Consideration of the aggregate behavior of the tribe leads to a simpler, low-dimensional, model. We first follow the example of Zeeman's cusp-catastrophic model for the arms race, shown in Fig. 2. (Zeeman, 1977)

4.1 Single-cusp models

For the state variable, in place of the 'tm'military actionº of Zeeman's model, we pose the accumulation of goods: domesticated animals, fields, tools, etc. For the control variables, in place of the perceived 'tm'threatº and 'tm'costº of a war, we take synergy and willingness to trade with other social groups. Assuming then, as in the Zeeman model, a gradient dynamical scheme with a cusp catastrophe, we obtain the model shown in Fig. 3. Within this model we have shown a curve of gradual decrease in social synergy, resulting in a catastrophic increase in hoarding.

While this is a close analog of the Zeeman model, a simple modification will provide another model which more closely fits the !Kung data. We now rearrange the model of Fig. 3, by exchanging two axes, into a cusp model in which a gradual increase in stored goods results in a catastrophic drop in social synergy, as shown in Fig. 4.

We now go on to consider the interaction of the !Kung and the Bantu, by combining two single cusp models, as in the Kushelman double-cusp model for an arms race.

4.2 A double-cusp model

We consider two similar models for the synergy of a tribe, as in Fig. 4. We connect them in a complex dynamical scheme by determining the trade-willingness parameter of each as an increasing sigmoid function of the synergy of the other. The wealth of each remains (for the moment) an uncoupled control parameter. Of course, there are many other possibilities for this mutual coupling. But with this agreement, we have a complex scheme with a two- dimensional state space (Bantu synergy, !Kung synergy) and a two-dimensional control space (Bantu wealth, !Kung wealth). As established by Kushelman, this model admits periodic attractors in a large region of the control space. The response diagram of Kushelman is shown in Fig. 5.

These control parameters could be further reduced by the introduction of a macroeconomic model, presumably of the North-South type. (See Chichilnisky, 1993) For example, a cone field for intertribal trading could be introduced on the control plane. In any case, we obtain a reasonable and simple model for the !Kung catastrophe.