2
.
4
. Proof By assumption H1, both of the autotrophs are
limited by their carrying capacities, and so from Eq. 2.3
x ;
i
5 x
i
b
i
N − l
i
− x
i
= x
i
g
i
x
i
; i = 1, 2. By the usual comparison theorem Hale, 1969,
we have x
i
B C
i
= N − l
i
Let us define, S
1
t = x
1
t + x
2
t. The time derivative along a solution of the system is
S : =x
1
g
1
x
1
+ x
2
g
2
x
2
− b
1
+ b
2
x
1
x
2
− yp
1
x
1
+ b
1
x
1
− yp
2
x
2
+ b
2
x
2
5 x
1
g
1
x
1
+ x
2
g
2
x
2
= − x
1
g
1
0 + x
2
g
2
0 + x
1
g
1
0 + x
2
g
2
+ x
1
g
1
x
1
+ x
2
g
2
x
2
or, S
: 5 −m
1
x
1
+ x
2
+ C
1
g
1
0 + C
2
g
2
0 + h
1
+ h
2
Therefore, S
:
1
+ m
1
S
1
B k
2
, where k
2
= C
1
g
1
0 + C
2
g
2
0 + h
1
+ h
2
Applying a theorem on differential inequalities Birkhoff and Rota, 1982, we obtain
0 5 S
1
5 k
2
m
1
+ e
− mt
S
1
x
1
0, x
2
0. As t , 0 B S
1
B k
2
m
1
. Define S
2
t = x
1
t + x
2
t + y. Then S
:
2
= x
;
1
+ x
;
2
+ y
;
2
5 x
1
g
1
x
1
+ x
2
g
2
x
2
− o
3
y = −
x
1
g
1
0 + x
2
g
2
0 + o
3
y + x
1
g
1
+ x
2
g
2
0 + x
1
g
1
x
1
+ x
2
g
2
x
2
. Therefore,
S :
2
B −
m
2
x
1
+ x
2
+ y + C
1
g
1
+ C
2
g
2
0 + h
1 +
h
2
, or S
:
2
+ m
2
S
2
B k
2
and 0 B S
2
B k
2
m
2
as t . Hence system 2.3 is dissipative with the
asymptotic bound k
2
m
2
. Thus there is a compact
neighbourhood B¤R
+ 3
such that for sufficiently large T = T x
,t , xt B for all t \ T, where
xt = {x
1
t, x
2
t
,
yt} is a solution of 2.3 that
initiates in R
+ 3
. This completes the proof of the lemma.
3. Existence of possible boundary equilibria
In system 2.3, when all species are absent, the trivial equilibrium E
0,0,0 always exists. The existence of other boundary equilibria depends
critically on the amount of total nutrient N in the environment. The axial equilibria E
1
N − l
1
, 0, 0 and E
2
N − l
2
, 0, 0 exist provided N \ l
1
and N \ l
2
, respectively. We now investigate the inte- rior equilibrium in the positive x
1
− y plane.Here
isoclines are x ;
1
= 0, or y = F
1
x
1
; Nwhere F
1
x
1
; N = b
1
N − l
1
− x
1
p
1
x
1
x
1
+ b
1
3.1 and y
; =0 is the vertical line x
1
= m
1
, where m
1
is finite.
From Eq. 3.1, F
1
N − l
1
; N = 0 and F
1
0; N = lim F
1
x
1
;N = b
N − l
1
p
1
0 + b
1
\ d F
1
0;N d x
1
= lim
x
1
0 +
F
1
x
1
;N = −
2b
1
p
1
0 + b
1
+ b
1
N − l
1
p ¦0
2{p
1
0 + b
1
}
2
a possible configuration of the autotroph isoclines is given in Fig. 1.
Hence the unique interior equilibrium E
13
m
1
,0, b
1
N − l
1
− m
1
p
1
m
1
m
1
+ b
1
exists in the x
1
− y plane provided
N \ l
1
+ m
1
3.2
Fig. 1.
Æ Ã
à Ã
È H
1
x
1
,x
2
,y −
b
1
x
1
− p
1
x
1
+ b
1
x
1
− x
2
b
2
H
2
x
1
,x
2
,y −
p
2
x
2
+ b
2
x
2
p
1
x
1
y p
2
x
2
y −
o + p
1
x
1
+ p
2
x
2
Ç Ã
à Ã
É where
H
1
x
1
,x
2
,y = b
1
N − l
1
− 2b
1
x
1
− b
1
x
2
− yp
1
x
1
+ b
1
and H
2
x
1
,x
2
,y = b
2
N − l
2
− b
2
x
1
− 2b
2
x
2
− yp
2
x
2
+ b
2
. 4.1
Evaluating V at different equilibria, we get their local stability properties.
4
.
1
. Case
4
.
1:
N B l
1
B l
2
E
o
is a sink i.e. E
o
is saturated. No organism can live in the environment.
4
.
2
. Case
4
.
2:
l
1
B N B l
2
E and E
1
exists and no other boundary equi- libria are feasible. E
is a hyperbolic saddle point or equivalently non-saturated. It has along each
of the y-axis and x
2
-axis a non-empty one dimen- sional local stable manifold.
The flows along the x
1
and x
2
axes are away from E
and approach E
1
. E
1
is stable or unstable locally along the y-direction according as − o
3 +
p
1
N − l
1
B or \ 0, i.e. N B or \ l
1
+ m
1
. So E
1
is unstable along y-direction and y always in- creases as its invasion parameter − o
3
+ p
1
N − l
1
\ 0 at the axial equilibrium E
1
if N \ l
1
+ m
1
. Hence E
1
becomes a saturated equilibrium for l
1
B N B l
1
+ m
1
B l
2
4.2 E
1
becomes non-saturated for l
1
B l
1
+ m
1
B N B l
2
. 4.3
We now have the following result.
4
.
3
. Theorem
4
.
1
Suppose that l
1
B N B l
1
+ m
1
B l
2
. Then any solution of system 2.3 with x
i
0 \ 0, y0 \ 0 satisfies
Similarly, under H1 E
23
0,m
2
, b
2
N − l
2
− m
2
p
2
m
2
m
2
+ b
2
exists in x
2
− y plane provided
N \ l
2
+ m
2
. 3.3
From Eq. 2.3, it follows that if the herbivore is
absent, two
autotrophs can
not attain
E
12
x
1
,x
2
,0 and they are in a dominance state. Existence of different axial and planar equi-
libria thus depends critically on the total nutrient N, l
i
and m
i
; i = 1,2. It is difficult to determine the interior equi-
librium Ex
1
,x
2
,y but one can gain informa- tion about whether the two autotrophs and the
herbivore can co-exist or not.
4. Stability of equilibria