Travelling Wave Solutions for the Spread of Farmers into a Region Occupied by Hunter–Gatherers

The geographical spread of an initially localized population of farmers into a region occupied by hunter–gatherers is modelled as a reaction–diffusion process in an infinite linear habitat. Hunter–gatherers who come in contact with farmers are converted at ratee. Growth of initial farmers, converted farmers, and hunter–gatherers is logistic with intrinsic ratesrf,rc, andrh. The carrying capacities of all farmers (i.e., initial and converted farmers combined) and hunter–gatherers areKandL. Individuals migrate at random, where the diffusion constant,D, is the same for all three groups. Under the above assumptions, we deduce the conditions under which wavefronts of initial or converted farmers are generated. Numerical work suggests that a travelling wave solution of constant shape always exists, comprising an advancing wavefront of all farmers and a retreating wavefront of hunter–gatherers. Linear analysis in phase space suggests that the speed is given by 2(Drf)1/2or 2[D(rc+eL)]1/2, whichever is greater. The composition of the expanding distribution of all farmers appears to depend on the relative magnitude ofrfversusrc+eLand ofeKversusrh. A wavefront of initial farmers is not generated ifrfrh. On the other hand, a wavefront of initial farmers is generated ifrf>rc+eL. In this case, the waveform is peaked with leading and trailing edges that converge to 0 ifeKrhso that there is substantial displacement of the indigenous hunter–gatherers by the initial farmers.