2.2.2 Multitype contact processes
If Γ
k α,β
= e Γ
k α,β
= 0, for all α, β ∈ X
2
, k ≥ 0, that is when no jumps of particles are present, all
rates are contact-type interactions. Such a process is called multitype contact process. Conditions 2.13–2.14 reduce to: for all
α, β ∈ X
2
, γ, δ ∈ X
2
, α, β ≤ γ, δ, h
1
≥ 0, j
1
≥ 0, i
X
k δ−β+ j
1
e Π
0,k α,β
≤ X
l j
1
Π
0,l γ,δ
; ii
X
k h
1
e Π
−k,0 α,β
≥ X
l γ−α+h
1
e Π
−l,0 γ,δ
. 2.19
Many different multitype contact processes have been used to study biological models. We propose some examples with the corresponding conditions. Since the state space Ω =
{0, 1, . . . , M}
Z
d
where M
∞ is compact, we refer to the construction in [11].
Spread of tubercolosis model [17]. Here M represents the number of individuals in a population at a site x
∈ Z
d
. The transitions are: P
1 β
= φβ1l
{0≤β≤M−1}
, R
0,1 α,β
= 2dλα1l
{β=0}
, P
−β β
= 1l
{1≤β≤M}
, px, y =
1 2d
1l
{x∼ y}
. where y
∼ x is one of the 2d nearest neighbours of site x. Given two systems with parameters
λ, φ, M and λ, φ, M , the proof of [17, Proposition 1] reduces to check Conditions 2.19:
φβ1l
{0≤β≤M−1}
+ 2dλα1l
{β=0}
≤φδ1l
{0≤δ≤M−1}
+ 2dλγ1l
{δ=0}
, if
β = δ, j
1
= 0 1l
{1≤α≤M,αh
1
}
≥1l
{1≤γ≤M,γγ−α+h
1
}
, if
γ ≥ α, h
1
≥ 0 which are satisfied if
λ ≤ λ, φ ≤ φ and M ≤ M. In the following examples we suppose f
S = S , that is we consider necessary and sufficient conditions for attractiveness.
2 -type contact process [14]. In this model M = 2. Since a value on a given site does not
represent the number of particles on that site, we write the state space {A, B, C}
Z
d
. The value B represents the presence of a type-B species, C the presence of a type-C species and A an empty site.
If A = 0, B = 1, C = 2 then the transitions are R
0,1 α,β
= 2dλ
1
1l
{α=1,β=0}
, R
0,2 α,β
= 2dλ
2
1l
{α=2,β=0}
, P
−β β
= 1l
{1≤β≤2}
, px, y =
1 2d
1l
{x∼ y}
. By taking h
1
= 0, Condition 2.19 is X
k δ−β
1l
{k=1}
2d λ
1
1l
{α=1,β=0}
+1l
{k=2}
2d λ
2
1l
{α=2,β=0}
≤ 2dλ
1
1l
{γ=1,δ=0}
+ 2dλ
2
1l
{γ=2,δ=0}
; By taking
β = 0, δ = 1, α = γ = 2 we get 2dλ
2
≤ 0, which is not satisfied since λ
2
0. As already observed, see [20, Section 5.1], one can get an attractive process by changing the order between
species: namely by taking A = 1, B = 0 and C = 2 the process is attractive.
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2.2.3 Conservative dynamics
Kategori