TRAVELING-WAVE SOLUTIONS FOR THE SPREAD OF FARMERS INTO A REGION OCCUPIED BY HUNTER-GATHERERS

The geographical spread of an initially localized population of farmer s 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 initi al farmers, converted farmers, and hunter-gatherers is logistic with i ntrinsic rates r(f), r(c), and r(h). The carrying capacities of all fa rmers (i.e., initial and converted farmers combined) and hunter-gather ers are K and L. Individuals migrate at random, where the diffusion co nstant, D, is the same for all three groups. Under the above assumptio ns, we deduce the conditions under which wavefronts of initial or conv erted farmers are generated. Numerical work suggests that a travelling wave solution of constant shape always exists, comprising an advancin g wavefront of all farmers and a retreating wavefront of hunter-gather ers. Linear analysis in phase space suggests that the speed is given b y 2(Dr(f))(1/2) or 2[D(r(c) + eL)](1/2), whichever is greater. The com position of the expanding distribution of all farmers appears to depen d on the relative magnitude of r(f) versus r(c) + eL and of eK versus r(h). A wavefront of initial farmers is not generated if r(f) < r(c) eL. In this case, the initial farmers disappear completely if eK < r( h), whereas they spread mainly by diffusion behind the wavefront of co nverted farmers if eK > r(h). On the other hand, a wavefront of initia l farmers is generated if r(f) > r(c) + eL. In this case, the waveform is peaked with leading and trailing edges that converge to 0 if eK < r(h), while it is flat behind the wavefront if eK > r(h) so that there is substantial displacement of the indigenous hunter-gatherers by the initial farmers. (C) 1996 Academic Press, Inc.