##### Model Description

This is a simple model with reciprocal feedbacks between social and ecological dynamics of farmland abandonment. With the rising urbanization, human migration to urban centers have increased significantly around the globe. One notable consequence of this migration pattern is that mountainous forests that had been traditionally cleared for farming are increasingly becoming abandoned. As a result, such lands likely become forests again through natural regeneration. These trends may induce two reinforcing feedbacks: (1) more migration means less farmlands due to forest regeneration and (2) less farmlands mean less economic incentives to stay in farming and more incentives to move to urban areas for better opportunities.

**Reference**

Figueiredo, J., & Pereira, H. M. (2011). Regime shifts in a socio-ecological model of farmland abandonment. Landscape Ecology, 26(5), 737-749. doi:10.1007/s10980-011-9605-3

##### Scenarios

The coupled system has five types of stable fixed points: (1) M=P and F=T, i.e., entire population migrates and all farmlands become forests, (2) M=0 and F=0, i.e., no migration happens and all forests are cleared for farming, (3) M=0 and F='a non-zero value', i.e., individuals do not migrate but do not deforest all of the area, (4) M=some value between 0 and 100 and F=0, i.e., a part of the population migrates but people still find a way to deforest all of the forests, and (5) M=some value between 0 and 100 and F=some value between 0 and 100, i.e., a part of the population migrates and a portion of the area still remains as forests. Hence, the possible system states encompass extreme cases, i.e., all of the population either migrates or stays and all of the forests remain or become clearned, as well as moderate cases, i.e., parts of the population and the forests still remain.

To explore the fixed points, a broad range of paramater combinations should be tested: (1) forest growth rate (epsilon) ranging from 0.1 to 5, residents' ability to deforest (lambda) ranging from 0 to 0.25, annual farmland utility (gamma) ranging from 1 to 5, and social bonding (s) ranging from 0.01 to 0.5. For the details, please see Figure 3 and Figure 4 in Figueiredo and Pereira (2011).

$\Large \frac{dF}{dt}=\epsilon F(1-\frac{F}{K})-\lambda RF$ |

Rate of change in area occupied by forests |

$\Large \frac{dM}{dt}=\omega\left ( \frac{1}{1+e^{\frac{\mu-\frac{M}{P}}{s}}}-\frac{M}{P} \right )P$ |

Rate of change in migration rate (from the agricultural area to urban areas) |

$\Large R=T-M$ |

Current population (total population minus those who migrated out) |

$\Large \mu=\frac{\frac{Ah}{P-M}}{2\gamma}$ |

Mean threshold of migration (population whose propensity for migration is below this level migrate to urban areas). This relationship is derived by the equation below. |

$\Large \frac{Ah}{P-M}=\gamma \ \ \textrm{holds when} \ \ \mu=0.5$ |

This means that the utility from farming should equal the utility from living in urban areas. At this point the mean threshold of migration (mu) should equal to 0.5 |

$\Large A=T-F$ |

Currently cultivated area (forested area cleared for farming) |

par s=0.05, epsilon=0.1, gamma=1, h=1, K=100, lambda=0.02 par omega=1, P=100 aux A=T-F aux R=P-M dF/dt=epsilon*F*(1-F/K)-lambda*(P-M)*F dM/dt=omega*(P/(1+exp((((((K-F)*h)/(P-M))/(2*gamma))-M/P)/s))-M) #=============initial data init M=50, F=50 @ meth=qualrk @ bound=1000 @ total=100 @ dt=0.5 @ xp=F,yp=M @ xlo=-5,xhi=105,ylo=-5,yhi=105 done

Yu JHD, Arizona State University.

Bozicevic M, Arizona State University.