##### Model Description

This is a model for the coupled dynamics of conflict between two different user groups regarding their socioeconomic choice (between cooperative and non-cooperative collective action) and nutrient loading input level into a lake water system. Suzuki and Iwasa (2009),gives the following overview of the model. "Conflict among multiple groups is a major source of difficulty in environmental conservation. People are often divided into various groups that have different social factors, sometimes leading to differences in the degree to which they cooperate in environmental conservation. This obstructs the social consensus needed to solve the environmental problems. Here we study the coupled dynamics of human socio-economic choice and lake water pollution, and examine the magnitude of the difference in cooperation levels between two groups. In the model, many players choose between a costly but cooperative option and a selfish option. The former results in a reduced phosphorus discharge into the lake. Each player's choice is affected by an economic cost and social pressure. Social pressure is a psychological factor that promotes cooperation: it becomes stronger when more players in the society are cooperative (conformist tendency) and when the problem at hand is a greater concern to society. In the model, two groups sometimes show large differences in their cooperation levels even when both have exactly the same social factors. However, cooperation levels are more likely to differ between groups that have different social factors. Enhancement of the cross-group conformist tendency is the most effective way to minimize differences in cooperation levels and to mitigate conflict between groups".

**Reference**

Suzuki, Y., & Iwasa, Y. (2009). Conflict between groups of players in coupled socio-economic and ecological dynamics. Ecological Economics, 68(4), 1106-1115. Elsevier B.V. doi:10.1016/j.ecolecon.2008.07.024

##### Scenarios

Here, four cases are illustrated. In the first case, there is no cross-group conformist tendency (zeta12=zeta21=0) and the two user groups are symmetric (i.e., all of the parameters have same values). The interesting aspect here is that a strong conflict (i..e, different cooperation levels) between two groups can emerge even if they have identical social and ecological parameter configurations. This divergence depends on initial conditions. For some initial conditions, two groups have the same cooperation level; in other cases, they differ much in the cooperation level. To see this, set *x _{1}*(0)=0.1,

*x*(0)=0.9,

_{2}*y*(0)=0.2, s=0.1, beta=1, gamma=2, c

_{1}=c

_{2}=7, zeta

_{1}=zeta

_{2}=2, k

_{1}=k

_{2}=1, PH

_{1}=PH

_{2}=0.08, PL

_{1}=PL

_{2}=0.02, alpha=0.4, m=1, and r=0.7. In this set of values, two groups have the same cooperation level. Now, set

*x*(0)=0.1,

_{1}*x*(0)=0.9,

_{2}*y*(0)=1.0, and the parameter values the same as above. Under this initional condition, two groups differ in the level of cooperation.

In the second case, there is no cross-group conformist tendency (zeta12=zeta21=0) and the two user groups are asymmetric (i.e., some of the parameter values differ between two groups). Naturally, two user groups exhibit different levels of cooperation. To see this, set parameter values in one of the following ways: c_{1}=6 & c_{2}=8, zeta_{1}=4 &zeta_{2}=2, or k_{1}=2 & k_{2}=1. Rest of the parameters have the same values as those of the first case.

In the third case, there is a same cross-group conformist tendency (zeta12=zeta21>0 ) and the two user groups are symmetric. Here, the cooperation levels differ only when the cross-group conformist tendency is small and different from the within-group conformist tendency. Set the parameter values accordingly (other than the cross- and within-group conformist tendencies, rest of the parameters have the same values as those in the first case).

In the fourth case, there is a different cross-group conformist tendency (zeta12!=zeta21>0) and the two user groups are symmetric. Under this setting, different cooperation levels or conflict between groups arise. A very interesting observation here is that when both zeta12 and zeta21 have a same value, the resulting effect mitigates the effects of differences in other parameter values and induce a same level of cooperation (i.e.., elimiate conflict betweeng groups). Set the parameter values accordingly to see this phenomenon.

$\Large x_{1}(t+1)=(1-s)x_{1}(t)+\frac{s}{1+e^{-\beta F_{1}}}$ |

Change in cooperation level in user group 1. |

$\Large x_{2}(t+1)=(1-s)x_{2}(t)+\frac{s}{1+e^{-\beta F_{2}}}$ |

Change in cooperation level in user group 2. |

$\Large y(t+1)=(1-\alpha)y(t)+P(x_{1}(t),x_{2}(t))+\pi(y(t))$ |

Change in water pollution level (amount of phosphorus in lake water). |

$\Large F_{1}=\gamma(1+\zeta_{1} x_{1}(t)+zeta_{12} x_{2}(t))(1+\kappa_{1} y_{t})-c_{1}$ |

Difference between social pressure and cost of cooperation in user group 1. |

$\Large F_{2}=\gamma(1+\zeta_{2} x_{2}(t)+zeta_{21} x_{1}(t))(1+\kappa_{2} y_{t})-c_{2}$ |

Difference between social pressure and cost of cooperation in user group 2. |

$\Large P(x_{1}(t),x_{2}(t))=(\rho_{H1}(1-x_{1}(t))+\rho_{L1}x_{1}(t))n_{1}+(\rho_{H2}(1-x_{2}(t))+\rho_{L2}x_{2}(t))n_{2}$ |

Aggregate amount of phosphorus discharged by both cooperators and non-cooperators from user group 1 and 2. |

$\Large \pi(y(t))=\frac{ry(t)^{q}}{m^{q}+y(t)^{q}}$ |

Amount of recycled phosphorus in lake water (internal dynamics of lake water system). |

#=====define parameters par s=0.1, beta=1, gamma=2, zeta1=2, k1=1, c1=7 par zeta2=2, k2=1, c2=7 par alpha=0.4, ph1=0.08, pl1=0.02, r=0.7, q=2, m=1 par ph2=0.08, pl2=0.02 par zeta12=0, zeta21=0 par n1=0.5, n2=0.5 #=====define some hidden variables===== #============economic functions #==================demographicfunctions #=====auxiliary quantities================= #======right hand sides x1(t+1)=(1-s)*x1+s/(1+exp(beta*(c1-gamma*(1+zeta1*x1+zeta12*x2)*(1+k1*y)))) x2(t+1)=(1-s)*x2+s/(1+exp(beta*(c2-gamma*(1+zeta2*x2+zeta21*x1)*(1+k2*y)))) y(t+1)=(1-alpha)*y+(ph1*(1-x1)+pl1*x1)*n1+(ph2*(1-x2)+pl2*x2)*n2+(r*y^q)/(m^q+y^q) #=============initial data init x1=0.2, y=1 @ meth=qualrk @ bound=1000 @ total=1000 @ dt=0.5 @ yp=x2 @ xlo=0,xhi=1,ylo=0,yhi=3 done

Yu JHD, Arizona State University.

Bozicevic M, Arizona State University.