The Cauchy problem for the nonlinear integro-partial differential equation in quantitative sociodynamics

doi:10.1016/S0096-3003(01)00210-7

Applied Mathematics and Computation

Volume 132, Issues 2–3, 10 November 2002, Pages 537-552

https://doi.org/10.1016/S0096-3003(01)00210-7 Get rights and content

Abstract

The master equation is a nonlinear integro-partial differential equation, which describes the evolution of various quantities in quantitative sociodynamics. For example, the master equation can describe interregional migration. The purpose of this paper is to obtain asymptotic estimates for solutions to the Cauchy problem for the equation.

Introduction

Large free economic unions such as EU and NAFTA have been established recently. In such free trade unions, goods are traded freely, but interregional labor mobility is restricted at a certain level of rigidity. However, there is now a move to abolish the restriction entirely. If no restriction is imposed on the regional labor mobility, and if there exists regional economic disparity, then workers will move so as to achieve a higher income. This phenomenon is called interregional migration motivated by regional economic disparity, and it is known in [3], [4] and [11], [12] that the master equation can describe such a phenomenon (see, e.g., [1], [2], [5], [13], [14] for the theory of interregional migration). The equation plays very important roles in quantitative sociodynamics (see, e.g., [4]). Furthermore, the master equation approach is taken also in nonlinear evolutionary economics (see, e.g., [10]).

The master equation is a nonlinear integro-partial differential equation, which has the form: $∂ ν(t,x)/ ∂ t=−w(t,x)ν(t,x)+∫_{y∈D} W(t;x|y)ν(t,y) d y,$ where D is a bounded Lebesgue measurable set included in the 2-dimensional Euclidean space. By ν=ν(t,x) we denote the unknown function which represents the density of population at time t⩾0 and at a point x∈D. By W=W(t;x|y) we denote the transition rate at time t⩾0 and from a point y∈D to a point x∈D. The coefficient w=w(t,x) is defined from the transition rate as follows: $w=w(t,x)≔∫_{y∈D} W(t;y|x) d y.$ The master equation has its origin in statistical physics, and has been fully studied in mathematical physics. However, the transition rate of the master equation studied in quantitative sociodynamics is completely different from that treated in statistical physics. Hence we cannot apply various methods developed in statistical physics to the master equation studied in quantitative sociodynamics. There have been few studies on the master equation treated in quantitative sociodynamics except for [6], [7], [8]. Therefore it is important to investigate the master equation treated in quantitative sociodynamics (we simply call the master equation studied in quantitative sociodynamics “the master equation”).

In the same way as [4, pp. 137–138] and [12, pp. 81–100], we will impose the following assumption on the transition rate W=W(t;x|y) in this paper.

Assumption 1.1

The transition rate W=W(t;x|y) has the following form: W(t;x|y)=θ(t)exp{U(t,x)−U(t,y)−E(x,y)}, where θ=θ(t) denotes the flexibility at time t⩾0, U=U(t,x) is the utility at time t⩾0 and at a point x∈D, and E=E(x,y) denotes the effort from a point y∈D to a point x∈D.

See, e.g., [4, p. 137] for the sociodynamic definitions of flexibility, utility, and effort. In the same way as [8], in this paper we make the following assumption (see [8] for the reasons for making this assumption).

Assumption 1.2

The flexibility θ=θ(t) and the effort E=E(x,y) are identically equal to positive constants.

Let us discuss the utility. In a real world we often observe that the utility increases with the population density. If such a phenomenon is observed, then we say that imitative process works. In order to assume that imitative process works in interregional migration, in [8] we impose the following assumption on the utility (by this assumption, in [8] we fully investigate the asymptotic behavior of solutions to the Cauchy problem for the master equation).

Assumption 1.3

The utility U=U(t,x) has the form $U(t,x)=c_{1.1} ν (t,x)+c_{1.2}$ , where $ν = ν (t,x)≔ν(t,x)/∥ν(t,·)∥_{L^{1}(D)}$ (we denote the norm of L¹(D) by ∥·∥_L¹(D)), c_1.1 is a positive constant, and c_1.2 is a real constant.

It is plausible to assume that imitative process works at a certain degree. However, in a real world, we observe that if the density of population is sufficiently large, then the utility does not increases with the population density, and moreover we find that over population makes the utility decrease. If such a phenomenon is observed, then we say that avoidance process works. In [8] we assume that only imitative process works, but in this paper we take not only imitative process but also avoidance process into account. Hence, for example, we need to assume that the utility U=U(t,x) is a strictly concave function of $ν = ν (t,x)$ which monotonously increases (decreases, respectively) with $ν ∈[0,k)$ $(ν ∈(k,+∞)$ , respectively), where k is a positive constant. Therefore in the present paper we will make the following assumption in place of Assumption 1.3.

Assumption 1.4

The utility has the form $U(t,x)≔−(α_{1} ν (t,x)−α)^{2} +α_{2}$ , where α and α₁ are positive constants, and α₂ is a real constant.

We will impose Assumption 1.1, Assumption 1.2, Assumption 1.4 on this paper. In the same way as [6], [7], [8], we can prove that the Cauchy problem for the master equation has a unique positive-valued local solution (Proposition 2.1). Combining this result and a priori estimates for solutions (Lemma 4.1), we can demonstrate that the Cauchy problem has a unique positive-valued global solution (Theorem 4.2). The purpose of this paper is to prove that if certain assumptions are made, then each global solution to the Cauchy problem converges to a stationary solution (Theorem 4.3). This paper has six sections in addition to this section. In Section 2 we give preliminaries. In Section 3 we obtain all the stationary solutions of the master equation. In Section 4 we present the main result, which will be proved in 5 Estimates for, 6 Estimates for solutions, 7 Proof of the main theorem.

Remark 1.5

(i) In [12, pp. 92–96] Assumption 1.4 is proved in the sociodynamic level of rigor (see [12, (4.15)–(4.19)]).
(ii) We can apply the results of this paper and [8] to economics. This subject will be discussed in [9].

Section snippets

Preliminaries

Integrating both sides of (1.1) with respect to x∈D, in the same way as [6], [7], [8] we obtain the conservation law of total population, ∥ν(t,·)∥_L¹(D)=∥ν(0,·)∥_L¹(D) for each t⩾0. Hence, $ν (t,x)=ν(t,x)/∥ν(0,·)∥_{L^{1}(D)}$ (see Assumption 1.3 for ∥·∥_L¹(D) and $ν = ν (t,x)$ ). Assumption 1.1, Assumption 1.2, Assumption 1.4 give $W(t;x|y)=α_{3} exp −(α_{1} ν (t,x)−α)^{2} +(α_{1} ν (t,y)−α)^{2},$ where α₃ is a positive constant. Let us rewrite (1.1) with (2.1) by introducing the new unknown function $u=u(t,x)≔α_{1} ν (t/α_{3} |D|,|D|^{1/2} x)$ in

Stationary solutions

Let us rewrite Eq. (2.2) as follows: $∂ u(t,x)/ ∂ t=a(u(t,·))g_{α} (u(t,x)){−f_{α} (u(t,x))+b(u(t,·))/a(u(t,·))},$ where f_α=f_α(z)≔ze^2(z−α)² and g_α=g_α(z)≔e^{−(z−α)²}. See Assumption 1.4 for α. Noting that a(u(t,·))g_α(u(t,x))>0, we see that the following equation is a sufficient and necessary condition for u=u(x) to be a stationary solution of (2.2): $f_{α} (u(x))=b(u(·))/a(u(·)).$

We can easily prove the following lemma (hence we omit the proof).

Lemma 3.1

(i) f_α(0)=0,f_α(z)>0 for each z>0, lim_z→+∞f_α(z)=+∞.
(ii) If 0<α⩽1, then f_α=f_α(z

The main result

Let us prove a priori estimates for solutions of (CP).

Lemma 4.1

If (CP) has a solution u=u(t,x) in [0,T], where T is a positive constant, then the solution satisfies the following (i)–(ii):

(i) If 0<α⩽1, then $u_{0,−} ⩽ ess inf_{(t,x)∈[0,T)×Ω} u(t,x), ess sup_{(t,x)∈[0,T)×Ω} u(t,x)⩽u_{0,+}$ .
(ii) If α>1, then $min {u_{0,−},β_{1}}⩽ ess inf_{(t,x)∈[0,T)×Ω} u(t,x), ess sup_{(t,x)∈[0,T)×Ω} u(t,x)⩽ max {u_{0,+},β_{4}}$ .

Proof

See Assumption 1.4, 2 Preliminaries, 3 Stationary solutions for α, u_0,±, and β_j, j=1,4. We will prove only the second inequality of (ii), since

Estimates for R(t)

From (4.8) (see Remark 4.4(i)) we have $c_{5.1} ≔ ess inf_{t⩾0,x∈Ω} a(u(t,·))g_{α} (u(t,x))>0.$ Applying this inequality to (3.1), we find the need to inspect the supremum and the infimum of R=R(t) (see proof of Lemma 4.1 for R=R(t)).

Lemma 5.1

γ₁<R(t)<γ₂ for each t⩾0.

Proof

If the equal sign of (4.3) holds at some t⩾0, then by performing calculations similar to those done in proof of Lemma 4.1, we deduce that u₀=u₀(x) is a stationary solution. By Remark 4.4(ii), (4.4), Lemma 3.1, Lemma 3.2, Lemma 3.3 and Proposition 3.4, we see

Estimates for solutions

Making use of Proposition 5.2, we will seek estimates for u=u(t,x). The purpose of this section is to prove Lemma 6.1, Lemma 6.2.

Lemma 6.1

If there exist positive-valued sequences ${t_{n,1}}_{n∈N}$ and ${t_{n,2}}_{n∈N}$ such that lim_n→+∞t_n,j=+∞ and lim_n→+∞R(t_n,j)=R_j, j=1,2, then lim_n→+∞|B_j(t_n,j,δ)|=0 for each δ>0, j=1,2, where $B_{j} (t,δ)≔{x∈Ω; (−1)^{j} f_{α} (u(t,x))⩾(−1)^{j} R_{j} +δ}$ , j=1,2.

Proof

Let us prove this lemma only when j=2 by contradiction (the lemma with j=1 can be obtained in the same way). Assume that the negation holds. Applying

Proof of the main theorem

We employ the following proposition:

Proposition 7.1

(i) $|Ω_{2,∞} |=0$ , where $Ω_{2,∞} :=⋂_{t⩾0} Ω_{2} (u(t,·))$ .
(ii) R₁=R₂.

Proof

Let us prove (i) by contradiction. Assume that $Ω_{2,∞}$ is not a null set. Noting that $Ω_{2,∞} ⊆Ω_{2} (u_{0} (·))$ (see (4.9)), and making use of (4.5), we deduce that there exist $X_{1},X_{2} ∈Ω_{2,∞}$ such that u₀(X₁)<u₀(X₂) and |S_j(0)|>0, j=1,2, where $S_{j} (t):={x∈Ω_{2,∞};(−1)^{j} u(t,x)⩾(−1)^{j} u(t,X_{j})}$ , t⩾0, j=1,2. By Proposition 2.1(iv), we obtain $u(t,X_{1})<u(t,X_{2}) and S_{j} (t)=S_{j} (0), j=1,2, for each t⩾0.$

Replace x by X_j in (2.2), j=1,2. Subtracting one

References (14)

M. Tabata et al.
The Cauchy problem for the system of equations describing migration motivated by regional economic disparity
Appl. Math. Comput.
(1998)
M. Tabata et al.
The nonlinear integro-partial differential equation describing the logistic growth of human population with migration
Appl. Math. Comput.
(1999)
M. Tabata et al.
The behavior of solutions to the Cauchy problem for the master equation
Appl. Math. Comput.
(2000)
A.E. Åndersson et al.
Economic models of migration
D.S. Dendrinos et al.
Chaos and Socio-spatial Dynamics
(1990)
G. Haag et al.
A dynamic migration theory and its evaluation of concrete systems
Reg. Sci. Urban Econ.
(1986)
D. Helbing
Quantitative Sociodynamics
(1995)

There are more references available in the full text version of this article.

Cited by (0)

View full text

The Cauchy problem for the nonlinear integro-partial differential equation in quantitative sociodynamics

Abstract

Introduction

Section snippets

Preliminaries

Stationary solutions

The main result

Estimates for R(t)

Estimates for solutions

Proof of the main theorem

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Economic models of migration

Chaos and Socio-spatial Dynamics

A dynamic migration theory and its evaluation of concrete systems

Reg. Sci. Urban Econ.

Quantitative Sociodynamics