text:
\section{Introduction}\label{sec:intro}

The coexistence of competing  selfish individuals is an ubiquitous  issue in the study of systems  described by
the modern Darwinian paradigm, known as the ``Evolutionary Synthesis''  \citep{Maynard_95,Mayr_01,Mayr_02,Mayr_04,Nowak_04}.
In the context of prebiotic or chemical evolution, this matter surfaced
with the observation by \citet{Eigen_71} that, due to the  finite fidelity of replication,
the information content of a single self-replicating macromolecule (a template for short)
is far too few to permit the coding  of macromolecules with any functional complexity.
A way out of this difficulty, so-called the information crisis of prebiotic evolution,
 is to assume the information is distributed among a number of distinct template types and
enforce cyclic cooperative interactions among them -- the hypercycle --  to guarantee coexistence \citep{Eigen_78,Eigen_80}.

Alternatively, coexistence between distinct template types can be achieved by confining the templates in  packages
or protocells and  requiring that the survival or the reproduction chances of a
protocell be dependent on its template composition \citep{Bresch_80,Niesert_81,Szat_87}.  The study of
this two-level selection problem can be carried out by introducing  minor changes on the
mathematical models developed to address the efficiency of group selection to maintain an altruistic trait
\citep{Eshel_72,Aoki_82,Donato_97}.
In particular,  in a recent paper we have used a diffusion model of group selection  \citep{Kimura_83}
to study analytically the conditions for the coexistence of two template types which differ on their replication rates
\citep{Fontanari_13}. However,  that  study   left out a crucial characteristic of the primitive protocell populations, namely,
the elevated exchange flux of templates among protocells, known as lateral or horizontal gene transfer. In fact,
the acceptance of  the operation of this process in the early history of microbial life has wiped out completely the
familiar Darwinian notion of a universal  ancestor \citep{Woese_98,Doolittle_00}.

Here we model the process  of template swapping among protocells by the classic migration process of
Wright's island model \citep{Wright_51}. We find that introduction of migration renders the evolutionary
process ergodic
in the sense that the steady state does not depend  on the initial set-up of the population. In addition, migration
allows  a steady-state solution corresponding to protocells carrying  both template types (coexistence regime)
or a solution where  the more efficient  template type is fixed in all protocells
(homogeneous regime). There is a smooth transition between these two regimes   provided that the two template types
exhibit distinct replication rates. In the degenerate case, where the template types  have identical replication efficiencies, only the  coexistence
regime is stable. This contrasts with the results obtained  in the absence of migration, for which there is a non-ergodic segregation regime  characterized
by a mixture of two types of protocells, each type  carrying solely one of the template types \citep{Fontanari_13}.

The remainder of the paper is organized as follows. In Sec.\ \ref{sec:model} we describe the  three
evolutionary processes -- template competition, migration and intercell competition --  that comprise the
dynamics of our two-level selection model, and derive the partial differential equation that governs the time evolution of
the fraction of protocells carrying a given template composition.
Sec.\  \ref{sec:Steady} is devoted to the numerical and analytical study of the steady-state
solutions of that equation. In particular, our numerical approach relies on the
interpretation of the steady-state ordinary differential equation  as an eigenvalue  problem
whose eigenvalue corresponds to the  mean group selection pressure.
 Our concluding remarks are presented in Sec.\ \ref{sec:Conc}. In Appendix A
we present the analytical calculation of the probability that a template type fixates
in a given protocell in the non-ergodic segregation regime for the case migration is
not allowed. This calculation generalizes that presented in \citet{Fontanari_13} by
taking into account the different replication efficiencies of  the template types.

\section{The model}\label{sec:model}

Following \cite{Kimura_83}, we consider a hypothetical population divided into an infinite number of
competing protocells, each of which containing exactly $N$ templates. There are two types of templates
which differ only by their replication efficiency: type 1 templates have a selective disadvantage $s$ relative to type 2
templates, where $s \geq 0$ is a parameter on the order of $1/N$. More pointedly, type 1 templates are assigned fitness $1-s$  and
type 2 templates
fitness $1$.  In addition, we assume that
$N$ is large enough so that the frequency of  type 1 templates  within a protocell, denoted by  $x$,
can be viewed as a continuous variable in the interval $ \left [ 0,1 \right ]$. Of course, the frequency of
type 2 templates within the same protocell is $1-x$.
The population   is
described by  the fraction of protocells $\phi \left ( x, t \right ) \Delta x$ whose frequency of type 1 templates lies in the range $\left ( x , x + \Delta x \right )$  at time $t$. Our goal is to determine  how the probability density $\phi \left ( x, t \right )$ is affected
by the three  evolutionary processes: individual template competition within a protocell, migration of templates between protocells
and competition between protocells.

The template competition process  within each  protocell takes place according to the rules of the standard
Wright-Fisher model of population genetics \citep{Crow_70}.  In particular, assuming that a protocell contains
$j$ type 1 templates and  $N-j$ type 2 templates, the probability that  there will be exactly $i$ type 1 templates after
template competition  is given by the  Wright-Fisher process
\begin{equation}\label{rij}
r_{ij} = \left( {\begin{array}{*{20}c} N \\ i \\ \end{array}} \right) w_j^i \left ( 1 - w_j \right )^{N-i},
\end{equation}
where $w_j = j \left ( 1 - s \right )/\left ( N - j s  \right )$ is the relative fitness of the subpopulation of
type 1 templates in the protocell.
To determine how this process affects the probability density $\phi \left ( x, t \right )$ we resort to the   diffusion
approximation of population genetics \citep{Crow_70}, which consists essentially on the calculation of
the jump moments $\left \langle \left ( x' - x \right ) \right \rangle_r$ and $\left \langle \left ( x' - x \right )^2 \right \rangle_r$
where $x = j/N$ and $x'= i/N$  are the frequencies of type 1 templates before and after  template competition, respectively. Here
$\langle \ldots \rangle_r$ stands for an average using the transition probability $r_{ij}$. These moments
contribute to the  drift  and the diffusion  terms of a forward Kolmogorov-like equation for $\phi \left ( x, t \right )$. More pointedly,
direct evaluation of the jump moments to first order in  $1/N$ using the transition probability (\ref{rij}) yields
\begin{equation}\label{d1r}
\left \langle \left ( x' - x \right ) \right \rangle_r = w_j - x \approx - s x \left ( 1 - x \right )
\end{equation}
and
\begin{equation}\label{d2r}
\left \langle \left ( x' - x \right )^2 \right \rangle_r =  \frac{1}{N} w_j \left ( 1 - w_j \right ) + \left ( w_j - x \right )^2  \approx \frac{1}{N}  x \left ( 1 - x \right ),
\end{equation}
where  we have used that the fitness disadvantage $s$ of the type 1 templates  is on the order of $1/N$.

Migration follows Wright's island model \citep{Wright_51} that posits  that  $J$  templates of each protocell are replaced by migrants
in the time interval $\Delta t$  and that the frequency of type 1 templates among the migrants is  equal to the average frequency
of type 1 templates in the entire protocell population, i.e., $\bar{x} = \int_0^1 x \phi \left ( x, t \right ) dx$.  The probability
that a  protocell with $j$ type 1 templates ($x = j/N$) becomes a protocell with $i$ type 1 templates ($x' = i/N$) due to the migration process is then
\citep{Aoki_82}
\begin{equation}
m_{ij} = \sum_{k=k_l}^{k_u} \frac{ \left( {\begin{array}{c} j \\ k \\ \end{array}} \right) \left( {\begin{array}{c} N -j\\ J-k \\ \end{array}} \right)}
{\left( {\begin{array}{c} N \\ J \\ \end{array}} \right)}  \left( {\begin{array}{c} J \\ i -j + k\\ \end{array}} \right)
\bar{x}^{i-j+k} \left ( 1 - \bar{x} \right )^{J-i+j-k},
\end{equation}
where $k_l = \max \left (j-i,0,J-N+j\right)$ and $k_u = \min \left ( j,J-i+j,J \right )$.
Here the hyper-geometric component  yields the probability that exactly
$k$ type 1 templates and $J - k$ type 2 templates are eliminated from the protocell to make room for the $J$ migrants, whereas the binomial part yields the probability that
there are exactly $i-j+k$ type 1 templates among the $J$ migrants. After migration the number of type 1 templates in the protocell is given by the sum of
the type 1 templates originally in the protocell  $\left ( j-k \right )$  and the number of type 1 templates among the migrants $\left ( i-j+k \right )$.
The first two jump moments are given by
\begin{equation}\label{d1m}
\left \langle \left ( x' - x \right ) \right \rangle_m = m \left ( \bar{x} - x \right )
\end{equation}
and
\begin{equation}\label{d2m}
\left \langle \left ( x' - x \right )^2 \right \rangle_m=  \frac{m}{N} \bar{x} \left ( 1 - \bar{x} \right ) + m^2 \left ( \bar{x}- x \right )^2
+ \frac{m \left ( 1 - m \right )}{N-1}  x \left (1 - x \right ),
\end{equation}
where $\langle \ldots \rangle_m$ stands for an average using the transition probability $m_{ij}$ and
 $m = J/N$ is the fraction of the protocell population that is replaced by migrants. Assuming that $m$ is on the  order of
$1/N$, i.e., that the number of migrants $J$  remains finite and limited when $N$ grows large,  we can
neglect the second jump moment which is $O \left ( 1/N^2 \right )$.

Finally, the competition between protocells is taken into account as follows. Denoting  by $c \left ( x \right )$ the
selection coefficient of a protocell with a fraction $x$ of type 1 templates  we have
\begin{equation}
\phi \left ( x , t + \Delta t \right ) = \left [ \phi \left ( x , t \right ) + c \left ( x \right )  \phi \left ( x , t \right )  \Delta t \right ] \zeta,
\end{equation}
where $\zeta$ is such that $\int_0^1  \phi \left ( x , t + \Delta t \right ) dx  =1$, i.e,
$\zeta = 1/\left [ 1 +   \bar{c}\left( t \right )  \Delta t \right ]$ with
\begin{equation}\label{cbar}
\bar{c} \left ( t \right ) = \int_0^1 c \left ( x \right ) \phi \left ( x , t  \right )  dx .
\end{equation}
Taking the limit $\Delta t \to 0$ we obtain the change in the fraction of protocells due to intercell
selection, $\Delta \phi = \left [ c \left ( x \right ) - \bar{c} \left ( t \right ) \right ]  \phi \left ( x , t  \right ) \Delta t$.

Combining the changes in $\phi$  due to the three processes described above and introducing the rescaled variables $\tau = t/2N$, $S= 2Ns \geq 0$, $M=2Nm \geq 0$ and $C \left ( x \right ) =2N c \left ( x \right ) \geq 0$ we obtain \citep{Kimura_83}
\begin{equation}\label{eqphi}
\frac{\partial}{\partial \tau} \phi \left (x,\tau \right )  = \frac{\partial^2}{\partial x^2} \left[ x \left (1-x \right )\phi \left (x, \tau \right ) \right]
-\frac{\partial}{\partial x} \left[ b \left (x, \tau \right ) \, \phi \left (x,\tau \right ) \right]
+ \left  [ C \left ( x \right )-\bar{C} \left (\tau \right ) \right ] \phi  \left (x, \tau \right ),
\end{equation}
where
\begin{equation}\label{drift}
b \left (x,\tau \right ) = -S x \left (1-x \right ) - M \left [ x- \bar{x} \left (\tau \right ) \right ]
\end{equation}
is the drift term,
\begin{equation}
\bar{x} \left (\tau \right )=\int_0^1 x \, \phi \left (x,\tau \right ) \, dx
\end{equation}
is the mean number of type 1 templates in the
protocell population,
and
\begin{equation}
\bar{C} \left (\tau \right ) =  \int_0^1 C \left (x \right ) \, \phi \left (x,\tau \right )\, dx
\end{equation}
is the mean group selection pressure.  The constraint $\int_0^1 \phi \left (x,\tau \right ) \, dx = 1$
holds for all times $\tau$.

We note that whereas the linear forward Kolmogorov equation is the standard output in the case  of random drift and  individual selection
\citep{Crow_70}, eq.\ (\ref{eqphi}) is nonlinear because of the presence of $\bar{x}\left (\tau \right )$ and $\bar{C} \left (\tau \right )$, which are
associated to migration and group selection. In addition, the  singularities (if any) of the solution of eq.\ (\ref{eqphi})  must be integrable
so as to guarantee that it is normalizable for all times.

Kimura's choice for the intercell selection coefficient, $C(x) \propto x$, aimed at  exploring
the efficiency of group selection to maintain  an altruistic character -- the type 1 template  in that case --
which has a selective disadvantage $s$ relative  to  its competitor  but
whose presence would boost  the protocell reproduction rate, which increases linearly with the frequency of altruists
inside it.
We refer the reader to  \cite{Ogura_87}  for a rigorous analysis of
the  linear intercell selection model introduced by \cite{Kimura_83} and to \cite{FS_13} for the
analysis of the nonlinear variant of Kimura's model.
Here we consider the coexistence problem instead,
which is more burdensome to group selection than the altruistic version, since the fixation of a template
type  through the effect of random drift, regardless of its selective advantage or disadvantage,
acts against coexistence \citep{Fontanari_06}. According to  the so-called metabolic model of template cooperation
\citep{Bresch_80,Niesert_81,Szat_87,Czaran_00,Silvestre_08}, in order to favor coexistence we choose the intercell selection coefficient
\begin{equation}\label{presc}
C \left ( x \right ) = C x \left ( 1 - x \right )
\end{equation}
which is maximum for well-balanced protocells at which $x=1/2$. Here $C$ is a parameter on the order of $1$
that measures the intensity of the group selection pressure towards coexistence. The idea behind
eq.\  (\ref{presc}) is that
 the two functional template types coded for a small piece of a modular enzyme which then promoted  protocell replication  \citep{Manrubia_07}.
Since the hookup of the
replicase requires products from the two template types, its production rate is proportional to the concentration of the rare  type, hence
the requirement that $c \left ( x \right )$ is maximized by well-balanced protocells.

The model has three  parameters, namely, $S$ that measures the selective disadvantage of type 1 templates  in
the within cell competition process,
$M$ that measures the strength of migration, and $C$ that measures the strength of the group selection pressure towards
template coexistence. The scale of these parameters is given by the coefficient of the diffusion term which is set to $1$ in
eq.\  (\ref{eqphi}).

\section{The steady-state solutions}\label{sec:Steady}

The steady-state  protocell probability density  $ \phi = \phi \left ( x \right ) = \lim_{\tau  \to \infty} \phi \left (x,\tau \right )$ satisfies
\begin{equation}\label{stat1}
 \frac{d^2}{d x^2} \left[ x \left (1-x \right )\phi  \right]
+\frac{d}{d x} \left[ S x \left (1-x \right ) \, \phi  + M \left ( x- \bar{x}  \right ) \, \phi  \right]
+ \left  [C  x \left (1-x \right )-\bar{C}  \right ] \phi  = 0
\end{equation}
with $\bar{x} = \lim_{\tau  \to \infty} \bar{x} \left (\tau \right )$,  $\bar{C} = \lim_{\tau  \to \infty} \bar{C} \left (\tau \right )$,
and $\int_0^1  \phi  \left ( x \right ) dx = 1$.

For $M > 0$, eq.\ (\ref{stat1})  is satisfied both by $\phi = \delta \left ( x \right )$ and    $\phi = \delta \left ( x-1 \right )$, and it may also  be
satisfied by a regular function $\phi = \phi_r \left ( x \right )$.  By a regular solution of eq.\ (\ref{stat1}) we intend a non-vanishing continuous function
$\phi_r$ in the interval $\left [0,1 \right ]$ which is of class $C^2$ in
$\left (0,1 \right )$ where it satisfies (\ref{stat1}).
Since $\phi_r$ is a probability density we additionally restrict to
normalizable functions, i.e. functions such that
$\int_0^1 \phi_r \left (x \right ) dx$ is finite.
We note that in the absence of the coexistence pressure $C=0$, the regular solution is missing
\citep{Crow_70}. However, one can easily verify  that the migration term prohibits solutions which are combinations of
the three possibilities (i.e., the  deltas at $x=0$ and $x=1$ and the regular solution), since in that case
eq.\  (\ref{stat1}) would be violated in one of the two extremes, $x=0$ or $x=1$. Clearly, each possibility corresponds to  protocell  populations with
distinct characteristics. In particular,  $\phi = \delta \left ( x \right )$ describes a population composed of type 2 templates only,
$\phi = \delta \left ( x-1 \right )$ a population of type 1 templates only, and  $\phi = \phi_r \left ( x \right )$ describes the desired
situation   where the
different templates cohabit  a same protocell.

It is instructive to note that if a regular solution exists,  then integration of eq.\ (\ref{stat1}) over the interval $\left [ 0, 1 \right ]$
yields
\begin{equation}
\left. \frac{d}{dx} \left ( x \phi_r \right ) - M \bar{x} \phi_r \right |_{x=0} = 0
\end{equation}
and
\begin{equation}
\left. \frac{d}{dx} \left [ \left (1- x \right )\phi_r \right ] - M \left ( 1 - \bar{x} \right ) \phi_r \right |_{x=1} = 0
\end{equation}
which imply that for $x$ close to $0$ one has $\phi_r \sim x^{M \bar{x} -1}$, whereas for $x$ close to $1$ one has
$\phi_r \sim \left ( 1 - x \right )^{M \left ( 1-\bar{x}\right ) -1}$. Hence, in spite of the fact that $\phi_r$ describes a regime of
coexistence, this coexistence can be very unbalanced in the sense that the majority of the protocells may be populated by
essentially a single template type.  This unbalance is typical in the case $M<1$.

\subsection{Numerical analysis}

The steady-state solutions  of a diffusion model of intergroup selection for the maintenance of an altruistic trait were obtained numerically by
\cite{Kimura_83} in the simple case of a linear group selection pressure $c \left ( x \right ) \propto x$, i.e., $\bar{C} \propto \bar{x}$. In that case
eq.\ (\ref{stat1}) exhibits  only one non-local term
   and a straightforward self-consistent iterative approach yields the correct solution.
In our case such a direct approach is doomed to failure, as it will become clear below.

Following \cite{Kimura_83} we write the regular solution of  eq.\ (\ref{stat1}) in the form
$ \phi_r \left ( x \right )= \kappa  \phi_0 \left ( x \right )  \psi \left ( x \right ) $ where $\phi_0$ is the solution
 in  the absence of group selection ($C=0$) and for fixed $\bar{x} \neq 0,1$, namely,
\begin{equation}\label{phi0}
\phi_0 = \exp \left ( -Sx  \right ) x^{ M \bar{x} - 1} \left ( 1 - x \right ) ^{ M(1-\bar{x}) -1} ,
\end{equation}
and $\kappa$ is  the normalization constant. Hence the equation for $\psi$ reads
\begin{equation}\label{psi}
x \left  (1-x \right ) \frac{d^2\psi }{d x^2} -  \left  [ Sx \left(1-x \right ) + M \left(x-\bar{x} \right ) \right ] \frac{d \psi }{d x} +
C x \left  (1-x \right ) \psi =  \bar{C}  \psi,
\end{equation}
which, as already pointed out, for fixed $\bar{x}$ can be viewed as an eigenvalue problem without boundary conditions that can be
solved by requiring the regularity of $\psi \left ( x \right )$ in $\left [ 0,1 \right ]$  only \citep{Chalub_09}.
In addition, according to the expected behavior of $\phi_r$ in the vicinities of $x=0$ and $x=1$ we can guarantee
that $\psi$ is bounded at these extreme values. Of course, the attempt to solve  eq.\ (\ref{psi}) numerically
for an arbitrary  value of $\bar{C}$ using, say the Runge-Kutta algorithm,   results in divergences at the extremes,
which ruins  any self-consistent iterative approach to solve this equation.

Next we define $\psi = \exp(y)$ and get the following nonlinear equation
\begin{equation}\label{eqy}
x \left  (1-x \right ) \left [ y'' + \left ( y' \right )^2 \right ] -  \left  [ Sx \left (1-x \right ) + M \left (x-\bar{x} \right ) \right ] y'
+ C x \left  (1-x \right )  =  \bar{C},
\end{equation}
where the primes indicate derivatives with respect to $x$.  This is really a first order equation
for $ z \equiv y'$,
\begin{equation}\label{z}
x \left  (1-x \right ) \left [ z' + z^2 \right ] -  \left  [ Sx \left (1-x \right ) + M \left (x-\bar{x} \right ) \right ] z
+  C x \left  (1-x \right )  =  \bar{C},
\end{equation}
with
\begin{equation}\label{z0}
z (0) = \frac{\bar{C}}{M \bar{x}}
\end{equation}
and
\begin{equation}\label{z1}
z (1) = - \frac{\bar{C}}{M \left ( 1 - \bar{x} \right )} .
\end{equation}
At this stage the problem is ready for a numerical approach. For fixed $\bar{x}$ and $\bar{C}$ we solve
eq.\ (\ref{z}) by propagating the Runge-Kutta algorithm from $x=0$ to $x=1$ using the initial condition
(\ref{z0}).
Of course, the choice of an arbitrary value of $\bar{C}$ will not satisfy the boundary condition (\ref{z1}) so we
adjust $\bar{C}$ in order that condition is satisfied. This is essentially an application of the well-known shooting method to solve boundary values
problems \citep{Press_92}. Once this is achieved, we have solved the problem  for a fixed $\bar{x}$.
Explicitly, $y$ is obtained from
\begin{equation}
y = \int_0^x  z (\xi ) d\xi ,
\end{equation}
where we have defined $y \left ( 0 \right ) = 0$ (hence $\psi \left ( 0 \right ) = 1$). This choice is inconsequential since the physical quantities
are given by ratios of integrals involving $\psi = e^y$. In fact, we can then calculate $\bar{x}$,
\begin{equation}
\bar{x} = \frac{\int_0^1 x \phi_0 (x) e^y dx}{\int_0^1  \phi_0 (x) e^y dx} ,
\end{equation}
return to eq. (\ref{z}) and repeat the process until  we reach the convergence for $\bar{x}$. In particular, we
assume that convergence occurs whenever the change in $\bar{x}$ is less than $10^{-6}$ in two consecutive iteration steps. This  iterative
scheme is extremely efficient since it involves the numerical solution of a single first-order ordinary differential
equation  and the iteration over a single quantity only, namely $\bar{x}$.

\begin{figure}
\includegraphics{bmb_1}
\caption{Eigenvalue  $\bar{C}$  of the eigenvalue problem (\ref{psi}) as function of  the coexistence pressure  $C$ for
the degenerate case $S=0$ and migration parameter $M$  as indicated in the figure.  A phase transition takes place at $C_c = \pi^2$ in the case $M=0$. }
\label{fig:1}
\end{figure}
\begin{figure}
\includegraphics{bmb_2}
\caption{Regular normalized steady-state solution  $\phi_r$ giving the  proportion of protocells  that contain a fraction $x$ of type 1 templates
for the degenerate case  $S=0$, coexistence pressure $C=5$ and
migration rates $M=3, 2, 1$ as indicated in the figure.}
\label{fig:2}
\end{figure}

In Fig.\ \ref{fig:1} we show the dependence of the eigenvalue $\bar{C}$ on the coexistence pressure parameter $C$ in the
degenerate case   $S=0$ and for a variety
of values of the migration parameter. In this case, the symmetry of eqs.\  (\ref{stat1}) and (\ref{psi}) with respect to the interchange of $x$
and $1-x$  yields $\bar{x} = 1/2$ regardless of the values of $M$ and $C$.  This is illustrated in Fig.\ \ref{fig:2} where the regular solution
$\phi_r$ is shown for representative values of the migration parameter.
Interestingly, the phase transition between the coexistence
($\bar{C} > 0$)
and the  segregation ($\bar{C} = 0$) phases that takes place at $C= \pi^2$ for $M=0$  and $S=0$ \citep{Fontanari_13} disappears altogether
when the process of migration is included in  the model. The segregation phase, which is characterized by
a well-balanced mixture of protocells composed of either type 1 or type 2 templates, is eliminated in this case. Hence, in the degenerate case
where there is no  selective advantage at the template level  ($S=0$), migration promotes coexistence (see Fig.\ \ref{fig:2}).

\begin{figure}
\includegraphics{bmb_3}
\caption{Eigenvalue  $\bar{C}$  of the eigenvalue problem (\ref{psi}) as function of  the coexistence pressure $C$ for a non-degenerate
template competition scenario  with $S=1$ and  values
of the migration parameter $M$ as indicated in the figure. The transition point jumps from   $C_c = \pi^2 + 1/4$ for  $M=0$ to
$C_c \approx 2$ for $M \to 0$.  }
\label{fig:3}
\end{figure}

\begin{figure}
\includegraphics{bmb_4}
\caption{Mean frequency of type 1 templates $\bar{x}$  as function of  the coexistence pressure  $C$ for $S=1$ and
$M$ as indicated in the figure. For $M=0$, the segregating phase for $C < \pi^2 + S^2/4 \approx 10.12$ is non-ergodic
and the  result exhibited was obtained with the initial probability density $\phi \left ( x,0 \right ) = \delta \left ( x -1/2 \right )$ (see Appendix A). }
\label{fig:4}
\end{figure}

The scenario becomes more interesting when  the replication rates of the template types
 are allowed to differ, as illustrated in Figs.\ \ref{fig:3} and \ref{fig:4} for $S=1$. The first
noteworthy result exhibited in these figures is the appearance of a phase transition separating the homogeneous regime dominated by the
more efficient template type and  characterized by $\bar{x}=\bar{C} =0$, from the coexistence regime, $\bar{C} > 0$.
We note that  $\bar{x} > 0$ does not imply coexistence,
since this condition  holds true in the segregating phase that exists for $M=0$ and is characterized by an unbalanced mixture of
delta functions at the extremes $x=0$ and $x=1$.  Hence the eigenvalue $\bar{C}$ is the order parameter of our group selection
diffusion model.
Fig.\ \ref{fig:4}   offers a better view of the  transition and highlights the
singular nature of the segregation phase for $M=0$.
Overall the effect of migration for $S>0$ is to hamper coexistence, as indicated by the need of a larger coexistence
pressure to establish the coexistence regime as $M$ increases. However, the transition from $M=0$  to an arbitrarily small
migration  value $M \to 0$ results  in a discontinuous jump  on the value of the minimal
coexistence
pressure needed to stabilize the coexistence phase (e.g., from $C \approx 10.12$ to $C \approx 2$ for $S=1$).
As pointed out in Sec. \ref{sec:Steady}, this is so because the $M=0$ non-ergodic
segregating phase, characterized by the combination of delta functions $\phi = A _0 \delta \left ( x \right ) + A_1 \delta \left ( x - 1 \right )$,
with $A_0 + A_1 = 1$, is unstable to the effect of migration $M > 0$.   In this phase, $\bar{x} = A_1$ depends on the initial probability density
(see Appendix A) and for $\phi \left ( x, 0 \right ) = \delta \left ( x - 1/2 \right )$ we find $A_1 = 1/\left [ 1 + \exp \left ( S/2 \right ) \right ]$ which is
depicted in Fig.\ \ref{fig:4}.
In the ergodic  phase (i.e., $C \geq \pi^2 + S^2/4$), however, the value  of $\bar{x}$ at $M=0$ is approached  smoothly in the limit
$M \to 0$. The same is true for the order parameter $\bar{C}$  (see Fig.\ \ref{fig:3}), except that in this case the behavior is
continuous for all values of $C$.

\subsection{The critical line}

The critical line separates the homogeneous from the coexistence regime. Since  at this line  $\bar{x}=\bar{C} = 0$, eq. (\ref{z})
reduces to
\begin{equation}\label{zc}
\left  (1-x \right ) \left [ z_c' + z_c^2 \right ] -  \left  [ S \left (1-x \right ) + M \right ] z_c
+ C  \left  (1-x \right )  =  0
\end{equation}
with
\begin{equation}\label{z_0}
z_c (0) \equiv z_0 = \lim_{\bar{C}, ~\bar{x} \to 0}  \frac{\bar{C}}{M \bar{x}}
\end{equation}
and
\begin{equation}\label{z_c1}
z_c (1) = 0 .
\end{equation}
For fixed values of  the model parameters $S$, $C$ and $M$,  eq.\ (\ref{zc})  can be solved numerically
by propagating the solution from $x=1$ to $x =0$ using  the Runge-Kutta algorithm. Thus,
given an arbitrary set of model parameters, eq.\ (\ref{zc})  has a unique solution under condition (\ref{z_c1}), which then
determines  $z_0$  univocally.  However,  since eq.\ (\ref{zc}) is valid  at the critical line  only we need another condition
to  constraint the values of the model parameters. Of course, this supplementary condition is  provided by  eq. (\ref{z_0}), which
reads
\begin{eqnarray}\label{critical}
z_0  & = & \frac{C}{M} \frac{\int_0^1 dx \exp \left ( - S x + y_c \right ) \left ( 1-x \right )^M}{\int_0^1 dx \exp \left ( - S x + y_c \right )
\left (1-x \right )^{M-1}} \nonumber \\
&  = & \frac{ C \int_0^1 dx \exp \left ( - S x + y_c \right ) \left ( 1-x \right )^M}{1 + \int_0^1 dx \exp \left ( - S x + y_c \right )
\left (1-x \right )^{M} \left ( -S + z_c  \right )},
\end{eqnarray}
where $y_c = \int_0^x z_c (\xi) d\xi$. The second line of this equation  is derived from the  first line by integration by parts and its
sole purpose is to emphasize the fact that $z_0$ is finite for $M \to 0$. The limits $\bar{x} \to 0$ and $\bar{C} \to 0$ were omitted
in eq. (\ref{critical}), so it is left implicit that this expression must be evaluated for values of  $S$, $C$ and $M$ at the
critical line. The critical line is then obtained by fixing $S$ and $M$ and adjusting  $C$ such that the value of  $z_c$
at the $x=0$ boundary of eq.\ (\ref{zc}) coincides with the value obtained using expression (\ref{critical}).
This procedure is illustrated in  Appendix B for  the limit $M \to 0$,  where we can obtain  the analytical solution of
eq.\ (\ref{zc}) as well as carry out explicitly the integrals in eq.\ (\ref{critical}).

The final outcome of the self-consistent iterative procedure described above is summarized in Fig.\ \ref{fig:5}. On the one hand, these
results support the conclusion that  for fixed $S>0$ increasing the migration rate $M$ hinders coexistence since
it is then  necessary to increase the coexistence pressure $C$  to guarantee the onset of the coexistence phase.
On the other hand,  a vanishingly small migration rate, represented by the curve $M \to 0$ in Fig.\ \ref{fig:5}, constitutes a
huge benefit to coexistence, as compared with the no-migration situation $M=0$ when the onset of the coexistence phase happens for
$C> C_c = \pi^2 + S^2/4$ only \citep{Fontanari_13}.
The  reason is that for $M=0$ both template types are present in the  population but reside in distinct protocells, and so
a vanishingly small migration rate allows their meeting in a same protocell.

\begin{figure}
\includegraphics{bmb_5}
\caption{Critical coexistence pressure $C_c$ as function of the selective advantage $S$ of type 2 templates.  For fixed $M$, the
lines separate the homogeneous regime ($C \leq C_c$) where
the population is dominated by type 2 templates  from the coexistence regime
($C > C_c$) where both template types cohabit the same  protocell.
 }
\label{fig:5}
\end{figure}

\subsection{Analytical approximation}

In the case $C$ and $S$ are small we can easily derive explicit expressions for the order parameter $\bar{C}$, as well as for
$\bar{x}$, and so obtain an analytical expression for the critical lines shown in Fig.\ \ref{fig:5}.  As pointed out, the assumption that
$C \ll 1 $ and $S \ll 1$ amounts to saying that these two selective pressures are small with respect to random drift and migration.
Considering the regular solution $\phi = \phi_r \left ( x \right )$ of eq.\ (\ref{stat1}),
we begin by multiplying that equation by $x$ and then integrating over the interval $\left [ 0, 1 \right ]$,
yielding
\begin{equation}\label{ap1}
-S \int_0^1  \phi_r \left ( x \right )  x \left ( 1 - x \right )  dx + C \int_0^1  \phi_r \left ( x \right )  x^2 \left ( 1 - x \right )  dx -
\bar{x} ~C \int_0^1  \phi_r \left ( x \right )   x \left ( 1 - x \right )  dx =0.
\end{equation}
Next, to obtain results  that are correct to first order in  $S$ and $C$, we need only to replace $\phi_r$ by its
expression for $S=C=0$ [see eq.\ (\ref{phi0})], namely, the Beta distribution
\begin{equation}\label{ap2}
\hat{\phi}_0 = \frac{x^{M\bar{x} -1} \left ( 1 - x \right )^{M\left ( 1 - \bar{x} \right ) -1}}{ B \left [ M \bar{x} , M \left ( 1 - \bar{x} \right )\right ]}
\end{equation}
where $B \left ( x,y \right )$ is the standard Beta function \citep{Abramowitz_72}. The final result is simply
\begin{equation}\label{ap3}
\bar{x} = \frac{1}{2} \left [ 1 -  \left ( M + 2 \right )  \frac{S}{C} \right ] ,
\end{equation}
from where we get $C_c = \left ( M + 2 \right )  S$ which agrees with the curves shown in Fig.\ \ref{fig:5} for small $S$.
For $S=0$ eq. (\ref{ap3}) yields $\bar{x} =1/2$ which is actually valid for all $C$ since
$\phi_r \left ( x \right ) = \phi_r \left (1 - x \right )$ in this case.

Finally, to first order in $S$ and $C$ the order parameter $\bar{C}$ is given by
\begin{equation}\label{ap4}
\bar{C} = C \int_0^1 \hat{\phi}_0 \left ( x \right ) x \left ( 1 - x \right ) dx  = \frac{C M}{M+1} \bar{x} \left ( 1 - \bar{x} \right )
= \frac{CM}{4\left ( M+1 \right )} \left [ 1 - \left ( M+2 \right )^2 \frac{S^2}{C^2} \right ] ,
\end{equation}
which fits  very well  the curves  of Fig.\ \ref{fig:1} in the small $C$ regime, but fails to describe
the results of Fig.\ \ref{fig:3}  for $S=1$ since in that case the condition of small $S$  is not
satisfied.

\subsection{Discussion}\label{sec:Disc}

Here we address two issues that  were somewhat glossed over in the previous sections. The first issue
is the difference between the limit $M \to 0$ and the case $M=0$. From the physical perspective, that
difference is clear: in the absence of migration ($M=0$) there appears a segregation phase for $C < \pi^2 + S^2/4$ which is
unstable to the effect of a vanishingly small migration rate ($M \to 0$). However, from the mathematical
perspective that difference is blurred by the fact that   the limit $M \to 0$ is obtained simply by setting $M=0$ in
our equations. The key point here is that  by writing the regular solution of  eq.\ (\ref{stat1}) in the form
$ \phi_r \left ( x \right ) \propto \phi_0 \left ( x \right )  \psi \left ( x \right ) $ with  $\phi_0$  and $\psi$
given by eqs.\ (\ref{phi0}) and (\ref{psi}), respectively, we constrained the subsequent analysis to the region
$M >0$ only, since  in that form $\phi_r$ is not normalizable for $M=0$.
We note that for $M=0$ the regular solution of eq.\ (\ref{stat1}), which exists for  $C \geq \pi^2 + S^2/4$,
is  finite at the extremes $x=0$ and $x=1$ and so it  is always  normalizable, as expected \citep{Fontanari_13}. Thus setting $M=0$  in
eqs.\ (\ref{zc}) and (\ref{critical}) actually  means taking the limit $M \to 0$ of eq.\  (\ref{stat1}).

The second issue concerns the uniqueness of the eigenvalue $\bar{C}$ of  the second-order differential equation  for
$\psi$, eq.\ (\ref{psi}). In fact,  if  there were no constraints on $\psi$ then there would be an infinity of admissible values
for  the eigenvalue $\bar{C}$ as well as for the eigenfunctions $\psi$.  It is the condition that $\psi$ be positive and normalizable
that reduces the acceptable solutions to a single one.  We note that by writing $\psi = e^y$ and solving numerically
for  $y$ (see  eq.\ (\ref{eqy})) we have automatically restricted the numerical analysis to the valid regime $ \psi > 0$ only.

\section{Conclusion}\label{sec:Conc}

Contrary to the acrimony that has accompanied  the group selection accounts of  altruism
 and eusociality since  the 1960s \citep{Vero_62,Williams_66,Nowak_10,Rousset_11},
 group selection ideas have been mainstream in the prebiotic evolution  context \citep{Michod_83,Alves_01} since
 there is  a consensus that the compartmentalization of  templates was an essential stage in the
process of molecular evolution \citep{Bresch_80,Eigen_80}. In addition,  compartmentalization offers a solution to
 the problem
of the coexistence between different templates \citep{Niesert_81,Silvestre_08}, which is the topic we address in this paper.
We should mention, however, that within the  context of the maintenance of cooperation the group selection or, more generally,
the multilevel selection approach
has been applied to the study of the dynamics of cancer, which may be viewed as a result of the breakdown of cooperation
between cells in the body \citep{Michor_04,Bellomo_08,Bellouquid_13}.

In this contribution we build on the  seminal paper by \citet{Kimura_83}, which  presented a diffusion model incorporating group
selection, and study a group selection pressure towards
the coexistence of two types of templates  that are differentiated by their replication rates. Our focus is on
the effect of template swapping (migration)  among protocells. This is
a key process within the modern prebiotic scenario, which is  based on the radical notion of an ancestral community
of cell lines lacking long-term genetic history and individuality, rather than of a single ancestral organism \citep{Woese_98}.

We find that the  progression of the template type that exhibits the selective advantage  at the individual level is greatly  promoted by
migration, in the same manner that an antibiotic resistant gene spreads among a population comprising different bacterial species.
In that sense, migration hinders coexistence. Nevertheless, migration is very effective to counterweight the homogenizing effect
of random drift (i.e., the fixation of a template type) so that in the degenerate case, where there is no selective advantage at the individual level, coexistence  is
the only possible outcome of the evolutionary  process. In addition,  even in the non-degenerate case, a small amount of template swapping
increases greatly the parameter range for which coexistence is stable in comparison with the case where there is no migration at all.

An interesting aspect of the diffusion model of  group selection  is the existence  of a continuous transition  between a homogeneous regime dominated by the
more efficient template type and a coexistence regime where the two template types cohabit a same protocell. The order
parameter that characterizes these regimes is the eigenvalue $\bar{C}$ of the eigenvalue problem (\ref{stat1}), whose eigenfunction is
the fraction of protocells with a given template composition at the the steady state.
In particular, we find $\bar{C}>0$  in the coexistence regime,  and $\bar{C}=0$  in the homogeneous regime
with $\bar{C}$ vanishing linearly with the distance to  the critical line that separates those regimes.

A simplifying feature of the model with migration is that the evolutionary dynamics is ergodic, i.e., the steady-state solution does not
depend on the details of the initial distribution of templates among the protocells, provided the two template types are present in
the population at the initial time. In fact, in the homogeneous phase  the more efficient template type fixates in all protocells with probability one,
whereas in the coexistence phase the distribution of template compositions inside the protocells are described univocally by the regular
solution  of eq.\ (\ref{stat1}).  The dynamics is non-ergodic  only in the  segregating phase  that appears for low
coexistence pressure values  in the case migration is not allowed \citep{Fontanari_13}. For that case, we derive in  Appendix A
exact analytical expressions for the probability that one of the two template types fixates in a given protocell. Most interestingly,
this kind of local fixation  occurs both in the ergodic and in the non-ergodic phases of the model in the absence of migration
and so this model  offers a rare instance of subdivided population where the (local) fixation probabilities can  be calculated  exactly
\citep{Slatkin_81,Blythe_07}.

To conclude, a word is in order about the stability of  the steady-state solutions of the non-linear (and non-local) partial differential equation
that determines the time evolution of the protocell population, eq.\ (\ref{eqphi}). On physical grounds one expects the existence of a coexistence regime for
large values of the coexistence group selection pressure  $C$ and so the stability of the steady-state regular solution
$\phi = \phi_r \left ( x   \right)$, which
satisfies eq.\ (\ref{stat1}). In addition, in the absence of the coexistence pressure ($C=0$) the only steady-state solution is the homogeneous
one, i.e., $\phi = \delta \left ( x  - 1 \right) $. Whereas the regular solution exists  for $C > C_c \approx \left ( M + 2 \right ) S$ only, the homogeneous solution exists
for all $C \geq 0$ and so a possible instability of the regular solution at a finite value of  $C > C_c$ would shift the transition point as well as turn the
transition from continuous to discontinuous, in the sense that  the eigenvalue $\bar{C}$ would jump to zero at the new hypothetical transition point.
The analysis of the stability of the steady-state solutions by  techniques
such as the spectral theory in infinite dimensions \citep{Engel_00} is a most interesting and challenging enterprise that could reveal the influence of the
parameters $S$, $C$ and $M$  on the   relaxation time to  equilibrium as well as confirm the steady-state prediction of  the critical point
$C_c$
separating the homogeneous and coexistence regimes. We hope our paper will motivate further studies on this research line.

\section*{Appendix A: Local fixation probability for the $M=0$  non-ergodic segregation regime}

As shown by \citet{Fontanari_13}, setting $M=0$ in eq.\ (\ref{stat1}) yields two possible steady-state solutions:
the solution corresponding to the ergodic coexistence phase, which is a combination of two Delta functions and a regular
function,
$\phi   \left ( x \right ) = A_0 \delta \left ( x \right ) + A_1 \delta \left ( x - 1\right ) + B \phi_r \left ( x \right ) $, with $A_0 + A_1 + B = 1$,
and the solution corresponding to the non-ergodic segregation phase, which is a combination of the two Delta functions,
$\phi   \left ( x \right ) = A_0 \delta \left ( x \right ) + A_1 \delta \left ( x - 1\right )$, with $A_0 + A_1 = 1$.
The non-ergodic regime, which is our focus here, occurs for $C < \pi^2 + S^2/4$.
Note that  in both regimes $A_1$ may be interpreted as the probability that
the type 1 template fixates in a given protocell and  a similar interpretation holds for $A_0$ as well. However,  the result  $\bar{x} = A_1$,
which we used to draw the curve for $M=0$ in fig.\ \ref{fig:4}, holds in the
segregation regime only.
 In \citet{Fontanari_13} we have calculated the dependence of the weight $A_1$ on the
initial probability density $\phi \left ( x, 0 \right) $ for $S=0$ only, and  in this appendix we generalize that calculation for $S \geq 0$.

We begin by rewriting eq.\ (\ref{eqphi}) for $M=0$,
\begin{equation}\label{A1}
\frac{\partial}{\partial \tau} \phi \left (x,\tau \right ) = \frac{\partial^2}{\partial x^2} \left[ x \left (1-x \right )\phi \left (x, \tau \right ) \right]
+ S \frac{\partial}{\partial x} \left[ x \left (1 - x \right ) \, \phi \left (x,\tau \right ) \right]
+ \left  [ C x \left (1-x \right )-\bar{C} \left (\tau \right ) \right ] \phi  \left (x, \tau \right )
\end{equation}
and introducing the  abbreviation
 $\left \langle f  \left ( x \right ) \right \rangle_\tau = \int_0^1 f \left ( x \right ) \phi \left ( x , \tau \right ) dx$ for
the expected value of  a regular function  $f \left ( x \right )$ at time $\tau$. Hence
\begin{eqnarray}\label{s_f1}
\frac{d}{d\tau} \left \langle f  \left ( x \right ) \right \rangle_\tau  & = & \left \langle x \left ( 1- x \right ) \frac{\partial^2 f (x)}{\partial x^2} \right \rangle_\tau
-S \left \langle x \left ( 1- x \right ) \frac{\partial f (x)}{\partial x} \right \rangle_\tau
  \nonumber \\
&  &  + C \left \langle x \left ( 1- x \right ) f  \left ( x \right ) \right \rangle_\tau
- \bar{C} \left ( \tau \right ) \left \langle f  \left ( x \right ) \right \rangle_\tau
\end{eqnarray}
with  $\bar{C} \left ( \tau \right )  = C \left \langle x \left ( 1- x \right ) \right \rangle_\tau$.   The idea is to choose
a function $f(x)$ such that the first three  terms of the right hand side of eq.\ (\ref{s_f1})  cancel  out.  This choice
depends on the value of the parameter  $\Gamma  \equiv C -  S^2/4$ as discussed next. We note that $\Gamma < \pi^2$
in the non-ergodic regime.

\paragraph{Region $0 < \Gamma < \pi^2$.}

In this region we choose $f \left ( x \right ) = e^{S x/2} \sin \left ( \sqrt{\Gamma} x + \theta \right) $ where
$\theta$ is an arbitrary constant. Then eq.\ (\ref{s_f1}) rewrites
\begin{equation}
\frac{d}{d\tau} \left \langle  e^{S x/2} \sin \left ( \sqrt{\Gamma} x + \theta \right) \right \rangle_\tau
= - \bar{C} \left ( \tau \right ) \left \langle  e^{S x/2} \sin \left ( \sqrt{\Gamma} x + \theta \right) \right \rangle_\tau
\end{equation}
which has the formal solution
\begin{equation}
\frac{\left \langle  e^{S x/2} \sin \left ( \sqrt{\Gamma} x + \theta \right) \right \rangle_\tau}{\left \langle  e^{S x/2} \sin \left ( \sqrt{\Gamma} x + \theta \right) \right \rangle_0} = \exp \left [ - \int_0^\tau \bar{C} \left ( \eta \right ) d \eta \right ] .
\end{equation}
As the right hand side of this equation does not depend on $\theta$, neither does the ratio in its left hand side. Hence, equating
the ratios evaluated at $\theta = 0$ and $\theta = \pi/2 - \sqrt{\Gamma}/2$ yields
\begin{equation}
\frac{\left \langle  e^{S x/2} \sin \left ( \sqrt{\Gamma} x \right) \right \rangle_\tau}
{\left \langle  e^{S x/2} \cos \left [ \sqrt{\Gamma} \left ( x - 1/2 \right ) \right] \right \rangle_\tau}
= \frac{\left \langle  e^{S x/2} \sin \left ( \sqrt{\Gamma} x  \right) \right \rangle_0}
{\left \langle  e^{S x/2} \cos \left [ \sqrt{\Gamma} \left ( x - 1/2 \right ) \right] \right \rangle_0} .
\end{equation}
In the limit $\tau \to \infty$ we have
\begin{equation}
 \left \langle  e^{S x/2} \sin \left ( \sqrt{\Gamma} x \right) \right \rangle_\infty =
A_1 e^{S/2} \sin \left ( \sqrt{\Gamma}  \right)
\end{equation}
 and
\begin{eqnarray}
\left \langle  e^{S x/2} \cos \left [ \sqrt{\Gamma} \left ( x - 1/2 \right ) \right] \right \rangle_\infty & = &
\left ( A_0 + A_1 e^{S/2} \right ) \cos \left (\sqrt{\Gamma}/2 \right ) \nonumber \\
& = &
\left [ 1 + A_1\left ( e^{S/2} - 1 \right )  \right ] \cos \left (\sqrt{\Gamma}/2 \right )
\end{eqnarray}
which leads to
\begin{equation}
A_1 = \frac{1}{1 +  e^{S/2} \left ( \Xi_0 -1 \right)}
\end{equation}
where
\begin{equation}
\Xi_0 = 2 \sin \left (\sqrt{\Gamma}/2 \right ) \frac {\left \langle  e^{S x/2} \cos \left [ \sqrt{\Gamma} \left ( x - 1/2 \right ) \right] \right \rangle_0}
{\left \langle  e^{S x/2} \sin \left ( \sqrt{\Gamma} x  \right) \right \rangle_0} .
\end{equation}
In the limit  $\Gamma \to  \pi^2$, we have $\Xi_0 \to 2$ regardless of  the initial probability density $\phi \left ( x, 0 \right )$ and so
$A_1 \to A_1^c =  1/\left (1 +  e^{S/2}  \right )$. In addition, for the initial probability density $\phi \left ( x, 0 \right ) = \delta \left ( x - 1/2 \right )$ used to
calculate $\bar{x}$ at $M=0$ in Fig.\ \ref{fig:4}, the dependence on $\Gamma$ (and hence on $C$) disappears and so
$A_1 = A_1^c$.

\paragraph{Region $-S^2/4 < \Gamma < 0$.}

In this region the choice $f \left ( x \right ) = e^{S x/2} \left ( e^{ux}  + \theta e^{-ux} \right ) $ with
$u = \sqrt{-\Gamma}$ and $\theta$ arbitrary leads to the canceling
of the  first three  terms of the right hand side of eq.\ (\ref{s_f1}) yielding
\begin{equation}
\frac{\left \langle  e^{S x/2} \left ( e^{ux}  + \theta e^{-ux} \right )  \right \rangle_\tau}{\left \langle  e^{S x/2} \left ( e^{ux}  + \theta e^{-ux} \right ) \right \rangle_0} = \exp \left [ - \int_0^\tau \bar{C} \left ( \eta \right ) d \eta \right ] .
\end{equation}
The same argument used in the analysis of the  $\Gamma > 0$ region allows us to equate the ratio that appear in the left hand side of this
equation for $\theta = -1$ and $\theta =0$,
\begin{equation}
\frac{\left \langle  e^{S x/2} \sinh \left ( u x \right) \right \rangle_\tau}
{\left \langle  e^{S x/2 +ux}  \right \rangle_\tau}
= \frac{\left \langle  e^{S x/2} \sinh \left ( u x \right) \right \rangle_0}
{\left \langle  e^{S x/2 + ux} \right \rangle_0} .
\end{equation}
Finally, taking the limit $\tau \to \infty$ yields
\begin{equation}
A_1 = \frac{1}{1 +  e^{S/2} \left [ \Omega_0\sinh \left ( u \right ) -e^u  \right]}
\end{equation}
where
\begin{equation}
\Omega_0 = \frac {\left \langle  e^{S x/2 + ux } \right \rangle_0}
{\left \langle  e^{S x/2} \sinh \left ( u x  \right) \right \rangle_0} .
\end{equation}
By taking the limit $u \to 0$ we can easily verify that  $A_1$ is continuous at the boundary of
the two regions.  In addition, in the limit $C \to 0$, i.e., $ u \to S/2$ we recover the classical  formula
for the fixation of an allele with selective disadvantage $S$ \citep{Crow_70},
\begin{equation}
A_1 = \frac{\left \langle e^{Sx} \right \rangle_0 - 1}{e^S - 1 }.
\end{equation}
Similarly to our finding in the analysis of the previous region,  the initial probability density $\phi \left ( x, 0 \right ) = \delta \left ( x - 1/2 \right )$
yields $A_1 = 1/\left (1 +  e^{S/2}  \right )$ regardless of the value of $C$, as shown in Fig.\ \ref{fig:4}.

\section*{Appendix B: Critical line for the limit $M \to 0$}

Setting  $M = 0$ in eq.\ (\ref{zc})  yields
\begin{equation}\label{zcq}
z'_c + z^2_c - S z_c +C=0
\end{equation}
with the condition
\begin{equation}\label{zc1}
z_c \left ( 1 \right )=0 .
\end{equation}
In the region  $C- S^2 /4 > 0$  its solution is
\begin{equation}\label{zcx}
z_c \left (x \right )= \frac{S}{2} -\gamma \tan \left ( \gamma x + \theta \right )
\end{equation}
where $\gamma=  \sqrt{C- S^2 /4}$ and $\theta = \theta \left (\gamma, S \right )$
is fixed by condition (\ref{zc1}) as
\begin{equation}\label{zcf}
\frac{S}{2} -\gamma \tan \left ( \gamma + \theta \right )=0 .
\end{equation}
We note that the critical value   $C_c \left  (S \right )$  is in the region $C- S^2 /4 > 0$ (see Fig.\ \ref{fig:5}).
To evaluate eq.\ (\ref{critical}) we use the equality
\begin{equation}\label{eyc}
\exp \left ( -Sx + y_c \right ) =
\frac{\cos \left( \gamma x + \theta \right )}{\cos \left (\theta \right )}
 \; \exp \left ( -S x/2 \right ).
\end{equation}
which follows directly from the definition $y_c \left ( x \right ) = \int_0^x z_c \left ( \xi \right ) d \xi$ with
$z_c$ given by (\ref{zcx}). Now the integrals in eq.\ (\ref{critical}) can be readily evaluated yielding
\begin{equation}\label{tli}
\frac{S}{2} -\gamma \tan \left ( \theta \right )=
\frac{
\gamma \left [ e^{-\frac{S}{2}}\sin \left( \gamma+ \theta \right ) -\sin \left ( \theta \right ) \right ]
- \frac{S}{2} \left [ e^{-\frac{S}{2}}\cos \left (\gamma+ \theta \right )-\cos \left ( \theta \right) \right ]
}
{e^{-\frac{S}{2}}\cos \left (\gamma+ \theta \right )} .
\end{equation}
This equation can be further simplified using the equalities
 $\sin \left ( \gamma + \theta \right )= S/ \left ( 2\sqrt{C} \right )$
and $\cos \left (\gamma + \theta \right )=\gamma/\sqrt{C}$ that follow from eq.\ (\ref{zcf}).
The final result is simply
\begin{equation}\label{tl2}
\gamma = \sqrt{C} e^{\frac{S}{2}}\cos \left ( \theta \right ) .
\end{equation}
Finally, we rewrite eq.\  (\ref{zcf}) as
\begin{equation}\label{tl3}
\theta = \arctan\left( \frac{S}{2\gamma}\right) - \gamma
\end{equation}
in order to make clear that eq.\ (\ref{tl2}) yields a relation
$ C = C_c  \left ( S \right )$,
which is  the critical line $M \to 0$ depicted in Fig.\ \ref{fig:5}.

\begin{acknowledgements}
The research of J.F.F. was supported in part by Conselho Nacional de Desenvolvimento
Cient\'{\i}fico e Tecnol\'ogico (CNPq) and the research of M.S. was
partially supported by PRIN 2009 protocollo n.2009TA2595.02.
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


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text:
Q: Good practice to store Data in $_SESSION from inside a function Is it good  practice  to  store  Data in $_SESSION  from inside a function,I know  echoing  from  inside  a  function  is a  bad  practice.

A: A good practice is to create Handler that handles all the Session data, with methods to pull and push data to and from the session.

redpajama_set_name: RedPajamaStackExchange


text:
Biografia
Dai dati biografici pervenutici si sa che fu giullare presso diverse corti dell'Italia settentrionale: nel 1363 operò presso Cansignorio della Scala a Verona, nel 1371 a Padova, nel 1372 a Venezia e in seguito a Bologna.

Nel 1381 venne invitato dal consigliere di Cansignorio, nuovamente a Verona e in questa città rimase fino alla caduta della città sotto il dominio dei Visconti nell'anno 1387.

Rifugiatosi a Padova scrisse otto sonetti raccolti nell'opera La cantilena pro comite virtutum nei quali egli immagina che siano le città italiane ad invocare in prima persona la protezione di Gian Galeazzo Visconti.

Si trasferì in seguito a Milano dove le ultime notizie riguardanti la sua persona risalgono al 1389, anno probabile della sua morte.

Il canzoniere da lui composto, pubblicato a Bologna nel 1928 sotto il titolo di Le Rime di Francesco di Vannozzo, presenta stile e contenuti molto diversi tra loro.

Scrisse alcune lettere di argomento poetico ad alcuni rimatori dell'epoca, tra i quali Marsilio di Carrara, Gidino di Sommacampagna e Nicolò del Bene e fu autore di due sonetti inviati al Petrarca che lo lodò non tanto per la sua bravura di poeta ma per quella di musico.

La produzione poetica di Francesco di Vannozzo è importante come testimonianza della diffusione del toscano nell'Italia settentrionale. Essendo padovano figlio di genitori aretini, Francesco ha completa familiarità con il toscano, il padovano e il veneto. Nei suoi componimenti sono pertanto presenti forme in vernacolo toscano accanto a venetismi. Produce inoltre un sonetto caudato in dialetto veneziano, responsivo ad un altro sonetto anch'esso in padovano di Marsilio da Carrara, e due sonetti e una frottola in dialetto veneziano. La frottola, dal titolo Se Dio m'aide, scritta in occasione della celebrazione di nozze ("mariazo"), successivamente e variamente rielaborata in un componimento drammatico, porterà alle opere Cinquecentesche di Ruzante e di Andrea Calmo.

Opere
Le rime di Francesco di Vannozzo, a cura di Antonio Medin, Bologna: Commissione per i testi di lingua, 1928, p. XXIX, 325.

Note

Voci correlate
Storia della letteratura italiana

Altri progetti

Collegamenti esterni

Poeti in lingua veneta

redpajama_set_name: RedPajamaWikipedia


Just Children Child Care Centers incorporate the most current childcare methods into each classroom. State-of-the-art equipment and playgrounds are the standard at each location, and special programs are offered year-round to enrich your child's experience.  Through creative exploration, children experience situations that promote decision-making.  Parents have grown to appreciate the unique atmosphere at Just Children. Along with a bright, stimulating environment, children learn as they play.  Our professional staff carefully plans each child's daily activities, considering each child's uniqueness.


Today Juventus.com recalls the appointment of Giovanni Agnelli, one of the most important figures in Bianconeri history, who was officially confirmed club president 67 years ago on this day.
Affectionately known as 'L'Avvocato', he would go on to occupy the role of president for seven years, having taken over the reins from Pietro Dusio in 1947.
Giovanni was the second member of the Agnelli family to step into the presidential hotseat after his father Edoardo had been in charge from 1923 to 1935.
He left his post on 18 September 1954 after overseeing the Bianconeri's two Scudetto triumphs in 1950 and 1952 but continued to have the club in his heart right up until his death on 24 January 2003.
Juventus Football Club will forever be indebted to Giovanni Agnelli for his work.


text:
What Is a Tree? Live Podcast with In Defense of Plants
S1:E1 The start of the journey
S1:E2 Early steps: first experiences in science
S1:E3 Choosing a direction
S1:E4 Merging interests
S1:E5 Navigating multiple opportunities
S1:E6 North stars: the leaders in your journey
S1:E7 Final destinations
S1:E8 What adventure is next?
S1:E9 Meet the hosts
S2:E1 Collaboration
S2:E2 Innovation
S2:E3 Versatility
S2:E4: Vision
S2:E5 Communication
S2:E6 Risk-Assessing
S2:E7 Leadership
S2:E8 Passion
S2:E9 Adventure
Carmen, the Conservation Scientist
Chloe, the Research Assistant
Danny, the Arboriculture Scientist
Dina, the Conservation Advocate
Felix, the Research Leader
Leo, the Research Scientist
Monica, the Conservation Leader
Sasha, the Horticulturist
Pursue a graduate degree
Teacher classes and workshops
Schedule and location
Thursday, July 11, 2019, 7:00 to 9:00 p.m.
Arbor Room, Thornhill Education Center
$20.00 nonmember
(Fees include admission to the Arboretum)
ONLINE: REGISTER NOW
BY PHONE: 630-719-2468 ( Monday through Friday, 9:00 a.m. to 4:00 p.m.)
We see trees every day - but what are trees, really? Experts from The Morton Arboretum and Matt Candeias, host of the In Defense of Plants Podcast, will take you on an exploration of what it means to be a tree. This radio show style discussion will touch on why we have forests, why trees are under threat, what The Morton Arboretum is doing to ensure we have more trees in the future. Answer the questions about trees you never knew you had, and get inspired about what all of us can do to help trees.
Matt Candeias, host, In Defense of Plants Podcast
Host: Matt Candeias, host of the In Defense of Plants Podcast, Matt was raised in the forests of northeastern North America and is obsessed with botany. He earned a master's degree in community ecology from SUNY Buffalo State and is currently working on a PhD in community ecology with the University of Illinois. When he is not reading, writing, or talking about plants, he is usually hiking and photographing them or trying to grow them in and around his home.
Matt Lobdell is the Curator of Living Collections at The Morton Arboretum, where he has worked since 2014. He holds a B.A. in Anthropology from Kenyon College, a M.S. in Plant and Soil Science from the University of Delaware, and is a member of Magnolia Society International's Board of Directors. Matt specializes in the selection and evaluation of plant species for cultivation across the grounds of the Arboretum. He is working on building the Arboretum's Nationally Accredited Collections in Acer (maples), Magnolia, Malus (crabapples), Quercus (oaks), Tilia (lindens), and Ulmus (elms) through collection and other related activities. He is also interested in utilizing the Arboretum's collections as a "living gene bank", cultivating endangered or rare species through ex-situ conservation. For this purpose, he has completed projects targeting both Magnolia fraseri var. pyramidata (pyramid magnolia) and Quercus oglethorpensis (Oglethorpe oak).
Murphy Westwood is the Director of the Global Tree Conservation Program at The Morton Arboretum. The mission of the program is to save trees from extinction through global collaborations. Murphy works closely with colleagues from botanical gardens, universities, and other sectors in strategic locations like China, Europe, and Mexico, as well as throughout the United States, to develop conservation projects targeting key threatened tree species, especially oaks (genus Quercus). She also creates tools, guidelines, publications and other resources to build capacity and catalyze action for tree conservation. Additionally, Murphy is a Global Tree Conservation Officer for Botanic Gardens Conservation International and works to build support for the Global Trees Campaign, the only global conservation program dedicated to saving all of the world's threatened tree species. Murphy also manages ArbNet, the interactive, global community of arboreta and tree-focused professionals. She facilitates the ArbNet Arboretum Accreditation Program and develops resources and content for the ArbNet website. She holds a BS in Environmental Policy and Behavior from the University of Michigan-Ann Arbor, an MS in Advanced Methods in Taxonomy and Biodiversity from Imperial College London, and a PhD in Plant Developmental Biology, from the University of Cambridge.
Notes: Held entirely indoors. This program will be recorded and distributed as an In Defense of Plants Podcast. Limit 170.
Arboretum class weather, cancellation and refund policies
2020 Midwest Perennial Plant Association Symposium
On Saturday, February 1, horticulture professionals and garden enthusiasts can meet well-known authors, discover great plants, and get ideas and inspiration for the 2020 growing season.
On Saturday, January 18, bring seeds to share, browse a wide selection, and take home a variety to plant this spring. Included in Arboretum admission.
Explore a wide selection of courses designed to train volunteers who want to make a difference by caring for natural ecosystems.
Planted Podcast: Season 2
Season 2 of the Arboretum's podcast, "Planted: Finding Your Roots in STEM Careers," is now available! The podcast will introduce students to plant professionals with amazing careers. Download and subscribe today.
URL: http:// www.mortonarb.org/courses/what-tree-live-podcast-defense-plants Copyright 2020 The Morton Arboretum

redpajama_set_name: RedPajamaCommonCrawl


text:
Thymus serpylloides ist eine Pflanzenart aus der Gattung der Thymiane (Thymus) in der Familie der Lippenblütler.

Beschreibung
Thymus serpylloides ist ein kleiner Strauch mit schlanken, hochgebogenen bis niederliegenden, holzigen Stängeln, in deren Achseln Büschel aus Laubblättern und blütentragende, in Reihen stehende Stängel mit 1 bis 8 cm Höhe stehen. Die blütentragenden Stängel besitzen nur an der Basis ein Büschel aus kleinen Blättern, in den Achseln fehlen sie. Die Laubblätter sind 5 bis 8 mm lang und 0,5 bis 1 mm breit. Sie sind linealisch-lanzettlich, dicht drüsig gepunktet, flaumhaarig oder unbehaart. Der Blattrand ist zurückgerollt oder nahezu zurückgerollt und an der Basis bewimpert.

Die Blütenstände sind fast köpfchenförmig und bestehen aus locker stehenden, aus zwei bis sechs Blüten bestehenden Scheinwirteln. Die Tragblätter sind 1,5 bis 2 mm breit, grün, ähnlich den Laubblättern, aber nahezu flach. Der Kelch ist 3,5 bis 4,5 mm lang und behaart, die oberen Zähne sind so lang wie breit und 0,5 bis 0,7 mm breit. Die Krone ist 5 bis 6 mm lang.

Die Chromosomenzahl beträgt 2n = 58.

Vorkommen und Standorte
Die Art ist in den Bergen im  Süden Spaniens verbreitet. Sie wächst dort meist in Höhenlagen über 2000 m.

Systematik
Man kann zwei Unterarten unterscheiden:
 Thymus serpylloides subsp. gadorensis : Sie kommt im südlichen und im südöstlichen Spanien vor.
 Thymus serpylloides subsp. serpylloides: Sie kommt in der Sierra Nevada von Spanien vor.

Literatur
 Thomas Gaskell Tutin u. a. (Hrsg.): Flora Europaea. Volume 3: Diapensiaceae to Myoporaceae. Cambridge University Press, Cambridge 1972, ISBN 0-521-08489-X.

Einzelnachweise

Thymian
serpylloides


text:
Root Board
Root Board Game Mod Apk v1.28.0 (MOD, All Unlocked) 2022
MOD, All Unlocked
The boardgame is still developing and is expected to be one of the leading categories in the market for gaming. This means that it didn't provide the same level of impact as fighting games. Board games require more planning and a lot of intelligence to win. The players must be patient when playing this kind of game, as it gradually will take over the player's mind in a unique manner.
Root Board Game is a brand-new product developed by the manufacturer Dire Wolf Digital. It's a digital version of the game board of the identical name. If you're a huge lover of the game, you'll want to test this version.
Attractive graphics, original features, parallel development
The basic idea is that the original version was built using exceptionally high-quality images. Also, between the both versions, you will find some similarities but not many deviations. The game lets you an experience in the woods, where animals are abound. With 3D-generated graphics, everything seems real.
Additionally the color scheme makes players feel as if they're in a fairytale; it's a little nostalgia and basic. In terms of creating, the characters of the game create an impression of photographs in a different manner. They appear like medieval characters, wearing costume made up of weapons and armor made from primitive materials. They exhibited the same manner as human society in the past.
An original and captivating story that provides a fresh perspective on the world of the forest.
As previously mentioned The forest in Root Board Game acts like an ancient human society. In complete contrast to the adorable characters, the story is about the bloody conflict between various forces in order to control the forest. The game will be split into four different groups to battle and make use of each other. The Cat Empire is an imposing force that has an expansionist mentality. We have found a way to rule this forest. The Eyries were the first owners of this forest.
They exert a certain influence on the territories of physical power over other powerhouses. To regain their kingdom away from the cats empire the Eyries have to build for a stronger. Woodland Alliance is a group who wants to claim their own righteousness but isn't quite sufficient in strength. They sought to create conflicts between different factions within the forest. The last group was the knights who wandered Vagabond. They are neutral and militarily powerful, yet without a clear political direction. So long as the group has a solid financial backing They will fight with determination.
The cards are the device that allows you to make orders
When there's four people who are playing together, each player will be split into two separate areas to fight. Each faction has distinct advantages. For instance, if you select a cat, you'll receive the resources needed to manage your army, construct buildings and also recruit more troops to be ready for the battle. The empire of Eyries attempt to expand the most in order to take on the remaining shortage of power. Eyries are rewarded for successfully fulfilling tasks.
Woodland Alliance receives support from other powers. Woodland Alliance gathers sympathy from other powers. This means it can receive support cards each time it is its turn. Vagabonds fight for money therefore their earnings comes from trading, and also serving other powers. It is only possible to claim victory once you've reached the first 30 points of victory.
The players that play the game are certain to include a variety of characters that have special capabilities. They are clearly defined by the names of the cards which means you'll have to spend a significant amount of time figuring out how their actions impact the game. The game is believed to balance two factors. The first is to determine the best way to get maximum points as quickly as is possible. Furthermore, you must take out the enemy and they are deducted points.
The incompatibility between the two forces makes this game take a lengthy duration. In reality, every card has an ability that can alter the strategy in the play. When you combine these abilities together with advantages from the faction it can create a variety of unique scenarios.
Because of its distinctiveness, the game has tremendous replay value. So long as you be aware of the unique strategies for combination it is possible to still play the game. Additionally, over the course of a couple of months ago, the content creation team has added more elements. This is why you can play the game without feeling bored. Both version for offline as well as online are constantly updated at the same at the same time.
Download Root Board
You are now ready to download Root Board for free. Here are some notes:
Locus Map Pro
3.56.5 + 42.50MB
ProShot APK
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Latest, Unlocked
Photo Lab PRO
3.12.6 + n/a
MOD Patched
Shadow Knight Premium
1.19.12 + 153.90MB
No Skill CD, Immortality
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MOD Premium Skins Unlocked
Monopoly MOD APK
1.7.13 + n/a
(Unlimited Money/All Unlocked) 2022


Lee Stone, a longtime resident of Santa Barbara passed away on November 5th, 2016 at the age of 67.
Lee was born in Town Creek, Alabama. However, over the course of her life she lived in many states. In 1966, after graduating from high school in Ridgecrest California, she attended the College of Southern Idaho in Twin Falls. While a student there she earned an AA degree, was inducted into Phi Beta Kapa and received honors in English. She then attended Abilene Christian College in Abilene Texas, where in 1968 she earned her B.A. in English Literature. While at A.C.C. she was a member of their "Big Purple" marching band, and again received honors in English as well as membership in Sigma Tau Delta, a nationwide English honors society. Always committed to expanding her knowledge and possibilities for career advancement, Lee went on to obtain a Master's Degree in English from Idaho State University.
After college Lee returned to California where she worked at several jobs, most notably in sales and marketing for Cox Cable. Shortly after joining Cox in 1987, Lee was in an automobile accident that left her permanently wheelchair bound. This accident put an end to a highly athletic life of skiing, bodybuilding and jogging. Undaunted by her disabilities, she continued to work for Cox Cable for well over the next decade.
Her heroic adjustment to her injuries was emblematic of how Lee lived all facets of her life. She was a fervent defender of women's rights with a passionate focus on women's advancement in the workplace. She was active in the Santa Barbara Professional Women's club, where in 1977 she earned the designation of Young Career Woman of the Year. That same year she was honored on a state level as well. Lee was happily surprised to learn that she had been awarded 3rd place as the California Federation of Business Professional Women's, Young Career Woman of the Year.
Politically engaged as a fighter, she, in 1979, was elected to the board of directors of the Democratic Women of Santa Barbara. Lee was also a champion of animal rights and an outspoken supporter of Dawg and Peta. Lee found support for these convictions and deep spiritual comfort in the Unitarian Universalist Church of which she was a member and regular attendee.
As her health continued to decline her active participation in these organizations diminished as well. Her uncompromised view of the importance of fairness and compassion for all living creatures will remain an inspiration to all those who knew her.
Lee leaves behind several fine and loving friends; sisters Linda Hayhurst of Idaho, Margaret Hicks of Colorado, and brother Bob of California. She will be honored with a private celebration of life.


text:
Tag: CenterPlan
Center Plan Policies Released
The city has released the policy directions portion of Center plan.The remaining portions of the plan will be released on the 19th and 26th in Public presentations. The Heritage and Culture section begins on P23.
I have Excerpted the heritage policies below.
There is reference to the Standards and Guidelines for Historic Places in Canada. In it Standard 11 states:
Conserve the heritage value and character-defining elements when creating any new additions to an historic place or any related new construction. Make the new work physically and visually compatible with, subordinate to and distinguishable from the historic place.
You can find the Standards and Guidelines for Historic Places in Canada Here.
2.4 Culture & Heritage
The celebration and preservation of cultural heritage makes a vital contribution to good city building connecting citizens with a diverse and shared past. The wealth of historic resources and the range of traditional cultural activities that have evolved over time define the Centre's distinctive maritime character.
Cultural heritage includes tangible resources and intangible traditions, values and beliefs. The protection of these assets enhances the character of the Centre's urban districts and supports the valued diversity of its unique neighbourhoods. The preservation and celebration of the Centre's cultural heritage contributes to the HRM's goals for economic development and environmental sustainability and is integral to a planning approach that supports this value in all aspects of municipal planning in order to balance growth with the protection and enhancement of community character.
The Plan recognizes the important contribution its diverse and vibrant arts and cultural community makes to the vitality of the Regional Center and the character of the Municipality. Centre Plan policies will support a full range of cultural heritage preservation initiatives including protection of historic buildings and their adaptive re-use, support for community arts and cultural expressions and celebrations and the encouragement of the diversity of emerging artists that continue to shape the community's future legacy.
Celebrating and preserving historic resources contributes to good city building by connecting people with the past, enhancing unique aspects of the urban character, and adding interest to the streetscape. At the same time, culture and historical legacies transcend physical buildings, and are sometimes best captured through public art, exhibitions and festivals, stories, and the beliefs and lived experience of the Regional Centre's diverse residents. The Centre Plan recognizes the vitality that culture and heritage contribute to the Regional Centre, and includes policies that support historical preservation, sensitive adaptive re-use, cultural expression, as well as the community of artists and craftspeople that are helping shape the community's future legacy.
To provide innovative and holistic heritage and land use planning policies to ensure community identified heritage values are considered in decisions affecting land use.
To develop an evidence-based inventory of potential heritage
properties in the Regional Centre to inform land use and development decisions affecting cultural heritage.
To enable the definition of potential heritage conservation districts and cultural landscapes within the Regional Centre.
To establish a range of monetary and nonmonetary incentives and develop partnerships for the encouragement of public and private sector investments in cultural heritage conservation and arts and culture initiatives.
To encourage and support the on-going growth of a diverse range of publicly accessible arts and cultural venues, cultural events and seasonal celebrations.
Policy Directions
2.4.1 HERITAGE
The Regional Centre is a unique complex of military, commercial and residential precincts whose evolution spans over 250 years of settlement around Halifax Harbour, the historic Gateway to Canada. The citizens who built these districts brought their own distinct cultures and responded over time to a range of economic and social challenges that continue to be reflected today in the physical form and character of the Regional Centre's many diverse neighbourhoods. Heritage policies address the need to ensure growth and change is accommodated and is compatible with the protection and enhancement of built form and the enhancement of community character, vitality and sense of place.
Application of Regulatory and Policy Tools
a)Ensure the consistent application of available regulatory and policy tools in order to balance the need for new development, adaptive re-use, and growth with heritage conservation values and the protection of community character.
Viewplane Protection
b)Continue to protect existing viewplanes and corridors.
Potential Heritage Properties, Heritage Conservation Districts,
and Cultural Landscapes
c) Consider potential heritage properties, heritage conservation districts, and cultural landscapes recommended by the Heritage Advisory Committee for protection under the Nova Scotia Heritage Properties Act.
Cultural Heritage Conservation
d) The Culture and Heritage Priorities Plan should, in accordance with Council's direction at the time of its adoption, be the guiding document for the implementation of an evolving cultural heritage conservation program strategy for the Regional Centre.
Registration of Heritage Conservation Districts
e) Give priority consideration to registration of South Barrington (Old South District), Schmidtville and Historic Properties NHS as Heritage Conservation Districts under the Heritage Properties Act.
f)In recognition of the high concentration of neighbourhoods developed prior to 1918 and ongoing growth pressures within the Regional Centre, undertake a GIS block survey of existing buildings, by age and historic use, to identify and prioritize potential heritage conservation districts and cultural landscapes for their consideration by Regional Council. As a first priority the Municipality should conduct a survey by block of buildings in Downtown Dartmouth and in contiguous areas of Halifax North End as outlined in the Urban Structure.
Identification and Registration of Potential Heritage Conservation
Districts and Cultural Landscapes
g)As a priority the Municipality should conduct block surveys within in the Regional Centre and prepare studies in accordance with the Heritage Properties Act for the future registration of Heritage Conservation Districts and Cultural Landscapes within areas identified to the left.
Partnerships with Educational Institutions
h)Actively seek mutually beneficial partnerships with senior educational institutions to undertake GIS-based historic resource block surveys of growth areas within the Regional Centre to identify potential heritage conservation districts and cultural landscapes.
i)Consult with the diverse communities of the Regional Centre regarding the survey of potential heritage conservation districts and cultural landscapes to jointly identify the diversity of community values.
j) Develop a publicly accessible inventory of potential Heritage Conservation Districts and Cultural Landscapes and their component physical structures and elements.
Heritage Conservation Initiatives
k)Investigate heritage conservation supports and initiatives that include, but are not limited to, grants for restoration, rehabilitation and enhancements within Heritage Conservation Districts and Cultural Landscapes, and community heritage interpretation initiatives.
Establishment of Cash Contributions for Density Bonus Provision
l)Support, within the Centre Plan Area, a cash contribution option for the provision of a negotiated Density Bonus under Development Agreement for development within or abutting Heritage Conservation Districts, where such Heritage Conservation Districts allow.
Adaptive Reuse and Height of New Construction
m)Support the adaptive re-use of Heritage Properties in accordance with the Standards and Guidelines for the Conservation of Historic Places in Canada and investigate the creation of an alternate standard to supersede Standard 11, stating that the height of new construction on or abutting a Heritage Property, within or outside of a Heritage Conservation District, is an important factor affecting the protection of a Heritage Property 's cultural value as defined in its Statement of Significance.
Statements of Significance
n)In recognition of the predominantly low to mid-rise character of Heritage Properties and Heritage Conservation Districts in the Regional Centre, revise the requirements for Statements of Heritage Significance to include a detailed description of the significance of a building's scale and context.
Centre Plan
Centre Plan is Halifax's overarching planning strategy for the urban core. Its goal is to consolidate and update the four existing plans that cover this area. Planifax offers a good overview of the Plan in their video below.
The city has a Centre Plan website at http://www.halifax.ca/centreplan/ and also solicits feedback at http://shapeyourcityhalifax.ca/centre-plan/
AGBANS is primarily concerned with Area 4 – Culture and Heritage.
Right now, a heritage building outside a heritage conservation district has no protection. A building owner can apply for a demolition permit, and in the third year after applying, is allowed to demolish the building as-of-right.
One possible way to address is to introduceland controls under the HRM Charter as part of the Centre Plan project. This would amend Provencal legislation and would allow the city to have more control on demolitions.
If you have feedback on Heritage issues that should be addressed in the Centre Plan, please leave a comment below.
AGBANS will be meeting with the city in September.
Janes Walk 2018 is coming up the first weekend in May. Walks will again be taking place on May 5/6 this year. The full walk listing can be found at https://www.facebook.com/JanesWalkHalifax/events/
On 28 February, 2018, MIRCS filed a third-party application for municipal heritage registration of the Dennis Building. This application was supported with a Heritage Assessment Report on the building prepared by MIRCS.
Dylan was on the Sheldon MacLeod Show today talking about actual history being destroyed in Halifax. Give the interview a listen
The September 15 issue of The Coast will have comments on Interim Development Controls from Peter Ziobrowski.
We emailed out our first newsletter recently. You can read it Here.


SunPumps SDS-D series submersible pumps are one of the most highest quality, low votage DC pumps on the market. They were the first diaphragm submersible introduced to the solar industry in 1988 and have an excellent performance record.
The SDS series pumps are easy to install by hand without any special equipment. The pump includes a double shrink splice kit and a 1/2" brass discharge fitting.
SunPumps SDS series pumps can be installed below the water level in a pond, river or cistern, or installed by hand into a ground water well. They can be used to fill an open tank or in a pressurized water delivery system.
Simplicity is the key feature of the SDS series pumps. They are easy to install, require very little maintenance and are repairable.
and quiet. It is the ideal way to provide water for remote homes, campsites, livestock, small farms or for many other needs beyond the commercial power grid.
Fits in a 4" diameter well.
The SDS-D-228 pump now uses a HD santoprene diapgragm instead of the EPDM fabric reinforced diaphragm. It is much more chemical and abrasion resistant.


text:
Fantasy-FactionFantasy Book Reviews & Community
A Wind from the Wilderness
SPFBO #6 Finals Review
Fantasy-Themed Cookbooks
Multi-Book Review
Our Most Anticipated Books for 2021
Iain M. Banks Interview – Part One
I recently met Iain M. Banks for a long ol' chat that ranged from his latest Culture novel The Hydrogen Sonata and Gravitas ships to The Algebraist and his porno name. When I say a long ol' chat I mean a HUGE chat. We sat for almost an hour, and half way through I put my questions to the side and we just bounced the conversation around. I couldn't include everything we chatted and still it was so long that we've had to split it in to two pieces, but I can tell you we had a blast!
I cannot begin to explain just how much of an impressive chap Mr Banks is so I won't, I'll let him do that himself.
The obvious opening question is: what can we expect from your latest novel The Hydrogen Sonata?
Oh more Culture stuff. It's the usual mix of starships with ridiculous names and humans with names which aren't much more sensible either and a few sarcastic drones floating around and just lots of the standard Culture stuff. But this time it's sort of about Subliming which is this thing that I've mentioned in the past about what can happen to entire civilisations when they've, well, not when they've got to the end of their allotted span but when they've got bored with life. Then they can go in to the Sublime which is what's happening to this civilisation called the Gzilt in the book.
The Culture novels span huge periods of time, where in the chronology does The Hydrogen Sonata sit?
Well, it's not really that gigantic a chronological sweep compared to the millions and billions of years that some writers have set their series of books within. Really the whole thing takes place within about 2,000 years. Now in this one [Hydrogen Sonata] there's mention of stuff that happened 10000 years earlier but the vast majority of action in almost all the books with one exception, takes place in the space of about a 2000 year lifespan. The standard candle that I use to identify a spot in the books is when the Idiran War happened and in this case I think it's about 800 or 1100 years earlier, I can't remember. But it's certainly centuries before the action in Surface Detail the last Culture novel. You're told quite clearly then that the Idiran War took place 1500 years ago.
By this point the Contact has split up in to largely different specialisations one of which is Quietus which is to do with the dead and the virtual realities. And there's another one which deals with the Sublime or which tries to anyway, and the joke is – at least it's a joke that I found somewhat funny – where there's a ship in Hydrogen Sonata thinking, "I wish we had a specialist agency that dealt with this kind of stuff instead of us having to." Well, it's kind of the other ships thinking about this idea that leads to that all happening in later novels. So yeah it's significantly before Surface Detail but I think, I think it comes after all the others. I have got a document back at home that makes all this clear and I have to refer to that nowadays.
This one is for people who've never read your novels before, how best would you describe The Culture?
Well, I didn't come up with the phrase but it's probably the most apt description, it's a post-scarcity society or civilisation. Basically it's all humans but it's really run by the artificial intelligences, especially the ones called the Minds with a capital M that are mostly embodied in the ships, the spaceships. The Culture has been space faring for so long that when you say ship it always means spaceship and if I was to talk about something that floats on water I'd have to say seaship. That's how prevalent the whole idea of space faring is, you wouldn't automatically make the assumption. And they're scattered throughout the galaxy. They're not actually one of the biggest civilizations in the galaxy as there are lots and lots of other ones but they're one of the most interfering, always trying to do the right thing and help other civilisations. Not always successfully it must be said.
Would you live in the Culture if you could?
Good grief yes! I am always kind of stunned at people who ask, "but would you really though?" And I'm like, duh! It's paradise! It's a secular heaven. I think you'd have to be slightly mad not to want to live in the Culture. You can do anything you damn well please as long as you don't harm anyone else. You never have to worry about money or having to work. You live for at least 3-400 years and can travel to other planets. You've got drug glands embedded in your body and you can decide to be as stoned or whatever as you like with no side effects. And minute long orgasms! What's not to like?
So if you could have one of the genetic enhancements, from the Culture what would it be?
Just one? That's a very good question. I'm not sure I've ever said there is such a thing, well it's never come up in a plot, but if there is an intelligence enhancer then I'd definitely have that. Otherwise then yeah, the full suite of drug glands definitely! I'd probably never write a book again, that's the only problem.
Going back to those famously ostentatious names of yours, if you were a ship what would your name be?
Oh blimey. Well, technically ships have been known to take on the names of genuine people so if I was a ship I might be Iain Banks or Iain M. Banks but I don't know, I've kind of used up all the good ones already. I think I'd be one of the Gravitas series because there's a whole succession of them. The idea is that at some point some other civilisation rather sniffily said it's all very well having these comical names for your ships but they are somewhat lacking in gravitas so immediately one of the ship manufacturers started churning out ships with names like, Stood Far Back When The Gravitas Was Being Handed Out and Gravitas, Gravitas Can You Spell it For Me. Stuff like that. A Bit Low In The Gravitas Department At The Moment. I'd probably be one of the gravitas ships. Thoroughly Lacking In Gravitas.
I was always a fan of the warship names…
Attitude Adjustor!
Have you ever thought of contacting NASA and saying, 'Look guys, your names are boring, sort it out.'?
Haha, no, I think that would be too much of a 'Do you know who I am?' moment. You can't do that no but it would be lovely to think that someone at NASA was a fan and asked to use me as a consultant. The thing is I think their concern is getting it through congress so you can't start giving it a trendy and lefty and liberal socialist name like Smart Alec. They want them to be glorious and wonderful names like Endevour and Enterprise. So yeah I wouldn't hold your breath.
When it comes to writing a non-Culture science fiction novel does it come from a want to do something different or is it purely that an idea pops into your head and it isn't Culture?
It's a mixture of both. There's always a bias on the part of my publishers as they'd like to put "Culture novel" on the cover as they just sell more because they're a part of a series and like fantasy, SF is set up for trilogies and beyond.
For me there is two things, one is that I do feel the need not to be just a one trick pony when it comes to science fiction, I do want to be able to do other stuff as well as the Culture and then sometimes you do just come up with an idea that could only work in a non-Culture universe.
Certainly with the Algebraist that was the case. The whole thing with the choke points represented by the worm-hole network, you just couldn't have that in the Culture. The Culture universe is much freer, the ships can kind of go anywhere they want but in The Algebraist you can't do that, it makes it much more easy to control so you can have a big proper space empire or the equivalent thereof which you just can't have with the technical and the entirely physical law flouting that applies to the Culture universe.
In the Culture novels you approach 'what comes next' in interesting ways. As mentioned, there's Subliming and in Surface Detail you deal quite graphically with the idea of Hell and how horrific it is. Even living in the Culture you don't have to die if you don't want to and when you do you get to choose which way you go, like having your mind put in to an AI. So, straight in to a personal question, where do you the author sit with the idea of the afterlife?
I think it's nonsense. I think it's oblivion for us! No, I just think it's just a comforting myth that we've come up with. It's very, very highly unlikely and I really don't see it. I'm prepared to be proved wrong but yeah, there's just no proof, nothing. I'm utterly sceptical. You can never be absolutely, totally sure about anything but at the same time you have to get on with your own life and what you believe so yeah, I'm pure atheist and I think it's all bollocks basically.
One of my favourite villains of any book I've ever read is Luseferous and I have got to ask you, is there any chance we'll ever see him again in a sequel to The Algebraist?
Well up until a year ago I would have said absolutely not. And then about a year ago I was beginning the process of starting to think about the next Culture novel and there's always a lot of stuff flying around, lots of bits and pieces and ideas flying around. Usually I have to think of a specific plot to carry the whole thing so obviously I was in plot thinking up territory because my brain suddenly came up with two sequel ideas, neither of them to do with the Culture at all. One was an idea for a sequel to Against A Dark Background which is the first none-Culture one going way back and the second was for a sequel to the Algebraist. So, I mean really just initial ideas, nothing else.
The thing is with the Algebraist I had so much stuff that I never got around to using, there was a lot of extra detail for that universe that I created while I was thinking about it so it wouldn't be that difficult to go back. I suppose it's technically possible that Luseferous could show up again, he's kind of a bastard that way so you know, that is actually a possibility I guess yeah.
That book definitely has one of my favourite introductions to a villain.
It was particularly horrible wasn't it.
And the way he poisons people with his…
He really is an unpleasant character. If I did bring him back I'd have to kill him this time and have to do it properly like to [redacted] at the end of Surface Detail. A proper Hollywood death.
Check back tomorrow for the second half of this interview.
Check Out Our Related posts:
Iain M. Banks Interview – Part Two
The Player of Games by Iain M. Banks
Consider Phlebas by Iain M. Banks
Wesley Chu Interview
Tags: Iain M. Banks, interview, science fiction, The Algebraist, The Culture, The Hydrogen Sonata
Posted in Interviews
Paul Wiseall One day, Paul Wiseall intends on growing up and getting a real job as a superhero or a dinosaur but for the moment, he is quite happy with this writing malarkey as it is far too much fun. He does have a degree in History but please don't judge him too harshly as he really isn't that boring. Honest. For those who are interested, he is a film buff, a chronic comic collector and inhales anything written by Neil Gaiman, China Mieville and Terry Pratchett. Paul tends to live in his head more than anywhere else but his tangible self can often be found frequenting coffee shops or living behind a laptop somewhere in Italy.
Nyki Blatchley says:
Great. Iain's one of my favourite authors, with or without the M, and it's good to get his thoughts. I'm not quite up to date, and I'm currently reading Surface Detail (so I was pleased about the [redacted]. And I agree, Luseferous is a fantastic villain.
Geek Media Round-Up: October 30, 2012 – Grasping for the Wind says:
[…] Fantasy Faction interviews Iain M. Banks of Hydrogen Sonata (Pt […]
The Week In The World Of SFF | Fantasy Faction says:
[…] be cricket for an egomaniac like myself to pass up the opportunity to link to my recent two-part interview with the Banks himself where we talked everything from drug glands and Gravitas ships to his porno […]
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