##### Model Description

Here, we present a model of the coupled dynamics between human socioeconomic choice (between cooperative and non-cooperative collective action) and nutrient loading input level into a lake water system. Suzuki and Iwasa (2008) explains the model as the following. "In the model, many players choose one of the two options: a cooperative and costly option with low phosphorus discharge, and an economical option with high phosphorus discharge. The choice is affected by an economic cost, a social concern about water pollution, and a conformist tendency. The pollution level in the lake is determined by total phosphorus discharge by the players, the sedimentation and the outflow of phosphorus, and the recycling of phosphorus from the sediment. The model has two sources of nonlinearity: the cooperation level tends to be bistable due to con-formist tendency of people (social hysteresis) and pollution level tends to be bistable because phosphorus recycling occurs faster in more eutrophic lakes (ecological hysteresis). The combination of these two sources may cause multiple stable equilibria or oscillations with a long periodicity. Small economic cost and strong social concern about pollution level can decrease the pollution level, but may not be very effective in enhancing the cooperation level. In contrast, strong conformist tendency produces a stable state with a high cooperation level and a low pollution level".

**Reference**

Suzuki, Y., & Iwasa, Y. (2008). The coupled dynamics of human socio-economic choice and lake water system: the interaction of two sources of nonlinearity. Ecological Research, 24(3), 479-489. doi:10.1007/s11284-008-0548-3

##### Scenarios

The model exhibits three qualitatively different regimes. In the first regime, both social and ecological hysterisis effects are strong, causing the system to have many fixed points (i.e., two isoclines intersecting at multiple points). Set the x-axis as the fraction of cooperators and the y-axis as the level of pollution in a lake. In one extreme case, there are nine equilibria, four of which are stable fixed points (see Figure 4 in Suziki and Iwasa 2008). Hence, the system has four basins of attraction. To reproduce this case, enter the following parameter values: s=0.1, gamma=4, c=9, zeta=2, k=0.25, alpha=0.4, PH=0.04, PL=0.0012, r=0.8, m=1, and q=2.

In the second regime, two isoclines intersect at only one point, and the system exhibits an oscillating dynamics (see Figure 5 (b) in Suzuki and Iwasa 2008). To see this behavior, set gamma=2, c=8, zeta=2, k=1.6, PH/alpha=0.4 (i.e., PH divided by alpha), PL/alpha=0.04, r/alpha=1.4, m=1, q=2, s=0.1, and alpha=0.05.

In the third regime, two isoclines intersect at only one point, and the system converges to a steady state (see Figure 5 (c) in Suzuki and Iwasa 2008). To replicate this behavior, set amma=2, c=8, zeta=2, k=1.6, PH/alpha=0.4 (i.e., PH divided by alpha), PL/alpha=0.04, r/alpha=1.4, m=1, q=2, s=0.01, and alpha=0.5.

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

Change in cooperation level |

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

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

$\Large F=\gamma(1+\zeta x_{t})(1+\kappa y_{t})-c$ |

Difference between social pressure and cost of cooperation |

$\Large P(x_{t})=\rho_{H}(1-x_{t})+\rho_{L}x_{t}$ |

Aggregate amount of phosphorus discharged by both cooperators and non-cooperators |

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

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

#=====define parameters par s=0.1, beta=1, gamma=4, zeta=2, k=0.25, c=9 par alpha=0.4, ph=0.04, pl=0.0012, r=0.8, q=2, m=1 #=====define some hidden variables===== #============economic functions #==================demographicfunctions #=====auxiliary quantities================= #======right hand sides x(t+1)=(1-s)*x+s/(1+exp(-beta*(gamma*(1+zeta*x)*(1+k*y)-c))) y(t+1)=(1-alpha)*y+(ph*(1-x)+pl*x)+(r*y^q)/(m^q+y^q) #=============initial data init x=0.2, y=1 @ meth=qualrk @ bound=1000 @ total=1000 @ dt=0.5 @ xp=x,yp=y @ xlo=0,xhi=1,ylo=0,yhi=3 done

Yu JHD, Arizona State University.

Bozicevic M, Arizona State University.