Libro de Resúmenes / Book of Abstracts (Español/English)

Resumenes 56
Syncronism in population networks with non linear coupling
The goal of this work is to study the sincronism of populations
coupled by density-dependent migration. We consider a metapopulation
with n identical patches with local dynamics (reproduction and survival)
j
j
given by x t+
1 = f ( xt
) , where j
x t is the number of individuals in patch j at
time t. After the local dynamics stage, the migration process between
patches takes place. We let ( )
i
µ xt
be the fraction of individulas that leave
patch i at time t, while ( )
i
c ji µ xt
is the fraction of individuals leaving patch i
and ariving at patch j. The dynamics of the population network is given by
the system of n equations:
x
n
j
j j
t + 1 = t t ∑
i=
1
[ 1−
µ ( f ( x ))] f ( x ) + c µ ( f ( x )) f ( x ) , j = 1,
2,
K,
n .
We study the stability of a syncronous orbit x t = xt
for all
j = 1, 2,
K,
n , where x t+
1 = f ( xt
) . These orbits lie on the diagonal of the
phase space which is an invariant manifold in this case. Following the
temporal evolution of a small transversal perturbation of an orbit in the
invariant manifold (syncronous orbit), we can obtain conditions that
garantee the stability of attractors in the invariant manifold (syncronous
attractors). For typical orbits the stability condition is L Λ < 1 , where
1
L lim f ′ ( x ) K f ′ ( x ) f ′ ( x ) t
ji
= τ −1
τ →∞
is the Liapunov number and the quantifyer Λ is given by
lim I ϕ ′ ) A)
K ( I −ϕ
′ ( x ) A)(
I −ϕ
′ ( x ) ,
Λ = ( − ( xτ
−1
τ →∞
where ϕ ( x ) = xµ
( x)
is the function that gives the number of individulas that
migrate in each patch, A is a ( n −1) x(
n −1)
matrix obtained from the
coefficients c ij and I is the ( n −1) x(
n −1)
identity matrix. The analysis of the
above expression allows us to conclude that
Λ ≤ lnσ
( H ϕ x ) dρ(
x)
,
exp ∫ −1
′ ( )
where H ϕ′ (x)
is a nxn matrix with λ = 1 as an eigenvalue associated to the
eigenvector [ ] T
1 1 L 1 which is obtained from ϕ ′ and the coefficients c ij ,
σ − 1( H ϕ′
( x)
) is the spectral radius of the matrix (x)
i
t
1
1
i
t
0
0
1
τ
j
H ϕ′ once the eigenvalue
λ = 1 is removed, and ρ is the natural invariant measure. We show that in
some special cases the above inequality is in fact an equality.