Airline System Dynamics II - Chaos

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WHAT IS IT?

Evolutionary game model of four airline competition for market share employing a Lotka-Volterra model of competition leading to chaos, the model’s equations were adapted from Vano, JA, Wildenberg, JC, Anderson, MB, Noel, JK and Sprott, JC (2006) “Chaos in low-dimensional Lotka–Volterra models of competition” Nonlinearity 19, 2391-2404.

The model was used as classroom material for the Mathematics I and II lectures taught by the author at the Bachelor in Aeronautical Management (https://www.ulusofona.pt/en/undergraduate/aeronautical-management) at Lusophone University of Humanities and Technologies’s “School of the Economic Sciences and Organizations”.

The full lecture materials, including webinars (in Portuguese) are available at: https://sites.google.com/view/systemdynamics/

HOW IT WORKS

The model works with four competing airline companies. Each company is characterized by a fitness level, and their respective market shares are proportional to the fitness, so that for i = 1,2,3,4, we get:

market share(i,t) = fitness(i,t) / Total Fitness(t)

Total Fitness = fitness(1,t) + fitness(1,t) + fitness(3,t) + fitness(4,t)

The dynamics for each company's fitness is given by:

dfitness(i,t)/dt = market share growth(i,t) - competition(i,t)

market share growth(i,t) = growth rate(i) * fitness(i,t)

competition(i,t) = growth rate(i,t) * fitness(i,t) * competition factor(i,t)

The competition factor for each company is, in turn, given by:

competition(i,t) = fitness(i,t) + weighted competition strength(i,t)

weighted competition strength(i,t) = sum(wij*fitness(j,t)), for j different from i.

The model is implemented using Netlogo's system dynamics funcionality, with the stock/level variables Company1, Company2, Company3, Company4 representing the fitness for each company.

The growth rates are parameters that the user can change within the system dynamics tab, while the competition weights wij are provided as sliders.

All the parameters default values are set to Vano et al. (2006)'s example that leads to chaos.

The plots show the market share for each airline company along with the phase space projections for the market share of company 1 versus each of the other companies.

A slider for transient time was set so that only the results after a transient period are plotted.

HOW TO USE IT

Try changing the slider values and growth rates for the companies and study the resulting nmarket dynamics.

Are there dominant companies/market leaders that emerge from the game?

What results do we get from changing the parameters in terms of both market dynamics and dominance?

NETLOGO FEATURES

The model produces an interesting example of chaos in an economics game, showing how Netlogo's system dynamics features can be used as a tool for both research and teaching on chaos theory in economics and in evolutionary game theory.

RELATED MODELS

The model is a second in a series of two models featured in lectures for the research and educational project https://sites.google.com/view/systemdynamics. The first model is the airline ticket sales model showing logistic growth:

Airline Ticket Sales (System Dynamics) http://modelingcommons.org/browse/one_model/6899#model_tabs_browse_info

More models are also made available at Modeling Commons project:

http://modelingcommons.org/projects/181

CREDITS AND REFERENCES

Vano, JA, Wildenberg, JC, Anderson, MB, Noel, JK and Sprott, JC (2006) “Chaos in low-dimensional Lotka–Volterra models of competition” Nonlinearity 19, 2391-2404.

The model is available at: http://modelingcommons.org/projects/181

Comments and Questions

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Click to Run Model

;; system-dynamics-setup, system-dynamics-do-plot, and system-dynamics-go are automatically
;; generated by the System Dynamics Modeler.  The code can be viewed in the
;; Code Tab of the System Dynamics Modeler.

globals [market_share_1
         market_share_2
         market_share_3
         market_share_4
         Total_Fitness]

to setup
  clear-all
  system-dynamics-setup
end 

to go
  system-dynamics-go
  set Total_Fitness Company1 + Company2 + Company3 + Company4

  set market_share_1 Company1 / Total_Fitness
  set market_share_2 Company2 / Total_Fitness
  set market_share_3 Company3 / Total_Fitness
  set market_share_4 Company4 / Total_Fitness

  if ticks >= transient

  [ set-current-plot "Company 1"
    plot market_share_1
    set-current-plot "Company 2"
    plot market_share_2
    set-current-plot "Company 3"
    plot market_share_3
    set-current-plot "Company 4"
    plot market_share_4

   set-current-plot "Company 1 vs Company 2"
   plotxy market_share_1 market_share_2
   set-current-plot "Company 1 vs Company 3"
   plotxy market_share_1 market_share_3
   set-current-plot "Company 1 vs Company 4"
   plotxy market_share_1 market_share_4
  ]
end 


; Copyright 2005 Uri Wilensky.
; See Info tab for full copyright and license.

There is only one version of this model, created over 3 years ago by Carlos Pedro S. Gonçalves.

Attached files

File	Type	Description	Last updated
Airline System Dynamics II - Chaos.png	preview	Preview for 'Airline System Dynamics II - Chaos'	over 3 years ago, by Carlos Pedro S. Gonçalves	Download

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