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Simple non-local models of population dynamics.
Continuum description and influence of advection.
Cristóbal López
Instituto Mediterráneo de Estudios
Avanzados (IMEDEA)
Palma de Mallorca, Spain.
Emilio Hernández-García, Francesco D’Ovidio
Two research directions:
1) Basic processes in fluid flows such as stirring, mixing, chemical or
biological reacivity, pattern formation, motion of non-ideal tracers.
Special focuss on chaotic advection flows.
2) Applications of these methods and concepts to geophysical
settings, mostly ocean dynamics: plankton structures, ocean
forecasting, data analysis, transport modelling...
MOTIVATION
The discrete nature of organisms or chemical molecules are missed
in general when a continuum approach is used to model processes in
Nature.
Continuum descriptions have many advantages:
stability analysis and pattern formation
Therefore, there is the need of deriving continuum equations of
microscopic particle systems that still remain discreteness effects
Derivation and study of
advection-reaction-diffusion
equations for simple models of
interacting particles.
Brownian bug model
Young, Roberts, and Stuhne, Nature 412, 328 (2001).
N particles performing brownian motion and in the presence of an
external flow. They die with probability p, and reproduce, given rise
to a new bug close to the parent, also with probability p
This model is almost independent of the external flow.
Total
number of
particles
For different
values of the
typical velocity
of the flow
We introduce a new model that depends critically on the external flow by
changing the reproduction and birth rates.
Birth rate
Death rate
1
 ( i )  0 
Ns
 (i )   0
N
Neigh.
j  neighb
Meaning a

distance

N j (i )

less than R
N s j  neighbo
j
(i )
Spatial patterns in the absence of flow
Neighbourhood-dependent
Here we have taken
 0
Continuum description
Master equation (no flow) in a lattice
dP ( N 1 ,..., N  )
   W ( N i  1  N i ) P ( N 1 ,.. N i  1,.. N  )
dt
i
 W ( N i  1  N i ) P ( N 1 ,.. N i  1,.. N  )  W ( N i  N i  1) P ( N 1 ,.. N i ,.. N  )
 W ( N i  N i  1) P ( N 1 ,.. N i ,.. N  )
W ( N i  N i  1)   ( i ) N i
W ( N i  N i  1)   ( i ) N i

Using the Fock space representation
a i N 1 ,.., N i ,.., N   N i N 1 ,.., N i  1,.., N 
a i N 1 ,.., N i ,.., N   N 1 ,.., N i  1,.., N 
a , a   
i

j
ij
ai 0  0
Defining the many-particle state
 

N1 ,.., N 

P ( N 1 ,..., N  ) (a i ) N i 0
i 1
One can obtain a Schrodinger-like equation for this state vector, which
defines a Hamiltonian for the model. Then make the continuum limit
and from the action of the system obtain a Langevin equation for the
density of particles.
Langevin equation for the density of particles without diffusion and
external flow
 ( x , t )
1
 1
 ( 0   0 
) 
  dr ( r , t )   ( x , t )
t
Ns
N s r R
1
1
 ( x , t ) ( x , t )  2 ( x  x )  ( x , t ) 0 

Ns Ns
'
'
 dr (r , t ) 
r R
 1

 ( x , t )  ( x ' , t ) (0  x  x '  R )
Ns
What about advection and diffusion?
One can show that considering both in a lattice like a bith-death
process gives rise to no corrections in the fluctuations (noise) and
just add two new terms in the density equation, the advective and
the diffusion terms.
Pattern formation for the continuum equation.
Simplest case: no flow. For comparisons 
 ( x , t )
1
1
2
 D   (0   0 
) 
  dr ( r , t )
t
Ns
N s r R
0
In order to
study patterns
just the
deterministic
equation is
needed.
Two homogenous steady-state solutions:
0   0 
0  0
1 
1
0   0 
0
Ns
0   0
1
Ns
2R
1

 0
Ns
2
Ns
Stability analysis around
  1   
1
   e t  ikx
One obtains the dispersion relation
1
0   0 
Ns
2
   Dk 
J1 (kR)
kR
REMARKS
The noise term does not change the nature of the patterns. Just
changes the values of the parameters at which the transitions
take place.
At the level of the discrete numerical
simulations, advection does not change the
nature of the patterns. Just elongates the
clusters. STILL WORKING ON.