# Dynamic redundancy as a mechanism to optimize collective random searches

Daniel Campos and Vicenç Méndez  Grup de Física Estadística, Departament de
Física. Facultat de Ciències), Universitat Autònoma de Barcelona, 08193
Bellaterra (Barcelona), Spain

###### Abstract

We explore the case of a group of random walkers looking for a target randomly
located in space, such that the number of walkers is not constant but new ones
can join the search, or those that are active can abandon it, with constant
rates $r_{b}$ and $r_{d}$, respectively. Exact analytical solutions are
provided both for the fastest-first-passage time and for the collective search
time required to reach the target, in the seminal case of Brownian walkers
with $r_{d}=0$. We prove that even for such a simplified situation there
exists an optimal rate $r_{b}$ at which walkers should join the search to
minimize the collective effort required to reach the target. We discuss how
these results open a new line to understand the optimal regulation of
cooperative random searches, e.g. for the case of biological foraging in
social species.

##  I Introduction

In the last years, there has been an increasing interest about the statistical
properties of the so-called redundancy principle and the role it may play in
biochemical and other biological processes [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].
This principle is based on the notion that ’many copies of a single object
such as molecules, cells, etc… is not a waste, but it has a specific function
in living systems’ [11]. This has been explored in particular as a mechanism
to explain how chemical activation rates at the cellular level can be
regulated or facilitated by the presence/creation of a very large (redundant)
number of activators. When this happens, the statistical properties of the
activation process will be governed by the fastest activator to reach the
receptor, so providing an attractive link to extreme value statistics [11, 12,
13, 14, 15, 16, 17].

There is actually a relatively rich literature, which extends over the last 40
years, about the problem of the fastest-first-passage time (FFPT) of a group
of $N$ random searchers/particles through a target located at a distance
$\mathbf{x}_{0}$ from their initial position (see, e.g., [18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]). For the case of
Brownian particles in 1d infinite domains, for example, it is known that the
mean fastest-first-passage time (MFFPT) becomes finite for $N\geq 3$ [24]
(remember that the mean first-passage time for a single-walker, i.e. $N=1$,
diverges), and it has a leading term $\langle
T\rangle=\mathbf{x}_{0}^{2}/(4D\log{N})$ for $N\rightarrow\infty$ [19, 23],
with $D$ the diffusion coefficient of the particles. For two and higher
dimensions, instead, the MFFPT remains infinite for any finite value of $N$
[18, 24]. Also, some approximations and series expansions have been proposed
for the corresponding FFPT distribution [23, 37, 38], but exact results are
extremely difficult to obtain.

The fact that redundant walkers/trajectories can modify the first-passage
statistics of the process, or they can even turn (at least in 1d) the MFFPT
finite, is also reminiscent of the case of random trajectories with stochastic
resetting and other stochastic processes with spatial memory [39, 40, 41, 42,
43]. There, the possibility to restart the trajectory anew gives the walker
the option to rectify strong departures from the target location. Stochastic
resetting has also been studied for the $N$-walkers case, but this has been
limited to simplified situations or approximated results (some examples are
those in [44, 41] and the references therein).

In this Letter we present a more general version of these problems that (as we
shall show) still allows an exact analytical treatment in order to explore how
the idea of redundancy applies to cooperative random searches. We introduce
the notion of dynamic redundancy, which corresponds to the case where the
number of searchers $N$ is not fixed but changes with time, so that each
searcher can join the foraging process and/or abandon it later. This is
inspired in the case of eusocial biological species, like honeybees or ants,
where the rate at which new individuals emerge out of the nest for foraging is
regulated internally by the colony (according to some communication rules and
other mechanisms). Surprisingly, this case of a non-constant number of
searchers has received relatively low attention in the literature of random
searches; a few exceptions are those works in which walkers are assumed to
become inactivated/vanished after some time (see, e.g, [30, 35, 17]), or they
can aggregate [45]. Instead, the case where new searchers can join the process
at $t>0$ apparently has not been considered to date. Therefore, in the
following we introduce formally the problem, and then study some seminal cases
to illustrate its interest.

Figure 1: Schematic representation of the dynamic redundancy problem, where
the number of walkers searching for a random target, $n(t)$, evolves in time
according to a birth-death dynamics between a quiescent (where the individual
stays quiet in the nest) and an active (random-walk) state.

Dynamic redundancy problem. We consider a group of walkers which are assumed
to be initially at the same position $\mathbf{x}_{0}$ (which represents their
nest) in a quiescent state, so they cannot move from that location. Then, we
define $t=0$ as the time at which one walker suddenly switches to the active
state, which means that it starts a random walk trajectory from
$\mathbf{x}_{0}$. From then on, the number of walkers in the active state,
$n(t)$, is assumed to follow a birth-death stochastic dynamics such that (i) a
new active individual can emerge from the nest at some random time $t>0$, and
(ii) any of the active walkers can, with a certain probability, become
quiescent again (so its position is instantaneously reset to
$\mathbf{x}_{0}$). For simplicity, we will assume that the transition between
the two states is Markovian, so the birth-death process is governed by
constant rates $r_{b}$ and $r_{d}$ (see Fig. 1).

Our specific aim in the present work is twofold. First, we want to understand
the properties of the FFPT (denoted by $T$) that it takes for any walker of
the group to reach a target located at $\mathbf{x}=0$, as a function of the
birth-death dynamics described above and the distance to the nest
$|\mathbf{x}_{0}|$. Second, we want to check whether there is an optimal shape
of $n(t)$ (that is, optimal rates $r_{b}$ and $r_{d}$) that can be used to
reach the target with a minimum effort. For the latter, we introduce the
concept of collective search time (CST) as

$T_{c}(t)=\int_{0}^{t}n(t^{\prime})dt^{\prime}.$ (1)

This can be interpreted as the sum of the individual search times spent by
every active walker up to time $t$. Then, $T_{c}(t=T)$ represents the CST up
to the FFPT. Formally, $T_{c}(T)$ represents a first-passage functional
subordinated to the birth-death process $n(t)$ [46, 47]. Note that $T$ itself
depends explicitly on $n(t)$, so this makes the computation of $T_{c}(T)$
nontrivial. Minimizing the CST requires reducing $T$ but without involving too
many walkers in the process, so a proper balance between keeping both $T$ and
$n(t)$ small is required; here we will focus on understanding the specific
statistical properties of that balance.

Connection to previous search problems. The dynamic redundancy problem here
presented can be reduced to some previously studied cases of interest.
Obviously, the case $r_{b}=r_{m}=0$ corresponds to the classical first-passage
problem for a single walker, while the case $r_{b}=0$, $r_{m}\neq 0$ would
correspond to the case of mortal random walkers [48, 49, 50, 28, 17, 51].
Also, the situation where $r_{b}\ll r_{d}$ can be mapped into the case of
stochastic resetting, with $r_{d}$ representing then the reset rate. In this
case, the birth-death process will take most of the time the value $n(t)=0$,
and from time to time a single active walker will emerge from the nest
($n(t)>1$ being extremely unlikely). Since the quiescent periods where
$n(t)=0$ do not contribute to the integral in (1), the statistics of the CST
$T_{c}(T)$ will then correspond to that of the first-passage time for the
stochastic resetting problem. This means, for instance, that its average value
will tend to $\langle
T_{c}\rangle=\left(e^{x_{0}\sqrt{r_{d}/D}}-1\right)/r_{d}$ for the case of 1d
Brownian walkers with a diffusion coefficient $D$ [39].

Exact solution for $r_{d}=0$. In the following we will focus in the case where
individual search trajectories are never terminated until the target is found
by the group, so $r_{d}=0$. Then, the stochastic process $n(t)$ becomes
strictly increasing. As we shall show, this particular case suffices to
investigate the aforementioned balance necessary for minimizing the CST on
average.

We will define $f(t)$ as the probability distribution for the FFPT, and the
corresponding survival probability of the target is denoted as
$S(t)=\int_{t}^{\infty}f(t^{\prime})dt^{\prime}$. For convenience, we will
separate all the realizations of the stochastic process that lead to the
target detection at time $t$ into those which correspond to $n(T)=1$,
$n(T)=2$, and so on. So that, we define the survival probability for the case
$n(T)=N$ as $S_{N}(t)$, so this corresponds to the probability that the target
has not been reached yet at time $t$, provided that there are $N$ active
walkers by that time. Note that the expression for $N=1$ will satisfy
$S_{1}(t)=S_{sw}(t)e^{-r_{b}t}$, where $S_{sw}(t)$ represents the survival
probability for the classical case of a single walker (it is, for
$r_{b}=r_{d}=0$), and the exponential factor $e^{-r_{b}t}$ is the probability
that no additional active walkers have emerged in the interval $(0,t)$.

Through these definitions, we can then write the recurrence relation (see
Supplementary Information for details)

$S_{N}(t)=r_{b}e^{-r_{b}t}\int_{0}^{t}S_{sw}(t)S_{N-1}(t-t_{1})dt_{1},$ (2)

where the integration variable $t_{1}$ represents the random time at which the
second walker becomes active, it is, the time at which $n(t)=1$ switches to
$n(t)=2$. So that, in (2) the term $S_{1}(t)=S_{sw}(t)e^{-r_{b}t}$ is being
multiplied by $r_{b}\int_{0}^{t}S_{N-1}(t-t_{1})dt_{1}$, where the latter
provides the survival probability for the remaining $N-1$ active walkers over
all possible values of $t_{1}$. The previous relation allows us to find
recurrently the general expression for $S_{N}(t)$. Using this procedure a
simplified expression is obtained in the form

$S_{N}(t)=e^{-r_{b}t}S_{sw}(t)\frac{\left[r_{b}g(t)\right]^{N-1}}{(N-1)!}$ (3)

where we have introduced
$g(t)\equiv\int_{0}^{t}S_{sw}(t^{\prime})dt^{\prime}$. Now, the general
expression for the survival probability in our dynamic redundancy problem is
easily derived as

$S(t)=\sum_{N=1}^{\infty}S_{N}(t)=S_{sw}(t)e^{-r_{b}(t-g(t))},$ (4)

and the corresponding first-passage distribution is simply the time derivative
(with a minus sign) of $S(t)$; in the Supplementary Information file we
provide the whole expressions for this as well as additional discussions.

For computing $T_{c}$ one can follow a similar procedure to that above. So, we
will define $\overline{T_{c}^{(N)}}(t)$ as the contribution to the mean
collective search time (MCST) that comes from all the realizations of the
search process for which $n(T)=N$. Hence, the overall MCST (computed over all
possible values of $N$ and all possible values of the FFPT) is simply

$\langle
T_{c}\rangle=\int_{0}^{\infty}dT\overline{T_{c}}(T)=\int_{0}^{\infty}dT\sum_{N=1}^{\infty}\overline{T_{c}^{(N)}}(T).$
(5)

It is possible to check that the expressions for $\overline{T_{c}^{(N)}}(T)$
also satisfy a recurrent relation which allows us to write them in terms of
$S^{(N-1)}(t)$ and $\overline{T_{c}^{(N-1)}}(t)$, though in this case the
interpretation of this recurrence is much less straightforward . Using that
recurrent relation, one can derive again the expression for the integrand in
the MCST (5) (all technical details and discussions are provided in the
Supplementary Information file):

$\displaystyle\overline{T_{c}}(T)$ $\displaystyle=$ $\displaystyle
e^{-r_{b}\left[T-g(T)\right]}\left\\{r_{b}S_{sw}(T)\left(T+g(T)-2TS_{sw}(T)\right)\right.$
(6) $\displaystyle+$
$\displaystyle\left.r_{b}^{2}h(T)S_{sw}(T)\left(1-S_{sw}(T)\right)\right.$
$\displaystyle+$
$\displaystyle\left.f_{sw}(T)\left(T+r_{b}h(T)\right)\right\\}$

with $h(t)\equiv\int_{0}^{t}t^{\prime}S_{sw}(t^{\prime})dt^{\prime}$, and
where we define $f_{sw}(t)=-dS_{sw}/dt$ as the first-passage time distribution
for the classical (single-walker) case.

Brownian walkers in 1d. While we have now some exact expressions for the
survival probability and for the collective search times, it is not trivial in
general to compute their average properties. Let us consider the seminal case
in which all active walkers behave as Brownian particles moving in a 1d domain
with diffusion coefficient $D$. For this, it is well-known that
$S_{sw}(t)=\mathrm{erf}\left(\frac{x_{0}}{2\sqrt{Dt}}\right)$, with
$\mathrm{erf}(\cdot)$ representing the error function [52]. However,
introducing this expression into (4) or (6) does not allow to reach an exact
expression neither for the mean-first passage time $\langle
T\rangle=\int_{0}^{\infty}S(t)dt$ nor for the MCST (5). So, we will take the
two timescales driving the search process, i.e. the diffusive scale
$x_{0}^{2}/4D$ and the birth scale $r_{b}^{-1}$, and will focus on the
behavior for the limit cases where the dimensionless parameter $\chi\equiv
x_{0}^{2}r_{b}/4D$ becomes either small or large.

Integrating by parts and using (4), the MFFPT $\langle T\rangle$ can be
written in the scaling form

$\displaystyle\left\langle T\right\rangle=\frac{x_{0}^{2}}{2D}\phi(\chi)$ (7)

where $\phi(\chi)$ is defined in the Supplementary Information (SI) file.
Using this scaling, the leading term of $\langle T\rangle$ can be properly
computed in the limits $\chi\ll 1$ and $\chi\gg 1$ (the corresponding
derivation is also provided in the SI file). This leads to

$\left\langle
T\right\rangle\simeq\left\\{\begin{array}[]{cc}\frac{x_{0}}{\sqrt{Dr_{b}}},&r_{b}\ll
4D/x_{0}^{2}\\\
\frac{x_{0}^{2}}{4D\ln\left(\sqrt{\frac{2}{243\pi}}\frac{x_{0}^{2}r_{b}}{D}\right)},&r_{b}\gg
4D/x_{0}^{2}.\end{array}\right.$ (8)

Similarly, the CST also admits a scaling form

$\displaystyle\langle T_{c}\rangle=\frac{x_{0}^{2}}{2D}\phi_{c}(\chi),$ (9)

where $\phi_{c}(\chi)$ is also defined in the SI file. Its leading order
behavior in the two limits $\chi\ll 1$ and $\chi\gg 1$ above reads

$\left\langle
T_{c}\right\rangle\sim\left\\{\begin{array}[]{cc}\frac{x_{0}}{\sqrt{Dr_{b}}},&r_{b}\ll
4D/x_{0}^{2}\\\ \frac{\chi}{\left(\ln{\chi}\right)^{2}},&r_{b}\gg
4D/x_{0}^{2}.\end{array}\right.$ (10)

Eqs. (8) and (10), together with the general expressions (4) and (6) from
which they are derived, represent the main results of our work. To visualize
these results, in Figure 2 we plot both the MFFPT and the MCST as a function
of the birth scale. There, the solid lines correspond to the exact values of
these two quantities obtained from numerical integration of our exact
expressions above, and the symbols correspond to random-walk simulations that
we have carried out (averaging over $10^{4}$-$10^{5}$ realizations of the
collective search process) to confirm the validity of our derivations. From
the plots obtained, we see how the asymptotic results indicated above for the
limits $\chi\gg 1$ and $\chi\ll 1$ (represented through dotted and dashed-
dotted lines, respectively) are recovered in the appropriate regimes.

Figure 2: MFFPT (in blue) and the MCST (in orange) as a function of the birth
timescale $r_{b}^{-1}$. Solid lines correspond to the exact values computed
from numerical integration, and symbols correspond to random-walk simulations.
Dotted and dotted-dashed lines represent the analytical asymptotic behavior
predicted in (8) and (10). All the results shown correspond to the case
$x_{0}=5$, $D=1$.

It is interesting to note that in the regime $\chi\ll 1$ one finds $\langle
T\rangle\approx\langle T_{c}\rangle$. This is because for this regime, in most
of the realizations the target will be reached by the first active walker
before the second one becomes active. There is only a relatively small
probability that the first active walker departs from the target as time goes
by and misses the target. For $r_{b}=0$ this would lead to the divergence of
both $\langle T\rangle$ and $\langle T_{c}\rangle$, while for $r_{b}$ small
the target is found in a finite time thanks to the new active walkers
eventually appearing. From the properties of a diffusive flux to an absorbing
boundary in a 1d semiinfinite domain (see, e.g, [53]), we know that the
fraction of the trajectories that lead the first searcher far away from the
target decreases as $\sim t^{-1/2}$, so this explains the scaling $\langle
T\rangle\sim 1/\sqrt{r_{b}}$ obtained in (8).

The opposite situation, $\chi\gg 1$, would correspond to the case where a very
large number of active walkers emerge in a time much smaller than the
diffusive scale $x_{0}^{2}/D$. Such situation is then reminiscent of the case
of a set of $N$ walkers searching from the target starting from the same
initial position. As mentioned above, it is known for this case that in the
limit $N\rightarrow\infty$ the leading order term of the MFFPT satisfies
$\langle T\rangle\sim(\log{N})^{-1}$. Since on average the number of active
walkers appearing will be proportional to $r_{b}$, this explains the scaling
$\langle T\rangle\sim(\log{r_{b}})^{-1}$ we observe. Also, for this regime
where the process is dominated by large values of $N$ we can approximate
$\langle T_{c}\rangle=\langle\frac{1}{2}NT\rangle\sim\langle N\rangle\langle
T\rangle$, where $\langle N\rangle$ is the mean number of active walkers we
have at $t=T$. Note that the factor $\frac{1}{2}$ appears from the fact that
any of these walkers will be active on average during a time $T/2$. Then, we
note that on average $\langle N\rangle=r_{b}\langle T\rangle$ is satisfied. So
that, we realize that $\langle T_{c}\rangle\approx\frac{1}{2}r_{b}\langle
T\rangle^{2}$, and combining this with the scaling above for $\langle
T\rangle$ (see Eq. 8) we finally obtain $\langle
T_{c}\rangle\sim\chi/\left(\ln{\chi}\right)^{2}$, which confirms intuitively
the formal result in Eq. (10).

Anyway, the most relevant feature in Figure 2 is the fact that while $\langle
T\rangle$ decays monotonically with $r_{b}$, there is instead an optimum rate
which minimizes $\langle T_{c}\rangle$, so providing a criteria to optimize
these collective searches. To understand better this optimization mechanism,
in Figure 3 we show again the behavior of $\langle T_{c}\rangle$ as a function
of $r_{b}^{-1}$, but now for different values of $x_{0}^{2}/D$. This allows us
to see how the dynamic redundancy problem can be optimized as a function of
the distance from the nest to the target. If we denote by $(r_{b}^{*},\langle
T_{c}\rangle^{*})$ the coordinates of the point at which the MCST reaches its
minimum, we observe that (i) $r_{b}^{*}$ increases linearly with the diffusive
timescale, such that $r_{b}^{*}\approx 7.536D/x_{0}^{2}$, and (ii) the minimum
value of the MCST scales as $\langle T_{c}\rangle^{*}\approx
1.489x_{0}^{2}/D$, so this means that $\langle
T_{c}\rangle^{*}\sim(r_{b}^{*})^{-1}$ (additional justification for these
relations is provided in the Supplementary Information file).

Figure 3: MCST as a function of $r_{b}^{-1}$ for different values of the
distance between nest and target $x_{0}$. Full symbols denote the minimum
search intensity for each case and the dotted line provides a linear fit
between them to confirm the scaling $\langle
T_{c}\rangle^{*}\sim(r_{b}^{*})^{-1}$. In all cases $D=1$ is used.

Brownian walkers in higher dimensions. The mechanism of dynamic redundancy
presented here allows the continuous emergence of new active trajectories from
$\mathbf{x}_{0}$, so the target will be found in a finite time even for the
case of non-recurrent random walk trajectories, so both $\langle T\rangle$ and
$\langle T_{c}\rangle$ will be finite for Brownian particles in arbitrary
dimensions. In Fig. 4 we show the results for $\langle T_{c}\rangle$ for the
cases $d=2$ and $d=3$ to check whether the properties discussed above for the
1d case remain valid. From there, we confirm that there is still an optimum in
$\langle T_{c}\rangle$ as a function of $r_{b}$. Since for this case we do not
have a simple exact expression for $S_{sw}(t)$, we cannot predict analytically
the particular behavior observed in the limits $r_{b}\gg 4D/x_{0}^{2}$ and
$r_{b}\ll 4D/x_{0}^{2}$. However, generalizing our intuitive arguments above
for 1d we observe that the former still satisfies $\langle T\rangle\sim
1/\ln{N}$ for the case of $N$ fixed, and so the scalings $\langle T\rangle\sim
1/\ln{r_{b}}$ and $\langle T_{c}\rangle\sim\chi/\left(\ln{N}\right)^{2}$
derived for the 1d case should persist. On the other side, for $r_{b}\ll
4D/x_{0}^{2}$ the same reasoning above should lead to $\langle
T\rangle\approx\langle T_{c}\rangle\sim r_{b}^{-d/2}$ (taking into account
again the properties of a diffusive flux to an absorbing trap in $d$
dimensions [53]).

From Fig. 4 we observe that the convergence to the asymptotic regimes $\chi\ll
1$ and $\chi\gg 1$ becomes extremely slow, so it is not possible at practice
to fully observe the agreements with the scalings above. Also, we observe that
the scaling between $\langle T_{c}\rangle^{\ast}$ and
$\left(r_{b}^{\ast}\right)^{-1}$ becomes now nontrivial. In the insets of Fig.
4 we provide a power-law fit $\langle
T_{c}\rangle^{\ast}\sim\left(r_{b}^{\ast}\right)^{-\beta}$ to the results
obtained, which predict exponents $\beta=1.20\pm 0.01$ (for 2d) and
$\beta=2.43\pm 0.02$ (for 3d) for the region of situations explored.

Figure 4: MFFPT (in blue) and the MCST (in orange) for two and three
dimensions (upper and lower panels, respectively) as a function of the birth
timescale $r_{b}^{-1}$. Symbols correspond to random-walk simulations, and the
solid lines in this case are only provided as a visual guide. Dotted and
dashed-dotted lines are the limit behavior expected for $\chi\ll 1$ and
$\chi\gg 1$, respectively. All results shown correspond to the case $x_{0}=2$,
$D=1$. (Insets): minimum of the MCSTe when computed for different values of
$x_{0}$, as in Fig. 3. The fitted scaling $\langle
T_{c}\rangle^{\ast}\sim\left(r_{b}^{\ast}\right)^{-\beta}$ is provided, with
$\beta=1.20\pm 0.01$ and $\beta=2.43\pm 0.02$ for 2d and 3d, respectively.

Discussion and implications of our work. Despite the interest raised by the
redundancy principle, on one side, and stochastic resetting, on the other, in
the literature on stochastic processes, the notion of dynamic redundancy
introduced here represents a novel line of study which has remained unexplored
to date. While the results above have been restricted to a very particular
case for the sake of clarity, they show the potential interest that the
problem has in terms of understanding and optimizing collective random
searches, and they pave the way to explore more general situations. For
instance, we have carried out some preliminary work with Brownian active
trajectories that can be terminated at a rate $r_{d}>0$, and we observe that
including the additional timescale $r_{d}^{-1}$ in the process makes that the
MCST can reach minimum values for highly nontrivial combinations of $r_{b}$,
$r_{d}$ (this will be explored in detail in a forthcoming publication). Also,
the extension to non-Markovian birth-death mechanisms, or to more general
movement processes (e.g. persistent random-walks, Lévy flights,…) and/or
different geometries, offers a vast range of possibilities to explore.

On the other side, it is obvious that the results obtained above provide
direct theoretical predictions for $\langle T\rangle$ and $\langle
T_{c}\rangle$ that could be experimentally tested in real scenarios, e.g. in
group animal foraging. For such cases, it is reasonable to ask whether the
recruitment of new foragers can significantly reduce the search time, and/or
increase the average energy intake for the group. Regulation of $n(t)$ then
seems to be a simple mechanism that groups could use in order to orchestrate
their searches and minimize their search efforts, and one could hypothesize
this should actually play a role in the biological evolution and adaptation of
such groups to their specific environmental conditions. These prospective
ideas provide just some cues about the interest that the concept of dynamic
redundancy could potentially have.

Acknowledgements DC acknowledges the organizers of the program in ’Mathematics
of movement: an interdisciplinary approach to mutual challenges in animal
ecology and cell biology’ at the Isaac Newton Institute in Cambridge for
fruitful discussions during his stay there. The Authors acknowledge the
financial support of the Spanish government under grant PID2021-122893NB-C22.

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## Dynamic redundancy as a mechanism to optimize collective random searches
SUPPLEMENTARY INFORMATION FILE

## Derivation of the survival probability $S(t)$

In the main text we have already introduced the notation $S_{N}(t)$ for the
survival probability of the target up to time $t$, conditioned to $n(t)=N$;
this is, we divide all realizations of the stochastic redundancy process into
those which have the same value of the birth-death process $n(t)$ at a
particular time.

According to this, the explicit expression of $S_{N}(t)$ by definition is

$\displaystyle S_{N}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}dt_{N-1}\int_{0}^{t_{N-1}}dt_{N-2}\ldots\int_{0}^{t_{2}}dt_{1}S_{1}(t)S_{1}(t-t_{1})\ldots
S_{1}(t-t_{N-1})=$ (11) $\displaystyle=$
$\displaystyle\int_{0}^{t}dt_{N-1}\int_{0}^{t_{N-1}}dt_{N-2}\ldots\int_{0}^{t_{2}}dt_{1}\left[S_{sw}(t)e^{-r_{b}t_{1}}\right]\left[S_{sw}(t-t_{1})r_{b}e^{-r_{b}(t_{2}-t_{1})}\right]\ldots\left[S_{sw}(t-t_{N-1})r_{b}e^{-r_{b}(t-t_{N-1})}\right],$

where we have used $S_{1}(t)=S_{sw}(t)e^{-r_{b}t}$ in the last equality. The
variables $t_{i}$ represent the random times at which new walkers become
active and so the process $n(t)$ increases by one unit. Then,
$r_{b}e^{-r_{b}(t_{i}-t_{i-1})}$ represents the probability that the birth
process requires a time $(t_{i}-t_{i-1})$ to increase from $n(t)=i-1$ to
$n(t)=i$.

Thanks to the Markovian nature of the birth-death process, the expression (11)
conveniently simplifies to

$S_{N}(t)=r_{b}^{N-1}e^{-r_{b}t}\int_{0}^{t}dt_{N-1}\int_{0}^{t_{N-1}}dt_{N-2}\ldots\int_{0}^{t_{2}}dt_{1}S_{sw}(t)S_{sw}(t-t_{1})\ldots
S_{sw}(t-t_{N-1}).$ (12)

On the other side, instead of considering all the random times $t_{i}$
explicitly, we can divide the whole process into two periods: (i) the interval
$(0,t_{1})$ during which $n(t)=1$, and (ii) the rest of the process. This
leads to the recurrent expression

$S_{N}(t)=\int_{0}^{t}r_{b}S_{1}(t)S_{N-1}(t-t_{1})dt_{1},$ (13)

where the factor $r_{b}$ in the integral represents the rate at which the
process switches from period (i) to period (ii).

From this expression, now it is immediate to check that the recurrent relation
mentioned in the main text

$S_{N}(t)=r_{b}e^{-r_{b}t}S_{sw}(t)\int_{0}^{t}S_{N-1}(t-t_{1})dt_{1},$ (14)

holds, and it is also immediate to verify that the general survival
probability of the overall search process reads

$S(t)=\sum_{N=1}^{\infty}S_{N}(t)=S_{sw}(t)e^{-r_{b}(t-g(t))}.$ (15)

with $g(t)=\int_{0}^{t}S_{sw}(t)dt$.

Also, the fastest-first-passage time distribution through the target follows
by deriving this expression with respect to $t$:

$f(t)=-\frac{dS(t)}{dt}=\left(f_{sw}(t)+r_{b}S_{sw}(t)[1-S_{sw}(t)]\right)e^{-r_{b}(t-g(t))}$
(16)

where we define $f_{sw}(t)=-dS_{sw}/dt$ as the first-passage time distribution
for the classical (single-walker) case.

Alternatively, the fastest-first-passage time distribution could be also
derived by using recurrent arguments as above. So that, the recurrent relation
can be written

$f_{N}(t)=\int_{0}^{t}r_{b}f_{1}(t)S_{N-1}(t-t_{1})dt_{1}+\int_{0}^{t}r_{b}S_{1}(t)f_{N-1}(t-t_{1})dt_{1},$
(17)

for the contribution to the fastest-first-passage coming from those
realizations where $n(t)=N$. Here, the first term on the right-hand side
correspond to the case where it is the first active walker (the one appearing
at $t=0$) the one to reach the target first. The second term then corresponds
to the case where any other of the $N-1$ active walkers present at time $t$
reaches the target first. So that, taking into account that
$f_{1}(t)=f_{sw}e^{-r_{b}t}$ and working out the recurrent relation, one
finally obtains

$\displaystyle
f_{N}(t)=r_{b}^{N-1}e^{-r_{b}t}\left[f_{sw}(t)\frac{(r_{b}g(t))^{N-1}}{(N-1)!}+S_{sw}(t)(1-S_{sw}(t))\frac{(r_{b}g(t))^{N-2}}{(N-2)!}\right],$
(18)

and finally using that $f(t)=\sum_{N=1}^{\infty}f_{N}(t)$ one recovers the
expression in (16).

## Derivation of the collective mean search time $\langle T_{c}\rangle$

Again, we will consider by separate the realizations of the search process
depending on the specific value of the birth-death process, $n(t)=N$, at the
time at which the target is reached. By definition, if there are $N$ walkers
active, then the contribution of the first ﬂwalker to the collective search
time is $t$, and the contribution from any other active walker will be
$(t-t_{i})$ (where $t_{i}$ represents the time at which tihs walker became
active). So that, the collective search time, averaged over all realizations
of the process that lead to the target detection at time $t$, is

$\overline{T_{c}^{(N)}}(T)=r_{b}^{N-1}e^{-r_{b}T}\int_{0}^{t}dt_{N-1}\int_{0}^{t_{N-1}}dt_{N-2}\ldots\int_{0}^{t_{2}}dt_{1}[T+(T-t_{1})+\ldots+(T-t_{N-1})]G_{N}(T),$
(19)

where we have defined

$G_{N}(t)\equiv
f_{sw}(t)\prod_{j=i}^{N-1}S_{sw}(t-t_{j})+S_{sw}(t)\sum_{i=1}^{N-1}f_{sw}(t-t_{i})\prod_{j\neq
i}S_{sw}(t-t_{j}).$ (20)

Here, the first term on the right-hand side of (20) corresponds to the case
where the first active walker reaches the target at time $t$ before any of the
other walkers do, as $\prod_{j=i}^{N-1}S_{sw}(t-t_{j})$ is the joint survival
probability for all the other active walkers appearing at times $t_{1}$,
$t_{2}$, $t_{3}$,…,$t_{N-1}$. Similarly, $\prod_{j\neq i}S_{sw}(t-t_{j})$
represents the joint survival probability for all the active walkers except
the one appearing at time $t_{j}$; then, the last term in (20) gives the
contribution from the realizations where the first walker to reach the target
is the one becoming active at time $t_{i}$.

In order to find a simple expression for $\overline{T_{c}^{(N)}}(T)$ we need
to introduce now an additional function. So, we define $I_{N}(t)$ as the mean
value of the collective search time by time $t$ provided that the target has
not been reached yet by then. This satisfies equations analogous to (19-20)
which read

$I_{N}(t)=r_{b}^{N-1}e^{-r_{b}t}\int_{0}^{t}dt_{N-1}\int_{0}^{t_{N-1}}dt_{N-2}\ldots\int_{0}^{t_{2}}dt_{1}[t+(t-t_{1})+\ldots+(t-t_{N-1})]H_{N}(t),$
(21)

$H_{N}(t)\equiv
S_{sw}(t)\prod_{j=i}^{N-1}S_{sw}(t-t_{j})+S_{sw}(t)\sum_{i=1}^{N-1}S_{sw}(t-t_{i})\prod_{j\neq
i}S_{sw}(t-t_{j}).$ (22)

By its definition, the function $I_{N}(t)$ can be seen to satisfy the
recurrent relation

$I_{N}(t)=r_{b}e^{-r_{b}t}S_{sw}(t)\int_{0}^{t}dt_{1}\left(\frac{t-t_{1}}{t}I_{N-1}(t-t_{1})+S_{N-1}(t-t_{1})\right),$
(23)

so applying this recurrence from $N=1$ on, it is possible to find a closed
expression for it:

$I_{N}(t)=e^{-r_{b}t}S_{sw}(t)\left[\frac{t\left(r_{b}g(t)\right)^{N-1}}{(N-1)!}+\frac{r_{b}h(t)\left(r_{b}g(t)\right)^{N-2}}{(N-2)!}\right],$
(24)

where $h(t)\equiv\int_{0}^{t}tS_{sw}(t)dt$.

With the help of the auxiliary function $I_{N}(t)$, we can write now a
recurrence relation for $\overline{T_{c}^{(N)}}(T)$ which takes the form

$\displaystyle\overline{T_{c}^{(N)}}(T)$ $\displaystyle=$ $\displaystyle
r_{b}e^{-r_{b}T}S_{sw}(T)\int_{0}^{T}dt_{1}\left[(T-t_{1})\overline{T_{c}^{(N-1)}}(T-t_{1})+f_{N-1}(T-t_{1})\right]$
(25) $\displaystyle+$ $\displaystyle
r_{b}e^{-r_{b}T}f_{sw}(T)\int_{0}^{T}dt_{1}\left[S_{(N-1)}(T-t_{1})+(T-t_{1})I_{N-1}(T-t_{1})\right].$

Now, since we do have explicit expressions for $S_{N}(t)$, $f_{N}(t)$ and
$I_{N}(t)$, it is also possible to reach recurrently a closed expression for
$\overline{T_{c}^{(N)}}(T)$

$\displaystyle\overline{T_{c}^{(N)}}(T)$ $\displaystyle=$ $\displaystyle
e^{-r_{b}T}S_{sw}(T)\left[\frac{r_{b}(T+g(T)-2TS_{sw}(T))\left(r_{b}g(T)\right)^{N-2}}{(N-2)!}+\frac{r_{b}^{2}(1-S_{sw}(T))h(T)\left(r_{b}g(T)\right)^{N-3}}{(N-3)!}\right]$
(26) $\displaystyle+$ $\displaystyle
e^{-r_{b}T}f_{sw}(T)\left[\frac{T\left(r_{b}g(T)\right)^{N-1}}{(N-1)!}+\frac{r_{b}h(T)\left(r_{b}g(T)\right)^{N-2}}{(N-2)!}\right],$

and from this we get the final form for the mean collective search time,

$\displaystyle\overline{T_{c}}(T)=\sum_{N=1}^{\infty}\overline{T_{c}^{(N)}}(T)$
$\displaystyle=$ $\displaystyle
e^{-r_{b}(T-g(T))}S_{sw}(T)\left[r_{b}\left(t+g(T)-2tS_{sw}(T)\right)+r_{b}^{2}\left(1-S_{sw}(T)\right)h(T)\right]$
(27) $\displaystyle+$ $\displaystyle
e^{-r_{b}(T-g(T))}f_{sw}(T)\left[T+r_{b}h(T)\right],$

so now the MCST would be the integral of this expression over the possible
values of the MFFPT, this is,

$\langle T_{c}\rangle=\int_{0}^{\infty}\overline{T_{c}}(T)dT.$ (28)

## Limiting behaviour of the MFFPT

The MFFPT in 1d can be found from (4) and the survival probability for one
walker $S_{sw}(t)$ as

$\left\langle
T\right\rangle=\int_{0}^{\infty}S_{sw}(t)e^{-r_{b}t}e^{r_{b}\int_{0}^{t}S_{sw}(t^{\prime})dt^{\prime}}dt.$
(29)

Integrating by parts, it turns into

$\left\langle T\right\rangle=I-\frac{1}{r_{b}}$ (30)

where

$I=\int_{0}^{\infty}e^{-r_{b}t}e^{r_{b}\int_{0}^{t}S_{sw}(t^{\prime})dt^{\prime}}dt.$
(31)

The integral in the exponent of the integrand can be explicitly found in terms
of the error function erf($\cdot$)

$\int_{0}^{t}S_{sw}(t^{\prime})dt^{\prime}=-\frac{x_{0}^{2}}{D}\textrm{erfc}\left(\frac{x_{0}}{\sqrt{4Dt}}\right)+t\textrm{erfc}\left(\frac{x_{0}}{\sqrt{4Dt}}\right)+x_{0}\sqrt{\frac{t}{\pi
D}}e^{-\frac{x_{0}^{2}}{4Dt}}.$

Defining the dimensionless parameter

$\displaystyle\chi\equiv\frac{x_{0}^{2}r_{b}}{4D}$ (32)

and the new integration variable $y=\sqrt{\chi/r_{b}t}$, the integral (31) now
becomes

$I=\frac{2\chi}{r_{b}}\int_{0}^{\infty}e^{\varphi(\chi,y)}dy,$ (33)

where $\varphi(\chi,y)=\chi F(y)-3\ln y$ with

$\displaystyle
F(y)=-\left(2+\frac{1}{y^{2}}\right)\textrm{erfc}(y)+\frac{2}{y\sqrt{\pi}}e^{-y^{2}}.$
(34)

This suggests to write (29) in the scaling form

$\displaystyle\left\langle T\right\rangle=\frac{x_{0}^{2}}{2D}\phi(\chi)$ (35)

where

$\displaystyle\phi(\chi)=\int_{0}^{\infty}e^{\varphi(\chi,y)}dy-\frac{1}{2\chi}.$
(36)

The validity of this scaling is verified (for the case of Brownian walkers in
1d) by observing the collapse of the numerical results obtained for $\langle
T\rangle$ when we plot them in the nondimensional form $4D\langle
T\rangle/x_{0}^{2}$ as a function of the variable $\chi$ (see Fig. 5).

Figure 5: Nondimensional MFFPT as a function of the quotient between diffusive
and birth scales, $\chi$, for different values of $x_{0}$ (see legend).

We note that the function $e^{\varphi(\chi,y)}$ is positive and has a global
maximum at $y=y^{*}$ where $y^{*}$ satisfies $\varphi^{\prime}(\chi,y^{*})=0$
with $\varphi^{\prime\prime}(\chi,y^{*})<0$ and the prime symbol means
derivative with respect to $y$. Then, $y^{*}$ is solution of the transcendent
equation

$3y^{*2}=2\chi\textrm{erfc}(y^{*}),$ (37)

i.e., $y^{*}$ is function of $\chi$ and it not difficult to show that
$\partial y^{*}/\partial\chi>0$, so $y^{*}$ is a monotonically increasing
function of $\chi$. This transcendent equation also lacks of analytical
solution but we can find the scaling behaviour of $\chi$ with $y^{*}$ in the
limits of small and large $\chi$. When $\chi$ is small, $y^{*}$ approaches to
zero and $\textrm{erfc}(y^{*})\simeq 1-2y^{*}/\sqrt{\pi}+O(y^{*3})$, so that
$\chi\sim y^{*2}$. When $\chi$ is large, $y^{*}$ is also large and
$\textrm{erfc}(y^{*})\simeq e^{-y^{*2}}/y^{*}\sqrt{\pi}$, so that $\chi\sim
y^{*3}e^{y^{*2}}$.

The main contribution to the integral in Eq. (36) comes from the vicinity of
$e^{\varphi(\chi,y)}$ around $y=y^{*}$. To proceed further we consider two
limiting situations, small $\chi$ and large $\chi$, separately.

### $\chi\ll 1$

Let us consider the case $\chi\to 0$, this is, $r_{b}\ll 4D/x_{0}^{2}$. In
this limit $y^{*}$ approaches $y=0$ as $\chi$ tends to zero, i.e. the function
$\varphi(\chi,y)$ is peaked around $y=0$. To estimate the integral in Eq. (33)
we expand $\varphi(\chi,y)$ around $y=0$ and we find
$\varphi(\chi,y)\simeq-\frac{1}{y^{2}}+\frac{4}{y\sqrt{\pi}}-3\ln(y)+O(0)$.
Then

$\int_{0}^{\infty}e^{\varphi(\chi,y)}dy\simeq\frac{1}{2\chi}+\frac{e^{4\chi/\pi}}{\sqrt{\chi}}\left[\textrm{erf}\left(2\sqrt{\frac{\chi}{\pi}}\right)+1\right]\simeq\frac{1}{2\chi}+\frac{1}{\sqrt{\chi}}+O(0)$
(38)

where we have expanded the last equality for small $\chi$. Finally, from (36)
and (38)

$\phi(\chi)\simeq\frac{1}{\sqrt{\chi}}$

and from (35)

$\left\langle
T\right\rangle\simeq\frac{2\sqrt{\chi}}{r_{b}}=\frac{x_{0}}{\sqrt{Dr_{b}}}.$

### $\chi\gg 1$

Next we consider the opposite limit $\chi\to\infty$, i.e., $r_{b}\gg
4D/x_{0}^{2}$. In this case, $y^{*}$ is such that
$\int_{0}^{y^{*}}e^{\varphi(\chi,y)}dy\ll\int_{y^{*}}^{\infty}e^{\varphi(\chi,y)}dy$.
Since the main contribution to the integral comes in the vicinity of $y=y^{*}$
and $y^{*}$ is large, we approximate $\varphi(\chi,y)$ in the large $y$ limit,
where $F(y)\simeq-y^{-5}e^{-y^{2}}/\sqrt{\pi}$. Then,

$\displaystyle\int_{0}^{\infty}e^{\varphi(\chi,y)}dy=\int_{0}^{\infty}\frac{e^{\chi
F(y)}}{y^{3}}dy\simeq\int_{0}^{\infty}\frac{e^{-\frac{\chi}{\sqrt{\pi}}\frac{e^{-y^{2}}}{y^{5}}}}{y^{3}}dy\simeq\int_{y^{*}}^{\infty}\frac{e^{-\frac{\chi}{\sqrt{\pi}}\frac{e^{-y^{2}}}{y^{5}}}}{y^{3}}dy\sim\int_{y^{*}}^{\infty}\frac{dy}{y^{3}}\sim\frac{\text{1}}{y^{*2}}$
(39)

where we have considered
$e^{-\frac{\chi}{\sqrt{\pi}}\frac{e^{-y^{2}}}{y^{5}}}\simeq 1$, since along
the integration region $y$ is large. To find $y^{*}$ we solve (37) for $\chi$
and expand for large $y^{*}$

$\chi\simeq\frac{3\sqrt{\pi}}{2}y^{*3}e^{y^{*2}}+O(1/y^{*}).$

Solving this equation for $y^{*2}$ in terms of $\chi$ we find

$y^{*2}\simeq\frac{3}{2}W_{0}\left(\frac{2}{3}\left[\frac{2\chi}{3\sqrt{\pi}}\right]^{2/3}\right),$
(40)

where $W_{0}(x)$ is the principal branch of the Lambert function and can be
approximated as $W_{0}(x)\simeq\ln(x)$ for large $x.$ Then, (40) can be
approximated through

$y^{*2}\simeq\frac{3}{2}\ln\left(\frac{2}{3}\left[\frac{2\chi}{3\sqrt{\pi}}\right]^{2/3}\right)=\ln\left[\left(\frac{2}{3}\right)^{5/2}\frac{\chi}{\sqrt{\pi}}\right]\sim\ln\chi\quad\textrm{for}\quad\chi\to\infty.$
(41)

Additionally, from (39) and (41) one has

$\int_{0}^{\infty}e^{\varphi(\chi,y)}dy=\int_{0}^{\infty}\frac{e^{\chi
F(y)}}{y^{3}}dy\sim\frac{1}{\ln\chi},$ (42)

and then combining (36) and (42) one obtains the scaling

$\displaystyle\phi(\chi)\sim\frac{1}{\ln\chi}\quad\textrm{for}\quad\chi\to\infty.$
(43)

## Limiting behaviour of the MCST

The integrals in the definitions of $g(t)$ and $h(t)$ can be explicitly
written to express Eqs. (5-6) of the main text in terms of $T$. Introducing
again the new variable $y\equiv\sqrt{\chi/r_{b}T}$ the mean collective search
time can be written in the scaling form

$\displaystyle\left\langle
T_{c}\right\rangle=\frac{x_{0}^{2}}{2D}\phi_{c}(\chi),$ (44)

with

$\phi_{c}(\chi)=\int_{0}^{\infty}e^{\varphi_{c}(\chi,y)}dy$ (45)

and

$\varphi_{c}(\chi,y)=\chi F(y)-3\ln y+\ln\Phi(\chi,y).$

Again, we verify the validity of the scaling proposed (for the case of
Brownian motion in 1d) by plotting $4D\langle T\rangle/x_{0}^{2}$ as a
function of $\chi$ and checking how all curves obtained numerically collapse
(Fig. 6).

Figure 6: Nondimensional MCST as a function of the quotient between diffusive
and birth scales, $\chi$, for different values of $x_{0}$ (see legend).

Here, $F(y)$ is defined in (34) and
$\Phi(\chi,y)=\phi_{0}(y)+\chi\phi_{1}(y)+\chi^{2}\phi_{2}(y)$, with

$\displaystyle\phi_{0}(y)=\frac{y}{\sqrt{\pi}}e^{-y^{2}}$
$\displaystyle\phi_{1}(y)=-(2y^{2}-1)\left[\frac{\textrm{erf}(y)\textrm{erfc}(y)}{y^{2}}+\frac{e^{-2y^{2}}}{3\pi}\right]+\frac{2y^{3}e^{-y^{2}}}{3\sqrt{\pi}}\textrm{erfc}(y)+\frac{5e^{-y^{2}}}{2y\sqrt{\pi}}\textrm{erf}(y)$
$\displaystyle\phi_{2}(y)=\textrm{erf}(y)\textrm{erfc}(y)\left[-\frac{2y^{2}-1}{3y^{3}\sqrt{\pi}}e^{-y^{2}}+\frac{\textrm{erf}(y)}{2y^{4}}+\frac{2}{3}\textrm{erfc}(y)\right].$

The function $e^{\varphi_{c}(\chi,y)}$ has a global maximum at $y=y^{*}$ where
$y^{*}$ is solution to the equation $\varphi_{c}^{\prime}(\chi,y^{*})=0$, this
is

$\displaystyle y^{*}\Phi(y^{*},\chi)\chi
f^{\prime}(y^{*})+y^{*}\Phi^{\prime}(y^{*},\chi)=3\Phi(y^{*},\chi),$ (46)

where the prime symbol means derivative with respect to $y^{*}$. Since
$\varphi_{c}(\chi,y)$ increases monotonically with $\chi$, the position of the
maximum $y^{*}$ moves with $\chi$ as for MFFPT. This transcendent equation
also lacks of analytical solution but we can find the behaviour of $\chi$ with
$y^{*}$ in the limits of small and large $\chi$. Inserting the definitions of
$F(y)$ and $\Phi(y,\chi)$ into Eq. (46) we find, for large $\chi$, that

$\displaystyle\chi\sim y^{*5}e^{y^{*2}}.$ (47)

### $\chi\ll 1$

In this limit $y^{*}$ tends to zero, i.e., the main contribution of the
integral (45) comes from the vicinity of $e^{\varphi_{c}(\chi,y)}$ near $y=0$.
Expanding $F(y)$ and $\Phi(\chi,y)$ around $y=0$ we find

$F(y)\simeq-\frac{1}{y^{2}}+\frac{4}{y\sqrt{\pi}}+O(1),\quad\Phi(\chi,y)\simeq\frac{y}{\sqrt{\pi}}+\frac{2\chi}{y\sqrt{\pi}}+\frac{8\chi^{2}}{3\pi
y^{2}}$

where we have expanded $\phi_{0}(y)$, $\phi_{1}(y)$ and $\phi_{2}(y)$ to the
lowest order in $y$. Plugging these results into Eq. (45) we find

$\int_{0}^{\infty}e^{\varphi_{c}(\chi,y)}dy\simeq\int_{0}^{\infty}e^{\frac{4u}{\sqrt{\pi}}-\frac{u^{2}}{\chi}}\left(\frac{1}{\chi\sqrt{\pi}}+\frac{2u^{2}}{\chi^{2}\sqrt{\pi}}+\frac{8u^{2}}{3\pi\chi^{2}}\right)dy=\frac{1}{\sqrt{\chi}}+O(1)$

where we have introduced the new variable $u=\chi/y$. Finally, from this
result together with (45) and (44) we get

$\left\langle T_{c}\right\rangle\simeq\frac{x_{0}}{\sqrt{Dr_{b}}},$

and so the result $\langle T\rangle\approx\langle T_{c}\rangle$ is recovered
in this limit (see Eq. ($\chi\ll 1$)), as discussed in the main text.

### $\chi\gg 1$

First we rewrite Eq. (45) in the form

$\displaystyle\phi_{c}(\chi)=\int_{0}^{\infty}\Phi(\chi,y)\frac{e^{\chi
F(y)}}{y^{3}}dy$ (48)

and note that when $\chi$ is large, then $y^{*}$ is also large, as we argued
below Eq. (46). Since in this limit, the main contribution to the integral in
(48) is due to the vicinity of $y^{*}$ and it is large, we expand $F(y)$ and
$\Phi(\chi,y)$ for large $y$. In the large $y$ limit
$F(y)\simeq-y^{-5}e^{-y^{2}}/\sqrt{\pi}$ and expanding $\phi_{0}(y)$,
$\phi_{1}(y)$ and $\phi_{2}(y)$ we have

$\Phi(\chi,y)\simeq\frac{e^{-y^{2}}}{\sqrt{\pi}}\left(y+\frac{\chi}{2y}+\frac{\chi^{2}}{2y^{5}}\right).$

According to (47), the dominant term of the above expression is the third term
of the right hand side in the vicinity of $y^{*}$, so that

$\Phi(\chi,y)\simeq\frac{\chi^{2}}{2y^{5}}\frac{e^{-y^{2}}}{\sqrt{\pi}}.$

Hence, from (48)

$\phi_{c}(\chi)\simeq\frac{\chi^{2}}{2}\int_{0}^{\infty}\frac{e^{-y^{2}}}{\sqrt{\pi}y^{5}}\frac{e^{-\frac{\chi}{\sqrt{\pi}}\frac{e^{-y^{2}}}{y^{5}}}}{y^{3}}dy=-\frac{\chi^{2}}{2}\frac{\partial}{\partial\chi}\int_{0}^{\infty}\frac{e^{-\frac{\chi}{\sqrt{\pi}}\frac{e^{-y^{2}}}{y^{5}}}}{y^{3}}dy,$

but the last integral was estimated in (42) so that from (44)

$\langle
T_{c}\rangle\sim\phi_{c}(\chi)\sim-\chi^{2}\frac{\partial}{\partial\chi}\left(\frac{1}{\ln\chi}\right)=\frac{\chi}{(\ln\chi)^{2}}.$

## optimal MCST

The optimal value of $\left\langle T_{c}\right\rangle$ can be obtained by
solving the equation $\partial\left\langle T_{c}\right\rangle/\partial\chi=0$
for $\chi$. Taking the derivative of (44) with respect to $\chi$ leads us to
the transcendental equation

$\displaystyle\int_{0}^{\infty}\frac{F(y)\phi_{0}(y)+\phi_{1}(y)}{y^{3}}e^{\chi
F(y)}dy+\chi\int_{0}^{\infty}\frac{F(y)\phi_{1}(y)+2\phi_{2}(y)}{y^{3}}e^{\chi
F(y)}dy+\chi^{2}\int_{0}^{\infty}\frac{F(y)\phi_{2}(y)}{y^{3}}e^{\chi
F(y)}dy=0,$ (49)

which is an integral equation for $\chi$. Solving (49) numerically one finds
$\chi^{*}=1.8877$, which from (32) corresponds to the optimal birth rate
$r_{b}^{*}\simeq 7.551D/x_{0}^{2}$. Substituting $\chi=\chi^{*}$ in (44) one
obtains $\left\langle T_{c}\right\rangle^{*}\simeq 1.489x_{0}^{2}/D$.


as a weight, if the background ratio is constant for all bins.

Figure 25: The distribution of the average count rates of sources in the SEP
field, in eRASS1, 2, and 3. The average count rates were determined by Eq. 6.
Due to the high exposures in the SEP field, _eROSITA_ could detect sources at
much lower count rates than in the rest of the sky. This figure does not show
the six sources with a slightly negative average count rate.

Fig. 25, shows the distribution of the average source count rate measured by
_eROSITA_ for all sources in the SEP field, in the
$0.2-5.0\leavevmode\nobreak\ \mathrm{keV}$ band. Almost all detected sources
had average source count rates in the range $1.8\times
10^{-4}-4.2\leavevmode\nobreak\ \mathrm{cts/s}$. The most common average
source count rate, per logarithmic interval, was $8.9\times
10^{-3}\leavevmode\nobreak\ \mathrm{cts/s}$. The distribution increases
rapidly from low count rates towards the maximum. Above the peak, the
distribution decreases less steeply to higher count rates, in a manner that is
consistent with a power law with an index of $\approx-1.2$. Unlike previous
plots, the distribution of the average source count rate in eRASS1, 2, and 3
is very similar.

The brightest source in the SEP field is the supernova remnant SNR B0535-66.0,
for which we measured an average source count rate of $\approx
97.5\leavevmode\nobreak\ \mathrm{cts/s}$. For such a bright source, the
measured count rate is inaccurate, due to pileup effects (Merloni et al. 2024,
accepted).

We found six sources with slightly negative average source count rates. These
sources were extremely faint. Despite being observed for several ks, only very
few source counts were detected for them, slightly below the average
background count rate. These lowest count rate estimates are not only
unreliable, but are also from potentially spurious detections.

In Fig. 26, the distribution of the total measured counts in the source
extraction region for the $0.2-5.0\leavevmode\nobreak\ \mathrm{keV}$ energy
band is displayed. The distribution features a steep rise from the lowest
number of counts, of merely 4, to a peak at 25 counts. This is the value at
which there were the greatest number of sources per logarithmic interval.
Above the peak, the distribution is dominated by a decline that also
approximately follows a power law, with an index of $\approx-1.2$. This power
law continues until at least 5600 counts per source, per eRASS. The greatest
number of source counts measured for a single source in one eRASS was
$3.19\times 10^{5}$ for SNR B0535-66.0 in eRASS1. The strongly positively
skewed distribution in Fig. 26 results from the positive skew of both the
average count rate distribution, and the effective exposure time distribution.

Figure 26: The distribution of the total source counts for all sources in the
SEP field, in eRASS1, 2, and 3, for the energy band
$0.2-5.0\leavevmode\nobreak\ \mathrm{keV}$.

###  A.6 Background count rate

The background ratio describes the ratio of the source extraction area to the
background extraction area. Fig. 27 shows that the background extraction
region was always significantly larger than the source extraction region. The
most common value of the background ratio, per logarithmic interval, was
$0.0089$. Nevertheless, the distribution of background ratios is broad,
spanning the interval from $0.0017$ to $0.074$, and features a second
prominent peak at a value of $0.0040$.

Figure 27: The distribution of the average ratio of the area of the source to
the background extraction region, for each source detected in the SEP field,
in eRASS1, 2, and 3. The background ratio exhibits slight variations
throughout each eRASS, and across eRASSs, but usually remained close to the
average value.

Fig. 28 depicts the observed distribution of the background count rate for the
$0.2-5.0\leavevmode\nobreak\ \mathrm{keV}$ interval, measured in the
background extraction region for each detected source. Similar to the source
count rate, the average background count rate, $\overline{R_{B}}$, can be
estimated by:

$\overline{R_{B}}\approx\frac{\sum_{i=1}^{N_{\mathrm{b}}}B(t_{i})}{\sum_{i=1}^{N_{\mathrm{b}}}\epsilon(t_{i})\Delta
t}.$ (7)

Similar to the distribution of the observed average source count rate, this
follows a sharp rise from the lowest background count rate of
$0.186\leavevmode\nobreak\ \mathrm{cts/s}$ up to a peak at
$0.71\leavevmode\nobreak\ \mathrm{cts/s}$. However, this distribution is much
more positively skewed, almost reaching a plateau at the highest background
count rates of up to $73.7\leavevmode\nobreak\ \mathrm{cts/s}$. These
background count rates are often much larger than the source counts rates, as
they are obtained over a much larger area than the source count rate (See Fig.
27). The part of the SEP field that intersects within the LMC has a higher
background count rate than the rest of the field, which accounts for some of
the high background count rates observed. The width of this distribution is
exacerbated by the distribution of the background ratio. This graph does not
represent the distribution of the background count rate per solid angle within
this field.

Figure 28: The distribution of the average background count rate in the
background extraction regions used to determine the background subtracted
source count rates. The background extraction region is typically more than
$\approx 100$ times larger than the source extraction region.

###  A.7 Detection likelihood

The DET_LIKE parameter quantifies the reliability of the source detection. The
greater the DET_LIKE value is, the greater the certainty that a source is
genuine. In Fig. 29, we plot the distribution of the detection likelihood for
all detected sources in the SEP field in eRASS1, 2, and 3, for the
$0.2-5.0\leavevmode\nobreak\ \mathrm{keV}$ energy band. The maximum of the
distribution occurs at the lower limit of significantly detected sources, of
$\mathrm{DET\\_LIKE}=6$. This indicates that a not insignificant number of
sources are barely above the detection threshold, and therefore should still
be considered as potentially spurious sources. The distribution drops rapidly,
but extends to very high DET_LIKE, of up to $1.89\times 10^{6}$ in eRASS1. Low
DET_LIKE sources are observed with just a few counts above the expected
background level.

Variable sources have an almost uniform DET_LIKE distribution, as shown in
Fig. 29. Therefore, the higher the DET_LIKE of a source, the more likely it is
to be classified as variable.

Figure 29: The distribution of the detection likelihood parameter of all
sources lying within $3\degree$ of the SEP in eRASS1, 2, and 3, for the energy
band $0.2-5.0\leavevmode\nobreak\ \mathrm{keV}$.