1 Random-Walk Simulations of the Neolithic in 2 Dimensions Joaquim Pérez-Losada Univ. de Girona (Catalonia, Spain) FEPRE European Project 2 nd Annual Workshop St. Petersburg, Russia April 5-10, 2008

2 Overview 1. Derivation of sequential integro-difference equations to analyze the dynamics of two interacting populations in the Neolithic transition. 2. Derivation of an equation for the coexistence time between the invasive and invaded population. 3. Method to estimate the interaction parameter.

3 Evolution Equation 1-Population r r (x+r, y) (x, y-r) (x-r, y) (x, y+r) - p e persitence - R 0 net reproductive rate - r distance - T generation time - p(x,y,t) population number density Dispersion Sequential Non- sequential

4 Sequential Model versus Non-Sequential Model (a) Non-sequential model. Parents ( ) migrate away from their children ( ). (b) Sequential model. Parents ( ) migrate with their children ( ). (x+r, y+r, t + T) b) a) Non-Sequential Model Sequential Model (x+r, y+r, t+T) (x,y,t) (x,y,t+T)

5 Two Interacting Populations. Sequential Model

6 (1) Initial Population How Does the Algorithm Work? (2) Dispersion (1-p e )/4 pepe (3) Reproduction R o ·(1-p e )/4 R o ·p e (4) Dispersion

7 Front Speed for 2 Populations - γ interaction parameter Fort,Pérez-Losada,Suñol, Escoda and Massaneda (New J of Phys 2008)

8 Predicted Speeds versus Interaction Parameter T32years R 0p 1.8gen -1 pepe p maxp 0.064km -2 p maxn 1.28km -2

9 Equation to Estimate the Coexistence Time An equation for the coexistence time in terms only of the parameters appearing in the evolution equations Fort,Pérez-Losada,Suñol, Escoda and Massaneda (New J of Phys 2008)

10 Estimation of the Coexistence Time (t c ) t c ≈ 2t slope ≈ 6 gen t c = 6 gen t slope = 3 gen

11 A Method to Estimate the Value of γ

12 Conclusions 1. Sequential integro-difference equations are more realistic. 2. Front speed depends on γ 3. Coexistence time depends on γ

13 Questions?