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Robustness, institutions, and large-scale change in social-ecological systems: the Hohokam of the Phoenix Basin

Model Type
SES Library XPP model

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.

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$$\Large \frac{dx_{1}}{dt}=r_{1}(R)x_{1}(1-\alpha_{1}(R)x_{1})-\alpha_{11}Y_{11}- \alpha_{12}Y_{12}$$
$$\Large \frac{dx_{2}}{dt}=r_{2}(S)x_{2}(1-\alpha_{2}x_{2})-\alpha_{21}Y_{21}- \alpha_{22}Y_{22}$$
$$\Large Y_{1j}=hy_{1j}$$
$$\Large Y_{2j}=hy_{2j}$$
$$\Large y_{1j}= A_{1j}\ast x_{1}\ast l_{1j}$$
$$\Large y_{2j}= A_{2j}\ast x_{2}\ast l_{2}\ast K$$
$$\Large Minimize: l_{11}+l_{12}+l_{2}$$
$$\Large Subject \ to:$$
$$\Large y_{11}+y_{21}\geq y_{1_{min}}$$
$$\Large y_{12}+y_{22}\geq y_{2_{min}}$$
$$\Large l_{11}+l_{12}+l_{2}\leq l_{max}$$
robustness_institutions_and_large-scale_change_in_social-ecological_systems_th_YQ8uxH8.ode
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