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FEPRE: human migration and
anomalous transport
Sergei Fedotov
School of Mathematics
The University of Manchester
UK
2015/7/18
FEPRE-Manchester
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Contents
Main Challenge
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Demic diffusion and logistic growth
Fisher equation for the human density n:
No consensus:
demic diffusion vs cultural diffusion
The advance of agriculture from the Near East:
Ammermann, Cavalli-Sforza, 1971
The time-delayed demic model (Fort, Mendez):
The environmental effects on the spread of Neolithic (Newcastle group)
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How to estimate D for the farmers?
Einstein formula:
In general:
subdiffusion:
normal diffusion
Superdiffusion:
Challenge:
The migratory behavior of farmers is driven by their search for more
viable locations to live
We need a search model for transport rather than passive diffusion
The search is not a pure random walk (farmers are not molecules!!!)
It shold involve a strong bias in the direction of those locations with
favorable….
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Simple random walk model for the farmers
Bad limit!!!
What about the limit:
Fisher equation with convection:
For “active” random walk probabilty q should depend on
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Discrete random walk models
Generalization:
density for jump length
logistic growth
We need a search
model for transport
rather than passive
diffusion
is the cost of migration
in the limit:
Bad limit:
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Continuous time random walk (CTRW)
CTRW is based on the idea that
1) the length of a given jump x is random
2) the waiting time t between jumps is random
joint pdf for jump length and time
Important characteristics:
jump length variance
mean waiting time
subdiffusion:
generalized diffusion
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CTRW theory and governing equation for
the density of farmers
the joint pdf for jump lengths and waiting times
logistic growth
the survival probability
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the waiting time pdf
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Associated linear integro-differential
equation
the memory effect
logistic growth
the long-range interaction
trivial
non-trivial
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Fractional reaction-diffusion equation
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Demic and cultural diffusion
the density for the hunters of migratory phenotype
the density for the farmers of proliferating phenotype
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