1

Random-Walk SimulationsRandom-Walk Simulationsof the Neolithic inof the Neolithic in

2 Dimensions2 DimensionsJoaquim Pérez-Losada

Univ. de Girona (Catalonia, Spain)

FEPRE European Project2nd Annual WorkshopSt. Petersburg, Russia

April 5-10, 2008

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OverviewOverview

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.

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Evolution Equation 1-PopulationEvolution Equation 1-Population

rr

rr (x+r, (x+r, y)y)

(x, y-r)(x, y-r)

(x-r, y)(x-r, y)

(x, (x, y+r)y+r)

- pe persitence

- R0 net reproductive rate

- r distance

- T generation time

- p(x,y,t) population number density

, ,

, ,

( , , ) , ,14 , ,

, ,

o

e

e

p p x y t

p x r y t

p x y t T p x r y tpp x y r t

p x y r t

R

DispersionDispersion

SequentialSequential

, ,

, ,

, ,1( , , )

4 , ,

, ,

, ,o

e

e

p p x y t

p x r y t

p x r y tpp x y t T

p x y r t

p x y r t

pR x y t

Non-Non-sequentialsequential

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Sequential Model versus Non-Sequential ModelSequential Model versus Non-Sequential Model

(a) Non-sequential model. Parents ( ) migrate away from their children ( ).(a) Non-sequential model. Parents ( ) migrate away from their children ( ).(b) Sequential model. Parents ( ) migrate with their children ( ).(b) Sequential model. Parents ( ) migrate with their children ( ).

(x+r, y+r, t + T)

b)a)

Non-Sequential ModelNon-Sequential Model Sequential ModelSequential Model

(x+r, y+r, t+T)

(x,y,t)(x,y,t)(x,y,t+T)

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Two Interacting Populations. Sequential ModelTwo Interacting Populations. Sequential Model

0

, ,

, ,

, ,1( , , )

4 , ,

, ,

e N

N

NeN N

N

N

p p x y t

p x r y t

p x r y tpp x y t T R

p x y r t

p x y r t

I

0

, ,

, ,

, ,1( , , )

4 , ,

, ,

e P

P

PeP P

P

P

p p x y t

p x r y t

p x r y tpp x y t T R

p x y r t

p x y r t

I

, , , ,

, , , ,

, , , ,14 , , , ,

, , , ,

e P N

P N

P Ne

P N

P N

p p x y t p x y t

p x r y t p x r y t

I p x r y t p x r y tpp x y r t p x r y t

p x y r t p x r y t

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(1) Initial Population(1) Initial Population

How Does the Algorithm Work?How Does the Algorithm Work?

(2) Dispersion(2) Dispersion

(1-pe)/4

pe

(3) Reproduction(3) Reproduction

Ro·(1-pe)/4

Ro·pe

(4) Dispersion(4) Dispersion

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Front Speed for 2 PopulationsFront Speed for 2 Populations

0 max

0

1ln 1 cosh 12

min

eN P e

pR p p rc

T

0NR

-γ interaction parameter

Fort,Pérez-Losada,Suñol, Escoda and Massaneda (New J of Phys 2008)

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Predicted Speeds versus Interaction ParameterPredicted Speeds versus Interaction Parameter

T 32 years

R0p 1.8 gen-1

pe 0.38 ---

pmaxp 0.064 km-2

pmaxn 1.28 km-2

0 1 2 3 4 516,0

19,2

22,4

25,6

28,8

32,0

35,2

38,4

41,6

44,8

48,0

0,5

0,6

0,7

0,8

0,9

1,0

1,1

1,2

1,3

1,4

1,5

Fort, Pérez-Losada, Suñol, Escoda and Massaneda (New J Phys 2008)

Simulations Equation

c (k

m/y

r)

R0N=1.6

R0N= 3.0

c (k

m/g

en)

=/R0N

(km2)

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Equation to Estimate the Coexistence TimeEquation to Estimate the Coexistence Time

00 max

0

421 11 1 exp 1

4

c slope

e NN P

N

Tt tp RR p r

R DT

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)

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Estimation of the Coexistence Time (tEstimation of the Coexistence Time (tcc))

499 500 501 502 503 504 505 506 507 508 5090,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

1,2

1,3499 500 501 502 503 504 505 506 507 508 509

(b)pN(x,y,t+T)

pN(x,y,t-T)

pN(x,y,t)p

max N / 2

Fort, Pérez-Losada, Suñol, Escoda and Massaneda (New J Phys 2008)p N

(km

-2)

t (generations)

tc ≈ 2tslope ≈ 6 gen

tc = 6 gen

tslope = 3 gen

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0,0 0,1 0,2 0,3 0,41

2

3

4

5

6

7

8Fort, Pérez-Losada, Suñol, Escoda and Massaneda (New J Phys 2008)

t c (g

ener

atio

ns)

p max P (km-2)

Simulations = 0.1 km 2

Equation tc

Simulations = 0.7 km 2

Equation tc

A Method to Estimate the Value of A Method to Estimate the Value of γγ

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ConclusionsConclusions

1. Sequential integro-difference equations are more realistic.

2. Front speed depends on γ3. Coexistence time depends on γ

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Questions?Questions?