Abstract
Our starting point is an article by Aoki, Shida and Shigesada [1], who present travelling wave solutions of a reaction-diffusion system modelling the spread of farmers in a region occupied by hunter-gatherers. More precisely, they consider three different populations, namely the original farmer population, the converted farmers and the hunter-gatherers. We denote their densities at positionxand at timetbyF(x, t),C(x,t)andH(x,t)respectively.The dynamics of the populations is governed by the following laws:
-
1
All the individuals move randomly with constant diffusion coefficientD > 0, which is assumed to be the same for all the three groups
-
2
Hunter-gatherers who meet farmers become themselves farmers at the ratee>0
-
3
The growth of the three populations is logistic, that is to say that it is expressed by terms of the form F(1 — XF)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
K. Aoki, M. Shida and N. ShigesadaTravelling wave solutions for the spread of farmers into a region occupied by hunter-gatherers. Theor. Pop. Biol. 50 (1996), 117.
P.C. FifeAsymptotic states for equations of reaction and diffusion.Bull. Amer. Math. Soc. 84 (1978), 693–726.
F. RotheConvergence to the equilibrium state in the Volterra-Lotka diffusion equations. J. Math. Biol. 3 (1974), 319–324.
N. Dancer, D. Hilhorst, M. Mimura and L.A. PeletierSpatial segregation limit of a competition-diffusion system.Europ. J. of Applied Mathematics 10 (1999), 97–115.
S.-I. Ei and E. YanagidaDynamics of interfaces in competition-diffusion systems. SIAM J.Appl. Math. 54 (1994), 1355–1373.
R. WeidenfeldLarge time behavior and singular limits for a reaction-diffusion system arising in population dynamics.in preparation.
J. SimonCompact sets in the space LP (0 T; B). Ann. Mat. Pura Appl. 146 (1987), 65–96.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this paper
Cite this paper
Hilhorst, D., Mimura, M., Weidenfeld, R. (2003). On a Reaction-Diffusion System for a Population of Hunters and Farmers. In: Colli, P., Verdi, C., Visintin, A. (eds) Free Boundary Problems. ISNM International Series of Numerical Mathematics, vol 147. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7893-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7893-7_15
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9613-9
Online ISBN: 978-3-0348-7893-7
eBook Packages: Springer Book Archive