Crop Specialization, Exchange and Robustness in a Semi-arid Environment

Abstract

We compare the robustness of food supplies to annual variation in rainfall within two different agricultural systems: a generalist system with one type of agent who cultivates both maize and agave, and a specialist system composed of two types of agents who cultivate either maize or agave and are able to exchange. When mean rainfall is relatively high and less variable or relatively low and more variable, food supplies in the specialist system are more robust than in the generalist system. However, at intermediate levels of mean rainfall and variability, food supplies in the specialist system are less robust than those in the generalist system. Our analysis suggests that conflicts of interest and their associated costs constrain the development of specialization in some environments. When considering the robustness of social-ecological systems, it is important to consider “for whom a coupled social and ecological system is robust?”

Appendix: Mathematical Details

Maize yield: As discussed in the text, the yield of maize in our model is a function of rainfall.

If the yield of maize per unit area and mean annual rainfall in year t are defined as Y _t and r _t , respectively, then the productivity of maize, as shown in Fig. 1a is formalized as

$$ {Y}_t\left({r}_t\right)=\left\{\begin{array}{ll}0\hfill & {r}_t\le {r}_m^l\hfill \\ {}\alpha \left({r}_t-{r}_m^l\right)\hfill & {r}_m^l<{r}_t\le {r}_m^u\hfill \\ {}Y \max \hfill & {r}_t>{r}_m^u.\hfill \end{array}\right. $$

(7)

Where α is the slope of the line between r ^°l _m and r ^°u _m in Fig. 1a, i.e.,

$$ \alpha =\frac{Y \max }{r_m^u-{r}_m^l}. $$

(8)

Note, see Table 7 for parameter definitions.

Table 7 Model state variables and parameter definitions

Maize storage: The amount of stored maize available for agent i in period t + 1 is equal to that available in period t, M ⁱ _t , less what is consumed in period t, U ⁱ _c , plus what is acquired in period t (either through cultivation or exchange), P ⁱ _m (r _t ), less what spoils between period t and t + 1. This yields the mathematical expression:

$$ {M}_{t+1}^i=\left({M}_t^i-{U}_c^i+{P}_m^i\left({r}_t\right)\right)\left(1-{\delta}_m\right) $$

(9)

where 1 − δ _m is the proportion of maize that does not spoil from period t to t + 1. As defined above, the amount of maize available to each type of specialist depends on the amount produced in each period, P _m (r _t ). For the generalist in the Anderies et al. (2008) model, P _m (r _t ) is equal to the yield of maize, Y _t . For the maize specialist here, P _m (r _t ) is the yield of maize less what is exchanged (Y _t  − gift). For the agave specialist, P _m (r _t ) is the maize that can be “purchased” through the exchange of agave with the maize specialist.

Agave population dynamics: The biomass of agave per unit area in each age class alive in period t + 1 is the biomass of the immediately younger age class in t times the proportional increase in biomass through plant growth between t and t + 1, times the proportion that survive between between t and t + 1, as described in the text. We define the biomass of age class k at time t as x ^k _t . Thus, x ⁷ ₆₂ is the biomass of 7-year olds in year 62. We define two survival functions s _j (r _t ), and s _a (r _t ) where the subscripts j and a denote juvenile and adult, respectively. These are functions of r _t , the rainfall in period t and which have the shapes shown in Fig. 1b and c, respectively. Given our definition of the growth function, we can write

$$ {x}_{t+1}^k={s}_j\left({r}_t\right)\cdot {x}_t^{k-1}\cdot \exp \left(a\cdot \left[1-{x}_t^{k-1}\right]\right) $$

(10)

for the juveniles (k = 1,2,3) and

$$ {x}_{t+1}^k={s}_a\left({r}_t\right)\cdot {x}_t^{k-1}\cdot \exp \left(a\cdot \left[1-{x}_t^{k-1}\right]\right) $$

(11)

for adults (k = 4–15). Finally, we assume that the cultivator collects and transplants a constant number of new “pups” each year, i.e. x ⁰ _t  = c for some constant c and for all t.

Rainfall dynamics: The rainfall in period t is a stochastic process given by

$$ {r}_t= \max \left(\widehat{r}\left(1+{\sigma}_rw\right),0\right). $$

(12)

where w is a standard normal random variable. Expression (12) generates a positive random variable that is approximately normally distributed with mean ≈ $\widehat{r}$, and standard deviation ≈ $\widehat{r}$ σ _r for values of σ _r  < 1, as in our case. See text for detailed discussions of U ⁱ _c , P ⁱ _m , g(x ^k − 1), and s _k (r _t ).

Comparing systems: We are comparing two “systems” consisting of populations of agents. System one consists of identical agents, all of the same type. The agents in this system exercise a strategy of cultivating both agave and maize. We call this agent a “generalist”, and by association we call this the “generalist” system. System two consists of a population with two types of agents. The first type exercises a strategy of cultivating only agave and exchanging with agents of the other type. We refer to this type as the “agave specialists”. The second type exercises the strategy of growing maize only and exchanging with the agave specialists. We call this second type the “maize specialists”. Again, by association, we call system two the “specialist system”. Because of its special characteristic, the “generalist system” can be viewed from the perspective of a single agent, as agents do not interact and they are all identical. In the specialist system, this is not the case, the units are defined in terms of a certain group size in which some individuals grow maize others grow agave.