Breaking and Sustaining Bifurcations in S N -Invariant Equidistant Economy

This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group S N [Elmhirst , 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models conﬁrms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics.


Introduction
Spatial economics (or economic geography) aims to explain the spatial distribution of economic activities and how distance between locations affects the economic behavior of agents. New economic geography might be regarded as a subfield of spatial economics emphasizing how economic agglomeration patterns evolve as a result of the historical increases in economic integration (i.e. decreased worldwide transportation costs). Ever since the seminal study by Krugman [1991], numerous contributions have provided insights into the connection between transportation costs and the spatial organization of economic activities.
Equidistant economy, in which transport costs between all pairs of places are identical, is an important spatial platform in spatial economics. Bifurcation mechanism of this economy has been investigated to observe a complicated mesh-like network of equilibrium curves [Gaspar et al., 2019b]. This paper elucidates the network mechanism (1) using a group-theoretic analysis for the symmetric group S N [Golubitsky & Stewart, 2002;Elmhirst, 2004] and (2) referring to invariant patterns for the replicator dynamics (Sec. 3 and [Ikeda et al., 2018b;Ikeda et al., 2019a;Ikeda et al., 2019b]). It is customary in spatial economics to deal with bifurcations of steady states. Therefore, Hopf bifurcations are not considered in this paper. Hopf bifurcations with S N -symmetry (e.g. [Golubitsky & Stewart, 1985;Diaz & Rodrigues, 2007]) and applications of bifurcation theory to dynamic problems in economics (e.g. [Dercole et al., 2008;Dercole & Radi, 2020]) are to be consulted with the literature.
A break point and a sustain point, which play important roles in this discussion, were first studied in a two-place economy under replicator dynamics [Fujita et al., 1999;Baldwin et al., 2003]. A break point is associated with a symmetry-breaking bifurcation of two identical places, whereas a sustain point is located at the intersection of a solution curve for two identical places and a curve for two different places. Thereafter, economic agglomeration has been studied under various settings (spatial topologies), such as a line segment (e.g. [Fujita & Mori, 1997]), a racetrack (e.g. [Tabuchi & Thisse, 2011]), and a lattice (e.g. [Ikeda et al., 2018a]). The equidistant case has been handled mostly with three regions [Fujita et al., 1999;Castro et al., 2012;Commendatore et al., 2015]. Direct bifurcation from N identical places leads to a two-level hierarchy state comprising one large place and N − 1 small places [Gaspar et al., 2018]. Other works have considered an arbitrary number of equidistant places under different settings. Nevertheless, they provide only an incomplete account of the network of possible equilibria [Puga, 1999;Tabuchi et al., 2005;Oyama, 2009;Zeng & Uchikawa, 2014;Gaspar et al., 2019a].
In light of the discussion presented above, this paper aims to elucidate the bifurcation mechanism of an equidistant economy with N places. Bifurcation analysis of a symmetric field is a well-matured topic [Golubitsky et al., 1988;Ikeda & Murota, 2019]. In fact, the mechanism of direct bifurcation of this economy is readily available through bifurcation analysis of the symmetric group S N [Golubitsky & Stewart, 2002;Elmhirst, 2004]: all possible bifurcating solutions from the uniform state were obtained and were proven to be unstable immediately after each bifurcation. By contrast, few reports describe the secondary bifurcation under S N -symmetry. For instance, it is analyzed as a simple example [Rodrigues, 2007]. Its stability is briefly remarked upon [Elmhirst, 2004]. A major theoretical contribution of the paper is our analysis of secondary and further bifurcations.
The existence of special spatial distributions, called invariant patterns, has come to be acknowledged. Steady-state solutions that satisfy the static governing equation form solution curves parameterized by the bifurcation parameter (transportation cost in spatial economics). In general, the spatial pattern changes along a solution curve. By contrast, there can be a special solution curve that has a constant spatial pattern. Such pattern is called herein an invariant pattern. Invariant patterns of a racetrack economy and of a lattice economy under the replicator dynamics, which is the most popular in economics, were found and employed to elucidate the bifurcation mechanisms of these economies [Ikeda et al., 2018b;Ikeda et al., 2019a;Ikeda et al., 2019b]. Stability analysis of invariant patterns of an equidistant economy of spatial economic models was conducted [Gaspar et al., 2018;Gaspar et al., 2019a]. In this paper, invariant patterns of an equidistant economy under the replicator dynamics are found. Then the secondary (sustain) bifurcation mechanism is investigated theoretically for these invariant patterns. This theoretical stability analysis demonstrates that some of the branches are stable immediately after bifurcation, unlike the direct bifurcation.
For numerical bifurcation analysis of a symmetric system, it is customary to first obtain the uniform solution, next find direct, secondary, and further bifurcating solutions successively, and then assemble these solution curves. As described herein, because of the existence of invariant solutions, we can use the following innovative bifurcation analysis procedure [Ikeda et al., 2019a]. (1) Obtain all invariant patterns and find all bifurcation points on the equilibrium curves for these patterns.
(2) Obtain all direct, secondary, and further bifurcating equilibrium curves from these invariant patterns and find all bifurcation points on these curves. This procedure is put to use for numerical analysis of an equidistant economy with three, four, and eight places of two spatial economic models Pflüger, 2004]. There are direct bifurcating curves connecting the curves of invariant solutions, thereby displaying mesh-like networks resembling threads constituting the weft of invariant patterns and the warp of noninvariant ones. Results demonstrate that almost all bifurcating equilibria are unstable.
The theoretical bifurcation mechanism and numerical examples of networks presented in this paper might be of great assistance in the study of spatial economics. In fact, this mechanism for three places [ Fig. 1(a)] is applied successfully to the study of the agglomeration mechanism of a spatial economic model [Gaspar et al., 2019b].
This paper is organized as explained hereinafter. A spatial economic model under the replicator dynamics is presented in Sec. 2. Invariant patterns are obtained in Sec. 3. A bifurcation mechanism of an equidistant economy is put forth in Sec. 4. The stability of bifurcating branches is studied in Sec. 5. Numerical bifurcation analyses of spatial economic models are conducted as described in Sec. 6.

Spatial Economic Model Under Replicator Dynamics
Economic geography models share some common components: (i) a manufacturing sector operates under imperfect competition with increasing returns to scale at the firm level; (ii) inter-regional trade of manufactured goods is costly, with transport costs; and (iii) inter-regional mobility (migration) of some production factors is allowed. Increasing returns at the firm level foster agglomeration in a place, although fiercer competition in larger markets tends to drive firms to disperse across places. Although many economic geography models specifically examine two places, the analysis has come to be extended to multiple places, as described in the Introduction. The present study examines a setup with an arbitrary number of places that are pairwise equidistant.
Spatial economic models are presented. Their steady-state solutions under the replicator dynamics are classified. Whereas the theoretical framework of this paper is efficacious for analyzing general spatial economic models, detailed aspects of payoff functions are defined in accordance with two important models: the FO model ] and the Pf model [Pflüger, 2004]. These two models serve as concrete examples of spatial economic models to be used for investigating the progress of bifurcation (Sec. 6).

Spatial economic model
There are N (≥ 3) places and mobile agents (workers or firms, entrepreneurs) that can choose where to locate from N places. Denote by h = {h i | i = 1, . . . , N} the spatial distribution of agents. It is assumed that N i=1 h i = 1; accordingly, the state space is the probability simplex. The payoff (indirect utility or profit) for locating in place i is derived from a model-specific short-run general equilibrium. It is a function of goods' prices and an amount of income reflecting agents' preferences and market conditions. In equilibrium, it is given as a function v i of the spatial distribution of mobile agents h and a parameter φ ∈ (0, 1) that represents the freeness of transport between the places. The parameter φ is an inverse measure of transport costs. It can be regarded as signifying the degree of economic integration between the N places. A continuous C 1 function v : R N × (0, 1) → R N + therefore defines a general spatial economic model with N places. An equilibrium is defined as a spatial distribution of agents h = h * that satisfies the conditions of where v * denotes the equilibrium payoff level. It is noteworthy that (1) includes the case in which the payoff varies across places, because v i (h * , φ) < v * can possibly occur for h * i = 0. Economic backbones of the payoff function v for the FO and Pf models are summarized briefly below, whereas Appendix A presents related details.
Two factors, skilled and unskilled labor, are used in production along with the two sectors. There are two types of workers associated with two types of labor. The workers supply one unit of each type of labor inelastically (i.e. irrespective of the wage rate). The total endowments of skilled and unskilled workers are respectively H and L, with H normalized to unity (H = 1). Skilled workers are mobile across places; h i denotes the number of skilled workers located in place i. Unskilled workers are immobile and are equally distributed across all places (i.e. the number of unskilled workers in each place is = L/N ). The two sectors are agriculture (A) and manufacturing (M). The A-sector output is homogeneous. Each unit is produced using a unit of unskilled labor under perfect competition. Trade of the A-sector good is frictionless. Choosing this good as numéraire means that its price and the wage paid to an unskilled worker are set to unity everywhere. Appendix A presents additional details. The M-sector output is a horizontally differentiated good produced using both skilled and unskilled labor under increasing returns to scale and Dixit-Stiglitz monopolistic competition [Dixit & Stiglitz, 1977]. Three major parameters exist for the models: σ represents the constant elasticity of substitution between any two manufactured goods; μ denotes the constant expenditure share on industrial varieties; and L signifies the endowment of immobile workers. Inter-regional trade of M-sector goods incurs iceberg costs. That is, for each unit of M-sector goods transported from place i to j (j = i), only a fraction 1/τ ij < 1 arrives. Intra-regional transport is frictionless: τ ii = 1 for all i. The main assumption used for these discussions is that transport costs among all pairs of places are equal, i.e. the space economy is equidistant.
By this assumption, the freeness of transport parameter φ = τ 1−σ ∈ (0, 1) characterizes the inter-regional transport cost structure of FO and Pf models. As φ approaches 0 or 1, the trade cost increases or decreases, respectively. We employ φ as the bifurcation parameter. Derivation of the shortrun general equilibrium is detailed in Appendix A. We reproduce here only the expressions of interest to us.
The payoff functions for the FO and Pf models are given as respectively, where w i is the wage function and

Replicator dynamics
It is customary in economics to replace the problem to obtain stable spatial equilibria in (1) with another problem to find a set of stable steady-state solutions of the replicator dynamics [Taylor & Jonker, 1978] as where Here is the average utility. Steady-state solutions (rest points) (h(φ), φ) of the replicator dynamics (6) are defined as those points which satisfy the static governing equation The law of preservation of population N i=1 h i = 1 is satisfied for the steady-state solutions. Accordingly, the space of h is an (N − 1)-dimensional simplex, as explained in Remark 2.1 below. A steady-state solution is stable if every eigenvalue of the Jacobian matrix J(h, φ) = ∂F/∂h(h, φ) has a negative real part and is unstable if at least one eigenvalue has a positive real part. It is necessary to exclude an eigenvalue associated with an eigenvector that does not belong to the (N − 1)-dimensional simplex. A stable equilibrium, which is the main target of this paper, is defined as a stable steady-state solution of (8) with non-negative population h i ≥ 0 (1 ≤ i ≤ N ); it is known that such a solution satisfies the equilibrium condition (1) of an underlying spatial economic model [Sandholm, 2010].
Steady-state solutions are classifiable into interior solutions for which all places have positive population, and corner solutions for which some places have zero population. A corner solution can be expressed, without loss of generality, by an appropriate permutation of components of h, as , are used because the symmetry condition and the spatial population distribution can be expressed in a much more consistent manner.

Invariant Patterns
Invariant patterns of an equidistant economy are presented. Steady-state solutions that satisfy the static governing equation F(h, φ) = 0 in (8) form solution curves (h(φ), φ) parameterized by φ. In general, the spatial pattern h(φ) varies with φ along a solution curve. By contrast, by virtue of the product form (7) of the replicator dynamics, there can be a special solution curve (h(φ), φ) = (h, φ) that has a constant spatial pattern h(φ) = h along the curve. Such pattern h is called herein an invariant pattern. The curve of an invariant pattern exists for any φ ∈ (0, 1). On the other hand, a pattern h(φ) that varies with φ is called a noninvariant pattern and might or might not be a steady state for a given φ.
The uniform state and a series of core-periphery patterns (cf. Remark 3.1) Proof. Appendix B.1 presents an associated proof.
Remark 3.1. In economic geography, the pattern h = (1, 0) (N = 2), in which a (core) place has the whole population of 1 and another (periphery) place has no population, presents a core-periphery pattern. As described herein, this notion is extended. The pattern in (10) is called a core-periphery pattern. This pattern is a special form of the corner solution (9) with a two-level hierarchy: the identical population 1/m is agglomerated to m core places, whereas other n peripheral places have no population.

Bifurcation Mechanism
The bifurcation mechanism of secondary and further bifurcations of an equidistant economy is investigated as a novel contribution of this paper. The mechanism of the direct bifurcation of the uniform state [Golubitsky & Stewart, 2002;Elmhirst, 2004] is also included in Sec. 4.1 to make the discussion self-contained. Since it is customary in spatial economics to deal with bifurcations of steady states, this paper does not address Hopf bifurcations. In economic geography [Fujita et al., 1999], a sustain point is defined as the value of the transport cost parameter below which an economy with the core-periphery pattern h CP 1 = (1, 0) in (10) becomes stable. In light of bifurcation theory, we can consider a sustain point as the bifurcation point at which some zero component(s) of population distribution h become nonzero on a bifurcating path. This kind of bifurcation is called sustain bifurcation herein and the associated bifurcation point is called a sustain bifurcation point. Similarly, we use break bifurcation and break bifurcation point.

Direct bifurcation from a uniform state
The mechanism of the direct bifurcation from the uniform state h uniform = 1 N 1 N of an equidistant economy with N places was elucidated by the bifurcation analysis of a symmetric group S N labeling the symmetry of this economy [Golubitsky & Stewart, 2002;Elmhirst, 2004]. This analysis is presented briefly and consistently with our formulation.
The uniform state has the Jacobian matrix of the form That is, all diagonal entries of the Jacobian matrix J are a and all off-diagonal ones are b. When a = b, this state encounters a direct bifurcation point with (N −1)-times repeated zero eigenvalues of the Jacobian matrix. At this point, several two-level hierarchy states of the form branch simultaneously in the incremental directions as as explained in Proposition 2 below. Consequently, N places split into m places with equal population size and n places with another size. A branch is called symmetric if δh m and −δh m denote the same state up to a permutation of place numbers. It is designated as asymmetric if they do not.
Proposition 2. At a bifurcation point of the uniform (equidistant) state, the two-level hierarchy states in (12) branch in the directions of (13). The associated branch is symmetric if n = m = N/2 (N even); it is asymmetric otherwise.
Proof. Appendix B.2 and an earlier report of the literature [Elmhirst, 2004] present this proof.

Bifurcation from a two-level hierarchical state
The two-level hierarchy state h m = (u 1 1 m , u 2 1 n ) in (12) has break and sustain bifurcation points. At a sustain bifurcation point, where u 2 1 n vanishes, this state exits to a corner solution expressing the core-periphery pattern in (10): For the discussion of a break bifurcation point, we refer to the Jacobian matrix of the two-level hierarchy state, which takes the form of Therein, A m (a, b) and A n (c, d) are defined similarly to A N (a, b) in (11) (1 ≤ p ≤ n − 1; m + n = N ; branch simultaneously in the directions of as explained in Proposition 3. Consequently, n identical places split into p places with an identical population size and n − p places of another size (1 ≤ p ≤ n − 1). (12), the three-level hierarchy states in (14) branch in the directions of (15). The branch is symmetric if p = n/2 (n even) and asymmetric otherwise.

Proposition 3. At a break bifurcation point of the two-level hierarchy state in
Proof. Appendix B.3 presents the associated proof.
The three-level hierarchy state in (14) can exit to the corner solution at a sustain bifurcation point; alternatively, it can undergo further bifurcations to arrive at an aggregated interior state with an s-level hierarchy of (2 ≤ s ≤ N ).

Bifurcation from core-periphery patterns
In the discussion of bifurcation from the coreperiphery pattern h CP = 1 m (1 m , 0 n ) in (10), we refer to its Jacobian matrix as with I n being an n×n identity matrix and The critical point of this pattern is either a break bifurcation point for a = b with singular A m (a, b) or a sustain bifurcation point for c = 0 with singular cI n in (17). Before the main discussion, we refer to the half branch. One must recall that the branches for the break bifurcations presented above do exist in both directions of δh p and −δh p . By contrast, a branch exists in only one direction for a sustain bifurcation point because no negative population is allowed (Propositions 4 and 6 below); such a branch is called a half branch.
We start with the simplest core-periphery pattern: full agglomeration h FA = (1, 0 N −1 ), which is an invariant pattern (Proposition 1). The full agglomeration has only sustain bifurcation points, at which several bifurcating solutions of the form branch simultaneously in several directions as

) does not have a limit point or a break bifurcation point but has sustain bifurcation points with the half branches in (18).
Proof. Appendix B.4 presents an associated proof.
Other core-periphery patterns (m ≥ 2) have both break and sustain bifurcation points, which engender several bifurcating solutions as expounded in the following propositions, the proofs of which resemble that for Proposition 3. (10), several three-level hierarchy states branch simultaneously as

Proposition 5. At a break bifurcation point of the core-periphery pattern in
The branch is symmetric if p = m/2 (m even) and asymmetric otherwise. (1 ≤ p ≤ n; u 1 m + u 2 p = 1; u 1 , u 2 > 0).

Simple examples
To illustrate the bifurcation mechanism in equidistant economy, Figs  top is connected to that for corner solutions at the bottom [see (9) for the definition of a corner solution]. There is a recurrent property: the hierarchy of N = 3 becomes the subhierarchy of corner solutions for N = 4. For an arbitrary number N of places, the subhierarchy of corner solutions is given by the hierarchy of N − 1 places [see Fig. 1(c)]. By virtue of this recurrent property, the hierarchy grows rapidly as N increases. In turn, the bifurcation mechanism becomes progressively complicated.

Asymptotic Stability of Branches
In the previous section, branches of bifurcation points for geometrical patterns of interest, such as the uniform state, the full agglomeration, and coreperiphery patterns are investigated respectively in Propositions 2, 4 and 6. In this section, as a novel contribution of this paper, we investigate asymptotic stability of these branches in a close neighborhood of the associated bifurcation point. The stability of branches from the uniform state is already studied as explained below [Elmhirst, 2004].
Lemma 1. Under the assumption that the uniform state h uniform = 1 N 1 N is stable until reaching the break bifurcation point, all branches of this state are asymptotically unstable.
We hereinafter investigate the stability of half branches from a sustain bifurcation point of the core-periphery pattern h CP m = 1 m (1 m , 0 n ) (m + n = N ) in (10). We recall its Jacobian matrix in (17) as and consider its sustain bifurcation point at φ = φ c with a singular cI n (c = 0) and a nonsingular A m (a, b) (a = b). Define incremental variables (y, x, ψ) from this point as with y = (y 1 , . . . , y m ) and x = (x 1 , . . . , x n ). We obtain the bifurcation equation based on the procedure: (1) Express the static governing equation F(h, φ) = 0 in (8) in terms of these incremental variables (y, x, ψ).
(2) Express y = y(x, ψ) using the first m components of F = 0, because A m (a, b) is nonsingular.
(3) Eliminate y = y(x, ψ) from the last n components of F = 0. By virtue of a factored form (7) of the replicator dynamics, G i (x, ψ) takes the special form of Then solutions (x, ψ) ofĜ 1 = · · · =Ĝ n = 0 give bifurcating solutions via y = y(x, ψ) and (24). At a sustain bifurcation point of the coreperiphery pattern h CP m , there emerge several half branches in (21), i.e.
We see that these branches are associated with similarly to the proof of Proposition 3. By analysis of the bifurcation equation G = 0 in (25) described in Appendix B.5, it is readily apparent that asymptotic bifurcating ψ versus ε curves exist as with expansion coefficients α, β, and γ of the bifurcation equation. The following lemma serves in a pivotal role for the description of the stability of half branches of the core-periphery pattern h CP m .
Lemma 2. The eigenvalues of ∂G/∂x for the branches of the core-periphery pattern h CP m are given asymptotically as Proof. Appendix B.5 presents an associated proof.
The associated half branches are stable if all eigenvalues in (29) are negative. Note that p = 1 and p = n are exceptional cases where λ 2 or λ 3 is absent, respectively. For these exceptional cases, stable half branches can exist as expounded below. This makes a sharp contrast with a break bifurcation point for the uniform state, for which all half branches are unstable (Lemma 1). For the description of stability, we use the following assumption, which accords with the numerical results of sustain bifurcation of the full agglomeration h FA = (1, 0 N −1 ) of spatial economic models (Sec. 6).
Proof. Appendix B.6 presents an associated proof.
Among the plethora of half branches in (26), at most one of them is stable. Figure 2 depicts the classification of the parameter space (β, γ) into three distinct cases for the full agglomeration. For the full agglomeration state h FA = (1, 0 N −1 ), for example, the stability of half branches is readily available by setting n = N − 1 in Proposition 7 and in Fig. 2.   Fig. 2. Classification of stability of half branches for the core-periphery pattern h CP m = 1 m (1m, 0n) in the parameter space (β, γ).
The following innovative bifurcation analysis procedure is used [Ikeda et al., 2019a]. (1) Obtain all invariant patterns and find all bifurcation points on the equilibrium curves for these patterns.
(2) Obtain all direct, secondary, and further bifurcating equilibrium curves from these invariant patterns and find all bifurcation points on these curves. Using this procedure, one can obtain all possible bifurcating solutions connected to invariant solutions. Figure 3 reports bifurcation diagrams for the FO and Pf models with N = 3, 4, and 8 equidistant places. The horizontal axis is the freeness of transport φ and the vertical axis is taken as h max (h) = max i {h i }. A series of horizontal lines expresses equilibrium curves for invariant patterns (Proposition 1):

core-periphery pattern
The solid (broken) curves correspond to stable (unstable) steady-state solutions of the governing equation (8). The white circles (•) in the figures denote break bifurcation points, whereas the black disks (•) denote sustain bifurcation points. The double circle ( ) in Fig. 3(b) represents a limit point of φ. The stability analysis of the direct bifurcating curves was conducted to confirm that there are no bifurcation points on these curves. Accordingly, there are no secondary bifurcations from these curves. Figure Each of these states connects a break bifurcation point (•) with a sustain bifurcation point (•). Such pairs of break and sustain bifurcation points were encountered recurrently until reaching the full agglomeration h FA = (1, 0 N −1 ) at the sustain bifurcation point B which resides at the left end of the solid horizontal line for stable full agglomeration.
Direct bifurcating curves connecting invariant solutions thereby display mesh-like networks, which resemble threads constituting the weft of invariant patterns and warp of noninvariant ones. These networks are much clearer and more systematic than that for a hexagonal lattice [Ikeda et al., 2019a], possibly by virtue of a large symmetry of the symmetric group S N .
Among invariant patterns, only the uniform state and full agglomeration have some stable equilibria. All the branches from the uniform state h uniform = 1 N 1 N are unstable immediately after Bifurcations in S N -Invariant Equidistant Economy bifurcation (Lemma 1). Curve BC in Fig. 3 for a noninvariant branch of the Pf model is stable as predicted in Proposition 7(ii); other cases include no stable, noninvariant curve. Figure 4 depicts the hierarchy of geometrical patterns for the present numerical analyses. The hierarchy diagram of the N = 3 case is a "subset" of the N = 4 case. These hierarchies correspond to the subsets of theoretical hierarchies summarized by Fig. 1. As demonstrated by Figs. 3(e) and 3(f) for N = 8, the hierarchy grows rapidly as N increases.

Conclusions
A thorough study was conducted on the bifurcation mechanism and stability of an equidistant economy. As a novel contribution, we elucidated the bifurcation mechanisms of secondary and further bifurcations. The bifurcation mechanism of the direct bifurcation of the uniform state [Golubitsky & Stewart, 2002;Elmhirst, 2004] is also included to make the discussion self-contained. The equilibrium curves of this economy have complicated mesh-like networks comprising intersecting equilibrium curves of invariant and noninvariant patterns, similar to threads of warp and weft. The bifurcation mechanisms advanced in this study were of great assistance in obtaining these complicated curves. The theoretical bifurcation mechanisms and numerical examples of networks advanced in this paper are expected to be of great assistance in the study of multiple places in spatial economics.

Core-Periphery Models
Two core-periphery models, the FO model and the Pf model, are introduced. In these models, each worker chooses a consumption level that maximizes the worker's own utility, given the spatial distribution h of workers. This process, the so-called "shortrun equilibrium," determines the payoff v = (v i | i = 1, 2, . . . , N) of workers as a function of h.
Individuals share the same utility function U over the M-sector and A-sector goods. The utility function in place i is where μ is a constant parameter, C A i represents the consumption of the A-sector product in place i, and C M i is the manufacturing aggregate in place i, which is defined as . Therein, q ji (ξ) stands for the consumption in place i of a variety ξ ∈ [0, n j ] produced in place j, n j denotes the continuum range of varieties produced in place j, often called the number of available varieties, and σ > 1 expresses the constant elasticity of substitution between any two varieties. The budget constraint is given as where p A i represents the price of the A-sector good in place i, p ji (ξ) signifies the price of a variety ξ in place i produced in place j and Y i stands for the income of an individual in place i. The incomes (wages) of skilled workers and unskilled workers are denoted, respectively, by w i and w L i . An individual in place i maximizes (A.1) subject to (A.2). This maximization yields the following demand functions: Therein, Accordingly, this state always satisfies the static governing equation F(h * , φ) = 0 in (8). For the core-periphery Consequently, the core-periphery pattern is a steady-state solution for any φ.

B.2. Proof of Proposition 2
Consider the uniform state h uniform = 1 N 1 N , which is invariant to the symmetric group S N , and a state with the symmetry of an axial subgroup S m × S n (m + n = N ). Denote by δh = (α 1 , . . . , α m , β 1 , . . . , β n ) an incremental variable vector for this state with S m × S n symmetry. By S m and S n symmetries, we have α 1 = · · · = α m = α, β 1 = · · · = β n = β for some variables α and β. By virtue of the orthogonality between subspaces for S N and S m × S n (m + n = N ), we have Hence β = − m n α and δh = α 1 m , − m n 1 n (B.1) spans a one-dimensional space. Then by the equivariant branching lemma [Golubitsky et al., 1988;Ikeda & Murota, 2019], a bifurcating solution exists in the direction (B.1), i.e. (13). A bifurcating solution takes the form: The branch for (B.1) is symmetric if m = n because δh = α(1 m , −1 m ) and −δh = α(−1 m , 1 m ) (N = 2m) are identical up to the permutation of components. It is asymmetric otherwise because the number of positive components is different for δh and −δh.
Because this is spanned by a one-dimensional space, by the equivariant branching lemma, there exists a bifurcating solution in this direction δh, which leads to (15) by setting β = ε(n − p), n 1 = p, and n 2 = n − p. Equation (14) and symmetry/asymmetry of the branch can be proved similarly to the proof presented for Proposition 2.

B.4. Proof of Proposition 4
The eigenvalue of the Jacobian matrix of the full agglomeration h FA = (1, 0 N −1 ) for a limit point or a break bifurcation point is given by Because v 1 is assumed to be positive (Sec. 2.1), this eigenvalue is always nonzero (negative). Accordingly, a limit point or a break bifurcation point does not exist. The proof for the half branches of the sustain bifurcation point is similar to that presented for Proposition 3.

B.5. Proof of Lemma 2
The asymptotic forms of G i in (25) and its Jacobian matrix ∂G i /∂x j are given as explained below. By virtue of a factored form (7) of the replicator dynamics, G i (x, ψ) takes a special form of G i = x i ·Ĝ i (x, ψ) (1 ≤ i ≤ n). We can expandĜ i into a power series to arrive at for some constants α and β i . By the symmetry (equivariance) of the system of equations G i (1 ≤ i ≤ n), a permutation x i ↔ x j must lead to a permutation G i ↔ G j . This entails β j = β (j = i) and β j = γ (j = i) for some constants β and γ. Then we have x j (i = j), γx i (i = j).
In summary, a unique stable branch exists for each of the cases (i) and (ii), whereas there are no stable branches in other cases, designated as (iii). For (iii), we have the remaining parameter space of β > 0 or −(n − 1)γ < β < 0.