

































Modeling and Analysis of the Interaction of Neutral Populations 
and Violent Cop Populations: A Competing Species Model 

 

A. Kazmierczak 

Department of Computer Science 
Oklahoma State University 

Stillwater, OK 
akazmie@okstate.edu 

 
 

Abstract 
 

The rise of police violence in the United States has brought concern, debate, and 

contention to the modern world. Police violence is national and is no longer 
concentrated in a particular location. In this paper, we present a dynamical model of 

the interaction between neutral population and violence. The formulation is based on 
models of interactions between competing species 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 three parts. Part one analyzes the current 

level of violence. Part two analyzes the situation when police violence increases. Part 

three analyzes the situation when police violence decreases 
 

Keywords: Drug cartels, DEA, competing species model, equilibrium solutions, 
stability at equilibrium solutions. 

 
 

1. Introduction 

Police violenceare not a new phenomena. However, there is a marked and exponential increase in 
the growth of police violence. This violence wreak havoc to native citizens. This violence affects 
political and social policies and createsserious issues. Consequently, countries are faced with 

extremely difficult, complex, and contentious political and social decisions on the issues of police 
violence. 

The acceptance of continued violence provides a tolerance for moreviolence, and a failing belief 

that police are actually there to protect and serve. Hence, countries face the possibility of ever 
increasing police violence. Despite these impending threats, there is not much literature that takes 

a dynamical systems approach to understanding the spread of police violence. Our primary 
objective is to bridge the gap. 

In our framework, we let V represent the level of police violence.This police violence denoted by V: 
V can be viewed as the amount of police violence in a community. This paper is a first step in 

providing a mathematical modeling framework to study the evolution and interaction between this 
neutral population and growing police violence. The neutral population is modeled by standard 

population growth models and is represented by N. 

The paper is organized as follows. In section 2, we develop and analyze the time-dependent 
autonomous police violence ordinary differential equation (ODE) model. We examine the 

equilibrium solutions, the stability of the equilibrium solutions and investigate the dynamics 
numerically. In section 2, we consider the situation of the current level of police violence in the 
system.We examine the equilibrium solutions, the stability of the equilibrium solutions and 

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mailto:akazmie@okstate.edu


investigate the dynamics numerically for this situation also. In section 3, we consider the situation 
where more police violence is encountered. In section 4 we consider the affect of reducing police 

violence In part 5 we present our conclusions based on the analysis in sections 2, 3 and 4. 

 

2. Neutral Violence (NV) ODE Model 

Consider the mathematical model 

  N = a1N/(1+d1U) – anrVN/(1+d2N) – b1N
2
 = 0 = fN(N, V)   (1) 

  V= a2V/(1+d3V) – anrN/(1+d2N)    – b2V
2
 = 0 = fR(C, D)   (2) 

The populations N(t) and V(t) represent the populations of the neutral and violent 

policepopulations. More policeviolent are slowly coming into the police population. 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 Police population 
 a2  Growth rate of the Neutral population 
 b1  Population loss in N due to intra-species competition and natural mortality 
 b2  Population loss in V due to intra-species competition and natural mortality  
 aNR  Maximum per capita loss in N due to police violence 
 d1  Measures the effectiveness of V in disrupting the growth rate of N 
 d2  Measures the resilience of N to violent strategies by V 
 d3  Measures the effectiveness of V in avoiding neutral punishment 
 
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. Or instance, if d1>> 1 then the 
growth rate of V is reduced. This is interpreted as violent police population, which can greatly hinder the 
growth rate of N. Notice, that if d2>> 1 then the recruitment by V is small, Also, if d3>> 1, new cartels are 
introduced into the violent police population more slowly 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 
 
 

2.1 Neutral Violent (NV) ODE Model 
 
Consider the mathematical model 
 
  

 fN(N, V) = ( a1/(1+d1V) – aNRVV/(1+d2N) – b1N ) N = 0    (3) 

fR(N,V) = (a2/(1+d3N))   ) – aNRN/(1+d2N)   – b2V )  V = 0   (4) 

 

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

The Jacobian for this system is defined as  
 

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

2.2 Equilibrium Points 

 

Using the Maple CAS from Maplesoft on (3) and (4) and obtained the following real valued equilibrium 

points: 

{N = 0., V = 0.},  
{N = 0., V = 4.},  
{N = 4., V = 0.},  
{N = .4891955799, V = .6319394087},  
{N = -.4325627635, V = -.6082709305},  
{N = -.4345884397, V = .1197573734},  
{N = -3.074988235, V = -2.874675564} 
 

2.1.1 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. 
 
 
 

2.1.2 Summarization 
 

Table 2 summarizes the results for the current violence levels. 
 

Table 2 – Results for Current Violence Levels 
Equilibrium  

Point 
Eigen 
values 

Node Type Stability 

(N = 0., 
V = 0.) 

2,  
2 

Repelling Unstable 

(N = 0., 
V = 4.) 

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

Attracting Stable 

(N = 4. 
V = 0.) 

-2,  
-86/117 

Attracting Stable 

(N = .4891955799, 
V = .6319394087) 

.812643939172984,  
-1.05727942377298 

Saddle Unstable 

(N = -.4325627635, 
V = -.6082709305) 

29.1880923942500+55.5122280784733*I, 
29.1880923942500-55.5122280784733*I 

Repelling Unstable 

(N = -.4345884397,  
V = .1197573734) 

-6.003222251700+9.00214635655358*I,  
-6.00322225170-9.00214635655358*I 

Attracting Stable 

(N = -3.074988235, 
V = -2.874675564) 

2.15399521750000+.302594212184905*I, 
2.15399521750000-.302594212184905*I 

Repelling Unstable 

 

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3. Growth of the Violent Population 

 
In this section, we consider the situation where there is a 25% increase inviolence. The mathematical 
model now becomes 
 

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

fR(N, V) = ( a2/(1+d3N) - aNRN/(1+d2N) – b2(1.25V) )  (1.25V) = 0     (6)V 

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

 

3.1.1 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. 
 
{N = 0., V = 0.},  

{N = 0., V = 3.200000000}, 

 {N = 4., V = 0.},  

{N = .4891955799, V = .5055515270},  

{N = -.4325627635, V = -.4866167444},  

{N = -.4345884397, V = 0.9580589875e-1}, 

 {N = -3.074988235, V = -2.299740451} 

 
3.1.2 Summarization 

 
Table 3 summarizes the results for an increased violence level. 
 
 

Table 3 – Results for Increased Violence Levels 
Equilibrium  

Point 
Eigen 
values 

Node 
Type 

Stability 

(N = 0., 
V = 0.) 

2,  
2 

Repelling Unstable 

(N = 0., 
V = 3.200000000) 

-6.31260878122594,  
-1.01712094877406 

Attracting Asymptotically 
Stable 

(N = 4., 
V = 0.) 

-2,  
-86/117 

Attracting Asymptotically 
Stable 

(N = .4891955799, 
V = .5055515270) 

2.35085912240317,  
1.20431576659683 

Repelling Unstable 

(N = -.4325627635, 
V = -.4866167444) 

-40.8902957221704,  
-14.0745709068296 

Attracting Asymptotically 
Stable 

(N = -.4345884397, 
V = .09580589875 

-4.55934547537500+8.41236264515438*I,  
-4.55934547537500-8.41236264515438*I 

Attracting Asymptotically 
Stable 

(N = -3.074988235, 
V = -2.299740451) 

2.35085912240317,  
1.20431576659683 

Repelling Unstable 

 
 

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4. Decrease in the Violent Population 
 
In this section, we consider the situation where the violence is reduced by 25%. The mathematical model 
now becomes 
 

 fN(N, V) = ( a1/(1+d1(.75V)) – aNR(.75V)/(1+d2N) – b1N ) N = 0    (7) 

fR(N, V) =  ( a1/(1+d3(N) – aNR(N)/(1+d2N)  – b2(.75V) )  (.75V) = 0    (8) 

 

4.1 Equilibrium Points 

 

Using the Maple CAS on (7) and (8) and obtained the following real valued equilibrium points: 

{N = 0., V = 0.},  

{N = 0., V = 5.333333333},  

{N = 4., V = 0.},  

{N = .4891955799, V = .8425858783}, 

 {N = -.4325627635, V = -.8110279073},  

{N = -.4345884397, V = .1596764979},  

{N = -3.074988235, V = -3.832900753} 

 

 

4.2 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.  

  

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4.3 Summarization 
 
Table 3 summarizes the results for adecreased violence level. 
 
 

Table 3 – Results for Decreased Violence Levels 
Equilibrium  

Point 
Eigen 
values 

Node  
Type 

Stability 

(N = 0., 
V = 0.) 

2,  
2 

Repelling Unstable 

(N = 0., 
V = 5.3333333) 

-10.5817315137926,  
-3.24683990620742 

Attracting Asymptotically 
Stable 

(N = 4., 
V = 0.) 

-2,  
-86/117 

Attracting Asymptotically 
Stable 

(N = .4891955799, 
V = .8425858783) 

.651325662915316,  
-1.35285239051532 

Saddle Unstable 

(N = -.4325627635, 
V = -.8110279073) 

88.4262919206642,  
-1.53454758536415 

Saddle Unstable 

(N = -.4345884397, 
V = .1596764979) 

-8.40445002895+9.519407358943*I,  
-8.40445002895-9.519407358943*I 

Attracting Asymptotically 
Stable 

(N = -3.074988235, 
V = -3.832900753) 

2.72977251800000+.817145809380968*I, 
2.72977251800000-.817145809380968*I 

Repelling Unstable 

 
 
 

5 Conclusions 
 
As Table 2 indicates, at the current level of police violence, there is not much stability in the system. 
Table three indicates that more police violence keeps the system very unstable, where we interpret 
instability as those points where police action is inappropriate or even excessive. At these points, the 
neutral population begins to mistrust police actions and may engender fear that such actions could be 
turned on even otherwise peaceful citizenry. Even in Table 4, where police tone down the level of 
violence the system is still unstable. 
 
 

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