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Analyzing the Impact of Agave Cultivation on Famine Risk in Arid Pre-Hispanic Northern Mexico

Model Type
SES Library XPP model

Here, a simple model of a subsistence economy based solely on the cultivation of maize and agave is presented. While maize is an annual plant that humans can eat and store, agave is a perennial plant that can be used for multiple purposes: as edible materials yielding caloric values and as fiber materials for producing items like clothing, ropes, and baskets. 

This model tries to capture the essence of a cultivation strategy of a portfolio of plants that have differing levels of sensitivity toward common environmental disturbances, e.g., droughts. Because adult agave is less sensitive toward droughts than maize and are perennial, the combined cultivation strategy can deliver the complementarity effect – if the cropping of maize fails, societies can resort to consuming agave as the backup source of caloric values.

However, is agave cultivation alone sufficient for guaranteeing this complementarity effect? Can there be any 'hidden' biophysical geometry that determines whether or not the complementarity effect is actually realized? This model illustrates that the cultivation of agave can complement the shortage of maize only when climatic conditions meet a certain pattern. It turns out that this condition is met when the average rainfall is at intermediate levels with modest variance (e.g., standard deviation of 20% from mean precipitation). 

Scenarios

In the model, annual rainfall (rhat) is a random variable. Stochasticity is added to the standard deviation of rainfall (signmar) to generate the randomness in annual rainfall. Annual rainfall in turn affects the annual maize and agave harvests. Here, we can test two scenarios: (scenario 1) the standard deviation of rainfall is low such that droughts don't occur often; and (scenario 2) the standard deviation of rainfall is high such that droughts occur more frequently.

To test scenario 1, lower the standard deviation of rainfall (signmar) to around 0.2 and observe the dynamics of total maize stock and agave harvest (set the y-axis to total maize stock and then to agave harvests). The total maize stock (Mt) and agave harvest (AGt) will seldom fall down to zero levels (note: if they both fall down to zero levels simultaneously, it means that a famine has occurred).

To test scenario 2, try to increase the standard deviation of rainfall (signmar) to around 0.9 and observe the dynamics of total maize stock and agave harvest. Most likely, the total maize stock (Mt) and agave harvest (AGt) will simultaneously reach zero levels frequently. In these instances, even if societies cultivate agave for insurance values, the complementarity effect doesn't happen and thus cannot prevent famine.

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$$\Large r_{t}=\widehat{r}(1+\sigma _{r}w_{t})$$
$$\Large \mbox {Defining} \ \ Y_{t}: \\ \Large \mbox {If} \ \ r_{t} > \overline{r}_{m} \ \ \rightarrow \ \ Y_{t}=Y_{max} \\ \Large \mbox {If} \ \ r_{t} < \underline{r}_{m} \ \ \rightarrow \ \ Y_{t}=0 \\ \Large \mbox {If} \ \ \underline{r}_{m} < r_{t} < \overline{r}_{m} \ \ \rightarrow \ \ Y_{t}=Y_{max} imes \frac{(r_{t}-\underline{r}_{m})}{(\overline{r}_{m}-\underline{r}_{m})}$$
$$\Large \mbox {Defining} \ \ x^{j}_{long_{t}}\ (j=1,2,3): \\ \Large \mbox {If} \ r_{t} < \underline{r}_{j} \ \ \rightarrow \ \ x^{j}_{long_{t}}=0 \\ \Large \mbox {If} \ r_{t} > \underline{r}_{j} \ \ \rightarrow \ \ x^{j}_{long_{t}}=\overline{s}_{j} \\ \Large \mbox {Defining} \ \ x^{j}_{long_{t}}\ (j=4,..,15): \\ \Large \mbox {If} \ r_{t} < \overline{r}_{a} \ \ \rightarrow \ \ x^{j}_{long_{t}}=\overline{s}_{a}*\frac{r_{t}}{\overline{r}_{a}} \\ \Large \mbox {If} \ r_{t} > \overline{r}_{a} \ \ \rightarrow \ \ x^{j}_{long_{t}}=\overline{s}_{a}$$
$$\Large \mbox {Defining} \ \ x^{j}_{short_{t}}\ (j=1,2): \\ \Large \mbox {If} \ r_{t} < \underline{r}_{j} \ \ \rightarrow \ \ x^{j}_{short_{t}}=0 \\ \Large \mbox {If} \ r_{t} > \underline{r}_{j} \ \ \rightarrow \ \ x^{j}_{short_{t}}=\overline{s}_{j} \\ \Large \mbox {Defining} \ \ x^{j}_{short_{t}}\ (j=2,..,7): \\ \Large \mbox {If} \ r_{t} < \overline{r}_{a} \ \ \rightarrow \ \ x^{j}_{short_{t}}=\overline{s}_{a}*\frac{r_{t}}{\overline{r}_{a}} \\ \Large \mbox {If} \ r_{t} > \overline{r}_{a} \ \ \rightarrow \ \ x^{j}_{short_{t}}=\overline{s}_{a}$$
$$\Large x^{j}_{long_{t+1}}=s_{j}(r_{t}, \underline{r}_{j}, \overline{s}_{j})x^{j-1}_{long_{t}} exp\left [ a(1-x^{j-1}_{long_{t}}) \right ] \ \ \ \ for \ \ j=1,2,3 \\ \Large x^{j}_{long_{t+1}}=s_{j}(r_{t}, \underline{r}_{a}, \overline{s}_{a})x^{j-1}_{long_{t}} exp\left [ a(1-x^{j-1}_{long_{t}}) \right ] \ \ \ \ for \ \ j=4,...,15$$
$$\Large x^{j}_{short_{t+1}}=s_{j}(r_{t}, \underline{r}_{j}, \overline{s}_{j})x^{j-1}_{short_{t}} exp\left [ a(1-x^{j-1}_{short_{t}}) \right ] \ \ \ \ for \ \ j=1,2 \\ \Large x^{j}_{short_{t+1}}=s_{j}(r_{t}, \underline{r}_{a}, \overline{s}_{a})x^{j-1}_{short_{t}} exp\left [ a(1-x^{j-1}_{short_{t}}) \right ] \ \ \ \ for \ \ j=3,...,7$$
$$\Large U_{t}^{maize}=\mbox{min} \left ( U_{d}, \ M_{t}+Y_{t}) \right )$$
$$\Large U_{t}^{total}=\mbox{min} \left ( U_{d}, \ U_{t}^{maize}+EatAgave*AG_{t}) \right )$$
$$\Large AG_{t}=A_{long}*x^{15}_{long_{t}}+A_{short}*x^{7}_{short_{t}}$$
$$\Large M_{t+1}=(M_{t}+Y_{t})-U_{t}^{maize})(1-\delta _{c})$$
analyzing_the_impact_of_agave_cultivation_on_famine_risk_in_arid_pre-hispanic__0th6cIt.ode
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