The diffusive farmers and hunter-gatherers model with a free boundary in a heterogeneous environment

Abstract Ecologically, the evolutionary advantage of farmers over hunter-gatherers has led to their worldwide spreading. In this paper, we study a free boundary problem of a farmers and hunter-gatherers model featuring a type of prey-predator dynamics. The nonlinear diffusion of farmers is determined by the density of farmers in a way that higher population density accelerates the migration. We establish a spreading-vanishing dichotomy and provide some sufficient conditions in a heterogeneous environment. Moreover, in a homogeneous environment, we achieve exact asymptotic convergence to the constant equilibrium. Thereby, we verify that a strong sedentary lifestyle of farmers is advantageous for the spreading of farmers.

[1]  James Steele,et al.  Neolithic transition in Europe: the radiocarbon record revisited , 2003, Antiquity.

[2]  Yihong Du,et al.  Remarks on the uniqueness problem for the logistic equation on the entire space , 2006, Bulletin of the Australian Mathematical Society.

[3]  Yihong Du,et al.  Spreading and vanishing in nonlinear diffusion problems with free boundaries , 2013, 1301.5373.

[4]  Dongmei Xiao,et al.  The diffusive logistic model with a free boundary in heterogeneous environment , 2014 .

[5]  Jie Wang.  The selection for dispersal: a diffusive competition model with a free boundary , 2015 .

[6]  Jong-Shenq Guo,et al.  On a Free Boundary Problem for a Two-Species Weak Competition System , 2012 .

[7]  Ling Zhou,et al.  A free boundary problem of a predator-prey model with advection in heterogeneous environment , 2016, Appl. Math. Comput..

[8]  Weisong Dong.  Positive solutions for logistic type quasilinear elliptic equations on RN , 2004 .

[9]  Yihong Du,et al.  Spreading-Vanishing Dichotomy in the Diffusive Logistic Model with a Free Boundary , 2010, SIAM J. Math. Anal..

[10]  Yihong Du,et al.  Spreading speed revisited: Analysis of a free boundary model , 2012, Networks Heterog. Media.

[11]  Mingxin Wang,et al.  A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries , 2013, 1312.7751.

[12]  Jie Wang,et al.  Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment , 2015 .

[13]  Mingxin Wang.  On some Free Boundary Problems of the Prey-predator Model , 2013, 1301.2063.

[14]  Luigi Luca Cavalli-Sforza,et al.  The Neolithic Transition and the Genetics of Populations in Europe. , 2009 .

[15]  Yihong Du,et al.  Logistic Type Equations on RN by a Squeezing Method Involving Boundary Blow‐Up Solutions , 2001 .

[16]  Large Time Behaviour of the Solution of a Nonlinear Diffusion Problem in Anthropology , 2018, Journal of Mathematical Study.

[17]  Zongming Guo,et al.  A diffusive logistic model with a free boundary in time-periodic environment☆ , 2013 .

[18]  Yihong Du,et al.  Spreading–vanishing dichotomy in a diffusive logistic model with a free boundary, II☆ , 2011 .

[19]  Luigi Luca Cavalli-Sforza,et al.  Measuring the Rate of Spread of Early Farming in Europe , 1971 .

[20]  M. H. Kabir,et al.  On the well-posedness of a dispersal model for farmers and hunter–gatherers in the Neolithic transition , 2018 .

[21]  Mingxin Wang.  Spreading and vanishing in the diffusive prey-predator model with a free boundary , 2015, Commun. Nonlinear Sci. Numer. Simul..

[22]  W. Choi,et al.  Non-uniform dispersal of logistic population models with free boundaries in a spatially heterogeneous environment , 2019, Journal of Mathematical Analysis and Applications.

[23]  Zhigui Lin,et al.  The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor , 2013, 1303.0454.

[24]  M. H. Kabir,et al.  Spreading Waves in a Farmers and Hunter-Gatherers Model of the Neolithic Transition in Europe , 2018, Bulletin of mathematical biology.

[25]  Mingxin Wang,et al.  A free boundary problem of the ratio-dependent prey-predator model , 2015 .