Мазалов В.В. (ред.) - Математические методы в экологии. Тезисы докладов Третьей Всероссийской школы молодых ученых
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m σ v
m σ v
∂x
∂t
+ div
α
(µ ◦ x) = D(x) + F (t, α, x, u),
x(t, α) = x
0
(α),
x(t, α)|
α∈∂A
= ¯x(t, α),
R
T ×A
ϕ(t, α, x, u)dαdt → sup
u∈U
.
(1)
t t ∈ T = [t; t] ⊂ R α
α ∈ A ⊂ R
m
x(t, α)
x ∈ G ⊂ H
2
(T ×A; R
n
)
u(t, α) u ∈ U ⊂ L
2
(T × A; R
m
)
µ = (µ
jk
)
n,m
j,k=1
div
◦ µ◦x = (µ
jk
x
j
)
n,m
j,k=1
A = (a
ij
)
m,n
i,j=1
B = (b
ij
)
n,m
i,j=1
A∗B = c
c = (c
i
)
m
i=1
c
i
=
P
n
j=1
a
ij
b
ji
D(x)
D(x) = (D
j
(x
j
))
n
j=1
, D
j
(x
j
) =
m
X
k=1
∂
∂α
k
[d
jk
∂x
j
∂α
k
].
d = (d
jk
)
n,m
j,k=1
µ
µ(t, α)|
α∈∂A
= 0,
x ∈ H
2
(T ×
A; R
n
) u ∈ U ⊂ L
2
(T × A; R
m
) U
u t, α, x ϕ
u F u
λ(t, α)
λ(t, α) = (λ
j
(t, α))
n
j=1
, λ(t, α) = 0, λ(t, α)|
α∈∂A
= 0.
L(t, α, x, u, λ) = ϕ(t, α, x, u) + λ(t, α)F (t, α, x, u).
∂x
∂t
+ div
α
(µ ◦ x) = D(x) + F (t, α, x, bu),
∂λ
∂t
+ (µ ∗
∂λ
∂α
)
∗
+
¯
D(λ) = −
∂L(t,α,x,bu,λ)
∂x
,
x(0, α) = x
0
(α), x(t, α)|
α∈∂A
= ¯x(t, α),
λ(t, α) = 0, λ(t, α)|
α∈∂A
= 0,
bu(t, α) = arg max
u∈U
L(t, α, bx, u, λ)
α, u
D
∂x
∂t
+
∂(µx)
∂α
= D
∂
2
x
∂α
2
+ F (t, α, x, u),
x(0, α) = x
0
(α),
x(t, α) = x(t), x(t, α) = x(t),
R
T ×A
ϕ(t, α, x, u)dαdt → sup
u∈U
.
(2)
A = [α; α], µ(t, α) = µ(t, α) = 0.
∂x
∂t
+
∂(µx)
∂α
= D
∂
2
x
∂α
2
+ F (t, α, x, bu ),
∂λ
∂t
+ µ
∂λ
∂α
+ D
∂
2
λ
∂α
2
= −
∂L(t,α,x,bu,λ)
∂x
,
x(0, α) = x
0
(α), x(t, α) = x(t), x(t, α) = x(t)
λ(t, α) = 0, λ(t, α) = λ(t, α) = 0,
bu(t, α) = arg max
u∈U
L(t, α, x, u, λ).
α
α ∈ [0; 1] D = 0
∂x
∂t
+
∂(µx)
∂α
= F (t, α, x, u),
x(0, α) = x
0
(α), x(t, 0) = x
0
(t),
R
T ×A
ϕ(t, α, x, u)dαdt → sup
u∈U
.
(3)
µ(t, α) > 0, µ(t, α) 6 0
∂x
∂t
+
∂(µx)
∂α
= F (t, α, x, bu),
∂λ
∂t
+ µ
∂λ
∂α
= −
∂L(t,α,x,bu,λ)
∂x
,
x(t, α) = x
0
(α), x(t, α) = x(t),
λ(t, α) = 0, λ(t, α) = 0,
bu(t, α) = arg max
u∈U
L(t, α, x, u, λ).
− −
u
Q(w, u, y)
w
Q
∗
(w, u
∗
, y) = min
u∈U
Q(w, u, y) , (1)
Q y U
y = Ψ(w, u, p) (2)
R
1
(w, u) > 0
(3)
R
k
(w , u) > 0
L
1
(y) > 0
(4)
L
m
(y) > 0 .
p Ψ, R, i = 1, .., k,
L, i = 1, .., m
CO
2
O
2
N
2
i (i + 1)
i
x
i
< x
∗
x
i
> x
∗
(i+1) i
(i + 1)
˙x
i
= ua
1i
+
µ
i
s
i
x
i
k
i
+ s
i
− (b + u)x
i
,
˙s
i
= ba
2i
−
µ
i
s
i
x
i
Y
i
(k
i
+ s
i
)
− (b + u)s
i
, i = 1, ..., n(t),
i i
s
i
x
i
Y
i
k
i
b, a
2i
u, a
1i
µ
i
u
n(t)
n = N − k u
k
< u < u
k+1
,
k +1 = i
u
k
u k = 0, 1, ..., N −1
N
γ γ
i
γ
i
= 1 i
γ
i
= 0