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 rate e. Growth of initial farmers, converted farmers, and hunter-gatherers is logistic with intrinsic rates rf, rc, and r h. The carrying capacities of all farmers (i.e., initial and converted farmers combined) and hunter-gatherers are K and L. 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)1/2 or 2[D(r c+eL)]1/2, whichever is greater. The composition of the expanding distribution of all farmers appears to depend on the relative magnitude of rf versus rc+eL and of eK versus rh. A wavefront of initial farmers is not generated if rf<rc+eL. In this case, the initial farmers disappear completely if eK<rh, whereas they spread mainly by diffusion behind the wavefront of converted farmers if eK>rh. On the other hand, a wavefront of initial farmers is generated if rf>rc+eL. In this case, the waveform is peaked with leading and trailing edges that converge to 0 if eK<rh, while it is flat behind the wavefront if eK>rh so that there is substantial displacement of the indigenous hunter-gatherers by the initial farmers.