





























Modeling and Analysis of the Interaction of Neutral and Drug 

Populations: A Competing Species Model 

 

A. Kazmierczak 

T. H. E. Institute 

1111 E. Brooks St. 

Norman, OK 

Akazmierczak1949@gmail.com 

 

 

Abstract 
 

The rise of the drugpopulation in the United States has brought concern, 

debate, and contention to the modern world. Thestrategies of the drug 

cartels are national and are no longer concentrated in a particular location. 

In this paper, we present a dynamical model of the interaction between 

drug cartel and DEA population. The formulation is based on models of 

interactions between competitive 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 five parts. Part 

one analyzes the current populations. Part two analyzes the situation when 

an additional number of drug users are introduced into the drug population. 

Part three analyzes the situation when there is a decline in the drug 

population. Part four analyzes the situation when the drug population goes 

to zero. Part five presents conclusions based on parts one through four. 

 

Keywords: Drugs,competing species model, equilibrium solutions, stability 

at equilibrium solutions. 

 

Mathematic subject classification: 62J12, 62G99 

 

Computing Classification: I.4 

 

1. Introduction 

Drugsare not a new phenomena. However, there is a marked and exponential increase in 

the growth of drug users. Drug users wreak havoc to native citizens. These drug users 

affect all areas of the global economy, markets, and political and social policies. In 

addition, the strength and presence of drug organization, activities create issues. In 

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particular, the rise of the drug users has reached epidemic proportions. Consequently, 

countries are faced with extremely difficult, complex, and contentious political and social 

decisions on the issues of drug users. 

The acceptance of drugs provides a Trojan horse of issues, namely, violence, the 

popularity of drugs in the native country, and the continued growth of drug related 

problems. Hence, countries face the possibility offurther drug users. Despite these 

impending threats, there is not much literature that takes a dynamical systems approach 

to understanding the spread of drug users at a population level. Our primary objective is 

to bridge the gap. 

In our framework, we let D represent the drug population. The neutral population is 

denoted by N: N can be viewed as the total neutral population of a country. This paper is 

a first step in providing a mathematical modeling framework to study the evolution and 

interaction between this neutral and drug population. The neutral population is modeled 

by standard population growth models 

Also, we also consider the addition to the drug population of increased drug users.  The 

paper is organized as follows. In section 2, we develop and analyze the time-dependent 

autonomous refugee ordinary differential equation (ODE) model. We examine the 

equilibrium solutions, the stability of the equilibrium solutions and investigate the 

dynamics numerically. In section 3, we consider the situation when more cartels are 

introduced into the system.We examine the equilibrium solutions, the stability of the 

equilibrium solutions and investigate the dynamics numerically for this situation also. In 

section 4, we present consider the situation where there is a decline in drug popultion. In 

section 5 we present our conclusions based on the analysis in sections 2 and 3 and 4. 

 

2. Neutral Drug (N, D) ODE Model 

Consider the mathematical model 

  N = (a1/(1+d1D) – aNRD/(1+d2N) – b1N)  N = 0 = fN(N, D)   (1) 

  D = (a2/(1+d3N)  – aNRN/(1+d2N)   – b2D)(D) = 0 = gR(N, D)  (2) 

The populations N(t) and U(t) represent the populations of the neutral and  

undocumented populations. New undocumented aliens are slowly coming into the 

undocumented 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 neutral population 

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 a2  Growth rate of the drug population 
 b1  Population loss in N due to intra-species competition and natural 
mortality 
 b2  Population loss in D due to intra-species competition and natural 
mortality  
 aNR  Maximum per capita loss in N due to recruitment by Druggies 
 d1  Measures the effectiveness of CN in disrupting the growth rate of D 
 d2  Measures the resilience of N to recruitment strategies by D 
 d3  Measures the effectiveness of D in creating more cartels 
 
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 D is reduced. This is interpreted as: a highly effective 
DEA population, which can greatly hinder the growth rate of N. The growth rate of the cartel 
population depends on the successful recruitment from the neutral population.  Notice, that if 
d2>> 1 then the recruitment by D is small, Also, if d3>> 1, new drug users are introduced into 
the drugpopulation 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 Drug (N, D) ODE Model 
 
Consider the mathematical model 
 
  

 fN(N, D) = ( a1/(1+d1D) – aNRD/(1+d2N) – b1N ) N = 0    (3) 

fR(N, D) =  (a2/(1+d3N) - (aNRN/(1+d2N))  – b2D )  D = 0    (4) 

 

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

The Jacobian for this system is defined as  
 

│ ∂f/∂N   ∂f/∂D  │ 
 J =        │                                  │ 
  │∂g/∂N   ∂g/∂D │ 
 

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

 

 

2.2 Equilibrium Points 

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

points: 

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

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

Table 2 – Results for Current Population Levels 

Equilibrium  
Point 

Eigen 
values 

Node 
Type 

Stability 

(D = 0., 
N = 0.) 

2, 
2 

Repelling Unstable 

(D = 0., 
N = 4.) 

-2,  
-86/117 

Attracting Unstable 

(D = 4., 
N = 0.) 

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

Attracting Unstable 

(D = .8213492010,  
N = .4301871556) 

.631870324280523,  
-1.35022436018052 

Saddle Unstable 

(D = -1.121275136,  
N = -.4311081397) 

124.789757665452, 
 -7.28136719345222 

Saddle Unstable 

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(D = .1299378971,  
N = -.4346164212) 

-
6.62013265655+9.18652446854370*I,  
-6.62013265655-9.18652446854370*I 

Attracting 
Spiral 

Asymptotically 
Stable 

(D = -2.658090053,  
N = -3.952306486) 

3.45507685676904,  
1.47441380823096 

Repelling Unstable 

 
 

3. Growth of the Drug Population 
 
In this section, we consider the situation where 5000000 new drug users are added to the  
population. The mathematical model now becomes 
 

 fN(N, D) = ( a1/(1+d1(D+5000000)) – aNR(D+5000000)N/(1+d2N) – b1N ) N = 0 (3) 

gR(N, D) =  (a2/(1+d3N) - (aNRN/(1+d2N))  – b2(D+5000000 )  (D+5000000) = 0 (4) 

 

3.1 Equilibrium Points 

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

{D = -5.000000*10^6, N = 0.},  

{D = -4.999996*10^6, N = 0.},  

{D = -5.000000*10^6, N = 4.},  

{D = -4.999999179*10^6, N = .4301871556},  

{D = -5.000001121*10^6, N = -.4311081397},  

{D = -4.999999870*10^6, N = -.4346164212},  

{D = -5.000002658*10^6, N = -3.952306486} 

 

3.2 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. 
 

Table 3 summarizes the results for an increased undocumented population level. 
 
 

Table 3 – Results for Increased Drug Population Levels 

Equilibrium 
 Point 

Eigen 
values 

Node Type Stability 

(D = -4.999996*10^6, 
N = 0.), 

1.0000000*10^7, 
5.000002*10^6 

Repelling Unstable 

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(D = -5.000000*10^6, 
N = 0.) 

9.999992*10^6, 
4.999998*10^6 

Repelling Unstable 

(D = -5.000000*10^6, 
N = 4.), 

1.23452844960607*10^5, 
4.99999921013939*10^6 

Repelling Unstable 

(D = -
4.999999179*10^6, 
N = .4301871556) 

2.88934551996940*10^6, 
4.99999770403060*10^6 

Repelling Unstable 

(D = -
5.000001121*10^6, 
N = -.4311081397) 

5.26749692299691*10^8, 
4.99999006030869*10^6 

Repelling Unstable 

(D = -
4.999999870*10^6, 
N = -.4346164212) 

5.84793624029869*10^8, 
4.99998950513101*10^6 

Repelling Unstable 

(D = -
5.000002658*10^6, 
N = -3.952306486) 

2.09763520857376*10^5, 
5.00000121804262*10^6 

Repelling Unstable 

 
 

4. Decline of the Drug Population 
 
In this section, we consider the situation where 5000000 are removed from the drug 
population. The mathematical model now becomes 
 

 fN(N, D) = ( a1/(1+d1(D-5000000)) – aNR(D-5000000)/(1+d2N) – b1N ) N = 0 (7) 

gR(N, D) = -2D/(1+3N)^2 - aNRD/(1+d2N) – b2(D-5000000) )  (D-5000000) = 0  (8) 

 

4.1 Equilibrium Points 

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

{D = 5.000000*10^6, N = 0.},  

{D = 5.000004*10^6, N = 0.},  

{D = 5.000000*10^6, N = 4.},  

{D = 5.000000821*10^6, N = .4301871556},  

{D = 4.999998879*10^6, N = -.4311081397},  

{D = 5.000000130*10^6, N = -.4346164212}, 

 {D = 4.999997342*10^6, N = -3.952306486} 

 

4.2Analyzing equilibrium points for stability 
 

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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.  

 
Table 3 summarizes the results for an decreased undocumented population level. 
 
 

Table 3 – Results for Decreased Drug Population Levels 

Equilibrium 
 Point 

Eigen 
values 

Node 
Type 

Stability 

(D = 5.000004*10^6, 
N = 0.) 

-1.0000008*10^7,  
-5.000002*10^6 

Attracting Stable 

(D = 5.000000*10^6, 
N = 0.) 

-1.0000008*10^7,  
-5.000002*10^6 

Attracting Stable 

(D = 5.000000*10^6, 
N = 4.) 

-1.23460735239320*10^5,  
-5.00000078986068*10^6 

Attracting Stable 

(D = 
5.000000821*10^6, N 

= .4301871556) 

-2.88934355603247*10^6,  
-5.00000229596753*10^6 

Attracting Stable 

(D = 
4.999998879*10^6, N 

= -.4311081397) 

-5.26749434300308*10^8,  
-5.00000993969177*10^6 

Attracting Stable 

(D = 
5.000000130*10^6, N 

= -.4346164212) 

-5.84793632770131*10^8,  
-5.00001049486937*10^6 

Attracting Stable 

(D = 
4.999997342*10^6, N 

= -3.952306486) 

-2.09755171342874*10^5,  
-4.99999878195713*10^6 

Attracting Stable 

 
 

5. Elimination of Drug Population 
 
In this section, we consider the situation where a mere 300,000 new undocumented aliens  are 
added to the Radical population. The mathematical model now becomes 
 

 fN(N, D) = ( a1/(1+d1(0)) – aNR(0)/(1+d2N) – b1N ) N = 0    (9) 

gR(N, D) =  aNRD/(1+d2N) – b2(0) )  (0) = 0      (8) 

5.1 Equilibrium Points 

Using the Maple CAS on (9) and (10) we obtained the following real valued equilibrium points 

{N = 0., D = 0}, 

 {N = 4., D = 0} 

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5.2Analyzing 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 

 

Table 5 summarizes the results for a zero drug population level. 
 
 

Table 5 – Results for ZeroDrug Population Levels 

Equilibrium  
Point 

Eigen 
values 

Node Type Stability 

(N = 0., 
D = 0) 

2, 
2 

Repelling Unstable 

(N = 4., 
D = 0) 

-2,  
-86/117 

Attracting Asymptotically 
stable 

 
 
 

6. Conclusions 
 
In this paper we modeled and analyzed the interaction of neutral and drug populations. A 
comparison of the results in Table 2 indicates that the system already contains some instability, 
the entire system becomes more unstable and Table 3 indicates that with an increase in drug 
population the system becomestotally unstable Table 4 indicates that with a decline in drug 
population the system becomes more stable, While table 5 indicates that one node is stable 
while the other is unstable. We interpret this to the fact that drug population could once again 
rise  
 
 

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