Về sự tồn tại sóng chạy trong mô hình rời rạc của các quần thể sinh học

Q[u]
A a
AA, Aa, aa AA, aa
Aa
A
A A n u
n
(i)
a 1 − u
n
(i)
AA; Aa; aa
(u
n
(i))
2
: 2u
n
(1 − u
n
) : ((1 − u
n
(i))
2
(1 + s
i
) : 1 : (1 + t

i
)
(1 + s
i
)(u
n
(i))
2
: 2u
n
(1 − u
n
) : (1 + t
i
)((1 − u
n
(i))
2
.
p
i
l
ij
i
j
j
u
n+1
(j) =


i
m
ji
g
i
(u
n
(i))
g
i
(u) =
2(1 + s
i
)u
2
+ 2u(1 − u)
2[(1 + s
i
)u
2
+ 2u(1 − u) + (1 + σ
i
)(1 − u)
2
]
m
ji
=
l
ji

p
i

k
l
ji
p
i
j i
{u
n
(i) : i = 1; 2; }
u
n+1
= Q[u
n
], (1)
Q[u](
j
) =

i
m
ji
g
i
(u
n
(i)).
l

ij
i j
R
2
{(x; y)|(k −
1
2
)h < x < (k +
1
2
)h, (l −
1
2
)h < y < (l +
1
2
)h, k, l = 0; 1; 2 }
h h
H
h H s t p
l
ij
x
i
− x
j
s, t, p u

k
l

ij
=

k
l(x
i
− x
j
) =

i
l(x
i
) =

i
l
i0
= 1.
m
ij
= l
ij
≡ m(x
i
− x
j
).
Q[u](x) =


y∈H
m(x − y)g(u(y)),

y∈H
m
ji
(x) = 1
g(u) =
su
2
+ u
1 + su
2
+ σ(1 − u)
2
.
g(u) − u =
u(1 − u)[su − σ(1 − u)]
1 + su
2
+ σ(1 − u)
2
.
u u(x) 0  u  1
g 0 1 u 0 1 Q[u]
[0; 1]
g
s > 0 > σ AA aa
g(u) > u, 0 < u < 1.
s < 0 < σ g(u) < u, 0 < u < 1

u 1 − u A a
s σ
g(u) > u, 0 < u < π
1
,
g(u) < u, π
1
< u < 1.
π
1
=
σ
s + σ
.
σ
g(u) < u, 0 < u < π
0
,
g(u) > u, π
0
< u < 1.
π
0
=
σ
s + σ
.
1 + s 1 + σ
u p
s σ

u
p
u
ij i
x
i
− x
j
l
A
(x − y, u) A l
a
(x − y, u)
x y u
u
n+1
= Q[u
n
], (1)
Q[u](x) =

y∈H
l
A
(x − y, u(y))

y∈H
[l
A
(x − y, u(y)) − l

a
(x − y, u(y))]
.
u
n
(x)
x Q
Q[u](x) =

R
2
m(x − y)g(u(y)).
m(x) |x|
Q
u
u
n
(x) = u(nτ, x) u(t, x)
∂u
∂t
= D∆u + f(u).
u
n
u
n+1
= Q[u
n
], (1)
Q[v](x) u(τ, x) u(t, x)
∂u

∂t
= D∆u + f(u), u(0, x) = v(x).
H
R
n
x, y ∈ H x + y, x − y ∈ H
x, y ∈ H x + y, x − y ∈ H 0 ∈ H
H
B H
[0; π
+
] π
+
= ∞ B H
[0; ∞]
y H
T
y
u(x) ≡ u(x − y).
u
n+1
= Q[u
n
],
u
n+1
u
n
B Q B
Q

Q
Q u
(i) Q[u] ∈ B ∀ ∈ B
(ii) Q[T
y
[u]] = T
y
[Q[u]] ∀ ∈ B ∈
(iii) 0  π
0
< π
1
 π
+
Q[α] > α α ∈ (π
0
; π
1
), Q[π
0
] = π
0
, Q[π
1
] = π
1
, π
1
< ∞.
(iv) u  v Q[u]  Q[v]

(v) u
n
→ u n → ∞ H
Q[u
n
](x) → Q[u](x), x ∈ H.
Q
Q[α] < α, α ∈ (π
1
, π
+
) π
1
< π
+
.
K
1
K
2
R
n
ε
1
> 0 π
1
< π
+
π
1

= π
+
u < π
1
+ ε
1
H
u < π
1
K
1
K
2
Q[u](0) < π
1
.
b > 0
u(x) = 0, |x|  b ⇒ Q[u](0) = 0.
v
n
B v
n
 π
1
v
n
k
Q[v
n
k

] H
u
0
B n
u
n
u
n+1
= Q[u
n
].
u
n
(x) = W [x · ξ − nc],
W ξ
W (s)
s x · ξ − nc x ∈ H, n = 0, 1, 2,
Q[u](x
0
) = Q[T
−x
0
[u]](0), ∀x
0
∈ R
n
.
Q
α Q[α]
α

α(x − y) = α(x) = α
y ∈ H
T
y
[Q[α]](x) = Q[α](x − y).
Q[T
y
[α]](x) = Q[α(x − y)](x) = Q[α](x),
y ∈ H
T
y
[Q[α]](x) = Q[T
y
[α]](x),
Q[α](x − y) = Q[α](x) .
Q[α]
α
n
α
α Q Q[α
n
](x)
Q[α].
R B B
R
v  w ⇒ R[v]  R[w]
v
n
v
n+1

 R[v
n
].
w
n
w
n+1
 R[w
n
],
v
0
 w
0
v
n
 w
n
, ∀n ∈ N
v
0
 w
0
R[v
0
]  R[w
0
].
v
1

 R[v
0
], w
1
 R[w
0
]
v
1
 R[v
0
]  R[w
0
]  w
1
.
v
2
 R[v
1
]  R[w
1
]  w
2

v
n
 R[v
n−1
]  R[w

n−1
]  w
n
, ∀n ∈ N.
R B B R
R[w
0
]  w
0
w
n
w
n+1
= R[w
n
] w
n+1
 w
n
, ∀n ∈ N
v
n
= w
n+1
, ∀n ∈ N
v
0
= w
1
= R[w

0
]  w
0
,
R
v
1
= w
2
= R[w
1
]  R[w
0
] = w
1
= v
0
v
2
= w
3
= R[w
2
]  R[w
1
] = w
2
= v
1

v
n
= w
n+1
= R[w
n
]  R[w
n−1
] = w
n
= v
n−1
w
n+1
 w
n
∀n ∈ N
c
∗
(ξ) Q
c
∗
(ξ)
c
∗
(ξ)
ξ c
∗
(ξ)
ξ c

∗
(ξ) ϕ
ϕ(s)
ϕ(s)
ϕ(s)
(i) ϕ
(ii) ϕ(−∞) ∈ (π
0
, π
1
)
(iii) ϕ(s) = 0 ∀s  0.
c ξ
R
c,ξ
[a](s) ≡ max{ϕ(s), Q[a(c, ξ; x · ξ + s + c)](0)}.
R
Q
R
c,ξ
[α](s) > α(c, ξ; x · ξ + s + c) = α, α ∈ (π
0
, π
1
).
R
c,ξ
[α](s) ≡ max{ϕ(s), Q[α(c, ξ; x · ξ + s + c)](0)}
 Q[α(c, ξ; x · ξ + s + c)](0) = Q[α](0)
> α, Q,

α ∈ (π
0
, π
1
), ∀x ∈ H.
Q u  v
R
c,ξ
[u](s)  R
c,ξ
[v](s).
Q u  v
Q[u(c, ξ; x · ξ + s + c)](0)  Q[v(c, ξ; x · ξ + s + c)](0).
R
c,ξ
[u](s) ≡ max{ϕ(s), Q[u(c, ξ; x · ξ + s + c)](0)}
 max{ϕ(s), Q[v(c, ξ; x · ξ + s + c)](0)}
≡ R
c,ξ
[u](s).
u  v R
c,ξ
[u](s)  R
c,ξ
[v](s).
R
c,ξ
c ξ a
n
(c, ξ, s)

a
n+1
= R
c,ξ
[a
n
], a
0
= ϕ.
a
n
a
n
(c, ξ, s) n s c
c, ξ s
a
n
n
a
0
(c, ξ; s) = ϕ(s),
a
1
(c, ξ; s) = R
c,ξ
[a
0
](s)
= max{ϕ(s), Q[a
0

(c, ξ; x · ξ + s + c)](0)}
 ϕ(s) = a
0
(c, ξ; s).
R
c,ξ
a
n+1
(c, ξ; s)  a
n
(c, ξ; s).
a
n
(c, ξ; s) n
a
n
s c
n = 0 a
0
(c, ξ; s) = ϕ(s)
ϕ(s) ∀c

 c; ∀s

 s
ϕ(s

)  ϕ(s).
a
n

(c, ξ; s) c, s
∀c

 c; ∀s

 s
a
n
(c

, ξ; s

)  a
n
(c, ξ; s).
a
n+1
(c

, ξ; s

)  a
n+1
(c, ξ; s), ∀c

 c; ∀s

 s.
∀c


 c; ∀s

 s x · ξ + s

+ c

 x · ξ + s + c,
a
n
a
n
(c

, ξ; x · ξ + s

+ c

)  a
n
(c, ξ; x · ξ + s + c),
Q
Q[a
n
(c

, ξ; x · ξ + s

+ c)](0)  Q[a
n
(c, ξ; x · ξ + s + c)](0).

a
n+1
(c

, ξ; s

) = R
c

,ξ
[a
n
(c

, ξ; s

)]
= max{ϕ(s

), Q[a
n
(c

, ξ; x · ξ + s

+ c

)](0)}
 max{ϕ(s), Q[a
n

(c, ξ; x · ξ + s + c)](0)}
= a
n+1
(c, ξ; s).
a
n
(c, ξ; s) s c n
a
n
(c, ξ; s) c, s, ξ
n = 0 a
0
(c, ξ; s) = ϕ(s)
ϕ(s) a
0
(c, ξ; s) c, s, ξ
a
n
(c, ξ; s) c, s, ξ
(c
υ
, ξ
υ
, s
υ
) (c, ξ; s) υ → ∞ a
n
(c
υ
, ξ

υ
; s
υ
)
a
n
(c, ξ; s) υ → ∞
a
n+1
(c
υ
, ξ
υ
, s
υ
) a
n+1
(c, ξ; s) υ → ∞
x B
1
H R
|x|  R, ∀x ∈ B
1
(c
υ
, ξ
υ
, s
υ
) (c, ξ; s) υ → ∞ (c

υ
, ξ
υ
, s
υ
)
M
1
> 0; M
2
> 0; M
3
> 0 υ
0
> 0
∀υ > υ
0
|c|  M
1
, |ξ|  M
2
, |s|  M
3
,
|c
υ
|  M
1
, |ξ
υ

|  M
2
, |s
υ
|  M
3
.
|x · ξ + c + s|  |x|.|ξ| + |c| + |s| = M
1
+ RM
2
+ M
3
|x · ξ
υ
+ c
υ
+ s
υ
|  |x|.|ξ
υ
| + |c
υ
| + |s
υ
| = M
1
+ RM
2
+ M

3
.
M = [−M
1
; M
1
] × [−M
2
; M
2
] × [−(M
1
+ RM
2
+ M
3
); M
1
+ RM
2
+ M
3
]
M M
(c
υ
, ξ
υ
, x · ξ
υ

+ s
υ
+ c
υ
) (c, ξ, x · ξ + s + c) M
a
n
M a
n
M ∀ > 0, ∃δ > 0 ∀(c
1
, ξ
1
, s
1
), ∀(c
2
, ξ
2
, s
2
)
M
|(c
1
, ξ
1
, s
1
) − (c

2
, ξ
2
, s
2
)| < δ
|a
n
(c
1
, ξ
1
, s
1
) − a
n
(c
2
, ξ
2
, s
2
)| < .
(c
υ
, ξ
υ
, s
υ
) (c, ξ; s) υ → ∞ ∀δ > 0 ∃υ

0
> 0
∀υ > υ
0
|c
υ
− c| <
δ
3
, |ξ
υ
− ξ| <
δ
3R
, |s
υ
− s| <
δ
3
.
|x · ξ
υ
+ c
υ
+ s
υ
− (x · ξ + c + s)|  |x|.|ξ
υ
− ξ| + |c
υ

− c| + |s
υ
− s|
< R.
δ
3R
+
δ
3
+
δ
3
= δ.
∀ > 0, ∃δ > 0, ∃υ
0
> 0 ∀υ > υ
0
|c
υ
− c| <
δ
9
; |ξ
υ
− ξ| <
δ
9R
; |s
υ
− s| <

δ
9
,
∀x ∈ B
1
(c
υ
, ξ
υ
, x · ξ
υ
+ s
υ
+ c
υ
) (c, ξ, x · ξ + s + c) M
|(c
υ
, ξ
υ
, x · ξ
υ
+ s
υ
+ c
υ
) − (c, ξ, x · ξ + s + c)| < δ
|a
n
(c

υ
, ξ
υ
, s
υ
) − a
n
(c, ξ, s)| < ,
a
n
(c
υ
, ξ
υ
; x · ξ
υ
+ c
υ
+ s
υ
) a
n
(c, ξ; x · ξ + c + s)
H
Q[a
n
(c
υ
, ξ
υ

; x · ξ
υ
+ s
υ
+ c
υ
)](0) → Q[a
n
(c, ξ; x · ξ + c + s)](0)
υ → ∞
a
n+1
(c
υ
, ξ
υ
, s
υ
) = R
c
υ
,ξ
υ
[a
n
(c
υ
, ξ
υ
, s

υ
)]
= max{ϕ(s
υ
), Q[a
n
(c
υ
, ξ
υ
; x · ξ
υ
+ s
υ
+ c
υ
)](0)}.
ϕ(s
υ
) → ϕ(s)
Q[a
n
(c
υ
, ξ
υ
; x · ξ
υ
+ s
υ

+ c
υ
](0) → Q[a
n
(c, ξ; x · ξ + c + s)](0)
υ → ∞
lim
υ→∞
max{ϕ(s
υ
), Q[a
n
(c
υ
, ξ
υ
; x · ξ
υ
+ s
υ
+ c
υ
)](0)}
= max{ϕ(s), Q[a
n
(c, ξ; x · ξ + c + s))](0)},
lim
υ→∞
a
n+1

(c
υ
, ξ
υ
, s
υ
) = a
n+1
(c, ξ; s).
a
n
(c, ξ; s) c, s, ξ n ∈ N
α
n+1
= Q[α
n
], γ
n+1
= Q[γ
n
].
α
0
= ϕ(−∞), γ
0
= 0.
α
n
π
1

n → +∞
γ
n
γ
γ = Q[γ]
n → +∞
∀c, ∀ξ ∀n ∈ N
a
n
(c, ξ; −∞) = α
n
a
n
(c, ξ; +∞) = γ
n
.
Q[α
0
] > α
0
Q[γ
0
]  γ
0
.
α
n
γ
n
γ

n
 π
0
 α
n
 π
1
.
n → +∞
α
n
→ α, γ
n
→ γ
π
0
< α  π
1
0  γ  π
0
.
α = π
1
, γ = Q[γ].
π
0
 α < π
1
< +∞ Q
Q[α

n
] → Q[α] > α.
α
n
→ α n → +∞,
α
n+1
= Q[α
n
] → α
n → +∞.
α
n
Q[α] = α.
Q
Q[α] > α.
α = π
1
,
lim
n→+∞
α
n
= π
1
.
γ
n
γ
γ = Q[γ].

n = 0
a
0
(c, ξ; −∞) = ϕ(−∞) = α
0
.
a
n
(c, ξ; −∞) = α
n
,
a
n+1
(c, ξ; −∞) = α
n+1
.
a
n
(c, ξ; −∞) = α
n
lim
s→−∞
a
n
(c, ξ; s) = α
n
∀ > 0, ∃s
0
> 0( ) ∀s < −s
0

|a
n
(c, ξ; s) − α
n
| < .
∀x H ∃R > 0 |x|  R,
|x · ξ|  |x|.|ξ| = R.
∀ > 0, c ∃k
0
> 0 k
0
= [R + c + s
0
] + 1
∀k > k
0
x · ξ − k + c < −k
0
u
k
(x) ≡ a
n
(c, ξ; x · ξ − k + c)
|u
k
(x) − α
n
| = |a
n
(c, ξ; x · ξ − k + c) − α

n
| < .
u
k
(x) ≡ a
n
(c, ξ; x · ξ − k + c)
H k → +∞.
Q
lim
k→∞
Q[a
n
(c, ξ; x · ξ − k + c)] = Q[α
n
] = α
n+1
.
α
n+1
 α
0
 ϕ
lim
k→∞
a
n+1
(c, ξ; −k)] = α
n+1
.

a
n+1
s
a
n+1
(c, ξ; −∞) = α
n+1
.
a
n
(c, ξ; −∞) = α
n
, ∀n, ∀c, ∀ξ.
k −k
a
n
(c, ξ; +∞) = γ
n
, ∀n, ∀c, ∀ξ.