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Protester Populations: A Competing Species Model

A. Kazmierczak

Department of Computer Science
Oklahoma State University

Stillwater, OK
akazmie@okstate.edu

Abstract

The  rise  of  radicalized  terrorist  groups,  such  as  the  Islamic  State  of  Iraq  and  Syria  (ISIS), 
throughout  the  world  have  brought  concern,  debate,  and  contention  to  the  modern  world.  The 
recruitment  strategies  of  terrorist  networks  are  global  and  are  no  longer  concentrated  in  a 
particular location. In this paper, we present a dynamical model of the interaction and recruitment 
between  a  non-radicalized  or  neutral  and  radicalized  population.  The  formulation  is  based  on 
models of interactions between competing species [3] type dynamics. An exploration of the long- 
term dynamics and stability of homogeneous equilibrium solutions and their stability is given. The 
paper  is  given  in  two  parts.  Part  one  analyzes  the  current  populations.  Part  two  analyzes  the

situation when an additional number of radicals are introduced into the radicalized population.

Keywords: Terrorism,  competing  species  model,  equilibrium  solutions,  stability  at  equilibrium

solutions.

Mathematica subject classification: 62J12, 62G99

Computing Classification: I.4

1. Introduction

Protestorsare not  a  new phenomena.  However,  there  is  a  marked  and  exponential  increase  in  the 
frequency of protests since the inception of the civil rights act. Protestors can wreak havoc and spread fear 
and panic to native citizens. In addition, the strength and presence of protestor activities create emigration 
issues. Consequently, countries are faced  with  extremely  difficult, complex, and contentious  political  and 
social decisions on the activities that cause protest situations.

Despite these ongoing protests, there is not much literature that takes a dynamical systems approach to 
understanding the spread of terrorism, at a population. Our primary objective is to bridge the gap.

In  our  framework,  we  let N represent  the neutral population.  The protestor is  denoted  by P: P can  be 
viewed  as  the  protestor population  of  a certain  situation.  This  paper  is  a  first  step  in  providing  a 
mathematical  modeling  framework  to  study  the  evolution  and  interaction  between  this  cop and protestor 
population. The protestor population is modeled by standard population growth models. Also, we consider 
the addition to the protestor population from increased protestors.  The paper is organized as follows. In 
section two,  we  develop  and  analyze the  time-dependent  autonomous  protestor ordinary  differential

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mailto:akazmie@okstate.edu


equation (ODE) modeland the effect of a current populations. We examine the equilibrium solutions, the 
stability of the equilibrium solutions and investigate the dynamics numerically. In section three, we consider 

the situation when more protesters are in the system.We examine the equilibrium solutions, the stability of 
the equilibrium solutions and investigate the dynamics numerically for this situation also. In section four, 

we consider the scenario when the protestor population declines. In section 5 we present ur conclusions. 

 

2. Neutral Protester (N, P) ODE Model 

Consider the mathematical model 

  N = a1N/(1+d1C) – aNRNP/(1+d2N) – b1N
2
= 0 = fN(N, P)    (1) 

  P = a2P/(1+d3N) - aNRNP/(1+d2N) – b2P
2
 = 0 = fR(N, P)    (2) 

The populations N(t) and P(t) represent the populations of the neutral and protester populations.. The 

parameters are all assumed to be positive and their descriptions are given in Table 1a. 

Table 1a: List of parameters used in the differential equation model 

Symbols Meaning 

 a1  Growth rate of the protestor population 
 a2  Growth rate of the police population 
 b1  Population loss in N due to intra-species competition and natural mortality 
 b2  Population loss in P due to intra-species competition and natural mortality  
 aNR  Maximum per capita loss in N due to recruitment by protester groups 
 d1  Measures the effectiveness of N in disrupting the growth rate of N 
 d2  Measures the resilience of N to recruitment strategies by P 
 d3  Measures the effectiveness of Nin disrupting protester activities 
 
In the case of di = bi = 0, the mathematical model becomes similar to the competing species model. The 
parameters di influence the carrying capacity of the individual populations. For instance, if d1>> 1 then the 
growth rate of N is reduced. This is interpreted as: a highly effective protester population can greatly 
hinder the growth rate of N. The growth rate of the protester population depends on the successful 
recruitment from the neutral. Notice, that if d2>> 1 then the recruitment by P is small, Also, if d3>> 1, new 
protester are introduced into the protester population is smaller. The values chosen for the variables in 
this model are listed in Table 1b. 
 

Table1b: Values of parameters 
a1 a2 b1 b2 aNR d1 d2 d3 

2 2 0.5 0.5 2 2 2 3 
 
 
 

2.1Neutral Protester (N, P) ODE Model 
 
Consider the mathematical model 
 

 fN(N. P) = ( a1/(1+d1P) – aNRP/(1+d2N) – b1N) N = 0     (3) 

fR(N, P) = ( a2/(1+d3N) –anrN/(1+dPN) – b2P )  P = 0     (4) 

 

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Since this system is nonlinear, the first step is linearization using the Jacobian. 

The Jacobian for this system is defined as  
 

│ ∂f/∂N   ∂f/∂P  │ 
 J =        │                                  │ 
  │∂g/∂N   ∂g/∂P │ 
 
Taking the partial derivatives, simplifying and using the values in table for the parameters, the Jacobian 
becomes. 
 
  │2/(1+2P)-2P/(1+2N)^2-N              -2/(1+2P)^2-2N/(1+2N) │ 
 J  = │        │ 
  │ -6P/(1+3N)^2-2P/(1+2N)^2         2/(1+3N)-2N/(1+2N)-P    │ 
 

 

2.2 Equilibrium Points 

Using Maple CAS, on (3) and (4) weobtained the following real valued equilibrium points: 

{N = 0., P = 0.},  
{N = 0., P = 4.},  
{N = 4., P = 0.},  
{N = .4301871556, P = .8213492010},  
{N = -.4311081397, P = -1.121275136}, 
 {N = -.4346164212, P = .1299378971}, 
 {N = -3.952306486, P = -2.658090053} 
 
 

2.3Analyzing equilibrium points for stability 
 
In this section we use the equilibrium points to generate the eigenvalues for the system and establish 
whether the equilibrium point is stable or unstable. Substituting equilibrium points into the Jacobian and 
solving for eigenvalues, we get the results in Table 2. 
 
 
  

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2.4Summarization 
 
Table 2 summarizes the results for the current population levels. 
 

Table 2 – Results for Current Population Levels 
Equilibrium 

Point 
Eigen 
Values 

Node 
Type 

Stability 

{N = 0., 
P = 0.}, 

2.00, 
2.00 

Repelling Unstable 

{N = 0., 
 P = 4.} 

-44/9+(2/9)*sqrt(185),  
-44/9-(2/9)*sqrt(185) 

Attracting Asymptotically 
stable 

{N = 4., 
 P = 0}, 

-2,  
-86/117 

Attracting Asymptotically 
stable 

{N = .4301871556, 
 P = .8213492010}, 

.757984137794684,  
-1.31667584619468 

Saddle Unstable 

{N = -.4311081397, 
 P = -1.121275136}, 

124.789757665452,  
-7.28136719345222 

Saddle Unstable 

{N = -.4346164212,  
P = .1299378971} 

-6.620132656550+9.18652446854370*I,  
-6.620132656550-9.18652446854370*I 

Attracting  
spiral 

Asymptotically 
stable 

{N = -3.952306486,  
P = -2.658090053} 

3.45507685676904,  
1.47441380823096 

Repelling Unstable 

 
 

3. Growth of the Protester Population 
 
In this section, we consider the situation where a there is a 25% increase in the protester population. The 
mathematical model now becomes 
 

 fN(N, P) = ( a1/(1+d1(1.25P)) – aNR(1.25P)/(1+d2N) – b1N ) N = 0    (5) 

fR(N, P) = ( a2/(1+d3(N) - aNRN/(1+d2N) – b2(1/+1.25P )  (1.25P) = 0   (6) 

 

Using the Maple CAS, on (5) and (6) we obtained the following real valued equilibrium points 

{N = 0., P = 0.},  
{N = 0., P = 3.200000000},  

{N = 4., P = 0.},  
{N = .3156552235, P = 1.024389733},  

{N = -.4403859177, P = 1.855697361},  
{N = -.4325472689, P = -.4910205568},  
{N = 1.304975560, P = -.5057028313},  

{N = -5.192142042, P = -1.990030373} 
  

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3.1Analyzing equilibrium points for stability 

 
In this section we use the equilibrium points to generate the eigenvalues for the system and establish 
whether the equilibrium point is stable or unstable. 

 

3.2 Summarization 
 
Table 3 summarizes the results for an increased police population level. 
 
 

Table 3 – Results for Increased ProtesterPopulation Levels 
Equilibrium 

Point 
Eigen 
Values 

Type of 
Node 

Stability 

(N = 0., 
P = 0.) 

2.00, 
2.00 

Repelling 
 

Unstable 
 

(N = 0., 
P =3.200000000) 

-6.31260878122594,  
-1.01712094877406 

Attracting 
 

Asymptotically 
stable 

(N = 4., 
P = 0.) 

-2,  
-86/117 

Attracting 
 

Asymptotically 
stable 

(N = .3156552235,  
P = 1.024389733) 

-1.60703900385485,  
.793359729354847 

Saddle Unstable 
 

(N = -.4403859177,  
P = 1.855697361) 

-249.390466169796,  
-11.5240335472042 

Attracting Asymptotically 
stable 

(N = -.4325472689,  
P = -.4910205568) 

82.9709967044000+730.356706992694*I, 
82.9709967044000-730.356706992694*I 

Repelling Unstable 

(N = 1.304975560,  
P = -.5057028313) 

-156.658811051449,  
-19.7304049866513 

Attracting Asymptotically 
stable 

(N = -5.192142042,  
P = -1.990030373) 

4.53052768941096,  
.781943173589040 

Repelling Unstable 

 
 

4. Decline of the Protester Population 
 
In this section, we consider the situation where there is a 25% in the protester population. The 
mathematical model now becomes 
 

 fN(N, P) = ( a1/(1+d1(0.75P)) – aNR(P)/(1+d2(N) – b1(N ) N = 0         (7) 

fR(N, P) = ( a2/(1+d3(N)) -  anrN/(1+d2 N) – b2(0.75P) )  (0.75P) = 0         (8) 

 

Using the Maple CAS, on (7) and (8) we obtained the following real valued equilibriumpoints: 

{N = 0., P = 0.},  
{N = 0., P = 5.333333333},  
{N = 4., P = 0.}, 
 {N = .4301871556, P = 1.095132268},  
{N = -.4311081397, P = -1.495033515}, { 
{N = -.4346164212, P = .1732505294},  
{N = -3.952306486, P = -3.544120071} 

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4.1 Analyzing equilibrium points for stability 
 
In this section we use the equilibrium points to generate the eigenvalues for the system and establish 

whether the equilibrium point is stable or unstable  

. 

4.2 Summarization 
 
Table 4 summarizes the results for a decreased protester population. 
 
 

Table 4 _ Results for Decreased Protestor population 
Equilibrium 

Point 
Eigen 
Values 

Type of 
Node 

Stability 

(N = 0., 
P = 0.), 

2, 
2 

Repelling Unstable 

(N = 0., 
P = 5.333333333) 

-10.5817315235956,  
-3.24683990940437 

Attracting Asymptotically 
stable 

(N = 4., 
P = 0.) 

-2, 
 -86/117 

Attracting Asymptotically 
stable 

(N = .4301871556, 
P = 1.095132268) 

.561198866339937,  
-1.68177931653994 

Saddle Unstable 

(N = -.4311081397, 
P = -1.495033515) 

166.047756962696,  
-8.18559401169599 

Saddle Unstable 

(N = -.4346164212, 
P = .1732505294), 

-9.22574755770000+9.57258822194460*I, 
-9.22574755770000-9.57258822194460*I 

Attracting Asymptotically 
stable 

(N = -3.952306486, 
P = -3.544120071) 

3.45408293768524,  
2.53347890031476 

Repelling Unstable 

 
 

5. Conclusions 
 
In this paper we modeled and analyzed the interaction of protestor and neutral populations. A comparison 
of the results in Table 2, Table 3,and Table 4 seem to indicate that no matter the relative sizes of the 
populations, there will besome level of instability in the system. 
 
 

References: 
 

1.  "Thousands march against nuclear power in Tokyo". USA Today. September 2011. 

2. St. John Barned-Smith, "How We Rage: This Is Not Your Parents' Protest," Current (Winter 
2007): 17-25. 

3. Adam Roberts, Introduction, in Adam Roberts and Timothy Garton Ash (eds.), Civil Resistance 
and Power Politics: The Experience of Non-violent Action from Gandhi to the Present, Oxford 
University Press, 2009, pp. 2-3, where a more comprehensive definition of "civil resistance" may 
be found. 

4. Daniel L. Schofield, S.J.D. (November 1994). "Controlling Public Protest: First Amendment 
Implications". in the FBI's Law Enforcement Bulletin. Retrieved 2009-12-16. 

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https://www.usatoday.com/news/world/story/2011-09-19/japan-anti-nuclear-protest/50461872/1
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5. Kruszewski, Brent Baldwin, Jackie. "Why They Keep Fighting: Richmond Protesters Explain Their 
Resistance to Trump's America". Style Weekly. Retrieved 29 March 2017. 

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9. Mcgrath, Ben (November 13, 2006). "Holy Rollers". 

10. "Critical Mass London". Urban75. 2006. 

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13. Seaton, Matt (October 26, 2005). "Critical crackdown". London: The Guardian. Retrieved May 
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14. Rosi-Kessel, Adam (August 24, 2004). "[*BCM*] Hong Kong Critical Mass News". 

15. https://www.flickr.com Image of black bloc members during Iraq War Protest in Washington, D.C., 
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16. D. Parvaz, Iran's Silent Protests 

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18. Newman, Lily Hay. "How to Use Social Media at a Protest Without Big Brother 
Snooping". WIRED. Retrieved 2017-02-09. 

19. Deseret Morning News, 13 Nov. 2007 issue, p. E3, Coverage of protests hurts firms, Cornell-Y. 
study says, Angie Welling 

 

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