# Modelling Population Dynamics of Social Protests in Time and Space: The Reaction-Diffusion Approach

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## Abstract

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## 1. Introduction

## 2. Mathematical Models in the Non-Spatial Case

#### 2.1. Single Species Model

- The rate of change in the number of people attending the event is a result of the interplay between two processes, recruitment (people joining) and withdrawal (people quitting);
- Following earlier studies [18], we consider recruitment to be a collective phenomenon so that the recruitment rate is a nonlinear function of the number of people currently involved in the event;
- Decision of withdrawal is made individually, so that the withdrawal rate is a linear function of the number of people participating in the event, where the per capita withdrawal rate (say m) depends on time.

#### 2.2. Two Component Model

## 3. Spatially Explicit Model

#### 3.1. Single Species Model

#### 3.2. Two Component Model

## 4. Discussion and Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Properties of the model given by Equations (1) and (2) in the baseline case of constant m. (

**a**) An example of the relative position of the per capita recruitment rate (black curve) as a function of N (in abstract units) and the per capita withdrawal rate (red line). Any intersection between the two lines is a steady state. The two stable steady states are shown by the small black circles, and the unstable steady state is shown by the red circle. (

**b**) A sketch of the bifurcation diagram: The thick black curve shows the steady state value as a function of m. The two long-dashed vertical lines correspond to the bifurcation values of m; for any value of m such as ${m}_{\ast}<m<{m}^{\ast}$ (see the vertical dotted line), the system has three steady states, two stable and one unstable.

**Figure 2.**Solution of Equations (1) with (2) and (5) obtained for different parameter values of ${m}_{0}$ and ${m}_{1}$. (

**a**) ${m}_{1}=3$, the values of ${m}_{0}$ are given in the figure legend. (

**b**) ${m}_{0}=5.5$, the values of ${m}_{1}$ are given in the figure legend. Other parameters are ${\u03f5}_{0}=0.1,\phantom{\rule{3.33333pt}{0ex}}\u03f5=3,\phantom{\rule{3.33333pt}{0ex}}a=150,\phantom{\rule{3.33333pt}{0ex}}h=35$, and $b=0.01$, which corresponds to ${m}_{\ast}=3.22$ and ${m}^{\ast}=5.15$.

**Figure 3.**Phase plane of the system (7). The blue Curve 1 shows the N-isocline; note its part that lies close to the vertical axis. The red Curve 2 shows the solution of the system (obtained for parameters ${\u03f5}_{0}=0.1,\phantom{\rule{3.33333pt}{0ex}}\u03f5=1,\phantom{\rule{3.33333pt}{0ex}}a=115,\phantom{\rule{3.33333pt}{0ex}}h=20$, and $\beta =0.00025$) where red arrows indicate the direction of the system’s evolution along the trajectory. The black arrows show the generic direction of the phase flow in different parts of the plane.

**Figure 4.**Solutions of Model (7) obtained for different values of $\beta $: (

**a**) the number of protesters N vs. time and (

**b**) the per capita withdrawal rate m vs. time. Other parameters are ${\u03f5}_{0}=0.1,\phantom{\rule{3.33333pt}{0ex}}\u03f5=3,\phantom{\rule{3.33333pt}{0ex}}a=115$, and $h=20$.

**Figure 5.**Solution of Model (8) shown at different moments of time (as is explained in the figure legend) obtained for ${m}_{1}=3$ and different values of the final withdrawal rate: (

**a**) for ${m}_{0}=5.4$ and (

**b**) for ${m}_{0}=5.8$. Other parameters are $\u03f5=3,\phantom{\rule{3.33333pt}{0ex}}{\u03f5}_{1}=0,\phantom{\rule{3.33333pt}{0ex}}a=150,\phantom{\rule{3.33333pt}{0ex}}h=35,\phantom{\rule{3.33333pt}{0ex}}b=0.01$, and $D=1$.

**Figure 6.**Dependence on time of (

**a**) M and (

**b**) the speed of the front calculated in the course of the travelling front propagation. The parameters are the same as in Figure 5a. It is readily seen that the moment ($t\approx 123$) when the front changes the direction of its propagation (invasion changes to retreat) coincides with the moment when M changes its sign.

**Figure 13.**Dependence of the speed of the travelling peak on $\u03f5$; diamonds show the numerical results, and the blue line shows the analytical estimate (24).

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**MDPI and ACS Style**

Petrovskii, S.; Alharbi, W.; Alhomairi, A.; Morozov, A.
Modelling Population Dynamics of Social Protests in Time and Space: The Reaction-Diffusion Approach. *Mathematics* **2020**, *8*, 78.
https://doi.org/10.3390/math8010078

**AMA Style**

Petrovskii S, Alharbi W, Alhomairi A, Morozov A.
Modelling Population Dynamics of Social Protests in Time and Space: The Reaction-Diffusion Approach. *Mathematics*. 2020; 8(1):78.
https://doi.org/10.3390/math8010078

**Chicago/Turabian Style**

Petrovskii, Sergei, Weam Alharbi, Abdulqader Alhomairi, and Andrew Morozov.
2020. "Modelling Population Dynamics of Social Protests in Time and Space: The Reaction-Diffusion Approach" *Mathematics* 8, no. 1: 78.
https://doi.org/10.3390/math8010078