61 research outputs found

    Breaking and sustaining bifurcations in SN-Invariant equidistant economy

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    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 SN [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.info:eu-repo/semantics/publishedVersio

    Bifurcation theory of a square lattice economy: Racetrack economy analogy in an economic geography model

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    Bifurcation theory for an economic agglomeration in a square lattice economy is presented in comparison with that in a racetrack economy. The existence of a series of equilibria with characteristic agglomeration patterns is elucidated. A spatial period doubling bifurcation cascade between these equilibria is advanced as a common mechanism to engender fewer and larger agglomerations in both economies. Analytical formulas for a break point, at which the uniformity is broken under reduced transport costs, are proposed for an economic geography model by synthetically encompassing both economies

    Break and sustain bifurcations of S_N-invariant equidistant economy

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    This paper aims at the elucidation of the bifurcation mechanism of an equidistant economy in Economic Geography. An attention is paid to the existence of invariant solutions that retain their spatial patterns when the bifurcation parameter changes. Theoretical results on symmetrybreaking bifurcation of the symmetric group SN, which describes the symmetry of this economy, is combined with the mechanism of sustain bifurcation of invariant patterns that is inherent to the economy. The stability of bifurcating branches is investigated theoretically to demonstrate that most of them are asymptotically unstable. Among a plethora of theoretically possible spatial patterns, those which actually become stable for spatial economic models are investigated numerically. The solution curves of the economy are shown to display a complicated mesh-like structure, which looks like threads of warp and weft

    Group-theoretic Study of Economic Agglomerations on a Square Lattice

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    The present paper aims to elucidate the mechanism of economic agglomerations in two-dimensional economic spaces equipped with square road networks, which prosper worldwide (e.g., Chicago and Kyoto). A series of theoretical approaches provided in the present thesis makes it possible to investigate the spatial patterns of economic agglomerations on such spatial platforms systematically. The present paper focuses on square distributions on the square lattice economy, which has not somewhat been given much attention. We apply group-theoretic predictions to the investigation of bifurcation behavior of economic geography models. The present paper provides a systematic analysis procedure that is applicable to a wide range of economic geography models

    Break and sustain bifurcations of S_N-invariant equidistant economy

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    This paper aims at the elucidation of the bifurcation mechanism of an equidistant economy in Economic Geography. An attention is paid to the existence of invariant solutions that retain their spatial patterns when the bifurcation parameter changes. Theoretical results on symmetrybreaking bifurcation of the symmetric group SN, which describes the symmetry of this economy, is combined with the mechanism of sustain bifurcation of invariant patterns that is inherent to the economy. The stability of bifurcating branches is investigated theoretically to demonstrate that most of them are asymptotically unstable. Among a plethora of theoretically possible spatial patterns, those which actually become stable for spatial economic models are investigated numerically. The solution curves of the economy are shown to display a complicated mesh-like structure, which looks like threads of warp and weft

    Break and sustain bifurcations of S_N-invariant equidistant economy

    Get PDF
    This paper aims at the elucidation of the bifurcation mechanism of an equidistant economy in Economic Geography. An attention is paid to the existence of invariant solutions that retain their spatial patterns when the bifurcation parameter changes. Theoretical results on symmetrybreaking bifurcation of the symmetric group SN, which describes the symmetry of this economy, is combined with the mechanism of sustain bifurcation of invariant patterns that is inherent to the economy. The stability of bifurcating branches is investigated theoretically to demonstrate that most of them are asymptotically unstable. Among a plethora of theoretically possible spatial patterns, those which actually become stable for spatial economic models are investigated numerically. The solution curves of the economy are shown to display a complicated mesh-like structure, which looks like threads of warp and weft

    Evolutionary Game Theory: Theoretical Concepts and Applications to Microbial Communities

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    Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the self-organization principles underneath the complexity of these systems. A major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established. It ensures the viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social” behavior. A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems. We give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems. Intrinsic fluctuations, stemming from the discreteness of individuals, are ubiquitous, and can have an important impact on the stability of ecosystems. In the absence of speciation, extinction of species is unavoidable. It may, however, take very long times. We provide a general concept for defining survival and extinction on ecological time-scales. Spatial degrees of freedom come with a certain mobility of individuals. When the latter is sufficiently high, bacterial community structures can be understood through mapping individual-based models, in a continuum approach, onto stochastic partial differential equations. These allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds. We conclude with a perspective on the current challenges in quantifying bacterial pattern formation, and how this might have an impact on fundamental research in non-equilibrium physics

    Group-theoretic analysis of a scalar field on a square lattice

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    In this paper, we offer group-theoretic bifurcation theory to elucidate the mechanism of the self-organization of square patterns in economic agglomerations. First, we consider a scalar field on a square lattice that has the symmetry described by the group D4Zn×Zn)\textrm{D}_{4} \ltimes \mathbb{Z}_{n} \times \mathbb{Z}_{n}) and investigate steady-state bifurcation of the spatially uniform equilibrium to steady planforms periodic on the square lattice. To be specific, we derive the irreducible representations of the group D4(Zn×Zn)\textrm{D}_{4} \ltimes (\mathbb{Z}_{n} \times \mathbb{Z}_{n}) and show the existence of bifurcating solutions expressing square patterns by two different mathematical ways: (i) using the equivariant branching lemma and (ii) solving the bifurcation equation. Second, we apply such a group-theoretic methodology to a spatial economic model with the replicator dynamics on the square lattice and demonstrate the emergence of the square patterns. We furthermore focus on a special feature of the replicator dynamics: the existence of invariant patterns that retain their spatial distribution when the value of the bifurcation parameter changes. We numerically show the connectivity between the uniform equilibrium and invariant patterns through the bifurcation. The square lattice is one of the promising spatial platforms for spatial economic models in new economic geography. A knowledge elucidated in this paper would contribute to theoretical investigation and practical applications of economic agglomerations
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