˖ڍመڙጲ
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Next, we consider the boundary equilibria and their stability of (2). E
0
= (0 , 0, 0, 0) is the
trivial equilibrium. To find other boundary equilibria, we consider
u
1
[r
1
− (a
1
+ m
1
)u
1
] + m
2
u
2
2
= 0,
u
2
[r
2
− (a
2
+ m
2
)u
2
] + m
1
u
2
1
= 0,
(4)
which gives the equilibrium of mutant population in the absence of resident population. One
can verify that (4) has a unique positive solution (u
10
, u
20
) (see, for example, [19, 26]). Thus,
(2) admits a boundary equilibrium E
1
= (u
10
, u
20
, 0, 0). Similarly, the following system
v
1
[r
1
− a
1
v
1
] − d
1
v
1
+ d
2
v
2
= 0,
v
2
[r
2
− a
2
v
2
] − d
2
v
2
+ d
1
v
1
= 0
(5)
has a unique positive solution (v
10
, v
20
). Thus, (2) admits another boundary equilibrium E
2
=
(0, 0, v
10
, v
20
).
Evidently, E
0
is a repeller. Recall that the stability modulus of an n ×n matrix J, denoted
by s(J ), is defined by
s(J) := max{Reλ : λ is an eigenvalue of J }.
Linearizing (2) at E
1
, we see that the stability of E
1
is determined by
J
1
=
r
1
− d
1
− a
1
u
10
d
2
d
1
r
2
− d
2
− a
2
u
20
.
Thus, E
1
is asymptotically stable if s(J
1
) < 0 and is unstable if s(J
1
) > 0.
Linearizing (2) at E
2
, we see that the stability of E
2
is determined by the stability modulus
of
J
2
=
r
1
− (a
1
+ m
1
)v
10
m
2
v
20
m
1
v
10
r
2
− (a
2
+ m
2
)v
20
.
That is, E
1
is asymptotically stable if s(J
2
) < 0 and is unstable if s(J
2
) > 0.
Using the persistence theory [27, 28] and similar arguments to those in [19], we can state
Theorem 1. The resident population and mutant population are uniformly persistent if s(J
1
) >
0 and s(J
2
) > 0.
We cannot directly draw influences of population dispersals on coexistence of two popula-
tions from Theorem 1 because u
i0
and v
i0
are not explicitly given. For this reason, we consider
two cases where population dispersals take different time scales from population growthes.
First, we assume that the time scale for population migrations is much smaller than the time
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