# Equilibrium and Stability of Entropy Operator Model for Migratory Interaction of Regional Systems

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## Abstract

**:**

## 1. Introduction

## 2. Structure of Dynamic Entropy Model of Migration

- ${x}_{in}(t),\phantom{\rule{0.166667em}{0ex}}(i,n)=\overline{1,N},$ as the migration flows $(i\rightleftarrows j)$ within the system;
- ${a}_{in}(t),\phantom{\rule{0.166667em}{0ex}}(i,n)=\overline{1,N},$ as the prior probabilities of individual migration between regions $(i,j)$ of the system;
- ${\alpha}_{n},\phantom{\rule{0.166667em}{0ex}}n=\overline{1,N},$ as the shares of mobile population in system regions;
- ${y}_{kn},\phantom{\rule{0.166667em}{0ex}}k=\overline{1,M},n=\overline{1,N},$ as the immigration flows $(k\to j)$ from external regions to system regions;
- ${b}_{kn},\phantom{\rule{0.166667em}{0ex}}k=\overline{1,M},n=\overline{1,N},$ as the prior probabilities of individual immigration from external region k to system region j;
- ${c}_{kn},\phantom{\rule{0.166667em}{0ex}}k=\overline{1,M},n=\overline{1,N},$ as the normalized (Here, normalization means that the values of specific cost belong to the range [0, 1].) specific generalized cost of immigration to system regions;
- ${c}_{kn}^{(s)},\phantom{\rule{0.166667em}{0ex}}k=\overline{1,M},n=\overline{1,N},s=\overline{1,r},$ as the normalized specific cost of immigration to system regions by the types of resources $(r)$;
- $T({K}_{1},\dots ,{K}_{N})$ as the normalized (Here, the normalized supply functions are defined by$$T({K}_{1},\dots ,{K}_{N})=\frac{1}{c}\tilde{T}({K}_{1},\dots ,{K}_{N}),\phantom{\rule{2.em}{0ex}}c=max{c}_{kn},$$
- ${T}^{(s)}({K}_{1},\dots ,{K}_{N})$ as the normalized supply functions by $s=\overline{1,r}$ types of financial resources for immigration.

- the generalized Boltzmann information entropy$${H}_{M}(X)=-\sum _{(i,n)=1;\phantom{\rule{0.166667em}{0ex}}i\ne n}^{N}{x}_{in}ln\frac{{x}_{in}}{{a}_{in}}$$
- by the generalized Boltzmann information entropy$${H}_{I}(X)=-\sum _{k=1}^{M}\sum _{n=1}^{N}{y}_{kn}ln\frac{{y}_{kn}}{{b}_{kn}}$$

- the generalized cost of the form$$\begin{array}{c}Y\in {\mathcal{D}}_{G}(\mathbf{K})=\{Y:\sum _{k=1}^{M}\sum _{n=1}^{N}{c}_{kn}{y}_{kn}=T(\mathbf{K}(t))\},\hfill \\ {c}_{kn}\in [0,1];\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}T(\mathbf{K}(t))=\frac{1}{c}\tilde{T}(\mathbf{K}(t)),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}c=\underset{kn}{max}{c}_{kn};\hfill \end{array}$$
- the cost by the types of resources of the form$$\begin{array}{c}Y\in {\mathcal{D}}_{D}(\mathbf{K})=\{\sum _{k=1}^{M}\sum _{n=1}^{N}{c}_{kn}^{(s)}{y}_{kn}={\tilde{T}}^{(s)}(\mathbf{K}(t))\};\hfill \\ {c}_{kn}^{(s)}\in [0,1];\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\tilde{T}}^{(s)}(\mathbf{K}(t))=\frac{1}{{c}^{(s)}}{T}^{(s)}(\mathbf{K}(t)),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}^{(s)}=\underset{kn}{max}{c}_{kn}^{(s)},\phantom{\rule{0.166667em}{0ex}}s=\overline{1,r}.\hfill \end{array}$$

- for the interregional migration flows within the system,$${x}_{in}^{\ast}(t)=argmax\left({H}_{M}(X),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}X\in \mathcal{D}(\mathbf{K})\right),\phantom{\rule{2.em}{0ex}}(i,n)=\overline{1,N};$$
- for the immigration flows from external regions to the system (with the generalized cost),$${y}_{kn}^{\ast}(t)=argmax\left({H}_{I}(Y),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Y\in {\mathcal{D}}_{G}(\mathbf{K})\right),\phantom{\rule{2.em}{0ex}}k=\overline{1,M},n=\overline{1,N};$$
- for the immigration flows from external regions to the system (with the cost by the types of resources),$${y}_{kn}^{\ast}(t)=argmax\left({H}_{I}(Y),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Y\in {\mathcal{D}}_{D}(\mathbf{K})\right),\phantom{\rule{2.em}{0ex}}k=\overline{1,M},n=\overline{1,N}.$$

## 3. Analysis of Entropy Operators

#### 3.1. Entropy Operator $\mathbf{K}\to X$

#### 3.2. Entropy Operator $\mathbf{K}\to Y$

## 4. Analysis of Migratory Interaction Model of Regional Systems

#### 4.1. Existence of Unique Singular Point

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

#### 4.2. Stability of Singular Point

**Theorem**

**3.**

**Proof.**

## 5. Model of Migratory Interaction: An Example

#### 5.1. Data Arrays for Numerical Study of Model (38)

- ${\alpha}_{1},{\alpha}_{2},$ and ${\alpha}_{3}$ as the characteristics of population mobility in the system $\mathcal{GFI}$;
- $A=[{a}_{in},\phantom{\rule{0.166667em}{0ex}}(i,n)=\overline{1,N},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{a}_{nn}=0]$ as the matrix of the prior probabilities of individual migration in the system $\mathcal{GFI}$;
- $B=[{b}_{kn},\phantom{\rule{0.166667em}{0ex}}k=1,2;n=1,2,3]$ as the matrix of the prior probabilities of individual immigration from the system $\mathcal{SL}$;
- $C=[{c}_{kn},\phantom{\rule{0.166667em}{0ex}}k=1,2;n=1,2,3]$ as the matrix of normalized specific cost of the system $\mathcal{GFI}$ for maintaining the immigration from the system $\mathcal{SL}$.

- mobility $(m):{m}_{1}=7,{m}_{2}=8,{m}_{3}=5$;
- migration choice $(h):{h}_{12}=3,{h}_{13}=3,{h}_{21}=5,{h}_{23}=2,{h}_{31}=6,{h}_{32}=4$;
- immigration choice $(q):{q}_{11}=6,{q}_{12}=5,{q}_{13}=4,{q}_{21}=4,{q}_{22}=6,{q}_{23}=6$;
- immigration cost $(e):{e}_{11}=5,{e}_{12}=6,{e}_{13}=5,{e}_{21}=5,{e}_{22}=6,{e}_{23}=5.$

- the characteristics of population mobility in the system $\mathcal{GFI}$ as$${\alpha}_{n}=\nu {m}_{n},\phantom{\rule{2.em}{0ex}}n=1,2,3;$$
- the prior probabilities of individual migration in the system $\mathcal{GFI}$ as$${a}_{in}=\pi {h}_{in},\phantom{\rule{2.em}{0ex}}(i,n)=1,2,3;$$
- the prior probabilities of individual immigration from the system $\mathcal{SL}$ as$${\tilde{b}}_{kn}=\mu {q}_{kn},\phantom{\rule{2.em}{0ex}}k=1,2;\phantom{\rule{0.166667em}{0ex}}n=1,2,3;$$
- the normalized specific cost of the system $\mathcal{GFI}$ for maintaining the immigration from the system $\mathcal{SL}$ as$${c}_{kn}=T{e}_{kn},\phantom{\rule{2.em}{0ex}}k=1,2;\phantom{\rule{0.166667em}{0ex}}n=1,2,3.$$

#### 5.2. Equilibrium Distributions of Regional Population Sizes in System $\mathcal{GFI}$

#### 5.3. Stability of Equilibrium Distribution of Regional Population Sizes in System $\mathcal{GFI}$

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Notations | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{3}$ |
---|---|---|---|

${C}_{1}$ | 0.0105 | 0.0120 | 0.0075 |

${C}_{2}$ | 0.0070 | 0.0080 | 0.0050 |

${C}_{3}$ | 0.0140 | 0.0160 | 0.0100 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|

0.000 | 0.375 | 0.375 | 0.000 | 0.300 | 0.300 | 0.000 | 0.450 | 0.450 |

0.625 | 0.000 | 0.250 | 0.500 | 0.000 | 0.200 | 0.750 | 0.000 | 0.300 |

0.750 | 0.500 | 0.000 | 0.600 | 0.400 | 0.000 | 0.900 | 0.600 | 0.000 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|

0.900 | 0.750 | 0.600 | 0.750 | 0.625 | 0.500 | 0.600 | 0.500 | 0.600 |

0.600 | 0.900 | 0.900 | 0.500 | 0.750 | 0.750 | 0.400 | 0.600 | 0.600 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|

0.83 | 1.00 | 0.83 | 0.66 | 0.80 | 0.66 | 0.50 | 0.60 | 0.50 |

0.83 | 1.00 | 0.83 | 0.66 | 0.80 | 0.66 | 0.50 | 0.60 | 0.50 |

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Popkov, Y.S.; van Wissen, L. Equilibrium and Stability of Entropy Operator Model for Migratory Interaction of Regional Systems. *Mathematics* **2019**, *7*, 130.
https://doi.org/10.3390/math7020130

**AMA Style**

Popkov YS, van Wissen L. Equilibrium and Stability of Entropy Operator Model for Migratory Interaction of Regional Systems. *Mathematics*. 2019; 7(2):130.
https://doi.org/10.3390/math7020130

**Chicago/Turabian Style**

Popkov, Yuri S., and Leo van Wissen. 2019. "Equilibrium and Stability of Entropy Operator Model for Migratory Interaction of Regional Systems" *Mathematics* 7, no. 2: 130.
https://doi.org/10.3390/math7020130