Chaos in a predator–prey-based mathematical model for illicit drug consumption

Abstract

Recently, a mathematical model describing the illicit drug consumption in a population consisting of drug users and non-users has been proposed. The model describes the dynamics of non-users, experimental users, recreational users, and addict users within a population. The aim of this work is to propose a modified version of this model by analogy with the classical predator-prey models, in particular considering non-users as prey and users as predator. Hence, our model includes a stabilizing effect of the growth rate of the prey, and a destabilizing effect of the predator saturation. Functional responses of Verhulst and of Holling type II have been used for modeling these effects. To forecast the marijuana consumption in the states of Colorado and Washington, we used data from Hanley (2013) and a genetic algorithm to calibrate the parameters in our model. Assuming that the population of non-users increases in proportion with the demography, and following the seminal works of Sir Robert May (1976), we use the growth rate of non-users as the main bifurcation parameter. For the state of Colorado, the model first exhibits a limit cycle, which agrees quite accurately with the reported periodic data in Hanley (2013). By further increasing the growth rate of non-users, the population then enters into two chaotic regions, within which the evolution of the variables becomes unpredictable. For the state of Washington, the model also exhibits a periodic solution, which is again in good agreement with observed data. A chaotic region for Washington is likewise observed in the bifurcation diagram. Our research confirms that mathematical models can be a useful tool for better understanding illicit drug consumption, and for guiding policy-makers towards more effective policies to contain this epidemic.

Bifurcation analysis and chaos control of discrete prey–predator model incorporating novel prey–refuge concept

“…The second effect on the trade‐offs and by‐products of the hiding behavior of prey, which could be advantageous or detrimental for all interacting populations. Almost all the researchers consider the refuge as a constant number of prey or proportional to prey 13‐34 . This article presents a novel refuge term which is depending on the quantity of prey population, and the refuge is reducing based on the quantity of proportion to predator.…”

Impact of strong determination and awareness on substance addictions: A mathematical modeling approach

“…[9][10][11][12][13][14][15][16][17] Mathematical models are significant in understanding and analysing different aspects of human life. [18][19][20] Individuals' social activity or the transmission of infectious diseases may be studied using mathematical models. [21][22][23] In terms of alcohol, numerous statistical models have been developed and studied in order to further reduce the number of drinkers.…”

“…Control theory and dynamical systems are very helpful to solve many day to day life problems 9–17 . Mathematical models are significant in understanding and analysing different aspects of human life 18–20 . Individuals' social activity or the transmission of infectious diseases may be studied using mathematical models 21–23 .…”

Dynamics of a discrete-time system with Z-type control

“…By experimentally manipulating the adult mortality rate of flour beetle Tribolium, they also observed changes in the dynamics from stable fixed points to periodic cycles and aperiodic oscillations that corresponded to the prediction of the mathematical model. On the other hand, for showing chaotic behavior in a continuous-time autonomous model minimum three species are needed [13,14]. Discrete-time models are easy to understand, develop, and simulate.…”