The Dynamics of a Spatial Economic Model with Bounded Population Growth

We investigate a spatial economic growth model with bounded population growth to obtain the asymptotic behavior of detrended capital in a continuous space. The formation of capital accumulation is expressed by a partial diﬀerential equation with corresponding boundary conditions. The capital accumulation interacts with the morphology to aﬀect the optimal dynamics of economic growth. After redrafting the spatial growth model in the inﬁnite dimensional Hilbert space, we identify the unique optimal control and value function when the bounded population growth is considered. With nonnegative initial distribution of capital, the explicit solution of the model is obtained. The time behavior of the explicit solution guarantees the convergence issue of the detrended capital level across space and time.


Introduction
Distribution of economic activities across space and time has been widely investigated in many literature [1][2][3][4][5], where economic geographers apply economic growth models to analyze the relationship between production agglomeration and location choices of people, reasons for migration flows, and formation of cities. In early studies of regional economic growth models, Beckmann [6] and Isard and Liossatos [7] discussed flows of labor and pollution, capital accumulation, and individuals welfare in discrete and continuous spaces, respectively. Recently, as one important issue of economic research studies, optimal market allocation has been considered in economic growth models with capital accumulation. Among relevant economic literature, Brito [8] has investigated the optimal dynamics of spatial capital accumulation in a continuous space. In the work of Brito [8], optimal consumption distribution is associated with the capital accumulation, which drives the dynamic processes of spatial capital growth and distribution. Maximizing an objective function in a continuous dimension, Boucekkine et al. [9] illustrated how and why policy makers choose optimal trajectories of the capital and consumption in a spatial Ramsey model. Considering a cyclic space, Boucekkine et al. [10] described characterizations of the optimal capital dynamics in the spatial Ramsey model with AK production function, which indicates that technology adoption decisions are closely tied to technological embodiment, capital, and goods. In a generic geographic structure, Fabbri [11] generalized the spatial AK economic growth model presented in [10] to obtain a generalized qualitative behavior of the economy.
In global economic growth systems, several factors affect the formation of capital accumulation. ese factors include population growth, environmental protection, and different production processes. For example, Guerrini [12] studied the one-dimensional Ramsey model with a population growth factor. e population growth rate is considered to be a variable, which follows the logistic growth law. La Torre et al. [13] discussed the interplay of pollution diffusion and capital accumulation in economic geography models, where the dynamics of spatial capital is driven by the pollution diffusion. In Kamihigashi and Roy [14], a discontinuous production function is applied to show the convergence of the capital stock when an optimal growth problem with the condition of discrete time is considered. Brianzoni et al. [15] considered the optimal dynamics of capital accumulation in a discrete economic model with one sector, in which nonconcave production factors lead to the complexity of capital stock in local and global economic systems. Capasso et al. [16] analyzed the relationship between industrial dynamics and economic geography to enrich the theory of spatial economy.
In general, it is very hard to obtain the explicit solutions of spatial economic growth systems. us, the equilibrium distribution of spatial capital is widely considered in many research studies. Camacho and Zou [17] discussed existence of the solution for a spatial growth system with local diffusion and obtained convergence properties of capital distribution in the long run. Under assumption of a nonconcave production function, Capasso et al. [18] investigated the steady state of a spatial Solow model with technological diffusion. Brock et al. [19] applied a mathematical method to study spatial spillover in an economic growth model, in which capital mobility is excluded since the diffusion term disappears from the state equation. In the long run, Xepapadeas and Yannacopoulos [20] illustrated stable distribution of per capita capital across space. e stable capital distribution is related to the growth process of economy and shows nonhomogeneous properties. In autarkic and open regions, Yue and Wenyi [21] depicted the formation of capital distribution by applying the equilibrium solution of a generalized growth model, respectively. For more contributions to the study of growth models, the reader is referred to [22][23][24][25][26][27][28][29] and the references therein.
In this paper, we introduce a bounded population growth factor into the spatial economic growth model presented in [10,11] and analyze the optimal dynamics of the model in a continuous space. In our model, the optimal behavior of the economic growth is affected by the interaction between the morphology and the dynamics of spatial capital. e long-run convergence of the spatial capital is closely associated with the bounded population growth factor and geographical environment. After redrafting the spatial growth model in the infinite dimensional Hilbert space, we use the Hamilton-Jacobi-Bellman (HJB) equation and a linear production function to derive an explicit value function of the model. e value function is used to confirm the optimal control of the model in feedback form. Due to application of the optimal control, an explicit solution of the growth system is obtained in close form, which satisfies firstorder optimality conditions to describe the time behavior of per capita capital. With nonnegative initial distribution of capital, we apply the explicit solution and Fourier series to express spatial detrended capital of our model. e asymptotic behavior of the spatial detrended capital distribution is proved to show that the capital distribution converges to a homogeneous result in the long run.
In particular, we extend parts of works in [10,11] by embedding the bounded population growth factor into the spatial growth model. In our discussion, the bounded population growth is one determinant of the spatial capital dynamics. When the bounded population growth rate is considered, we identify the unique optimal control and value function, which are used to derive the explicit solution of the model. Compared with the works in [10,11], we apply the explicit solution and Fourier series to obtain an expression of the spatial detrended capital which helps to prove the longrun convergence of the spatial detrended capital. is is the main contribution in our work. In addition to referring the research methods in [10,11], our work is also related to the study of Chatterjee [30], where the optimal control problem of a growth model with heterogeneous initial endowments is solved.
is paper is organized as follows. Section 2 states the growth model with bounded population growth and special spatial setting. In Section 3, the spatio-temporal capital dynamics of the growth model is analyzed by identifying the unique optimal control and the explicit value function. In Section 4, we apply the time behavior of per capita capital in close form to prove convergent properties of the spatial detrended capital distribution. Some numerical results are illustrated in Section 5, and our conclusions are presented in Section 6.

A Model with Bounded Population Growth.
Let Ω be an d-dimensional geographic structure (d � 2 or 3) and without boundary. We consider that economic activities are distributed homogeneously in geography Ω. Assume that x ∈ Ω is a generic spatial variable and k(x, t) denotes spatial capital stock of the individual located at x and time t ≥ 0. e population stock L(x, t) is considered to follow a logistic process: e population growth rate is denoted by . us, logistic equation (1) implies the upper and lower limits to the population growth rate which satisfies a hypothesis of the bounded population growth rate [12].
In open regions, we ignore intertemporal adjustment costs caused by the flows of capital and goods. When there is no capital depreciation, we have an aggregate balance equation in the geography Ω, which is given by In equation (2), c(x, t) > 0 denotes the consumption of the individual at point x and time t, and i(x, t) ≠ 0 is the net trade balance of individual at x and time t. Note that, in this paper, the production function is considered as the linear 2 Discrete Dynamics in Nature and Society function (A is a positive constant).
e trade i(x, t) is symmetry of capital account balance that is matched by reallocations of the capital in different regions. In addition, we assume that there is no inter-regional arbitrage opportunities among regions. From equation (2), we obtain the budget constraint which is written as Following the idea in [8], we assume that there are no institutional barriers when capital and goods flow from one region to others. In this case, the net trade balance i(x, t) is determined by different capital intensities among regions.
en, for a region E that is a bounded open subset of Ω, the Applying the divergence theorem to equation (4), we get us, we obtain Substituting equation (6) into equation (3), we have the following system: Note that, different from the model presented in [10,11], the population growth factor (the term N(x, t)k(x, t)) in system (7) describes the impact of the population growth on capital accumulation.
In order to illustrate the optimal dynamics of economic activities, a control c(·, ·) is chosen to maximize utility functional: where the constants ρ > 0 and σ > 1. For a given initial distribution k 0 (·) of the capital, the value function of system (7) is defined as In equation (9), the supremum is associated with a suitable control c(·, ·) which ensures that spatial capital is nonnegative at any point x and time t.

e Hilbert Space
Setting. For convenience, we state the Hilbert space setting to rewrite the optimal control problem (8). Let L 2 (Ω) denote the Hilbert space, in which we describe the dynamics of spatial capital by the adjusted model. In L 2 (Ω), the scalar product of f and g is defined as and the norm of the function is given by In addition, the distance between two elements space variables k(·, t) and c(·, t) are considered as elements of the Hilbert space L 2 (Ω). In order to depict the second derivative of functions in L 2 (Ω), we express operator G as where G is the generator of the heat semigroup, and it is also a C 0 semigroup on L 2 (Ω) [31,32]. We denote a subset of [31]. us, the functions in D(G) have first and second derivatives in the domain L 2 (Ω).
Since the population growth rate N(x, t) satisfies the logistic equation allowing the limits to the population growth, for simplifying calculation, we denote the logistic population growth rate N(x, t) by n which is independent of t, and we verify the lower and upper limits of the population growth rate in Proposition 1 and eorem 3, respectively. Based on the previous analysis, in the Hilbert space L 2 (Ω), system (7) is rewritten as the following form: where c(t) and k(t) are elements of L 2 (Ω) for any t ≥ 0. Note that, c(t) and k(t) are considered as the functions of space variable x and expressed by c(t)(x) and k(t)(x), respectively. at is to say, c(t) and k(t) still denote consumption and capital at time t with space point x, respectively [10,11]. erefore, we write We apply the notation L 2 loc (L 2 (Ω); (0, +∞)) to denote a function space, which reads and express the set of admissible controls as Discrete Dynamics in Nature and Society 3 us, value function (9) is rewritten as Remark 1. Choosing c(·) ∈ U k 0 for an initial capital k 0 ∈ L 2 (Ω), we know that k(x, t) is the solution of system (7) if the solution of system (13) is denoted by k(t). In addition, it is not necessary to ensure that the initial distribution and the control are regular. In this case, the solution of system (7) is not required to be C 2 for the space variable and C 1 for the time variable. erefore, the unique solution of system (7) is obtained in other form, i. e., it belongs to C(L 2 (Ω); t(0, +∞)) [32,33]. We define function 1: Ω ⟶ R in L 2 (Ω), which is a constant equal to 1 in Ω, that is, us, utility functional (8) is rewritten as the form where function U(θ)(x) � θ(x) 1− 1/σ /1 − 1/σ for a given θ ∈ L 2 (Ω).

Spatio-Temporal Capital Dynamics
In this section, we use the dynamic programming method to analyze spatio-temporal dynamics of the capital in L 2 (Ω).
Since optimal control problem (8) is equivalent to maximizing (18) subject to system (13), we concentrate our attentions on HJB equation related to system (13) to discuss the optimal control and explicit value function of system (13). e HJB equation of system (13) is given by where the function v: L 2 (Ω) ⟶ R, i.e., for any k ∈ L 2 (Ω).
In equation (19), the term 〈k, G▽v(k)〉 equals 〈Gk, ▽v(k)〉 since G is self-adjoint. According to the concept of Gateaux derivative, the gradient of the function v(k) (denoted by ▽v(k)) is an element of L 2 (Ω). en, the gradient ▽v(k) at a certain point x is computed in k ∈ L 2 (Ω). Furthermore, since ▽v(k) ∈ L 2 (Ω), we obtain G▽v(k) at the point x when ▽v(k) is in the domain D(G) [34]. Similar results appear in the standard dimensional case.
is fact allows us to expect that the solution v(k) of the HJB equation equals the value function of the infinite dimensional problem.
Let M ≔ k ∈ L 2 (Ω): 〈k, 1〉 > 0 be an open set. We consider a continuous function v ∈ C 1 (M) with continuous differential, which means that the gradient ∇v: M ⟶ L 2 (Ω) is continuous in L 2 (Ω). Now, in order to obtain the optimal control of system (13) in feedback form, we identify an explicit solution (denoted by v) of equation (19) on M in the following proposition.
then there exists an explicit solution of HJB equation (19), which is given by where for vol(Ω) ≔ Ω 1dx.

Proof.
Considering v(k) � μ〈k, 1〉 1− 1/σ for a positive real number μ, we have Since ▽v(k) belongs to the domain D(G) and G▽v(k): M ⟶ L 2 (Ω) is continuous for any k ∈ M, then substituting v(k) � μ〈k, 1〉 1− 1/σ into equation (19), we obtain 4 Discrete Dynamics in Nature and Society Due to G1 � 0, the supremum is obtained if c � (μ(1 − 1/σ)) − σ 〈k, 1〉1 in equation (24). us, we write equation (24) as where vol(Ω) ≔ Ω 1dx. To simplify μ〈k, 1〉 1− 1/σ (a nonzero factor) in equation (25), we obtain which reads en, we obtain an explicit solution of HJB equation (19), which is of the form On the contrary, the trajectory k(·) related to a control c(·) in set (15) is the solution of the following system: where _ k(t) ≔ zk(t)/zt. At every time t and point x, we observe that k(·) remains below: where k(t) is the solution of the following system: en, we have 〈k(t), 1〉 ≥ 〈k(t), 1〉 for all t > 0. Since we have a bounded utility: us, the bounded utility Inequality (34) yields and otherwise, |e − ρt v(k(t))| ⟶ +∞. (20) is applied to guarantee the finiteness of the objective function. Indeed, we also obtain the finiteness of the value function with condition (20) because of explicit expression (21). In the following proposition, we illustrate that the feedback control is admissible and prove that the solution of HJB equation (19) obtained in Proposition 1 is the value function. Furthermore, the optimal control of problem (18) is derived by employing the value function.

Proposition 2.
Assume that the hypotheses of Proposition 1 are satisfied. en, the feedback control c � (μ(1 − 1/σ)) − σ 〈k, 1〉1 is the unique optimal control of system (13) and the value function of system (13) Proof. e feedback control linked to the solution of HJB equation (19) is Choosing k 0 ∈ L 2 (Ω) and replacing c(t) by the feedback control Φ(k(t)) in system (29), we obtain Discrete Dynamics in Nature and Society According to the definition of mild solution in Bensoussan et al. [32], we have a mild solution of system (38), which is given by where G ≔ G + A − n is the identity operator.
Note that, due to examples in [32], we know that equation (39) has a unique solution even if it is not an explicit form. We denote the unique solution of equation (39) by k * Φ (t), which is the related trajectory driven by the feedback control c * with c * (t) ≔ Φ(k * Φ (t)) for all t > 0. Under the hypothesis of positive k * Φ (t)(x), c * (t) is positive and admissible.
In general, we know that c * (·) is the optimal control if we obtain (40) In equation (40), c(·) denotes any other admissible control of (15). Supposing that k(·) is the trajectory linked to the control c(·) and considering function Regularizing properties of the heat semigroup, we know that k(t) ∈ D(G) for any t > 0. Using inequality (33), we know that w(T, k(T)) ⟶ 0, T ⟶ ∞. us, we obtain Considering the solution v(k) of equation (19), we have where c * (·) is a control defined with the feedback in (37). erefore, for all admissible control c(·), we obtain which leads to Inequality (47) shows that c * (·) is the optimal control.

Convergence Analysis of Capital Distribution
In the long run, the willingness of the planner induces the spatio-temporal dynamics to give a certain consumption which equalizes the capital level in different locations. us, the spatio-temporal dynamics eliminates the initial inequalities in capital endowments. is result on decreasing returns plays a key role in the traditional theory of 6 Discrete Dynamics in Nature and Society convergence. In this section, we discuss the uniform convergence of the detrended capital distribution over space by applying the explicit solution of system (29).

Theorem 1.
Assume that the hypotheses of Proposition 1 are satisfied. en, there exists a closed-form solution of system (29), which is given by where e optimal consumption dynamics is an elementary factor of the dynamics programming method, which is used to maximize the utility function and obtain the behavior of optimal capital distribution.
us, we apply the optimal consumption to complete proof of eorem 1.
According to the works in [10], we know that the behavior of spatial capital distribution implies composite transitional dynamics because of the second order term △ x k(x, t) in system (7). Precisely, the aggregate level of capital distribution in the spatial model is equivalent to that in corresponding form of the one-dimensional model. In the following theorems, we discuss the characterization of aggregate capital with the optimal control obtained in Proposition 2.

Theorem 2. Under the assumptions of Proposition 1, there exists per aggregate capital K(t) with the optimal control, which is given by
Proof. Using closed-form solution (48), we obtain According to the evolution of optimal aggregate capital stock in the spatial growth model, we find that the aggregate capital grows with an explicit growth rate β from K(0). By (51) and (53), we obtain the relationship between the optimal control c * (·) and the explicit growth rate β, which means that, for all individuals, the consumption level chosen by planners is independent of their location and generation. In the following theorem, we analyze the induced optimal spatio-temporal dynamics of detrended capital when the bounded population growth rate satisfies certain conditions. Theorem 3. Suppose that the hypotheses of Proposition 1 hold. en, for a real number r 1 , if the detrended capital converges uniformly to a constant K(0)/vol(Ω) when t tends to infinity.
In eorem 3, the optimal spatial dynamics of capital, which is induced by the optimal consumption obtained in Proposition 2, guarantees the asymptotic behavior of detrended capital. Inequalities in initial endowments are eliminated by the spatial dynamics of capital to equalize the level of capital across locations in the long run. is fact is due to the explicit expression of the capital in eorem 1 and a threshold value of the population growth rate (n < n � A + ρσ/1 − σ+ σr 2 1 /σ − 1). In the sequel, we provide the proof of eorem 3 by employing corresponding form of Fourier series.

Discrete Dynamics in Nature and Society
Proof. Firstly, we write k D (x, t) by using Fourier series. For j ∈ Z, we denote the eigenfunction of △ x by a regular function ϕ j : Ω ⟶ R, such that In equation (57), r j is a real number eigenvalue in the set of eigenvalues which is discrete as form of the sequence 0 � r 0 < r 1 < r 2 < · · · < r j < · ·· [35]. erefore, for fixed x and considering φ(x, t) � e Gt ϕ j (x), we obtain (58) Equation (58) yields which implies en, we obtain In order to determinate the Fourier coefficients, using equation (48), we take the scalar product with ϕ j for all j and obtain where the unique normalized function is Applying Fourier series to write functions on Ω, we have for every function k ∈ L 2 (Ω) [36]. In this case, we express k D (t) as Using |k| 2 L 2 (T) � j∈z |〈k, ϕ j 〉| 2 and expression of k D (t) as Fourier series, we have where S is a positive constant and satisfies |sup x∈Ω |ϕ j (x)| ≤ S. Since we have Considering ϵ ∈ (0, (A − n)(σ − 1) + σr 2 1 − σρ/σ), we obtain en, we know H < + ∞ due to condition (70). erefore, we rewrite (67) as which completes the proof.

Numerical Simulations
In the spatial economic model, the adjustment of optimal trajectory is not instantaneous when the spatial capital flows from rich regions to poor regions. In the rich area, the flow of capital brings a temporary production agglomeration, while a depressed condition appears in the poor area. After that, the spatial capital in the rich location moves to the poor location, and then, the optimal path is formed. In this section, we consider different values of relevant parameters to show the formation of the optimal path and illustrate results of the spatial capital distribution with exponential decays. We simulate two cases of initial capital distribution to illustrate approximate profiles of the physical capital across space and time when some values of relevant parameters are given. As shown in Figure 1, the optimal trajectory of spatial capital distribution corresponds to A � 0.1, n � 0.03, ρ � 0.05, σ � 3, and r 1 � 0.1, which satisfy conditions (20) and (55). In the initial stage, we observe that the spatial capital distribution is highly clustered in the center section of space interval, while it is very dispersed elsewhere. is case is consistent with the optimal spatio-temporal dynamics of the detrended capital since we know that, as a sum of different terms, the second term on the right of equation (66) is invisible because of the exponential factor e (σA− σn− r 2 j − σρ− β)t with condition (68). Precisely, the limit of detrended capital k D (t)(x) is the term K(0)/vol(Ω) when t ⟶ ∞. On the contrary, if we change values of the relevant parameters to satisfy n(1 − σ) � A(1 − σ) + ρσ − σr 2 1 , Figure 2 shows the corresponding profiles of the initial capital and spatial capital distribution. In this case, the detrended capital is not convergent to a spatially homogeneous distribution in the long run since the term j∈z,j≠0 e (σA− σn− r 2 j − σρ− β)t 〈k 0 , ϕ j 〉ϕ j (x) is involved in the limit. Discrete Dynamics in Nature and Society 9

Conclusions
In this paper, we have investigated the optimal dynamics of a spatial economic growth model with bounded population growth in a continuous space. Considering the growth model redrafted in the infinite dimensional Hilbert space, we apply the HJB equation and linear production function to identify the unique optimal control and the value function of the growth model. Under assumption of the bounded population growth, the explicit solution of our model is derived in close form by using the optimal control, which satisfies the first-order optimality conditions to show the time behavior of per capita capital. With nonnegative initial capital distribution, we apply the explicit solution and Fourier series to express the detrended capital of our model and then prove the convergence of the detrended capital level across space and time.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.