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\begin{document}

%********************************************************
\title
 {The Influence of Temporal Migration in the Synchronization of Populations\thanks{VM thanks CAPES for supporting his PhD at UFRGS;  The paper was presented at CMAC-CO 2013.}}

\author
    { Author 1%
     \thanks{Email Author 1.}\,, % vanderlei.manica@ufrgs.br
       Author 2%
     \thanks{Email Author 2.}\,, %jaqx@mat.ufrgs.br
     PPGMAp, Federal University of Rio Grande do Sul - UFRGS,
     Bento Gonçalves Avenue 9500, 91509-900, RS, Brasil.}

\criartitulo

\runningheads {Author 1 and Author 2}{The Influence of Temporal Migration in the Synchronization of Populations}

\begin{abstract}
{\bf Abstract}. A discrete metapopulation model with
temporal dependent migration is proposed in order to study the
stability of synchronized dynamics. During each time step, we
assume that there are two processes involved in the population
dynamics: local patch dynamics and migration process between the patches
that compose the metapopulation. We obtain an analytical criterion
that depends on the local patch dynamics (Lyapunov number) and on
the whole migration process. The stability of synchronized
dynamics depends on how individuals disperse among the patches.

{\bf Keywords}. Metapopulation, temporal
migration, synchronization.
\end{abstract}


%********************************************************
\newsec{Introduction}

    The forms of dispersion in a metapopulation system
(populations of single-species that live in fragments called
patches) can induce the whole system to multiple
behaviors~\cite{Allen,Doebeli,Earn,Heino,Silva06}. An interesting behavior
related to the dispersal process is the synchronized dynamics
where the populations in all patches evolve with the same
density~\cite{May}. Its importance lies in the fact that if the
whole metapopulation is not synchronized and a local population is
extinct, it can be recolonized by individuals that migrate from
neighboring patches ($``$rescue effect''), favoring the population
persistence~\cite{Blasius}. A considerable number of populations
that live in distinct regions tend to cycle in synchrony. A
well-documented example is the Canadian lynxes that presents
synchronized dynamics in its densities fluctuations due to the
weather conditions~\cite{Blasius,May}. Another example is the vole
populations in Norway that synchronize due to the dispersal and
birds predation~\cite{Ims}.

Systems of discrete equations are often used to model
metapopulations~\cite{Allen,Earn,Heino,Silva06}.  A
metapopulation model with patches linked by migration and
subjected to external perturbations was considered
in~\cite{Allen}. The model is a discrete-time system composed by single species where a constant
fraction disperses per generation. Through numerical simulations, it was shown that
chaos can prevent global extinctions when coupling is weak. In~\cite{Earn} was
obtained an analytical result for
the stability of synchronized trajectories by considering a model
with an arbitrary number of patches linked by dispersal. An
analytical result examining a special case of density-dependent
dispersal was obtained in~\cite{Silva06}, concluding that this
mechanism reduces the stability of the synchronous dynamics.
Nevertheless, density independent dispersal is observed in the dispersal of insects, while
density-dependent dispersal is observed in such widely different invertebrates as locusts,
snails and copepods~\cite{Hansson}. In this paper,
we present a metapopulation model similar to the ones described
in~\cite{Allen,Earn,Silva06}. The main difference is the
assumption of temporal dependent migration. This assumption can be used in order to describe the movements of species that
move to other areas in different periods due to weather
conditions or dependence of foraging resources.

In section 2 we present the metapopulation model with temporal
dependent migration. In section 3 we analyze the asymptotic local
stability of synchronized trajectories and obtain a criterion to
its stability based on the calculation of the transversal Lyapunov
numbers. In section 4 we present numerical simulations. Final
comments and discussion are done in section 5.

%********************************************************
\newsec{Metapopulation Model}

   The metapopulation model consists of $d$ equal patches labeled as
$1,2,\ldots,d$. We assume that the processes of survival and
reproduction which compose the local dynamics is described by a
map $f$ on $[0,\infty)$ of class $C^{1}$. In the absence of
dispersal between patches, the time evolution of the population is
given by
\begin{equation}
x_{t+1} = f(x_{t}), \hspace{0.8cm} t = 0, 1, 2, \ldots,
\label{SinglePatch}
\end{equation}

\noindent where $x_{t}$ represents the number of individuals at
time $t$.

We assume that a fraction $m$ leave patch $i$ and disperse to the
neighboring patches. We assume that the migration fraction is
temporal dependent, that is, the migration fraction is given by a
a map on [0,1] such that $m_{t+1}=g(m_{t})$. Thus, the density of
individuals that leave patch $i$ is given by $m_{t}f(x_{t}^{i})$,
where $x_{t}^{i}$ denote the population density in patch $i$ at
time $t$, for all $i=1,\ldots,d$, $t=0,1,\ldots$. Moreover, from
the individuals that disperse from the neighboring patches $k$, a
fraction $\gamma_{ik}$ reach patch $i$. We assume that there is no
loss of individuals and the individuals do not return to the
original patch, therefore $\sum_{k=1}^{d} \; \gamma_{ki}=1$ and
$\gamma_{ii}=0$ for all $i=1,\ldots,d$. Taking these into
consideration, we can write a system of equations describing the
dynamics of the metapopulation by
\begin{equation}
x_{t+1}^{i} =
(1-m_{t})f(x_{t}^{i}) + \sum_{k=1}^{d} \gamma_{ik}m_{t}f(x_{t}^{k}).
\hspace{0.8cm} \label{LocalMetapDyn}
\end{equation}

The first term in equation (\ref{LocalMetapDyn}) represents the
individuals that did not leave patch $i$ at time $t$, while the
second term is the sum of all contributions of individuals of the
neighboring patches.

\section{Synchronization and Transversal Stability}


We assume that synchronization is achieved if the population
density of all patches is the same, that is,
$x_{t}^{i}=x_{t}^{s}$, for all $i=1,2,\ldots,d$ and
$t=0,1,2,\ldots$. Substitution of $x_{t}^{i}=x_{t}^{s}$ in
equation (\ref{LocalMetapDyn}) leads us to the existence of such
synchronized solution provided $\sum_{k=1}^{d} \gamma_{ik} = 1,
i=1,2,...,d$. Furthermore, the dynamics of each patch in the
synchronized state satisfies $x_{t+1}^{s}= f(x_{t}^{s})$ which is
equivalent to equation (\ref{SinglePatch}), the single patch model
equation. In other words the metapopulation synchronizes with the
same dynamics of a single isolated patch.

We are interested in studying the local asymptotic stability of
the synchronized state, that is, whether orbits that initiate
close to the synchronized state will be attracted to it. In order
to achieve this goal, we linearize the equation
(\ref{LocalMetapDyn}) around the synchronized trajectory,
obtaining
\begin{equation}
\Delta_{t+1} = J(\textbf{x}_{t}^{s})\Delta_{t},
\label{LinearizedSystem}
\end{equation}

\noindent where $\Delta_{t} \in R^{d}$ is the perturbation of the
synchronized trajectory, and $J(\textbf{x}_{t}^{s})$ is the
$d\times d$ Jacobian matrix of system (\ref{LocalMetapDyn})
evaluated at $\textbf{x}_{t}^{s}$, where
$\textbf{x}_{t}^{s}=(x_{t}^{s},x_{t}^{s},\ldots,x_{t}^{s}) \in
R^{d}$. Notice that the Jacobian matrix $J(\textbf{x}_{t}^{s})$
has its entries given by

\begin{center} $\alpha_{ik}=
\left\{%
\begin{array}{ll}
    (1-m_{t})f^{'}(x_{t}^{s}), & \hbox{if i = k;} \\
    \gamma_{ik} m_{t} f^{'}(x_{t}^{s}), & \hbox{if i  $\neq$ k}. \\
\end{array}%
\right.$
\end{center}

Thus, it can be written as
\begin{equation}
D\textbf{f}(\textbf{x}^{s}_{t}) = (I-m_{t}B)f^{'}(x_{t}^{s}), \label{JacMatrixDiag}
\end{equation}
\noindent where $I$ is the identity matrix and $B = I-\Gamma$, and
$\Gamma$ is the coupling matrix between the patches.

It is important to notice that the connectivity matrix $\Gamma$ is
doubly stochastic (all rows and columns add up to one). A simple
application of the Gershgorin Theorem~\cite{Lancaster} leads to
the fact that $\lambda_{0}=1$ is the dominant eigenvalue of $C$.
Moreover, its eigenspace is spanned by the vector $\vec{1}=[1 \;\;
1 \;\; \ldots \;\; 1]^{T} \in R^{d}$ which correspond to the
in-phase state, that is, the diagonal of the phase space
($x_{t}^{1}=x_{t}^{2}=\ldots=x_{t}^{d}$).

We assume that $\Gamma$ is diagonalizable which allows us to
express the Jacobian matrix as a diagonal matrix and the local
stability of the synchronized state can be analyzed through the
diagonal terms. With this assumption, there exists an invertible
matrix $Q$ such that $B= Q
diag(1,\lambda_{1},\ldots,\lambda_{d-1}) Q^{-1}$. It allows us to
write the Jacobian matrix (\ref{JacMatrixDiag}) in the following
diagonal form
\begin{equation}
J(x^{s}_{t}) = Q\left(%
\begin{array}{cccc}
 1 & 0 & \ldots & 0 \\
 0 & 1-m_{t}+\lambda_{1}m_{t} & \ddots & \vdots  \\
 \vdots & \ddots & \ddots & 0  \\
 0 & \ldots & 0 & 1-m_{t}+\lambda_{d-1}m_{t}  \\

\end{array}%
\right)f^{'}(x_{t}^{s}) Q^{-1}. \label{JacMatrixCoarseGrained2}
\end{equation}

Thus, the synchronized state will be stable if transversal
perturbations to the synchronized state shrink to zero. To reach
this goal, we define the maximum transversal Lyapunov number, $K$,
by
\begin{equation}
K(x_{0}^{s},m_{0})= \displaystyle{\max_{i=1,\ldots,d-1}} \; \displaystyle{\lim_{\tau \rightarrow \infty}} \;\; \| P_{\tau -
1,i}\cdot ...\cdot P_{1,i}P_{0,i} \|^{1/\tau},
\label{TranLyapNumbCoarGrain}
\end{equation}
\noindent where $P_{t,i} =
f^{'}(x_{t}^{s})(1-m_{t}+\lambda_{i}m_{t})$. Consequently, the
transversal perturbation tends to zero if $K(x_{0}^{s},m_{0}) <
1$.

Observe that
\begin{equation}
 | P_{\tau - 1,i}\cdot \ldots \cdot
P_{0,i} | = (\displaystyle{\prod_{t=0}^{\tau-1}} \;\; |f^{'}(x_{t}^{s})|) |
 (1-m_{\tau-1}+\lambda_{i}m_{\tau-1})\cdot \ldots \cdot (1-m_{0}+\lambda_{i}m_{0})|,
\end{equation}

\noindent thus, we can write the maximum transversal Lyapunov
number as

\begin{equation}
K(x_{0}^{s},m_{0})=  L(x_{0}^{s}) \Lambda(m_{0}),\label{Llambda}
\end{equation}

\noindent where
\begin{equation}
L(x_{0}^{s}) = \displaystyle{\lim_{\tau \rightarrow \infty}} \;\;
(\displaystyle{\prod_{t=0}^{\tau-1}} |f^{'}(x_{t}^{s})|)^{1/\tau}
\label{LyapunovNumber}
\end{equation}
\noindent is the Lyapunov number of $f$ starting in $x_{0}^{s}$
and
\begin{equation}
\Lambda(m_{0})= \displaystyle{\max_{i=1,2,\ldots,d-1}} \;\; \displaystyle{\lim_{\tau \rightarrow \infty}} \;\;
(| (1-m_{\tau-1}+\lambda_{i}m_{\tau-1})\cdot \ldots \cdot (1-m_{0}+\lambda_{i}m_{0}) |) \label{LambdaUmaMet}
\end{equation}

\noindent is a quantifier that depends on the initial migration
rate.

Let $\rho$ be the natural probability measure for the local map
$f$. Let $\nu$ be the natural probability measure for map $g$.
Assuming the integrability of $ln^{+} |f^{'} (x)|$ and $ln^{+}
|1-\lambda m|$ with respect to $\rho$ and $\nu$, we can apply the
Ergodic Theorem of Birkhoff~\cite{Eckmann} to guarantee the
existence and uniqueness $\rho$-almost every $x_{0}^{s}$ of the
limit defining $L$, and $\nu$-almost every $m_{0}$ of the limit
defining $\Lambda$ and state a criterion for the local asymptotic
stability of an attractor in the synchronized invariant state
given by
\begin{equation}
K=L\Lambda < 1, \label{StabilityCriterion}
\end{equation}
\noindent where
\begin{equation}
L = exp(\int_{0}^{\infty} ln|f^{'}(x)|d\rho(x)).
\end{equation}
\noindent and
\begin{equation}
\Lambda = \displaystyle{\max_{i=1,\ldots,d-1}} \; exp(\int_{[0,1]} ln \mid 1-m \lambda_{i} \mid) d\rho(m).\label{CritLambda}
\end{equation}

Notice that $L$ depends on the local habitat dynamics while
$\Lambda$ depends on the whole migratory process. It is important
to observe that the evolution of the term that corresponds to the
value 1 in the Jacobian matrix (\ref{JacMatrixCoarseGrained2}) is
exactly the Lyapunov number and it gives the behavior of the
synchronized trajectory within the phase space diagonal, that is,
a periodic trajectory ($L<1$) or a chaotic trajectory ($L>1$).

In the following, we calculate the value of the integral given in
(\ref{CritLambda}) to different temporal migration rules.

\subsection{Temporal migration given by a Dirac measure}

A Dirac measure is a measure $\delta_{y}$ defined on a set $E$
such that
\begin{equation}
\delta_{y}=
\left\{%
\begin{array}{ll}
    1, & \hbox{$y \in E$;} \\
    0, & \hbox{c.c..} \\
\end{array}%
\right.
\end{equation}

Let $\delta_{m_{0}}$ denote the Dirac measure centered on the
fixed migration rate $m_{0}$. Thus, we have
\begin{equation}
\begin{array}{lcl}
 \Lambda & = & \displaystyle{\max_{i=1,\ldots,d-1}} \;
exp(\int_{[0,1]} ln \mid 1-m \lambda_{i} \mid d\delta_{m_{0}}) \\
  & = & \displaystyle{\max_{i=1,\ldots,d-1}} \; exp(ln \mid 1-m_{0}
\lambda_{i} \mid)\\
 & = &    \displaystyle{\max_{i=1,\ldots,d-1}} (\mid 1-m_{0} \lambda_{i} \mid).
\end{array}\label{FixedMigration}
\end{equation}

In this case, the criterion established in
(\ref{StabilityCriterion}) is the same established by Earn et
al.~\cite{Earn} that considered a metapopulation with any number
of patches arbitrarily connected.

If we assume that the probability measure is concentrated in two
points, $m_{0}$ and $m_{1}$, we have
\begin{equation}
\begin{array}{lcl}
 \Lambda & = & \displaystyle{\max_{i=1,\ldots,d-1}} \;
exp(\int_{[0,1]} ln \mid 1-m \lambda_{i} \mid d\delta_{m_{0},m_{1}}) \\
  & = & \displaystyle{\max_{i=1,\ldots,d-1}} \; exp(\frac{ln \mid
1-m_{0} \lambda_{i} \mid + ln \mid 1-m_{1} \lambda_{i}
\mid}{2})\\
 & = & \displaystyle{\max_{i=1,\ldots,d-1}} exp(ln (\mid 1-m_{0}
\lambda_{i} \mid \cdot \mid 1-m_{1} \lambda_{i}
\mid)^{\frac{1}{2}})\\
& = & \displaystyle{\max_{i=1,\ldots,d-1}}  (\mid 1-m_{0} \lambda_{i}
\mid \cdot \mid 1-m_{1} \lambda_{i}
\mid)^{\frac{1}{2}}.
\end{array}
\end{equation}

In this case, the quantifier $\Lambda$ is the geometric average of
$(1-m_{0} \lambda_{i})$ and $(1-m_{1} \lambda_{i})$. In fact, if
the migration rates are distributed in $p$ periodic points,
$m_{0}$, $m_{1}$, ..., $m_{p-1}$, the quantifier $\Lambda$ is
given by the following geometric average
\begin{equation}
\Lambda=\displaystyle{\max_{i=1,\ldots,d-1}} \;(\mid  1-m_{0} \lambda_{i} \mid \cdot \ldots \cdot \mid
1-m_{p-1} \lambda_{i} \mid)^{\frac{1}{p}}.\label{QuantPeriodicMig}
\end{equation}

\subsection{Temporal migration given by a uniform distribution}

Now we assume that the temporal migration rates are given by a
uniform distribution. In our case, the probability density
function with a uniform distribution on a set $[a,b] \subset
[0,1]$ is

\begin{equation}
p(m) = \left\{%
\begin{array}{ll}
    p_{1} = 0, & \hbox{0 $\leq$ m $<$ a;} \\
    p_{2} = \frac{1}{b-a}, & \hbox{a $\leq$ m $<$ b;} \\
    p_{3} = 0, & \hbox{b $\leq$ m $<$ 1.} \\
\end{array}%
\right.
\end{equation}

Observe that $\rho(m)=\int_{0}^{1} p(m) dm = \int_{a}^{b}
\frac{1}{b-a} dm = 1$. Besides that, we can write
(\ref{CritLambda}) as
\begin{equation}
\Lambda = \displaystyle{\max_{i=1,\ldots,d-1}} \;
exp(\frac{1}{b-a }\int_{[a,b]} ln \mid 1-m \lambda_{i} \mid dm).
\end{equation}

\indent The above integral can be solved analytically resulting
\begin{equation}
\Lambda=
\left\{%
\begin{array}{ll}
    \displaystyle{\max_{i=1,\ldots,d-1}} \;   \frac{1}{e}(\frac{(1-a\lambda_{i})^{(1-a\lambda_{i})}}{(1-b\lambda_{i})^{(1-b\lambda_{i})}})^{\frac{1}{\lambda_{i}(b-a)}}, & \hbox{if $0 \leq m < \frac{1}{\lambda_{i}}$;} \\
    \displaystyle{\max_{i=1,\ldots,d-1}} \;
\frac{1}{e}(\frac{(b\lambda_{i}-1)^{(b\lambda_{i}-1)}}{(a\lambda_{i}-1)^{(a\lambda_{i}-1)}})^{\frac{1}{\lambda_{i}(b-a)}}, & \hbox{if $ \frac{1}{\lambda_{i}} \leq m \leq 1$}. \\
\end{array}%
\right.\label{QuantUniformDist}
\end{equation}


\section{Numerical Simulations}

We perform numerical simulation to illustrate the behavior of our
network of patches connected by temporal dependent migration. In
order to determine whether synchronization occurs in the
metapopulation we define the synchronization error, $e_{t}$, by
\begin{equation}
e_{t}= \frac{1}{d}\sum_{i=1}^{d}
|x_{t}^{i}-x_{t}^{i+1}|
\end{equation}
\noindent where $x_{t}^{d+1}=x_{t}^{1}$. Synchronization is
detected when $e_{t} \rightarrow 0$.

We consider that the local patch dynamics is given by the
following Ricker function
\begin{equation}
f(x)=xe^{r(1-x)},
\label{FuncExpLog}
\end{equation}
\noindent where $x$ represents the patch density and $r$ is the
intrinsic growth rate ($r>0$). The dynamics of a local habitat
with the Ricker model is well-known and exhibits equilibrium,
periodic and chaotic dynamics depending on the growth
rate~\cite{May74}. For $0<r<2$, the local dynamics become a state
of equilibrium. For $2<r<2.526$, the equilibrium point is unstable
and a two-periodic trajectory takes its place. As $r$ is
increased, there appears a four-periodic trajectory and the
two-periodic become unstable and we can observe period doubling
bifurcations to chaos. In order to simulate chaotic within patch
dynamics we assume $r=3.1$, which implies that the isolated patch
model have typical chaotic trajectories with Lyapunov number
$L\approx 1.327$.

The configuration matrix $\Gamma$ can be defined in different
forms. Two well-known configurations are the nearest neighbor
coupling and the global coupling~\cite{Earn}. We illustrate our
results considering the patches linked in a ring format with the
two-nearest neighbors coupling,
whose configuration matrix, $\Gamma$, is given by
\begin{equation}
\Gamma = \left(%
\begin{array}{cccccc}
  0 & 1/2 & 0 & \ldots & 0 & 1/2 \\
  1/2 & 0 & 1/2 & 0 & \ldots & 0 \\
  0 & 1/2 & \ddots & \ddots & \ddots & \vdots \\
  \vdots & \ddots & \ddots & 0 & 1/2 & 0 \\
  0 & \ldots & 0 & 1/2 & 0 & 1/2 \\
  1/2 & 0 & \ldots & 0 & 1/2 & 0 \\
\end{array}%
\right).
\end{equation}

In this case, the eigenvalues of $\Gamma$ are given by
$\lambda_{0}=1$ and $\lambda_{i}=cos(\frac{2\pi i}{d})$,
$i=1,2,\ldots,d-1$. Of course, other local patch dynamics and
configuration network topologies could be used, but our main
concernment is to show the different behavior generated by
temporal migration rates.

\begin{figure}[!htb]
\centering
\includegraphics[width=14 cm]{MigracFixaEPS}
\caption{Synchronization error ((a), (b) and (c)) and respectably maximum transversal Lyapunov number ((d), (e) and (f)) $vs$ $m$. Local dynamics
is given by the Ricker function $f(x)=x exp(r(1-x))$ with $r=3.1$. The patches are coupled with the two-nearest neighbors coupling.
(a) $2$ patches. (b) $5$ patches. (c) $10$ patches.}
\label{FigMigFixa}
\end{figure}

Figure \ref{FigMigFixa} shows the bifurcation diagram of the
synchronization error and the respective maximum transversal
Lyapunov number versus the migration rate. In all cases,
individuals migrate with the same rate at time $t$. We can observe
that the non-synchronization region is characterized by weak
interaction. Moreover, the increase in the number of patches
decreases the synchronization region and increase the maximum
transversal Lyapunov number. In fact, the subdominant eigenvalue of the
matrix $\Gamma$ tends to one as $n\rightarrow +\infty$
($\lambda_{i}\rightarrow 1$ as $d\rightarrow +\infty$,
$i=1,2,\ldots,d-1$), thus the quantifier $\Lambda$ given in
(\ref{FixedMigration}) also tends to one as $d\rightarrow +\infty$
(see~\cite{Silva2000}). It means that the stability criterion will
approach to the Lyapunov number if we increase the number of
patches. Moreover, the synchronized attractors will be unstable
for any value of the migration rate if the local dynamics of a
single isolated patch is chaotic ($L>1$).

Figure 2 shows the metapopulation behavior with periodic
migration. We consider 5 patches and three different scenarios. In
all cases, individuals migrate according to a two periodic rule.
In the first case, individuals migrate with a periodic rate given
by 0.1 and $m$ (Figure 2(a)). In the second case, the migration
rates are given by 0.5 and $m$ (Figure 2(b)), while in the third
case it is given by 0.9 and $m$ (Figure 2(c)). We can observe that weak
interactions between the patches decreases the region of
synchronization, while intermediate and high migration rates have
an oppositive effect.

\begin{figure}[!htb]
\centering
\includegraphics[width=14 cm]{MigracPeriodo2EPS}
\caption{Synchronization error ((a), (b) and (c)) and respectably maximum transversal Lyapunov number ((d), (e) and (f)) $vs$ $m$. Local dynamics
is given by the Ricker function $f(x)=x exp(r(1-x))$ with $r=3.1$. Five patches are coupled with the two-nearest neighbors.
(a) $m_{0}=m$ and $m_{1}=0.1$. (b) $m_{0}=m$ and $m_{1}=0.5$.
(c) $m_{0}=m$ and $m_{1}=0.9$.}
\label{FigMigPeriodica}
\end{figure}

Table 1 shows different migration rules and the values of the
quantifier $\Lambda$. We observe that, if the temporal migration
rates are distributed around an average, the values of the
quantifier $\Lambda$ won't change its values significantly. In the
table 1, we show the case where we distribute the migrations
around an average of $m=0.3$.

\begin{table}[!htb]%[ph]
 {\footnotesize
\begin{tabular}{@{}crrrrr@{}}
\hline
{} &{} &{} &{} &{} &{} \\[-1.5ex]
{} \hspace{30pt}  & Fixed Point & Period 2 & Period 4 & Uniform & Uniform  \\[1ex]
\hline
{} &{} &{} &{} \\[-1.5ex]
m & \hspace{3pt} 0.3 & \hspace{3pt} 0.2 and 0.4  & \hspace{3pt} 0.15, 0.25, 0.35 and 0.45 & \hspace{3pt} [0.2, 0.4] & \hspace{3pt} [0.1, 0.5] \\[1ex]
$\Lambda$ & 0.7927 & 0.78968  & 0.78892 & 0.7916 & 0.7887 \\[1ex]
\hline
\end{tabular} }
\caption{Quantifier $\Lambda$ for different
temporal migration rates. Temporal migration
rates are distributed around $m=0.3$. We can observe that the values of $\Lambda$
are not changed significantly.}
\label{table2}%\vspace*{-13pt}
\end{table}

\section{Discussion}

In this paper we develop a model of a network of equal patches
linked by temporal dependent migration. The time evolution of the
system involves two processes: local patch dynamics and migration
between the patches. We then analyze the phenomenon of
synchronization between the patches. We obtain an analytical
criterion for the local asymptotic stability of synchronized
trajectories based on the computation of the transversal Lyapunov
numbers of attractors on the synchronous invariant manifold. The
criterion is obtained via linearization process around the
synchronized trajectories. The criterion is given by the product
of two quantifiers: the separation rate of two nearby orbits in
the single isolated patch measured by the Lyapunov number, $L$,
and a quantifier that depends on the whole migration process,
$\Lambda$ (eq. \ref{StabilityCriterion} ). We then calculate the
value of this quantifier to different migration rules. At first,
we describe $\Lambda$ assuming the migration rate with a periodic
behavior (eq. \ref{QuantPeriodicMig}), we then consider the
migration rate uniform distributed on the interval $[a,b]
\subseteq [0,1]$ (eq. \ref{QuantUniformDist}). The quantifier
$\Lambda$ in the case of migration rates with a periodic behavior
involves the migration rates and the eigenvalues of the matrix
that inform the network topology between the patches, while in the
case of uniform distribution also involves the size of the
interval.

Our observation based on theoretical results and on numerical
simulations reveals the importance of analyzing a metapopulation
model with temporal migration. We performed numerical simulation
assuming each patch dynamics given by Ricker map with chaotic
behavior. Then, we analyze the influence that the migration
process has over synchronized dynamics. We observe that an
increase in the number of patches decrease the stability regions
(Figure \ref{FigMigFixa}). Besides that, weak interactions between
the patches, decreases the changes of synchronization, while
intermediate an high migration rates favors have an oppositive
effect (Figure 2). We observe that, if the temporal migration
rates are distributed around an average, the values of the
quantifier $\Lambda$ won't change its values significantly (Table
1).


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