An Example for Convergence of Environment-Dependent Spatial Models

‒ 179 ‒

An Example for Convergence of Environment-Dependent Spatial Models

DÔKU, Isamu

Faculty of Education, Saitama University

Summary

In this paper we consider an environment-dependent spatial model. Actually, this random model is closely related to some of the stochastic interacting system in Liggett [23] (1999). We shall show that rescaled processes of the models converge to a Dawson-Watanabe superprocess with suitable parameters. Our formulation of measure-valued branching Markov processes [17] is greatly due to a martingale problem formalism. The first step toward a transformation of spatial model into a superprocess is based upon construction of related empirical measures.

Key Words: environment-dependent spatial model, convergence, superprocess, interaction, mar- tingale problem, measure-valued process.

1. Introduction

In this section we shall introduce an environment-dependent random model [25]. Let Z

^d

be a d-dimensional integer lattice, and we suppose that each site on Z

^d

is occupied by all means by ei- ther one of the two species. At each random time passed, a particle dies and is replaced by a new one, but the random time and the type chosen of the species are assumed to be determined by the environment conditions around the particle. The random function ξ

t

≡ ξ

t

( x) : Z

^d

→ {0, 1} denotes the state at time t, and each number of {0, 1} denotes the label of the type chosen of the two spe- cies. When we set ∥ y ∥

∞

: = max

i

y

i

for y = (y

₁

, . . . , y

d

) , then we define

(1) where R is a positive constant. For i = 0, 1, let f

i

(x, ξ) be a frequency of appearance of type i in the neighborhood N

x

of x for ξ . In other words,

(2)

For non-negative parameters α

ij

≥ 0, the dynamics of ξ

t

is defined as follows. The state ξ makes tran- sition 0 → 1 at rate

(3) and it makes transition 1 → 0 at rate

An Example for Convergence of Environment-Dependent Spatial Models

D ˆ OKU, Isamu

Faculty of Education, Saitama University

Summary

In this paper we consider an environment-dependent spatial model. Actually, this random model is closely related to some of the stochastic interacting system in Liggett [23] (1999). We shall show that rescaled processes of the models converge to a Dawson-Watanabe superprocess with suitable parameters. Our formulation of measure-valued branching Markov processes [17]

is greatly due to a martingale problem formalism. The first step toward a transformation of spatial model into a superprocess is based upon construction of related empirical measures.

Key Words: environment-dependent spatial model, convergence, superprocess, interaction, martingale problem, measure-valued process.

1. Introduction

In this section we shall introduce an environment-dependent random model [25]. Let Z

^d

be a d-dimensional integer lattice, and we suppose that each site on Z

^d

is occupied by all means by either one of the two species. At each random time passed, a particle dies and is replaced by a new one, but the random time and the type chosen of the species are assumed to be determined by the environment conditions around the particle. The random function ξ

t

≡ ξ

_t

(x) : Z

^d

→ { 0, 1 } denotes the state at time t, and each number of { 0, 1 } denotes the label of the type chosen of the two species. When we set ∥ y ∥

∞

:= max

i

y

i

for y = (y

1

, . . . , y

d

), then we define

N

x

:= x + { y : 0 < ∥ y ∥

∞

R } , (1) where R is a positive constant. For i = 0, 1, let f

i

(x, ξ) be a frequency of appearance of type i in the neighborhood N

x

of x for ξ. In other words,

f

i

(x) ≡ f

i

(x, ξ) := # { y : ξ

t

(y) = i ; y ∈ N

x

}

# N

^x

. (2)

For non-negative parameters α

ij

≥ 0, the dynamics of ξ

t

is defined as follows. The state ξ makes transition 0 → 1 at rate

λf

1

(f

0

+ α

01

f

1

) λf

1

+ f

0

, (3)

An Example for Convergence of Environment-Dependent Spatial Models

D ˆ OKU, Isamu

Faculty of Education, Saitama University

Summary

In this paper we consider an environment-dependent spatial model. Actually, this random model is closely related to some of the stochastic interacting system in Liggett [23] (1999). We shall show that rescaled processes of the models converge to a Dawson-Watanabe superprocess with suitable parameters. Our formulation of measure-valued branching Markov processes [17]

is greatly due to a martingale problem formalism. The first step toward a transformation of spatial model into a superprocess is based upon construction of related empirical measures.

Key Words: environment-dependent spatial model, convergence, superprocess, interaction, martingale problem, measure-valued process.

1. Introduction

In this section we shall introduce an environment-dependent random model [25]. Let Z

^d

be a d-dimensional integer lattice, and we suppose that each site on Z

^d

is occupied by all means by either one of the two species. At each random time passed, a particle dies and is replaced by a new one, but the random time and the type chosen of the species are assumed to be determined by the environment conditions around the particle. The random function ξ

_t

≡ ξ

t

(x) : Z

^d

→ { 0, 1 } denotes the state at time t, and each number of { 0, 1 } denotes the label of the type chosen of the two species. When we set ∥ y ∥

_∞

:= max

i

y

i

for y = (y

1

, . . . , y

d

), then we define

N

x

:= x + { y : 0 < ∥ y ∥

∞

R } , (1) where R is a positive constant. For i = 0, 1, let f

i

(x, ξ) be a frequency of appearance of type i in the neighborhood N

x

of x for ξ. In other words,

f

i

(x) ≡ f

i

(x, ξ) := # { y : ξ

t

(y) = i ; y ∈ N

x

}

# N

^x

. (2)

For non-negative parameters α

ij

≥ 0, the dynamics of ξ

t

is defined as follows. The state ξ makes transition 0 → 1 at rate

λf

1

(f

0

+ α

01

f

1

)

λf

₁

+ f

₀

, (3)

An Example for Convergence of Environment-Dependent Spatial Models

D ˆ OKU, Isamu

Faculty of Education, Saitama University

Summary

In this paper we consider an environment-dependent spatial model. Actually, this random model is closely related to some of the stochastic interacting system in Liggett [23] (1999). We shall show that rescaled processes of the models converge to a Dawson-Watanabe superprocess with suitable parameters. Our formulation of measure-valued branching Markov processes [17]

is greatly due to a martingale problem formalism. The first step toward a transformation of spatial model into a superprocess is based upon construction of related empirical measures.

Key Words: environment-dependent spatial model, convergence, superprocess, interaction, martingale problem, measure-valued process.

1. Introduction

In this section we shall introduce an environment-dependent random model [25]. Let Z

^d

be a d-dimensional integer lattice, and we suppose that each site on Z

^d

is occupied by all means by either one of the two species. At each random time passed, a particle dies and is replaced by a new one, but the random time and the type chosen of the species are assumed to be determined by the environment conditions around the particle. The random function ξ

t

≡ ξ

t

(x) : Z

^d

→ { 0, 1 } denotes the state at time t, and each number of { 0, 1 } denotes the label of the type chosen of the two species. When we set ∥ y ∥

∞

:= max

_i

y

_i

for y = (y

₁

, . . . , y

_d

), then we define

N

^x

:= x + { y : 0 < ∥ y ∥

∞

R } , (1) where R is a positive constant. For i = 0, 1, let f

i

(x, ξ) be a frequency of appearance of type i in the neighborhood N

^x

of x for ξ. In other words,

f

_i

(x) ≡ f

_i

(x, ξ) := # { y : ξ

_t

(y) = i ; y ∈ N

x

}

# N

x

. (2)

For non-negative parameters α

ij

≥ 0, the dynamics of ξ

t

is defined as follows. The state ξ makes transition 0 → 1 at rate

λf

1

(f

0

+ α

01

f

1

) λf

1

+ f

0

, (3)

埼玉大学紀要　教育学部，65（1）：179－186（2016）

‒ 180 ‒

(4) The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result ) with the particle of type 1. The density-dependent death rate f

i

+ α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being pro- portional to the weighted density between the two species, expressed by a parameter λ . Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher pro- liferation rate than the type 0. In this article we shall discuss some convergence result of the envi- ronment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N, and we put 

N

: = M

N

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

)

λf

₁

+ f

₀

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

_i

+ α

_ij

f

_j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+ α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

_N

) \ { 0 } is defined as a random vector satisfying (i) L (W

_N

) = L ( − W

_N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P(W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

_t^N

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

^N_t

by the symbol Res(p

_N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

₀

in M

_F

( R

^d

) (N → ∞ ), (7) , and S

N

: = Z

^d

/ 

N

. And also W

N

= (

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

_i

+ α

_ij

f

_j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying (i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

_N

(x) := P (W

_N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

_t^N

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

^N_t

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) , . . . ,

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

_i

+ α

_ij

f

_j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+ α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

_N

) \ { 0 } is defined as a random vector satisfying (i) L (W

_N

) = L ( − W

_N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

^N_t

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+α

₀^N

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) ) ∈ ( Z

^d

/

M

N

) \ {0} is defined as a random vector satisfying (i) L (W

N

) = L ( − W

N

); (ii) E(

and it makes transition 1 → 0 at rate

f

₀

(f

₁

+ α

₁₀

f

₀

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+ α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying (i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

_N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

^N_t

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+α

₀^N

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) ) →

δ

ij

σ

²

( ≥ 0 ) ( as N → ∞ ) ; ( iii ) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y. For the kernel p

N

(x) := P (W

N

/

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+ α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying (i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

^N_t

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

_x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) = x), x ∈ S

N

and ξ ∈ {0, 1}

^SN

,

we define the scaled frequency and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

_i

+ α

_ij

f

_j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

_N

) \ { 0 } is defined as a random vector satisfying (i) L (W

_N

) = L ( − W

_N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P(W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

_t^N

the state determined by the scaled frequency depending on α

_i^N

and p

N

. As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

^N_t

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+ α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

^N_t

by the symbol Res(p

_N

, α

_i^N

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) as

(5)

We denote by ξ

^Nt

the state determined by the scaled frequency depending on α

^Ni

and p

N

. As a matter of fact, the rescaled process ξ

^Nt

: S

^N

 x → ξ

^Nt

(x) ∈ {0, 1} is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

)

λf

₁

+ f

₀

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+ α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

_N

∈ N , and we put ℓ

_N

:= M

_N

√

N , and S

^N

:= Z

^d

/ℓ

_N

. And also W

_N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying (i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

_t^N

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

^N_t

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

₁^N

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7)

, or else it makes tran- sition 1 → 0 at rate

and it makes transition 1 → 0 at rate

f

₀

(f

₁

+ α

₁₀

f

₀

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+ α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying (i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

_N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

_t^N

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

^N_t

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7)

. We denote the rescaled process ξ

^Nt

by the symbol Res (p

N

, α

^Ni

) . On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

(6) For the initial value X

^N₀

, we assume that

(7) where M

F

( R

^d

) is the totality of all the finite measures on R

^d

, equipped with the topology of weak convergence. For a finite measure µ ∈ M

F

(E) with a topological space E, we use the notation  µ,

  = ∫

E

(x)µ(dx) for integral of a measurable function  over E with respect to a measure µ on E. Note that the convergence in (7) is that in the sense of weak convergence for measures [20].

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

_i

+ α

_ij

f

_j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+ α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

_N

) \ { 0 } is defined as a random vector satisfying (i) L (W

_N

) = L ( − W

_N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

_t^N

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

^N_t

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

_N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

_N

∈ N , and we put ℓ

_N

:= M

_N

√

N , and S

^N

:= Z

^d

/ℓ

_N

. And also W

_N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying (i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

_t^N

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

^N_t

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+ α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

^N_t

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

i

+ α

ij

f

j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying (i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

_ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

_N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

_t^N

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

^N_t

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

The above-mentioned rate can be interpreted as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. The density-dependent death rate f

_i

+α

_ij

f

_j

consists of the intraspecific and interspecific competitive effects. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent. When λ ≥ 1, then it means that the type 1 has a higher proliferation rate than the type 0. In this article we shall discuss some convergence result of the environment-dependent spatial models.

2. Scaling rule and the associated measure-valued process

For brevity’s sake we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

) ∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying (i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ ); (iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable. Here L (Y ) indicates the law of a random variable Y . For the kernel p

_N

(x) := P (W

_N

/ √

N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) We denote by ξ

_t^N

the state determined by the scaled frequency depending on α

^N_i

and p

N

. As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

^N_t

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+ α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

). On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

^N_t

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7)