Model Description
This is a model that illustrates the relationship among levels of (1) population, (2) human-made capital, (3) natural capital , and (4) resource consumption. The key insight to be gained from the model is that as the ratio of capitalization in human-made infrastructure over human population is varied in the parameter space, the dynamics of natural capital changes and becomes vulnerable to different disturbance regimes. That is, as humans grow in population and over-invest in capitalization/physical infrastructure (e.g., irrigation infrastructure) to become robust to one disturbance regime (e.g., local droughts), they necessarily become vulnerable to different disturbance regimes (e.g., regional droughts, collective action problems, etc.). The model is inspired by the historical case of Hohokam Cultural Sequence (see the case in the case library for more details).
The dynamics of the natural capital are represented by the two state variables: (1) harvestable wild resources biomass (resource type 1) and (2) soil fertility (resource type 2). The population depends on these two resource systems for subsistence. Resource type 1 is primairly used for obtaining protein-based nutrients. Resource type 2 is where people obtain carbohydrates. When there is little or no irrigation infrastructure, the population is mainly hunter-gatherers and they depend on resource type 1. If the society heavily invests in irrigation systems, the population depends more on resource type 2.
Scenarios
To mimic the Pioneer Period (AD 1–750) of the Hohokam Cultural Sequence, try to maintain the ratio of capitalization over population around 0.01 (i.e., little irrigation infrastructure). This can be done by setting the population level (h) to 0.5 and varying the infrastructure level (k) to 0.005 initially. Now, slowly increase the population level. As the population level is increased, resource type 1 becomes increasingly vulnerable to droughts. That is, the intensity of drought required to tip the dynamics of resource type 1 towards a complete collapse becomes progressively less as the population level is increased.
To mimic the Colonial Period (AD 750–900) and the Sedentary Period (AD 900–1150) of the Hohokam Cultural Sequence, try to maintain the ratio of capitalization over population around 0.2 (i.e., irrigation dependent society) and keep the population level relative low (h=1.0~1.5). At this range, both resource type 1 and 2 are robust to droughts due to the increased levels of irrigation infrastructure.
To mimic the Classic Period (AD 1150–1450) (i.e., the period in which the Hohokam declined and collapsed), try to maintain the ratio of capitalization over population around 0.2 but keep the population level relatively high ( h=2.4 or above). Now, resource type 1 and 2 both become very fragile. Even a small perturbation (e.g., a drought) can flip the system to be on the trajectory of collapse. The Hohokam society may try to switch between relying on irrigation and wild resources in order to rejuvenate their declining natural capital, but they cannot escape from the trajectory once they are set on this course.
$\Large \frac{dx_{1}}{dt}=r_{1}(R)x_{1}(1-\alpha_{1}(R)x_{1})-\alpha_{11}Y_{11}- \alpha_{12}Y_{12}$ |
| Rate of change in harvestable wild resources (resource type 1) |
$\Large \frac{dx_{2}}{dt}=r_{2}(S)x_{2}(1-\alpha_{2}x_{2})-\alpha_{21}Y_{21}- \alpha_{22}Y_{22}$ |
| Rate of change in soil fertility (resource type 2) |
$\Large Y_{1j}=hy_{1j}$ |
| Total output 'j' from the resource type '1' |
$\Large Y_{2j}=hy_{2j}$ |
| Total output 'j' from the resource type '2' |
$\Large y_{1j}= A_{1j}\ast x_{1}\ast l_{1j}$ |
| An individual person's harvest of 'j' from the resource type '1' |
$\Large y_{2j}= A_{2j}\ast x_{2}\ast l_{2}\ast K$ |
| An individual person's harvest of 'j' from the resource type '2' |
$\Large Minimize: l_{11}+l_{12}+l_{2}$ |
| Optimization of labor work (i.e., minimize work) while satisfying the constraints below. Here, L11 and L12 represent individual labor allocated to producing output type 1 and 2 respectively from resource type '1'. L2 means individual effort put into resource type '2' (note: we do not differentiate efforts for producing different types of output from L2). |
$\Large Subject \ to:$ |
$\Large y_{11}+y_{21}\geq y_{1_{min}}$ |
$\Large y_{12}+y_{22}\geq y_{2_{min}}$ |
$\Large l_{11}+l_{12}+l_{2}\leq l_{max}$ |
#simple bioeconomic model with individual entry. #define parameters-------------------------------------------------------- par al11=0.1,al12=0.1,al21=0.1,al22=0.1, r1=1, r2=1,A11=1,A12=1,A21=1,A22=1 par S=1,h=2.4,dk=0.05,I=0.05,spow=10,Lmax=20,kfact=0.6 init x1=0.25, x2=0.25 par a1=1, a2=1 #------------------------------------------------------------------------- #define some hidden varaibles and functions------------------------------- K=h*kfact f(x,a,b)=if(x>0)then(x**b/(a**b + x**b))else(0) l2max=min(1/(A21*x2*K), 1/(A22*x2*K)) svar=(A21/A11 + A22/A12)*x2*k/x1 l2opt=l2max*f(svar,1,spow) l11opt=(1 - A21*x2*l2opt*K)/(A11*x1) l12opt=(1 - A22*x2*l2opt*K)/(A12*x1) l2=min(l2opt, Lmax) l11=min(l11opt, Lmax-l2) l12=min(l12opt, Lmax-l2-l11) Y11=A11*x1*l11*h Y12=A12*x1*l12*h Y2=(A21+A22)*x2*l2*h*K Y21=A21*x2*l2*h*K Y22=A22*x2*l2*h*K #define auxiliary quantities------------------------------------------------- #define right hand sides-------------------------------------------------- dx1/dt=r1*x1*(1-x1*a1) - al11*Y11 - al12*Y12 dx2/dt=r2*x2*(1-x2*a2) - al21*Y21 - al22*Y22 #dh/dt= b(Y1,Y2) - d(Y1,Y2) #dK/dt=I - dk*K #define some auxiliary variables aux per1=100*(Y11 + Y21)/h aux per2=100*(Y12 + Y22)/h aux lab2=l2 aux lab11=l11 aux lab12=l12 aux totlab=l2+l11+l12 aux wop1=Y11 aux wop2=Y12 aux irrout=Y2 #aux maxl2=l2max aux switch=svar #aux tot1out=Y11 + Y21 #aux tot2out=Y12 + Y22 aux K_Ratio=kfact/h #-------------------------------------------------------------------------- @ xlo=0,ylo=0,xhi=1.1,yhi=1.1 @ xp=x1,yp=x2 @ bounds=1000 @ total=50 @ dt=0.01 done
Yu JHD, Arizona State University.
Bozicevic M, Arizona State University.
. 2006. Robustness, institutions, and large-scale change in social-ecological systems: the Hohokam of the Phoenix Basin. Journal of institutional economics. 2(2):133-155.
