##### Model Description

This is a simple model that integrates 1) resource-population dynamics, 2) population migration, and 3) spatial heterogeneity in biophysical conditions (i.e., soi fertility). The reference article, Anderies and Hegmon (2011), gives the following abstract of the model. "Migration is arguably one of the most important processes that link ecological and social systems across scales. Humans (and other organisms) tend to move in pursuit of better resources (both social and environmental). Such mobility may serve as a coping mechanism for short-term local-scale dilemmas and as a means of distributing organisms in relation to resources. Movement also may be viewed as a shift to a larger scale; that is, while it may solve short-term local problems, it may simultaneously have longer term and larger scale consequences. We conduct a quantitative analysis using dynamic modeling motivated by an archaeological case study to explore the dynamics that arise when population movement serves as a link between spatial scales. We use the model to characterize how ecological and social factors can lead to spatial variation in resource exploitation, and to investigate the circumstances under which migration may enhance or reduce the capacity of the system to absorb shocks at different scale".

**Reference**

Anderies, J. M., & Hegmon, M. (2011). Robustness and Resilience across Scales : Migration and Resource Degradation in the Prehistoric U . S . Southwest. Ecology And Society, 16(2).

##### Scenarios

Here, we explore the changes in resource density in region 1 (R1) and region 2 (R2) as the parameter for treshold of migration transaction cost (b_{m}) is varied. Region 1 is more fertile (i.e., higher resource regeneration rate) than region 2. An interesting phenomenon demonstrated by the model is that with extreme levels of migration (zero migration or infinite migration), more fertile and technologically-advanced regions (e.g., region 1) can have more severe environmental degradations than those regions with less fertile lands and back-dated technologies (e.g. region 2). However, with intermediate levels of migration, more fertile and advanced regions can spatially spread out the pressure for exploiting their resources to other regions and thus attain higher resource density than the less fertile regions. This is in essence shifting short-term local problems to a larger scale. In this model, region 1 has higher resource regeneration rate and harvest rate than region 2 (i.e., g_{1}>g_{2} and q_{1}>q_{2} ).

To observe this phenomenon, set the parameters as the following: g_{1}=1, g_{2}=0.33, g_{b}=0.1, q_{1}=0.1, q_{2}=0.07, q_{b}=0.03, k_{1}=100, k_{2}=100, k_{b}=100, a_{m}=4, b_{m}=1, r_{g}=0.01, a_{g}=1, and b_{g}=5. Set the y-axis to the auxiliary variable 'delta R_{12 } (difference in resource density between region 1 and 2) and the x-axis to time. Under this setting (b_{m}=1, i.e., migration is very easy), the long-run resource density of region 1 (R1) is less than that of region 2 (R1<R2) and so the difference is negative. Now increase 'migration transaction cost threshold' to 20 (b_{m}=20). Under this setting (b_{m}=20, i.e., migration is neither too easy nor too difficult), we get R1>R2 and so the difference is positive. Now increase 'migration transaction cost threshold' to 100 (b_{m}=100). Under this setting (b_{m}=100, i.e., strong resistence to migration), the difference (R1-R2) becomes negative again.

To know more about other interesting phenomenon (i.e., how the system responds to external shocks such as droughts), please read Anderies and Hegmon (2011).

$\Large \frac{dR_{1}}{dt}=g_{1}(R)_{1}(1-\frac{R_{1}}{K_{1}})-q_{1}R_{1}H_{1}-\sigma_{1}s(t)R_{1}$ |

Rate of change in resource density in region 1 (Mimbres valley). |

$\Large \frac{dR_{2}}{dt}=g_{2}(R)_{2}(1-\frac{R_{2}}{K_{2}})-q_{2}R_{2}H_{2}-\sigma_{2}s(t)R_{2}$ |

Rate of change in resource density in region 2 (eastern Mimbres area). |

$\Large \frac{dR_{b}}{dt}=g_{b}(R)_{b}(1-\frac{R_{b}}{K_{b}})-q_{b}R_{b}H_{b}-\sigma_{b}s(t)R_{b}$ |

Rate of change in resource density in region b (background/hinterland). |

$\Large \frac{dH_{1}}{dt}=r_{g}\frac {\tanh {a_{g}(q_{1}R_{1}-b_{g})}}{\pi}H_{1}+M_{21}+M_{b1}-M_{12}+M_{1b}$ |

Rate of change in human population in region 1 (Mimbres valley). |

$\Large \frac{dH_{2}}{dt}=r_{g}\frac {\tanh {a_{g}(q_{2}R_{2}-b_{g})}}{\pi}H_{2}+M_{12}+M_{b2}-M_{21}+M_{sb}$ |

Rate of change in human population in region 2 (eastern Mimbres area). |

$\Large \frac{dH_{b}}{dt}=r_{g}\frac {\tanh {a_{g}(q_{b}R_{b}-b_{g})}}{\pi}H_{b}+M_{1b}+M_{2b}-M_{b1}+M_{b2}$ |

Rate of change in human population in region b (background/hinterland). |

$\Large M_{ij}=0.5+\frac{\tanh {a_{m}(d_{ji}-b_{m})}}{\pi}H_{i}$ |

Migration rate from location i to location j (for i=1,2,b, j=1,2,b) |

$\Large d_{ji}=U(q_{i}R_{i})-U(q_{j}R_{j})$ |

Difference in utilities of consumption between the two regions i and j. |

$\Large W_{ci}=(q_{i}^{eq}R_{i}^{eq}-q_{i}R_{i})H_{i}$ |

Gain or loss in welfare in region i. |

#simple model of spatial population dynamics par g1=1,g2=0.33,gb=0.1 par k1=100,k2=100,kb=100 par q1=0.1,q2=0.07,qb=0.03 par am=4,bm=1,mea=0,mep=1,mepb=1,rg=0.01,ag=1,bg=5 #par ttime=100 par mf=1,mfb=1,p1=100,sharp=60,sf1=0,sf2=0,sfb=0 par welfm=1 par sh1=0,sh2=0,shb=0 #eqbm cons with migation; 5.155, 4.583,2.917 #factors: 0.031, -0.0834, -0.02767 par ef1=0.031,ef2=-0.0834,efb=-0.02767 #eqbm cons with migration & cultural factors 4.932,6.386,2.962 #factors:-0.0136, 0.2772, -0.0127 #par ef1=-0.0136,ef2=0.2772,efb=-0.0127 #Hidden variables and functions g(x,a,b)= atan(a*(x-b), 0)/pi f(x,a,b)= 0.5+g(x,a,b) U(x,a,b)=x*a+b # Consumption possibilities in the different regions C1=q1*R1 C2=q2*R2 Cb=qb*Rb # differences for migration decisions d12=U(C1,mep,mea)-U(C2,1,0) d1b=U(C1,mep,mea)-U(Cb,mepb,0) d2b=U(C2,1,0)-U(Cb,mepb,0) # migration M21=f(d12,am,bm)*H2 M12=f(-d12,am,bm)*H1 Mb1=mfb*f(d1b,am,bm)*Hb M1b=mfb*f(-d1b,am,bm)*H1 Mb2=mfb*f(d2b,am,bm)*Hb M2b=mfb*f(-d2b,am,bm)*H2 NM1=M21+Mb1-M12-M1b NM2=M12+Mb2-M21-M2b NMb=M1b+M2b-Mb1-Mb2 Wl1pc=max(5*(1+mf*ef1)-C1,0) Wl2pc=max(5*(1+mf*ef2)-C2,0) Wlbpc=max(3*(1+mf*efb)-Cb,0) Wl1=Wl1pc*H1 Wl2=Wl2pc*H2 Wlb=Wlbpc*Hb TP=H1+H2+Hb # Resource and population dynamics R1'=g1*R1*(1-R1/k1)-C1*H1-sf1*shock*R1 R2'=g2*R2*(1-R2/k2)-C2*H2-sf2*shock*R2 Rb'=gb*Rb*(1-Rb/Kb)-Cb*Hb-sfb*shock*Rb H1'=rg*g(C1,ag,bg)*H1+mf*NM1 H2'=rg*g(C2,ag,bg)*H2+mf*NM2 Hb'=rg*g(Cb,ag,bg)*Hb+mf*NMb Wc1p'=welfm*Wl1pc Wc2p'=welfm*Wl2pc Wcbp'=welfm*Wlbpc Wc1'=welfm*Wl1 Wc2'=welfm*Wl2 Wcb'=welfm*Wlb init R1=100,R2=100,H1=1,H2=1,Rb=100,Hb=1 # oscilator-shock dy/dt=y*(1-y^2-v^2)-(2*Pi/p1)*v dv/dt=v*(1-y^2-v^2)+(2*Pi/p1)*y shock= exp(sharp*(y-1)) init y=-1,v=0 aux cons1=C1+sh1 aux cons2=C2+sh2 aux consb=Cb+shb aux Wc=Wc1+Wc2+Wcb #aux netmig1=NM1 #aux netmig2-NM2 #aux netmigb=NMb aux totpop=TP #aux totcons= Wlt/TP aux totres= R1+R2+Rb #aux m2to1=M21 #aux m1to2=M12 #aux mbto1=Mb1 #aux m1tob=M1b #aux m2tob=M2b #aux mbto2=Mb2 aux shck=shock aux R1s=R1+sh1 aux R2s=R2+sh2 aux Rbs=Rb+shb aux R12gap=R1-R2 aux H12gap=H1-H2 #aux func=f(x,a,b) #@ ylo=0,yhi=100,meth=qualrk,total=100,xhi=100,bounds=10000 @ meth=qualrk @ bound=1000 @ total=1000 @ dt=0.1 done

Yu JHD, Arizona State University.

Bozicevic M, Arizona State University.