂H12
(H1#
2 ,H2#2 )
= −(b2)2α1β1W2[b2β2H2#
2 + µ2N2 + b2β1(N2 − H2#2 )]
(b2β1H1#2 + b2β2H2#
2 + N2µ2)2N2
< 0,
it follows that ∂F2∂H1
2|(H1#
1 ,H1#2 ,H2#
1 ,H2#2 ) < 0. By the implicit function theorem, there exists a continuously differentiable function
H12 (H
11 ,H
21 ,H
22 ) defined on a neighborhood∆′ of (H1#
1 ,H2#1 ,H
2#2 ) such that
H12 (H
1#1 ,H
2#1 ,H
2#2 ) = H1#
2
and
F2(H11 , H
12 (H
11 ,H
21 ,H
22 ),H
21 ,H
22 ) ≡ 0
for (H11 ,H
21 ,H
22 ) ∈ ∆′. Moreover, we have
∂H12
∂H11|(H1#
1 ,H2#1 ,H2#
2 ) = −m21
ϕ2(H1#2 ,H
2#2 )+ H1#
2∂ϕ2∂H1
2|(H1#
1 ,H2#1 ,H2#
2 )
> 0.
Substituting (5.4) and H12 = H1
2 (H11 ,H
21 ,H
22 ) into the expression Γ 1
1 (H11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ), we can obtain
Υ (H11 ,H
21 ,H
22 ) = H1
1ϕ1(H11 ,H
21 )+ m12H1
2 (H11 ,H
21 ,H
22 ).
Differentiating the function Υ (H11 ,H
21 ,H
22 )with respect to H1
1 , we have
∂Υ (H11 ,H
21 ,H
22 )
∂H11
(H1#
1 ,H2#1 ,H2#
2 )
=
ϕ1(H11 ,H
21 )+ H1
1∂ϕ1
∂H11
− m12m21
ϕ2(H12 ,H
22 )+ H1
2∂ϕ2∂H1
2
(H1#
1 ,H1#2 ,H2#
1 ,H2#2 )
=1
ϕ2(H12 ,H
22 )+ H1
2∂ϕ2∂H1
2
ϕ1(H1
1 ,H21 )H
12∂ϕ2
∂H12
+ ϕ2(H12 ,H
22 )H
12∂ϕ1
∂H11
+ H11H
12∂ϕ1
∂H11
∂ϕ2
∂H12
(H1#
1 ,H1#2 ,H2#
1 ,H2#2 )
< 0 (B.2)
since ϕ1(H1#1 ,H
2#1 ) < 0, ϕ2(H1#
2 ,H2#2 ) < 0, ∂ϕ2
∂H12|(H1#
2 ,H2#2 ) < 0 and
ϕ1(H1#1 ,H
2#1 )ϕ2(H1#
2 ,H2#2 )− m12m21 = 0,
∂ϕ1(H11 ,H
21 )
∂H11
H1#1 ,H1#
2
= −(b1)2α1β1W1[b1β1H2
1 + µ1N1 + b1β1(N1 − H21 )]
(b2β1H11 + b1β1H2
1 + N1µ1)2N1< 0.
From (B.1) and (B.2) we can easily see that (−1)6W6 > 0 sinceW5 < 0.
34 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
Appendix C. Proof of (−1)7W7 > 0
From the equations in (5.3), we have
Γ 12 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Θ11 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Θ12 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Θ21 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Θ22 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0,
Γ 11 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) = 0.
It is easy to verify that
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
22 ,−Θ
21 ,Γ
11 )
∂(H12 , V
11 , V
12 ,−V 2
2 ,−V 21 ,H
11 )
(H1#
2 ,V1#1 ,V1#
2 ,V2#1 ,V2#
2 ,H1#1 )
= W6 > 0.
The implicit function theorem then implies that there exist continuously differentiable functions
H12 (H
21 ,H
22 ), V
11 (H
21 ,H
22 ), V
12 (H
21 ,H
22 ), V
21 (H
21 ,H
22 ), V
22 (H
21 ,H
22 ),H
11 (H
21 ,H
22 )
defined on a neighborhood∆ of (H2#1 ,H
2#2 ) such that
(1) H12 (H
2#1 ,H
2#2 ) = H1#
2 , V11 (H
2#1 ,H
2#2 ) = V 1#
1 , V12 (H
2#1 ,H
2#2 ) = V 1#
2 , V21 (H
2#1 ,H
2#2 ) = V 2#
1 , V22 (H
2#1 ,H
2#2 ) = V 2#
2 ,
H11 (H
2#1 ,H
2#2 ) = H1#
1 ;
(2) For (H21 ,H
22 ) ∈ ∆, H1
2 (H21 ,H
22 ), V
11 (H
21 ,H
22 ), V
12 (H
21 ,H
22 ), V
21 (H
21 ,H
22 ), V
22 (H
21 ,H
22 ),H
11 (H
21 ,H
22 ) satisfy the equations
Γ 1i (H
11 (H
21 ,H
22 ),H
12 (H
21 ,H
22 ), V
11 (H
21 ,H
22 ), V
12 (H
21 ,H
22 ),H
21 ,H
22 , V
21 (H
21 ,H
22 ), V
22 (H
21 ,H
22 )) ≡ 0,
Θji (H
11 (H
21 ,H
22 ),H
12 (H
21 ,H
22 ), V
11 (H
21 ,H
22 ), V
12 (H
21 ,H
22 ),H
21 ,H
22 , V
21 (H
21 ,H
22 ), V
22 (H
21 ,H
22 )) ≡ 0,
where i, j = 1, 2;(3) For (H2
1 ,H22 ) ∈ ∆, we have
−∂H1
2 (H21 ,H
22 )
∂H22
= −
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
21 ,−Θ
22 ,Γ
11 )
∂(−H22 ,V
11 ,V
12 ,−V2
1 ,−V22 ,H
11 )
det
∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
21 ,−Θ
22 ,Γ
11 )
∂(H12 ,V
11 ,V
12 ,−V2
1 ,−V22 ,H
11 )
,∂V 2
2 (H21 ,H
22 )
∂H22
= −
det∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
21 ,−Θ
22 ,Γ
11 )
∂(H12 ,V
11 ,V
12 ,−V2
1 ,−H22 ,H
11 )
det
∂(Γ 1
2 ,Θ11 ,Θ
12 ,−Θ
21 ,−Θ
22 ,Γ
11 )
∂(H12 ,V
11 ,V
12 ,−V2
1 ,−V22 ,H
11 )
.On the one hand, substituting the functions H1
2 (H21 ,H
22 ), V
11 (H
21 ,H
22 ), V
12 (H
21 ,H
22 ), V
21 (H
21 ,H
22 ), V
22 (H
21 ,H
22 ),H
11 (H
21 ,H
22 )
for the expression Γ 22 (H
11 ,H
12 , V
11 , V
12 ,H
21 ,H
22 , V
21 , V
22 ) yields that
Υ ′(H21 ,H
22 ) := Γ 2
2 (H11 (H
21 ,H
22 ),H
12 (H
21 ,H
22 ), V
11 (H
21 ,H
22 ), V
12 (H
21 ,H
22 ),H
21 ,H
22 , V
21 (H
21 ,H
22 ), V
22 (H
21 ,H
22 )).
Differentiating the function Υ ′(H21 ,H
22 )with respect to H2
2 , we have
∂Υ ′(H21 ,H
22 )
∂H22
(H2#
1 ,H2#2 )
=
b2α2
V 22 (H
21 ,H
22 )
N2
∂H12
∂H22
+ b2α2N2 − H1
2 (H21 ,H
22 )− H2
2
N2
×∂V 2
2
∂H22
− (b2α2V 22 (H
21 ,H
22 )+ γ 2
2 + m12 + ν2)
(H2#
1 ,H2#2 )
=W7
W6. (C.3)
On the other hand, the first equation in (5.5) leads to
Ξ := det∂(F1,F2)
∂(H11 ,H
12 )
(H1#
1 ,H1#2 ,H2#
1 ,H2#2 )
=
ϕ1(H1#
1 ,H2#1 )+ H1#
1∂ϕ1
∂H11
(H1#
1 ,H2#1 )
m12
m21 ϕ2(H1#2 ,H
2#2 )+ H1#
2∂ϕ2
∂H12
(H1#
2 ,H2#2 )
Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36 35
= ϕ1(H1#1 ,H
2#1 )H
1#2∂ϕ2
∂H12
(H1#
2 ,H2#2 )
+ H1#1 ϕ2(H1#
2 ,H2#2 )
∂ϕ1
∂H11
(H1#
1 ,H2#1 )
+ H1#1 H1#
2∂ϕ1
∂H11
(H1#
1 ,H2#1 )
∂ϕ2
∂H12
(H1#
2 ,H2#2 )
> 0
since ϕ1(H1#1 ,H
2#1 ) < 0, ϕ2(H1#
2 ,H2#2 ) < 0, ∂ϕ2
∂H12
(H1#
2 ,H2#2 )< 0, ∂ϕ1
∂H11
(H1#
1 ,H1#2 )< 0 and
ϕ1(H1#1 ,H
2#1 )ϕ2(H1#
2 ,H2#2 )− m12m21 = 0.
By the implicit function theorem, there exists a continuously differentiable function H11 (H
21 ,H
22 ), H
12 (H
21 ,H
22 ) defined on a
neighborhood∆′ of (H2#1 ,H
2#2 ) such that
H11 (H
2#1 ,H
2#2 ) = H1#
1 , H12 (H
2#1 ,H
2#2 ) = H1#
2
and
Fi(H11 (H
21 ,H
22 ), H
12 (H
21 ,H
22 ),H
21 ,H
22 ) ≡ 0, i = 1, 2
for (H21 ,H
22 ) ∈ ∆′. Moreover, we have
∂H12
∂H21
(H2#
1 ,H2#2 )
:= −Ξ2
Ξ,
∂H12
∂H22
(H2#
1 ,H2#2 )
:= −Ξ1
Ξ,
where
Ξ1 =
ϕ1(H1#
1 ,H2#1 )+ H1#
1∂ϕ1
∂H11
(H1#
1 ,H2#1 )
0
m21 H1#2∂ϕ2
∂H22
(H1#
2 ,H2#2 )
> 0,
Ξ2 =
ϕ1(H1#1 ,H
2#1 )+ H1#
1∂ϕ1
∂H11
(H1#
1 ,H2#1 )
H1#1∂ϕ1
∂H21
(H1#
1 ,H2#1 )
m21 0
> 0.
Substituting H12 = H1
2 (H21 ,H
22 ) into the second equation in (5.5), we can obtain
Υ ′(H21 ,H
22 ) = H2
2ψ2(H12 (H
21 ,H
22 ),H
22 )+ m21H2
1 .
Differentiating the function Υ ′(H21 ,H
22 )with respect to H2
2 , we have
∂Υ ′(H21 ,H
22 )
∂H22
(H2#
1 ,H2#2 )
= ψ2(H1#2 ,H
2#2 )+ H2#
2
−∂ψ2
∂H12
(H1#
2 ,H2#2 )
Ξ1
Ξ+∂ψ2
∂H22
(H1#
2 ,H2#2 )
=1Ξ
ψ2(H1#
2 ,H2#2 )+ H2#
2∂ψ2
∂H22
(H1#
2 ,H2#2 )
Ξ − H2#
2∂ψ2
∂H12
(H1#
2 ,H2#2 )Ξ1
=1Ξ
ϕ1(H1#
1 ,H2#1 )H
2#2 ψ2(H1#
2 ,H2#2 )∂ψ2
∂H22
(H1#
2 ,H2#2 )
+H1#1∂ϕ1
∂H11
(H1#
1 ,H2#1 )
ϕ2(H1#
2 ,H2#2 )
×
ψ2(H1#
2 ,H2#2 )+ H2#
2∂ψ2
∂H22
(H1#
2 ,H2#2 )
+ H1#
2 ψ2(H1#2 ,H
2#2 )
∂ϕ2
∂H12
(H1#
2 ,H2#2 )
< 0, (C.4)
∂Υ ′(H21 ,H
22 )
∂H21
(H2#
1 ,H2#2 )
= −H2#2∂ψ2
∂H12
(H1#
2 ,H2#2 )
Ξ2
Ξ+ m21
> 0, (C.5)
since ϕ1(H1#1 ,H
2#1 ) < 0, ψ2(H1#
2 ,H2#2 ) < 0, ∂ψ2
∂H22
(H1#
2 ,H2#2 )< 0, ∂ϕ1
∂H11
(H1#
1 ,H2#1 )< 0, ∂ϕ2
∂H12
(H1#
2 ,H2#2 )< 0 andΞ > 0. The facts that
ϕ1(H1#1 ,H
2#1 )ϕ2(H1#
2 ,H2#2 )− m12m21 = 0
and∂ϕ2
∂H12
(H1#
2 ,H2#2 )
∂ψ2
∂H22
(H1#
2 ,H2#2 )
=∂ϕ2
∂H22
(H1#
2 ,H2#2 )
∂ψ2
∂H12
(H1#
2 ,H2#2 )
were used in the above calculations. From (C.3) and (C.5) we can easily see that (−1)7W7 > 0 since W6 > 0.
36 Z. Qiu et al. / J. Math. Anal. Appl. 405 (2013) 12–36
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