Modeling and Analysis of the Interaction of Neutral and 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 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 06 | June 2020 www.ijojournals.com 16 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) IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 06 | June 2020 www.ijojournals.com 17 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. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 06 | June 2020 www.ijojournals.com 18 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} IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 06 | June 2020 www.ijojournals.com 19 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} IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 06 | June 2020 www.ijojournals.com 20 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. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 06 | June 2020 www.ijojournals.com 21 https://www.usatoday.com/news/world/story/2011-09-19/japan-anti-nuclear-protest/50461872/1 https://en.wikipedia.org/wiki/Adam_Roberts_(scholar) https://books.google.com/books?id=BxOQKrCe7UUC&dq=Civil+resistance+and+power+politics&source=gbs_navlinks_s https://books.google.com/books?id=BxOQKrCe7UUC&dq=Civil+resistance+and+power+politics&source=gbs_navlinks_s http://www.thefreelibrary.com/Controlling+public+protest%3a+First+Amendment+implications.-a016473804 http://www.thefreelibrary.com/Controlling+public+protest%3a+First+Amendment+implications.-a016473804 https://en.wikipedia.org/wiki/Federal_Bureau_of_Investigation https://en.wikipedia.org/wiki/FBI_Law_Enforcement_Bulletin 5. Kruszewski, Brent Baldwin, Jackie. "Why They Keep Fighting: Richmond Protesters Explain Their Resistance to Trump's America". Style Weekly. Retrieved 29 March 2017. 6. Global Nonviolent Action Database 7. Dynamics of Collective Action Project 8. Ratliff, Thomas (2014). "Practicing the Art of Dissent: Toward a Typology of Protest Activity in the United States". Humanity & Science. 38 (3): 268–294. 9. Mcgrath, Ben (November 13, 2006). "Holy Rollers". 10. "Critical Mass London". Urban75. 2006. 11. "Pittsburgh Critical Mass". 12. "Critical Mass: Over 260 Arrested in First Major Protest of RNC". Democracy Now!. August 30, 2004. 13. Seaton, Matt (October 26, 2005). "Critical crackdown". London: The Guardian. Retrieved May 22, 2010. 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., March 21, 2009. 16. D. Parvaz, Iran's Silent Protests 17. Adam Roberts and Timothy Garton Ash (eds.), Civil Resistance and Power Politics: The Experience of Non-violent Action from Gandhi to the Present, Oxford: Oxford University Press, 2009. ISBN 978-0-19-955201-6.[1] 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. 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