Competition of Two Host Species for a Single-Limited
Resource Mediated by Parasites
Sze-Bi Hsu
Department of Mathematics,
National Tsing-Hua University
Hsinchu, Taiwan
&
I-Fan Sun
Department of Life Science and
Center of Trophical Ecology and Biodiversity
Tung-Hai University
Abstract:
In this paper we consider a mathematical model of two host species competing for a
single-limited resource mediated by parasites. Each host population is divided into susceptible and
infective population. We assume that species 1 which has the smallest break-even concentration,
outcompetes species 2 when there is no parasite. We analyze the model by finding the conditions for
the existence of various equilibria and doing their stability analysis. The bifurcation diagram shows
that species 1 still outcompetes species 2 when the contact rate 1 of the species 1 is small
regardless of the contact rate  2 of the species 2. However if 1 is large enough, both of species
1 and species 2 coexist either in the form of equilibrium or in periodic oscillation..
Key words: multi-host, mathematical model, parasite, competition, resource.
INTRODUCTION In ecology, ecologists are interested in understanding how and to what extent,
interspecific interactions influence community structure, species coexistence and biodiversity. In the
past two decades, people investigate the potential importance of parasites and pathogens in
determining the outcomes trophic interaction and community process. In the paper (Hatch & et.
2006 ) the authors give a review about the recent research on how parasites influence competitive
and predatory interaction between the species they infect. In this paper we shall investigate a
mathematical model of two host species competing for a single-limit resource mediated by parasites.
Especially we focus on the question: can infection produces coexistence of species. When there is
no infection mediated by parasites, it is a well-known fact that only the species with the smallest
break-even concentration survives (Hsu 1978, Herbert et. 1956, Powell 1958, Hsu et 1977, Smith &
Waltman 1994). With the infection by parasites we show that it is possible to have coexistence of
two species if the contact rate  1 of surviving species is large enough. We shall find conditions for
the existence of various equilibria and do their stability analysis. A bifurcation diagram is presented
with contact rates 1 ,  2 as two bifurcation parameters. We also present a bifurcation diagram by
using input concentration R (0) of the nutrient as a bifurcation parameter.
The model
The model of two hosts competing for a single-limited resource mediated by parasites leads to the
differential equations
dR
1 m1R
1 m2 R
 ( R ( 0)  R) D 
( S1  I1 ) 
(S2  I 2 )
dt
y1 a1  R
y2 a2  R

dS1  m1R
 
 d1  S1  1I1S1   1I1
dt  a1  R

dI1
 1I1S1  (d1  1   1 ) I1
dt

dS2  m2 R
 
 d 2  S 2   2 I 2 S 2   2 I 2
dt
 a2  R

(1)
dI 2
  2 I 2 S 2  (d 2   2   2 ) I 2
dt
R ( 0)  0 , S1 (0)  0 , I1 (0)  0 , S2 (0)  0 , I 2 (0)  0
Here we assume that each host population is divided into susceptible and infectious hosts; these are
designated S, I. R is the concentration of resource. The rest of the model’s parameters are defined in
table 1.
Table 1
Parameter
(0)
R
D
yi
mi
ai
di
i
i
Definition of parameters used in the model
Definition
input concentration of nutrient R
dilution rate
yield constant of i-th host
maximum growth rate of i-th host
half-saturation constant for i-th host
death rate of i-th host
per capita additional mortality of i-th host when infected
with parasite
per capita rate of recovery of i-th host from infection
Figure 1
When there is no parasites, the infective population I1 (t ) , I 2 (t ) satisfy I1 (t )  0 , I 2 (t )  0 . Then the
system (1) becomes
Parasite
Host 1 S 1 , I 1
Host 2 S 2 , I 2
nutrient R
Figure 1
R'  ( R (0)  R) D 
1 m1R
1 m2 R
S1 
S2
y1 a1  R
y 2 a2  R
 mR

S1 '   1  d1  S1
 a1  R

 mR

S 2 '   2  d 2  S 2
 a2  R

(2)
R (0)  0, S1 (0)  0, S 2 (0)  0
If the maximal growth rate mi is less than or equal to the death rate di of the i-th host, then
Si (t )  0 as t   , i.e. the i-th host cannot survive. Assume the break-even concentration of the
i-th host i 
ai
 0 , i  1,2 .
(mi / di )  1
If the break-even concentration i is greater than the input concentration R ( 0) then the i-th host
species goes to extinction due to the fact that input concentration R ( 0) is too small to support the
survival of i-th host species Si (Hsu 1978, Smith and Waltman 1994). Thereafter we assume that
(H)
0  1  2  R( 0)
Under assumption (H), we conclude that 1st host species S1 outcompetes the 2nd host species S 2
(Hsu 1978, Smith and Waltman 1994) and the solutions R (t ) , S1 (t ) , S2 (t ) satisfy
lim R(t )  1
t 
lim S1 (t )  S1* 
t 
( R ( 0)  1 ) Dy1
d1
lim S 2 (t )  0
t 
Equilibria and Their Stability Analysis of (1):
In the followings, we state the results for the existence of equilibria and their stability analysis. The
proofs are deferred to Appendix. Under the assumption (H), the equilibrium E1  (1 , S1 ,0,0,0) of
system (1) always exists. The equilibrium E1 is locally asymptotically stable if
1S1  (d1  1   1 )  0
(3)
1S1  (d1  1   1 )  0
(4)
If
then E1 is unstable.
If the instability condition (4) holds then the equilibrium E1I1  ( Rˆ1 , Sˆ1 , Iˆ1 ,0,0) exists, where R̂1 ,
Ŝ1 , Iˆ1 satisfy
d   1  1
Sˆ1  1
0
(5)
1
 m1 Rˆ1

 d1  Sˆ1  (d1  1 ) Iˆ1  0

ˆ
 a1  R1

( Rˆ1  1 )
 m1Rˆ1


 1 
1 m1Rˆ1  a1  Rˆ1
ˆ
( R ( 0)  Rˆ1 ) D 
S1

y1 a1  Rˆ1
d1  1 






(6)
(7)
It can be proved that if E1I 1 is locally asymptotically stable then
Rˆ1  2
(8)
We note that under condition (8), E1I 1 may be a stable or an unstable equilibrium.
If
Rˆ1  2
(9)
then E1I 1 is unstable.
It can be proved that if the instability condition (9) holds then the equilibrium
~ ~ ~
E1I1 2  (2 , S1 , I1 , S2 ,0) exists, where
~
d  
S1  Sˆ1  1 1 1
(10)
~
1  m12

 d1 S1  0
d1  1  a1  2

(11)
1
~ ~
and S 2 , I 1 satisfy
~
I1 

 m12

 1  


~
y
1  m   ~ a  2
  0
S 2  2 ( R ( 0 )  2 ) D   1 2  S1  1
d2 
y1  a1  2   d1  1 






(12)
If E1I1 2 is locally asymptotically stable then
~
 2 S 2  (d 2   2   2 )  0
(13)
We note that under condition (13), E1I1 2 may be a stable or an unstable equilibrium.
If
~
 2 S 2  (d 2   2   2 )  0
(14)
then E1I1 2 is unstable.
It can be shown that if the instability condition (14) holds then the equilibrium
E1I1 2 I 2  ( R , S1, I1, S2 , I 2 ) exists where
d  
S1  Sˆ1  1 1 1
(15)
d  2   2
S 2  Sˆ2  2
(16)
1
2
I1 

1  m1R

 d1   0
d1  1  a1  R

I2 

1  m2 R

 d 2   0
d 2   2  a2  R

( R  1 )
( R  2 )
(17)
(18)
R satisfies
 m1R

 m2 R

 1 
 2 


1 m1R
a R
  1 m2 R S  a2  R

( R ( 0)  R ) D 
S1  1
2
y1 a1  R  d1  1  y2 a2  R  d 2   2 








(19)
It is difficult to determine the local stability of E1I1 2 I 2 .
We verify it by extensive numerical simulation. We conjecture that Hopf bifurcation may occurs
and there exists periodic solutions in some parameter range.
Bifurcation Analysis:
The equilibrium E1  (1 , S1 ,0,0,0) is locally stable if (3) holds. We rewrite (3) as
(d     )d
1  ˆ1  1 ( 0 ) 1 1 1
( R  1 ) Dy1
(20)
When 1  ̂1 , the equilibrium E1 is unstable and the equilibrium E1I1  ( Rˆ1 , Sˆ1 , Iˆ1 ,0,0) exists. If
E1I1 is locally stable then (8) holds. However, if (8) holds then E1I1 may be locally stable or
unstable. It can be shown in the Appendix that (8) can be rewritten as
 m12


 1 
~ 1 m12  a1  2
 d1  1   1
1  1 
y1 a1  2
d1  1
( R ( 0)  2 ) D
(21)
~
and ˆ1  1 .
~
If the instability condition for E1I1 1  1 holds, then the equilibrium E1I1 is unstable and the
~ ~ ~
equilibrium E1I1 2  (2 , S1 , I1 , S2 ,0) exists. If E1I1 2 is locally stable then (13) holds. However if
(13) holds then the equilibrium E1I1 2 may be locally stable or unstable. It can be shown in the
Appendix that (13) can be rewritten as
2 
A1

A 
 A2  3 
1 

(22)
where
A1 
d2
(d 2   2   2 )
y2
(23)
A2  ( R( 0)  2 ) D
A3 
(24)

1 m12 d1  1   1  m12

 1 
y1 a1  2 1  d1  a1  2

We note that the vertical asymptote of the curve  2 
If the instability condition for E1I1 2  2 
A1

A 
 A2  3 
1 

A1

A 
 A2  3 
1 

is 1 
(25)
A3 ~
 1 .
A2
holds then the equilibrium
E1I1 2 I 2  ( R , S1, I1, S2 , I 2 ) exists. It is difficult to prove the local stability of E1I1 2 I 2 analytically. In
the Figure 2, using 1 ,  2 as bifurcation parameters, we plot a bifurcation diagram. It shows that
the Hopf bifurcation occurs and there are periodic oscillation for some parameter ranges.
Figure 2
In region I, the equilibrium E1 is locally stable.
In region II, the equilibrium E1I 1 exists, E1 is unstable.
In region III, the equilibrium E1I1 2 exists, E1 , E1I 1 are unstable.
In region IV, the equilibrium E1I 1 2 I 2 exists, E1 , E1I 1 , E1I1 2 are unstable.
Figure 3
In Figure 3 our numerical simulation study shows there is a Hopf bifurcation curve H in the region
IV. For (1 ,  2 ) inside the curve H, the region V, E1I 1 2 I 2 is unstable and periodic oscillation occur
(See Figure 4).
R vs T
S1 vs T
S2 vs T
I2 vs T
I1 vs T
R ( 0) =
10
=
0.4
D
m1  1
m2  3
a1  1
a2  2
d1  0.5
d2  2
1  0.5
 2  0.4
 1  0.1
 2  0.001
y1  4
y2  2
Figure 4
The time series for 1  60,  2  20 .
Under the basic assumption (H), species 1 outcompetes species 2 when there is no infection by the
parasites. When the contact rate 1 satisfies 0  1  ˆ1 regardless of magnitudes of the species
2’s contact rate  2 , the species 1 still outcompetes the species 2 and there is no infection
~
population of species 1. If we increase 1 such that ˆ1  1  1 , then regardless of the
magnitudes of  2 species 1 still outcompetes species 2, but there is some infective population for
~
the species 1 in this case. For 1  1 , species 1 and species 2 coexist. For the parameters 1 ,  2
if the point 1 ,  2  is under the curve  2  f ( 1 ) 
A1
( A2 
A3
1
in the 1   2 plane, then species
)
1 which has both susceptible and infective population, coexists with species 2 which has only
susceptible population. If 1 ,  2  is above the curve 2  f (1 ) in the 1   2 plane then
species 1 and species 2 both with infective population coexists in the steady state . For 1 ,  2  in
the region V, species 1 and species 2 both with infective population coexist in the form of periodic
oscillation.
If we vary the input concentration R ( 0) and fix the other parameters then from (20), (21), (22)
it follows that
(d     )d
1  ˆ1  R ( 0)  1  1 1 1 1  Rˆ ( 0) .
Dy11
~
1  1  R ( 0)
2 
 m12


 1 
(d     ) m12  a1  2
~ ( 0)
R
 2  1 1 1
Dy11
a1  2
d1  1
d (d   2   2 )
~
 R (0)  R (0)  2 2
 R (0)

Dy


A
2
1
 A2  3 
1 

A1
~
From the hypothesis (H), we have Rˆ ( 0)  R ( 0)  R ( 0) . The bifurcation diagram is the following
Figure 5:
Figure 5
Thus in order to have coexistence of species 1 and species 2, we need to have large input
concentration R ( 0) i.e. R ( 0 )  R̂ ( 0) .
Discussion:
In this paper we study the competition of two host species for a single-limited resource
mediated by parasite. Our basic assumption is
0  1  2  R ( 0 )
(H)
In the absence of infection 1 , 2 are the break-even concentration for species 1 and species 2
respectively. We note that the break-even concentration i of ith species is
ai
i 
(
mi
di
) 1
If mi  di or 0  R (0)  i then the i-th species goes to extinction ( Hsu 1978). That is, if the
maximal growth rate mi is less than or equal to the death rate d i or the input concentration R (0)
is too small to support the ith species, then the ith species goes to extinction. Thus we only consider
the case that 0  i  R (0) . Under the assumption (H) the competitive exclusion principle holds and
species 1 which has the smallest break-even concentration, wins the competition (Hsu 1978).
Next we consider the case that the infection is mediated by parasite. The equilibrium
E1  (1 , S1* , 0, 0, 0) is asymptotically stable if (3) holds. We may rewrite (3) as
( R0 )1 
)
1 ( R ( 0 
 1) Dy 1
1
(d1  1   1 )d1
(26)
where ( R0 )1 is the basic reproduction number of the infection for species 1. From (26) it is easy to
see that with d1 fixed if the contact rate 1 or the input concentration R (0) is small or the
additional death rate 1 or the recovery rate  1 of the infective population of species 1 are large
then species 1 still wins the competition independent of the contact rate  2 of species 2 and
species 1 contains only susceptible population. When the basic reproduction number ( R0 )1 is
greater than 1, the equilibrium E1 becomes unstable and a new equilibrium E1I1  ( Rˆ1 , Sˆ1 , Iˆ1 , 0, 0)
bifurcates.
From (20), ( R0 )1  1 is equivalent to 1  ˆ1 .
If the instability condition (9) holds,
not only E1I1 is unstable but also the new equilibrium E1I1 2  (2 , S1 , I1 , S2 ,0) bifurcates. We note
that the instability condition (9) is equivalent to condition 1  1 .where 1 is defined in (20).
When ̂1  1  1 , there are two equilibria, namely E1 and E1I1 E1 is unstable and E1I1 is
either stable or unstable. In this case competitive exclusion principle still holds. The susceptible
and infective population of species 1 either go to positive steady state or sustain oscillation when
E1I1 is stable or unstable. To explain the biological meaning , we rewrite ̂1  1  1 as follows:
m12
m
 1 ) ( 1 2 )
(d     )d
( R   ) a1  2
a2  2
ˆ1  1 (0) 1 1 1  1  ˆ1 (0) 1
 1  h(2 )
( R  1 ) Dy1
( R  2 )
d1  1
d1
(0)
where
m1
m
 1 ) ( 1 )
( R   ) a1  
a1  
h( )  (0) 1
(R  )
d1  1
d1
(0)
(
(
(27)
It is easy to verify h(1 )  1 by the definition of break-even concentration 1 .and h ( ) is
increasing in  , lim( 0 ) h( )   . Let h( * )  1 for 1  ˆ1 . Then (24) implies that
 R
 *  2  R (0) ,i.e. the gap between 1 and 2 is large. In this case the species 2 is a weaker
competitor. Hence competitive exclusion principle still holds.
If 1  1 then 1  2   * and E1I1 becomes unstable and equilibrium E1I1 2 bifurcates.
Biologically it says that if the gap between 1 and 2 is small then species 2 can coexist with
species 1.
If 1  1 and
2 
A1
( A2 
A3
1
(28)
)
then there are three equilibria, namely E1 , E1I1 , E1I1 2 ; E1 , E1I1 are unstable and E1I2 2 is either
stable or unstable.. From (23),(24),(25) we may rewrite (28) as follows:

m12  m12
 1 
)
R 
1 a1  2  a1  2

y2  2 [( R (0)  1 ) D][ (0) 2 
R  1 ( R0 )1
d1
d1  1
( R0 )12 
1
d 2 (d 2   2   2 )
(
(0)
Or
)
y2  2( R( 0 
 2) D( ( R0 )1 h( 2 ) )
 1
(29)
( R0 )1 d 2 (d 2  2 )2
is the basic reproduction number of the infection of species 2 under the condition
( R0 )1 2
where ( R0 )12
that the infection outbreaks for species 1. In order to have ( R (0) )12  0 we need
( R0 )1  h (
2
)
(30)
i.e., the basic reproduction number of infection of species 1 is large enough to have the infective
population of species 1.Assume (30) holds. If either the contact rate  2 is small or the additional
death rate  2 , recovery rate  2 of species 2 are large, then (29) holds. Thus the disease does not
outbreak for species 2. In this case we have coexistence for species 1 and species 2. Species 1 has
both susceptible and infective populations, species 2 has only susceptible population. Either the
populations go to a steady state if E1I1 2 is stable or sustain periodic oscillation if E1I1 2 is unstable.
If 1  1 and
1 
A1
( A2 
A3
1
(31)
)
then the equilibrium E1I1 2 becomes unstable and a new equilibrium E1I1 2 I 2 bifurcates. Thus there
are four equilibria, namely E1 , E1I1 , E1I1 2 , E1I1 2 I 2 ; E1 , E1I1 , E1I1 2 are unstable and E1I1 2 I 2 is either
stable or unstable. We may rewrite (31) as
( R0 )1 2 1
(32)
From the expression of ( R0 )12 in (29), if the contact rate  2 or the input concentration
R (0) is
large then (32) holds thus the disease outbreak for species 2 and we have coexistence for species 1
and species 2. Both of them have infective populations. Either the populations go to a steady state if
E1I1 2 I 2 is stable or sustain periodic oscillation if E1I1 2 I 2 is unstable.
Appendix
In Appendix we discuss the existence of equilibria E1I 1 , E1I1 2 , E1I 1 2 I 2 and their relations
with the instability of E1 , E1I 1 , E1I1 2 respectively. First we note that the equilibrium
E1  (1 , S1 ,0,0,0) of system (1) always exists. The variational matrix of the system (1) at
E  ( R, S1, I1, S2 , I 2 ) is
 a11 a12
a
 21 a22
J ( E )   0 a32

a41 0
 0
0
a13 a14
a23 0
a33 0
0 a44
0 a54
a15 
0 
0

a45 
a55 
where
a11   D 
1
1
f1 ' ( R)( S1  I1 ) 
f 2 ' ( R)( S 2  I 2 )
y1
y2
a12  
1
f1 ( R )
y1
a13  
1
f1 ( R )
y1
a14  
1
f 2 ( R)
y2
a15  
1
f 2 ( R)
y2
a21  f1 ' ( R)S1
a22  ( f1 ( R)  d1 )  1I1
a23   1S1   1
a32  1I1
a33  1S1  (d1  1   1 )
a41  f 2 ' ( R)S2
a44  ( f 2 ( R)  d2 )  2 I 2
a45   2 S2   2
a54   2 I 2
a55   2 S2  (d 2   2   2 )
f1 ( R) 
m1R
mR
, f 2 ( R)  2
a1  R
a2  R
The variational matrix J ( E1 ) of system (1) at equilibrium E1  (1 , S1 ,0,0,0) is
1


 D  y f1 ' (1 ) S1
1


f
'
(

1 1 ) S1
J ( E1 )  
0

0


0


d1
y1
0
d1
y1
 1S1   1
0
0
1S1  (d1  1   1 )
0
0
f 2 (1 )  d 2
0
0
0


1
f 2 ( R)
y2
0





0

2

 (d 2   2   2 )

1
f 2 ( R)
y2
0
The eigenvalues of J ( E1 ) are
1  (d2   2   2 )  0
(by (H))
2  f 2 (1 )  d2  0
3  1S1  (d1  1   1 )
4 and 5
1

D
f1 ' (1 ) S1

are the eigenvalues of
y1

f1 ' (1 ) S1

g ( )   (  ( D 

d1 
y1  which characteristic polynomial is

0 
1
d
f1 ' (1 ) S1 ))  1 f1 ' (1 ) S1 . Then Re 4 , Re 5  0 . Thus E1 is locally stable
y1
y1
if 3  0 i.e. 1S1  d1  1   1 and E1 is unstable if 3  0 , i.e. 1S1  d1  1   1 .
Next we shall show that if E1 is unstable then the equilibrium E1I1  ( Rˆ , Sˆ1 , Iˆ1 ,0,0) exists
where R̂ , Ŝ1 , Iˆ1 satisfy (5), (6), (7). From (6), (7) we have the following Figure 6.
Figure 6
Since 1  R̂ , from Figure 6 we have
 m11

 1 

1 m11  a1  1
 Sˆ  ( R (0)   ) D
1
1
y1 a1  1
d1  1

1 d1  1   1
d1
 ( R (0)  1 ) D
y1
1
 1S1  d1  1   1
Thus instability of E1 implies the existence of E1I1 . The variational matrix of (1) at
E1I1  ( Rˆ , Sˆ1 , Iˆ1 ,0,0) is
1
1
1
1
1


ˆ ˆ ˆ
ˆ
ˆ
ˆ

f 2 ( Rˆ ) 
  D  y f1 '( R)( S1  I1 )  y f1 ( R)  y f1 ( R )  y f 2 ( R )
y2
1
1
1
2


ˆ


I
f1 '( Rˆ ) Sˆ1
 11
 1Sˆ1   1
0
0


Sˆ1
J ( E1I1 )  



ˆ
0
1 I1
0
0
0




0
0
0
f 2 ( Rˆ )  d 2
2


0
0
0
0
( d 2   2   2 ) 

The eigenvalues of J ( E1I1 ) are
1  (d2   2   2 )  0
 2  f 2 ( Rˆ )  d 2
and 3 , 4 , 5 are eighenvalues of the 3 3 matrix
1
1
1

ˆ ˆ ˆ
ˆ
ˆ 
 D  y f1 ' ( R )( S1  I1 )  y f1 ( R )  y f1 ( R ) 
1
1
1


ˆ

I

1 1
ˆ
ˆ
ˆ
f1 ' ( R ) S1

 1S1   1 


Sˆ1


0
1Iˆ1
0




which characteristic polynomial is  3  a1 2  a2   a 3  0
where
a1 
 1Iˆ1
Sˆ1
 (D 
1
f1 ' ( Rˆ )( Sˆ1  Iˆ1 ))
y1
a2 
a3 
 1Iˆ1
Sˆ1
(D 
1
1
f1 ' ( Rˆ ))( Sˆ1  Iˆ1 )  f1 ' ( Rˆ ) Sˆ1 f1 ( Rˆ )  1Iˆ1 ( 1Sˆ1   1 )
y1
y1
1
1
f1 ( Rˆ ) f1 ' ( Rˆ ) Sˆ11Iˆ1  1Iˆ1 ( 1Sˆ1   1 )( D 
f1 ' ( Rˆ )( Sˆ1  Iˆ1 ))
y1
y1
From Routh-Hurwitz criterion, E1I 1 is locally stable if a1a2  a3 . It is rather difficult to verify this
condition. In order to simplify the computation, we consider the case  1  0 to justify the
condition a1a2  a3 . When  1  0 , a1a2  a3 becomes
(D 

1
1
f 1 ' ( Rˆ )( Sˆ1  Iˆ1 ))[( f 1 ' ( Rˆ ) Sˆ1
f 1 ( Rˆ ))   1 Iˆ1  1 Sˆ1 ]
y1
y1
1
1
f 1 ( Rˆ ) f 1 ' ( Rˆ ) Sˆ1  1 Iˆ1   1 Iˆ1  1 Sˆ1 ( D 
f 1 ' ( Rˆ )( Sˆ1  Iˆ1 ))
y1
y1
 D
1
f1 ' ( Rˆ )( Sˆ1  Iˆ1 )  1Iˆ1
y1
f ' ( Rˆ )
f ( Rˆ )  d1 ˆ
 D  ( R ( 0 )  Rˆ ) D 1
 1 1
S1  f1 ( Rˆ )  d1
d1  1
f1 ( Rˆ )
m Rˆ
m1a1
Substitute f1 ( Rˆ )  1 , f1 ' ( Rˆ ) 
into above, it follows that
a1  Rˆ
(a1  Rˆ ) 2
( 0)
aR
Rˆ  1
Rˆ
m d 
  1 1 ( Rˆ  1 )
 D 
If D  (m1  d1 ) , then E1I 1 is locally stable. For the case D  (m1  d1 ) , if Rˆ  Rˆ * then E1I 1 is
locally stable; if Rˆ  Rˆ  then E1I 1 is unstable with 1-dimensional unstable manifold where R̂ *
is the unique positive root of R 
a1 R ( 0)  m1  d1 

( R  1 ) .
R
 D 
We note that 2  0 if and only if Rˆ  2 . We shall prove that the instability condition Rˆ   2
~ ~ ~
~
implies the existence of the equilibrium E1I1 2  (2 , S1 , I1 , S2 ,0) . From (12), if S 2  0 then
 m12

 1 

1  m   a  2
.
( R ( 0)  2 ) D   1 2  Sˆ1  1
y1  a1  2   d1  1 




From Figure 6 the above identity is equivalent to Rˆ  2 . Thus from (10), (11), (12), Rˆ  2
implies the existence of the equilibrium E1I1 2 .
~ ~ ~
The variation matrix of (1) at E1I1 2  (2 , S1 , I1 , S2 ,0) is
1
~ ~

 D  y f1 ' (2 )( S1  I1 )
1
1


f1 (  2 )
~
y1
 1

f
'
(

)
S
 y 2 2 2
2
~
~
J ( E1I1 2 )  
f1 ' (2 ) S1
( f 1 (  2 )  d1 )   1 I 1

~

0
1 I 1

~
f 2 ' ( 2 ) S 2
0


0
0

1
1

f1 (  2 ) 
f 2 ( 2 )
y1
y2
~
 1S1   1
0
0
0
0
0
0
0






0


0

~
  2 S2   2

~
 2 S 2  (d 2   2   2 )
1

f 2 ( 2 )
y2
~
The eigenvalues of J ( E1I1 2 ) is 1   2 S 2  (d1  1   1 ) and 2 , 3 , 4 , 5 which are
eighenvalues of the 4  4 matrix
~ ~
~
1
1

 D  y f1 ' (2 )( S1  I1 )  y f 2 ' (2 ) S 2
1
2

~
f
'
(

)
S
M 
1
2
1

0

~
f 2 ' (2 ) S 2

1
1
1

f1 (2 )
 f1 (2 ) 
f 2 (2 )
y1
y1
y2

~
~
( f1 (2 )  d1 )  1I1  1S1   1
0

~

1I1
0
0

0
0
0


It is difficult to verify all eigenvalues of the matrix M have negative real parts by Routh-Hurwitz
criterion. In fact E1I1 2 may be stable or unstable, we note that 1  0 if and only if
~ d  2   2
S2  2
. It can be shown that the instability condition for ( E1I1 2 ) , 1  0 
2
~ d  2   2
S2  2
implies the existence of the equilibrium E1I1 2 I2  ( R , S1 , I1, S2 , I 2 ) . From (14) -
2
(18), R satisfies R  2  1 and
 m1R

 m2 R

 1 
 2 


1 m1R
a R
  1 m2 R S  a2  R
  h( R ) .
( R (0)  R ) D 
S1  1
2



y1 a1  R
d1  1
y2 a2  R
d2   2 








From the following Figure 7
Figure 7
We have
 m12

 1 

1
m

a  2
1 2
 1 d S .
( R ( 0)  2 ) D 
S1  1
2 2
y1 a1  2  d1  1  y2




Thus
 m12

 1 

1 m12 d1  1   1  a1  2
  1 d d2   2   2 .
( R ( 0)  2 ) D 
 d1  1  y2 2
y1 a1  2
1
2




2 
where A1 
A1
d2
(d 2   2   2 )
y2
A2  ( R( 0)  2 ) D
A3 

1 m12 d1  1   1  m12

 1 
y1 a1  2 1  d1  a1  2

( A2 
A3
1
)
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