Central-place foraging by humans: transport and processing

Abstract

The Darwinian approach to behavior generates models that are widely used by anthropologists and archeologists. In this paper, I concentrate on a particular group of models based on cases in which a forager (or group of foragers) brings resources to a location known as a central place. I examine two topics in detail: (1) the economics of transporting a load to the central place, and (2) the extent to which items should be processed before they are brought back to the central place. In addition to presenting new results and bringing out common themes in archeology and behavioral ecology, I discuss problems with some of the models that have been used in archeology and offer suggestions for further work.

Appendix: models of processing

Following Metcalfe and Barlow (1992) and Bettinger et al. (1997), let

x _u :

= time to procure a load of unprocessed resources

x _p :

= time to procure and process a load

y _u :

= utility of an unprocessed load

y _p :

= utility of a processed load

From the equal delivery rates condition, at the critical travel time τ _p

$$ \frac{{{y_u}}}{{{\tau_p} + {x_u}}} = \frac{{{y_p}}}{{{\tau_p} + {x_p}}}. $$

(14)

It follows that

$$ {\tau_p} = \frac{{{y_u}{x_p} - {y_p}{x_u}}}{{{y_p} - {y_u}}} $$

(15)

(This is essentially Eq. 1 of Metcalfe & Barlow.)

Now, assume that the forager encounters identical items that have an energy content e and has the choice of reducing their mass by processing them or leaving them unprocessed. (The logic is the same if processing reduces volume rather than mass.) Processing does not change the energy content but removes waste. The forager collects a load of mass M. This load can be made up of unprocessed items or processed items.

M _u :

= mass of an unprocessed item

and

M _p :

= mass of a processed item.

Processing removes useless material so that processed items have lower mass, i.e., M _u  > M _p .

If the forager does not process items, its load consists of ${n_u} = M/{M_u}$ items, whereas if items are process, the load consists of ${n_p} = M/{M_p}$ items. Let c be the time to collect an item and h be the time to processes it. Then

$$ {x_u} = Mc/{M_u} $$

and

$$ {x_p} = M(c + h)/{M_p}. $$

All items have an energy content e, so

$$ {y_u} = Me/{M_u} $$

and

$$ {y_p} = Me/{M_p}. $$

From Eq. 15, it follows that

$$ {\tau_p} = \frac{{Mh}}{{{M_u} - {M_p}}}. $$

(16)

If processing removes a mass W, then

$$ {\tau_p} = \frac{{Mh}}{W} $$

(17)

which is Eq. 12 of the main text.

Bettinger et al. (1997) present an analysis of a model in which there are various stages of processing. They find the critical travel time τ _j (z _j in their notation) for switching from processing to stage j − 1 to processing to stage j. Their derivation is based on the equal delivery rates condition at the critical travel which they give as

$$ \frac{{{z_j} + {x_{j - 1}}}}{{{y_{j - 1}}}} = \frac{{{z_j} + {x_j}}}{{{y_j}}}, $$

where x _j  = time to procure and process a load to stage jand

y _j :

= utility of a load at stage j.

Bettinger et al. describe their Eqs. 1 and 2 as rates of utility delivery. In fact, the equations are times to deliver a unit of utility, i.e., the reciprocal of the rates in Eq. 14. The subsequent analysis is correct because the condition for it to be worth processing to stage j is that the reciprocal of rate for stage j is less than the reciprocal of rate for stage j − 1, cf. the reciprocal of Eq. 14.

Bettinger et al. give the following equation for the critical travel times:

$$ {\tau_j} = {p_j}\left( {\frac{{{y_{j - 1}}}}{{{y_j} - {y_{j - 1}}}}} \right) $$

where p _j is the time to process a load from stage j − 1 to stage j. Let an item at stage j have mass M _j . Then

$$ {y_j} = \frac{{Me}}{{{M_j}}} $$

and

$$ \frac{{{y_{j - 1}}}}{{{y_j} - {y_{j - 1}}}} = \frac{{{M_j}}}{{{M_{j - 1}} - {M_j}}}. $$

Bettinger et al. give the following equation for p _j :

$$ {p_j} = {h_j}M/{M_j}. $$

This follows from the item-based view of processing. The number of items processed is $M/{M_j}$ and h _j is the time to process an item from stage j − 1 to stage j.

Thus,

$$ {\tau_j} = \frac{{M{h_j}}}{{{M_{j - 1}} - {M_j}}}, $$

(18)

cf. Eq. 16. Let W _j  = the reduction in mass when an item is processed from stage j − 1 to stage j.

Then,

$$ {\tau_j} = \frac{{M{h_j}}}{{{W_j}}} $$

(19)

cf. Eq. 17.