ArticlePublisher preview available

Two-Population Evolutionary Oligopoly with Partial Cooperation and Partial Hostility

Authors:
Fabio Lamantia at University of Calabria
Fabio Lamantia
Davide Radi at Università Cattolica del Sacro Cuore
Davide Radi
T. Tichy
T. Tichy
T. Tichy
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Read publisher preview
Download citation
To read the full-text of this research, you can request a copy directly from the authors.

Abstract and Figures

In this paper, we reconsider the model in Bischi and Lamantia (J Econ Interact Coord 17:3–27, 2022) and reformulate it in a two-population context. There, the Cournot duopoly market examined is in equilibrium (Cournot-Nash-equilibrium quantities are produced) conditionally to the players’ (heterogeneous) attitudes toward cooperation. To accommodate players’ attitudes, their objective functions partly include the opponent’s profit, resulting in greater (partial) cooperation or hostility toward the opponent than in the standard duopoly setting. An evolutionary selection mechanism determines the survival of cooperative or competitive strategies in the duopoly. The game is symmetric and Bischi and Lamantia (J Econ Interact Coord 17:3–27, 2022) assumes that the two players involved start the game by choosing the same strategic profile. In this way, the full-fledged two-population game simplifies in a one-dimensional map. In this paper, we relax this assumption. On one hand, this approach allows us to investigate entirely the dynamics of the model and the evolutionary stability of the Nash equilibria of the static game that is implicit in the evolutionary setup. In fact, the model with only one population partially represents the system dynamics occurring in an invariant subset of the phase space. As a remarkable result, this extension shows that the steady state of the evolutionary model where all players are cooperative can be an attractor, although only in the weak sense, even when it is not a Nash equilibrium. This occurs when firms have a very high propensity to change strategies to the one that performs better. On the other hand, this approach allows us to accommodate players’ heterogeneity (non-symmetric version of the game), whose analysis confirms the main insights attained in the homogeneous setting.
Staircase diagrams of map (22) with different initial conditions. Parameters: a=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=4$$\end{document}, b=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=0.15$$\end{document}, c=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.5$$\end{document}, θ̲=-0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\theta }=-0.7$$\end{document}. aΨ<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi <0$$\end{document} with θ¯=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.5$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. b0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with θ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.25$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. c0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with θ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.25$$\end{document} and β=3.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =3.5$$\end{document}. d0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with θ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.25$$\end{document} and β=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =6$$\end{document}. e0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with θ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.25$$\end{document} and β=20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =20$$\end{document}. fΨ>-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi >-\Gamma$$\end{document} with θ¯=-0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=-0.25$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}
Staircase diagrams of map (22) with different initial conditions. Parameters: a=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=4$$\end{document}, b=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=0.15$$\end{document}, c=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.5$$\end{document}, θ̲=-0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\theta }=-0.7$$\end{document}. aΨ<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi <0$$\end{document} with θ¯=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.5$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. b0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with θ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.25$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. c0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with θ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.25$$\end{document} and β=3.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =3.5$$\end{document}. d0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with θ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.25$$\end{document} and β=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =6$$\end{document}. e0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with θ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=0.25$$\end{document} and β=20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =20$$\end{document}. fΨ>-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi >-\Gamma$$\end{document} with θ¯=-0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }=-0.25$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}
… 
Bifurcation diagrams of map (22) when 0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with varying θ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }$$\end{document} and different intensities of choice. Parameters: a=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=4$$\end{document}, b=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=0.15$$\end{document}, c=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.5$$\end{document}, θ̲=-0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\theta }=-0.7$$\end{document}, θ¯∈-0.7,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta } \in \left( -0.7,1\right]$$\end{document}. aβ=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =6$$\end{document}. bβ=20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =20$$\end{document}
Bifurcation diagrams of map (22) when 0<Ψ<-Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\Psi <-\Gamma$$\end{document} with varying θ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }$$\end{document} and different intensities of choice. Parameters: a=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=4$$\end{document}, b=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=0.15$$\end{document}, c=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.5$$\end{document}, θ̲=-0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\theta }=-0.7$$\end{document}, θ¯∈-0.7,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta } \in \left( -0.7,1\right]$$\end{document}. aβ=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =6$$\end{document}. bβ=20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =20$$\end{document}
… 
Basins of attraction: In red the basin of attraction of the locally asymptotically stable equilibrium E1,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{1,0}$$\end{document}, in blue the basin of attraction of the locally asymptotically stable equilibrium E0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{0,1}$$\end{document}, in green the basin of attraction of the equilibrium E1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{1,1}$$\end{document}, in magenta the basin of attraction of the equilibrium E0,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{0,0}$$\end{document}, in yellow the basin of attraction of the chaotic set in the diagonal. Parameters: a=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=4$$\end{document}, b=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=0.15$$\end{document}, c=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.5$$\end{document} and θ̲=-0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\theta }=-0.7$$\end{document}. Panel aθ¯=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.5$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. Panel bθ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.25$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. Panel cθ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.25$$\end{document} and β=51\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =51$$\end{document}. Panel dθ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.25$$\end{document} and β=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =100$$\end{document}. Panel eθ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.25$$\end{document} and β=200\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =200$$\end{document}. Panel fθ¯=-0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= -0.25$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. In Panels b–e the common inner equilibrium is Em∗,m∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{m^{*},m^{*}}$$\end{document} with m∗≈0.235179\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m^{*} \approx 0.235179$$\end{document} and βf≈3.09308\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{f}\approx 3.09308$$\end{document}
Basins of attraction: In red the basin of attraction of the locally asymptotically stable equilibrium E1,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{1,0}$$\end{document}, in blue the basin of attraction of the locally asymptotically stable equilibrium E0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{0,1}$$\end{document}, in green the basin of attraction of the equilibrium E1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{1,1}$$\end{document}, in magenta the basin of attraction of the equilibrium E0,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{0,0}$$\end{document}, in yellow the basin of attraction of the chaotic set in the diagonal. Parameters: a=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=4$$\end{document}, b=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=0.15$$\end{document}, c=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.5$$\end{document} and θ̲=-0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\theta }=-0.7$$\end{document}. Panel aθ¯=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.5$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. Panel bθ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.25$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. Panel cθ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.25$$\end{document} and β=51\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =51$$\end{document}. Panel dθ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.25$$\end{document} and β=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =100$$\end{document}. Panel eθ¯=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= 0.25$$\end{document} and β=200\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =200$$\end{document}. Panel fθ¯=-0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }= -0.25$$\end{document} and β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =1$$\end{document}. In Panels b–e the common inner equilibrium is Em∗,m∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{m^{*},m^{*}}$$\end{document} with m∗≈0.235179\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m^{*} \approx 0.235179$$\end{document} and βf≈3.09308\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{f}\approx 3.09308$$\end{document}
… 
Basins of attraction of model (60) (asymmetric case). Parameter as in Fig. 3 but: aβ1=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=100$$\end{document} and β2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=1$$\end{document}; bβ1=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=1$$\end{document} and β2=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=2$$\end{document}; cβ1=170\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=170$$\end{document} and β2=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=3$$\end{document}; dβ1=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=100$$\end{document} and β2=50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=50$$\end{document}; eβ1=200\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=200$$\end{document} and β2=50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=50$$\end{document}; fβ1=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=100$$\end{document} and β2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=1$$\end{document}
Basins of attraction of model (60) (asymmetric case). Parameter as in Fig. 3 but: aβ1=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=100$$\end{document} and β2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=1$$\end{document}; bβ1=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=1$$\end{document} and β2=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=2$$\end{document}; cβ1=170\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=170$$\end{document} and β2=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=3$$\end{document}; dβ1=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=100$$\end{document} and β2=50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=50$$\end{document}; eβ1=200\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=200$$\end{document} and β2=50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=50$$\end{document}; fβ1=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=100$$\end{document} and β2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=1$$\end{document}
… 
Basins of attraction (asymmetric case): In red the basin of attraction of the locally asymptotically stable equilibrium E1,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{1,0}$$\end{document}, in blue the basin of attraction of the locally asymptotically stable equilibrium E0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{0,1}$$\end{document}, in green the basin of attraction of the equilibrium E1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{1,1}$$\end{document}. Panel a: Parameters as in Fig. 3b but θ¯1=-0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }_{1}= -0.1$$\end{document} and θ¯2=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }_{2}= 0.25$$\end{document}. Panel b: Parameters as in Fig. 3e but θ¯1=-0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }_{1}= -0.1$$\end{document} and θ¯2=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }_{2}= 0.25$$\end{document}. The common inner equilibrium is m1∗≈0.334136,m2∗≈0.518768\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_1^*\approx 0.334136, m_2^*\approx 0.518768$$\end{document}
+1
Basins of attraction (asymmetric case): In red the basin of attraction of the locally asymptotically stable equilibrium E1,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{1,0}$$\end{document}, in blue the basin of attraction of the locally asymptotically stable equilibrium E0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{0,1}$$\end{document}, in green the basin of attraction of the equilibrium E1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{1,1}$$\end{document}. Panel a: Parameters as in Fig. 3b but θ¯1=-0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }_{1}= -0.1$$\end{document} and θ¯2=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }_{2}= 0.25$$\end{document}. Panel b: Parameters as in Fig. 3e but θ¯1=-0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }_{1}= -0.1$$\end{document} and θ¯2=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\theta }_{2}= 0.25$$\end{document}. The common inner equilibrium is m1∗≈0.334136,m2∗≈0.518768\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_1^*\approx 0.334136, m_2^*\approx 0.518768$$\end{document}
… 
Figures - available from: Computational Economics
This content is subject to copyright. Terms and conditions apply.
A preview of this full-text is provided by Springer Nature.
Learn more
Content available from Computational Economics
This content is subject to copyright. Terms and conditions apply.
Vol.:(0123456789)
Computational Economics (2025) 65:763–794
https://doi.org/10.1007/s10614-023-10536-7
Two‑Population Evolutionary Oligopoly withPartial
Cooperation andPartial Hostility
F.Lamantia1,2 · D.Radi2,3· T.Tichy2
Accepted: 5 December 2023 / Published online: 10 January 2024
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature
2024
Abstract
In this paper, we reconsider the model in Bischi and Lamantia (J Econ Interact Coord
17:3–27, 2022) and reformulate it in a two-population context. There, the Cournot
duopoly market examined is in equilibrium (Cournot-Nash-equilibrium quantities
are produced) conditionally to the players’ (heterogeneous) attitudes toward coop-
eration. To accommodate players’ attitudes, their objective functions partly include
the opponent’s profit, resulting in greater (partial) cooperation or hostility toward
the opponent than in the standard duopoly setting. An evolutionary selection mecha-
nism determines the survival of cooperative or competitive strategies in the duopoly.
The game is symmetric and Bischi and Lamantia (J Econ Interact Coord 17:3–27,
2022) assumes that the two players involved start the game by choosing the same
strategic profile. In this way, the full-fledged two-population game simplifies in a
one-dimensional map. In this paper, we relax this assumption. On one hand, this
approach allows us to investigate entirely the dynamics of the model and the evolu-
tionary stability of the Nash equilibria of the static game that is implicit in the evo-
lutionary setup. In fact, the model with only one population partially represents the
system dynamics occurring in an invariant subset of the phase space. As a remarka-
ble result, this extension shows that the steady state of the evolutionary model where
all players are cooperative can be an attractor, although only in the weak sense, even
when it is not a Nash equilibrium. This occurs when firms have a very high pro-
pensity to change strategies to the one that performs better. On the other hand, this
approach allows us to accommodate players’ heterogeneity (non-symmetric version
of the game), whose analysis confirms the main insights attained in the homogene-
ous setting.
Keywords Oligopoly modeling· Partial cooperation· Evolutionary games· Multi-
population games· Nonlinear dynamics
Extended author information available on the last page of the article
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... An agent-based extension considering a large number of heterogeneous agents with different heuristics can be valuable. In addition, following Benhabib and Day (1981) endogenous values of the elasticity of the utility to the private good can be included in the analysis and/or the study of other decision-making processes, e.g. the mechanism developed by Brock and Hommes (1997) or evolutionary mechanisms such as the replicator dynamics (Clemens and Riechmann 2006;Lamantia et al. 2024) may be promising. Finally, a natural extension of this work is the study of a dynamic model with local public goods. ...
Nonlinear dynamics in a public good game
Article
Full-text available
  • Aug 2024
The present work aims to study the problem of individual voluntary anonymous contributions to the financing of public goods in a dynamic setting. To do this, the article departs from a textbook model à la Naimzada and Tramontana (2010) augmented with public goods. The article studies how bounded rationality and dependence on agents’ past decisions combine with the problem of voluntary contributions. This favours the emergence of nonlinear dynamics in individual behaviour as well as in the aggregate contribution to the financing of a public good project. The Nash equilibrium can be destabilised through a flip bifurcation when the agent reactivity increases. In addition, some Neimark–Sacker bifurcations can also occur although not around the steady-state equilibrium. A sufficiently high agent reactivity level can also lead to chaotic dynamics with possible multiple attractors. When the chaotic regime prevails, synchronisation phenomena in agent behaviour may occur but are rare. Thus, usually, even if agents are homogeneous, they behave as if they were heterogeneous by making non-synchronised decisions. The work also explicitly deepens the case of a heterogeneous economy in terms of both consumer preferences and income.
View
Complex dynamics in a nonlinear duopoly model with heuristic expectation formation and learning behavior
Article
Full-text available
  • Jul 2023
  • ANN OPER RES
We develop a nonlinear duopoly model in which the heuristic expectation formation and learning behavior of two boundedly rational firms may engender complex dynamics. Most importantly, we assume that the firms employ different forecasting models to predict the behavior of their opponent. Moreover, the firms learn by leaning more strongly on forecasting models that yield more precise predictions. An eight-dimensional nonlinear map drives the dynamics of our approach. We analytically derive the conditions under which its unique steady state is locally stable and numerically study its out-of-equilibrium behavior. In doing so, we detect multiple scenarios with coexisting attractors at which the firms’ behavior yields distinctively different market outcomes.
View
View
Necessary and sufficient conditions for the roots of a cubic polynomial and bifurcations of codimension-1, -2, -3 for 3D maps
Article
Full-text available
  • May 2021
We reconsider the well-known conditions which guarantee the roots of a third-degree polynomial to be inside the unit circle. These conditions are important in the stability analysis of equilibria and cycles of three-dimensional systems in discrete time. A simplified set of conditions determine the boundary of the stability region and we prove which kind of bifurcation occurs when the boundary is crossed at any of its points. These points correspond to the existence of one, two or three eigenvalues equal to 1 in modulus, real or complex conjugate. We give the explicit expressions of the eigenvalues at each point of the border of the stability region in the parameter space. The results are applied to a system representing a housing market model that gives rise to a Neimark–Sacker bifurcation, a flip bifurcation or a pitchfork bifurcation.
View
Evolutionary oligopoly games with cooperative and aggressive behaviors
Article
Full-text available
  • Jan 2022
We propose an oligopoly model where players can choose between two kinds of behaviors, denoted as cooperative and aggressive, respectively. Each cooperative agent chooses the quantity to produce in order to maximize her own profit as well as the profits of other agents (at least partially), whereas an aggressive player decides the quantity to produce by maximizing his own profit while damaging (at least partially) competitors’ profits. At each discrete time, players face a binary choice to select the kind of behavior to adopt, according to a proportional imitation rule, expressed by a replicator equation based on a comparison between accumulated profits. This means that the behavioral decisions are driven by an evolutionary process where fitness is measured in terms of current profits as well as a weighted sum of past gains. The model proposed is expressed by a nonlinear two-dimensional iterated map, whose asymptotic behavior describes the long-run population distribution of cooperative and aggressive agents. We show under which conditions one of the following long-run behaviors prevails: (i) all players choose the same strategy; (ii) both behaviors coexist according to a mixed stationary equilibrium; and (iii) a self-sustained (i.e. endogenous) oscillatory (periodic or chaotic) time pattern occurs. The influence of memory and that of the levels of cooperative/aggressive attitudes on the dynamics are analyzed as well.
View
Residential segregation: The role of inequality and housing subsidies
Article
  • Oct 2020
  • J ECON BEHAV ORGAN
Residential segregation is a key public policy issue that is driven by economic factors on the one side, and individual attitudes towards ethnic diversity on the other side. We assume a modeling framework that consists of a population of two ethnic groups, a rental market for each neighborhood, and household’s utility which depends on consumption and housing. Accounting for income disparities and heterogeneous preferences for living in ethnically diverse neighborhoods, we examine the residential segregation patterns that occur when households make their neighborhood choice by taking economic and diversity related aspects into account. The investigation reveals that ethnic income disparities and heterogeneous preferences are antagonistic forces such that a certain level of income stratification is the price for residential integration. In light of these findings, we discuss to which extent and under which conditions housing subsidy policies can favor residential integration.
View
An evolutionary Cournot oligopoly model with imitators and perfect foresight best responders
Article
  • Jul 2019
We consider the competition among quantity setting players in a linear evolutionary environment. To set their outputs, players adopt, alternatively, the best response rule having perfect foresight or an imitative rule. Players are allowed to change their behavior through an evolutionary mechanism according to which the rule with better performance will attract more followers. The relevant stationary state of the model describes a scenario where players produce at the Cournot‐Nash level. Due to the presence of imitative behavior, we find that the number of players and implementation costs, needed to the best response exploitation, have an ambiguous role in determining the stability properties of the equilibrium and double stability thresholds can be observed. Differently, the role of the intensity of choice, representing the evolutionary propensity to switch to the most profitable rule, has a destabilizing role, in line with the common occurrence in evolutionary models. The global analysis of the model reveals that increasing values of the intensity of choice parameter determine increasing dynamic complexities for the internal attractor representing a population where both decision mechanisms coexist.
View
Some reflections on past and future of nonlinear dynamics in economics and finance
Article
  • Dec 2018
This paper offers an overview of the literature on the economic and financial applications of theory of nonlinear dynamics, especially bifurcation theory. After a short introductory discussion of the first nonlinear dynamic models in social sciences and the economic relevance of the zoo of bifurcations and complicated dynamics that such models can generate, we present an overview of the literature on nonlinear dynamic models in the areas of underdevelopment, environmental poverty traps, the management of common goods, industrial organization and financial markets. The review of the literature is enriched by reflections and ideas for future research.
View
Evolutionary technology adoption in an oligopoly market with forward-looking firms
Article
  • May 2018
In this paper, we propose an evolutionary oligopoly game of technology adoption in a market with isoelastic demand and two possible (linear) production technologies. While one technology is characterized by lower marginal costs, the magnitude of fixed costs entails that a technology does not necessarily dominate the other. Firms are forward-looking as they assess the profitability of employing either technology according to the corresponding expected profits. The dynamics of the system is studied through a piecewise-smooth map, for which we present a local stability analysis of equilibria and show the occurrence of smooth and border collision bifurcations. Global analysis of the model is also presented to show the coexistence of attractors and its economic significance. This investigation reveals that firms can fail to learn to adopt the more efficient technology.
View
The persistence of social strategies under increasing competitive pressure
Article
  • Mar 2018
  • J ECON DYN CONTROL
In this paper we study the long run outcome if firms are repeatedly matched to play a N-firm Cournot oligopoly game and can select their type of business objective. Firms can either be profit-maximizing or socially concerned, also pursuing consumer welfare. We use an indirect evolutionary approach to distinguish between the endogenous, profit-maximizing choice of the firms’ business objectives and their corresponding market behavior. Our main result is that the prospects of long run survival of firms pursuing social strategies strongly depends on the number of firms in the industry. If a social strategy entails higher marginal production costs, it is best suited if the number of firms in the industry is not too large. In particular, we demonstrate that the relation between long survival and the level of competitiveness of the industry is inverse U-shaped. We confirm this result by using the concept of coalition stability and further show that if a high number of firms adopt a social strategy, welfare might be lower than with just profit-maximizing firms.
View
Oligopoly Game: Price Makers Meet Price Takers
Article
  • Mar 2018
  • J ECON DYN CONTROL
The paper studies an oligopoly game, where firms can choose between price-taking and price-making strategies. On a mixed market price takers are always better off than price makers, though the profits of both types decline in the number of price takers. We investigate and confront two possibilities of firms’ decisions about their types: forward-looking equilibrium reasoning and backward-looking individual learning. We find that the Cournot outcome is the only equilibrium prediction and it is learnable if firms are sufficiently sensitive to profit differences. However, with a larger number of firms, a unilateral deviation from Cournot behavior becomes profitable. Under learning this incentive creates a space for permanent oscillations over different markets with a positive but low number of price takers.
View