# Communicating extraterrestrial intelligence (CETI) interaction models based
on the Drake Equation

Reginald D. Smith
Supreme Vinegar LLC
3430 Progress Dr. Suite D
Bensalem, PA 19020, USA
<EMAIL_ADDRESS>
###### Abstract

The Drake Equation has proven fertile ground for speculation about the
abundance, or lack thereof, of communicating extraterrestrial intelligences
(CETIs) for decades. It has been augmented by subsequent authors to include
random variables in order to understand its probabilistic behavior. However,
in most cases, the emergence and lifetime of CETIs are assumed to be
independent of each other. In this paper, we will derive several expressions
that can demonstrate how CETIs may relate to each other in technological age
as well as how the dynamics of the concurrent CETI population change under
basic models of interaction, such as the Allee Effect. By defining interaction
as the change in the expected communication lifetime with respect to the
density of CETI in a region of space, we can use models and simulation to
understand how the CETI density can promote or inhibit the longevity and
overall population of interstellar technological civilizations.

##  1 Introduction

The Drake equation has been a useful tool for thought experiments about the
presence (or lack thereof) of intelligent, communicating extraterrestrial
intelligences (hereafter “CETI”) for over fifty years. In recent years, the
Drake equation, long viewed as a deterministic type relation, has been
supplemented with the tools of the probability and statistics of random
variables [1, 2, 3, 4]. Recognizing that the terms in the Drake equation are
random variables, this allows us to express the Drake equation results as
first moments of a distribution with confidence intervals.

In [3], the Drake equation is refined by defining the arrival rate of
communicating civilizations as a Poisson distribution with their lifetime
modeled as a random variable. Monte Carlo simulations [5, 6, 7] have allowed
the approximation of various parameters for the emergence and permanence of
CETI depending on underlying models of fundamental factors. The possibility of
interaction between earlier, or first, civilizations and their predecessors
was broached by [7] by using Monte Carlo simulations to model the distribution
of times of the arrival of a first civilization and later CETI inter-arrival
times. One of the results is that there was likely a huge interval between the
arrival of the first technological civilization and its successors which
theoretically could allow it a long time to leave a legacy that would have a
disproportionate influence on subsequent societies.

Many papers have modeled the probabilities behind the likelihood of emergence
as well as durability of CETI. Few have gone into similar quantitative detail
on the nature of the interactions between CETI. In this paper we will address
the questions of the average time between successive civilizations though
unlike [7] we are looking at just the steady state, not the time between the
first one or first few CETI. Second, we will measure the probability that
CETIs will be able to communicate at all given their random emergence in a
volume of space. Finally, we will derive basic expressions for the interaction
of CETIs where the expected lifetime of a CETI depends on the density of
concurrent CETIs it can interact with. This imputes a covariance between $N$
and $L$ for concurrent CETI that could speak in general terms about the
typical fate of interaction between interstellar technological civilizations.

##  2 Expected technological ‘age’ gap

As stated earlier, the emergence of communicating civilizations can often be
approximated as a Poisson distribution. Assume $k$ is the number of CETIs that
emerge in a given interval, and $\lambda$ is the average arrival rate for
CETIs per unit time. The Poisson distribution of the CETI emergence,
$N_{b}(k,\lambda)$ , is given by

$N_{b}(k,\lambda)=\frac{\lambda^{k}}{k!}e^{-\lambda}$ (1)

Note this assumes that the Poisson distribution is homogenous so that
$\lambda$ is a constant that does not change over time. In queueing theory,
given a homogenous Poisson arrival process, one of the key findings of
operations research is Little’s Law, that states the average number of items
in a queue is equal to the arrival rate times the queue handling time. In [8]
it was shown that previous work by Brian Tung (private communication) at his
website Astronomy Corner showed that the Drake Equation was a special case of
Little’s Law [9]. Where $N$ is the average number of items in a system,
$\lambda$ is the average (stationary) arrival rate and $t$ is the average time
an item is in the system, Little’s Law is expressed as

$N=\lambda t$ (2)

Given the Drake equation [10]

$N=R^{*}\times f_{s}\times L$ (3)

where $R^{*}$ is the average production rate for stars ‘suitable’ for planets
and eventually intelligent life, $f_{s}$ is the probability of the emergence
of an intelligent and communicating civilization around one such star, and $L$
is the average lifetime of such a CETI, the variable $\lambda=R^{*}\times
f_{s}$ and $t=L$.

Therefore, in its simplest form, the Drake Equation and the average number of
concurrent civilizations in a region of space can be seen as analogous to some
queueing problems. A related question that can be explored from this type of
reasoning is the average difference in “age” between communicating
civilizations given as the difference in the amount of time each has been
broadcasting. This would be based on the average emergence time for a
civilization and the contingency that there are at least two existing
civilizations at any given time in range of contact.

The average interarrival period for a homogenous Poisson process is given by
the inverse of the arrival rate, $1/\lambda$. Likewise, on average, at least
two civilizations will exist at the same time if $L\geq 1/\lambda$. Given,
equations 2 and 3, where $N\geq 2$, the average gap in civilization
technological age, $\langle t_{gap}\rangle$ can be approximated by

$\langle t_{gap}\rangle=\frac{1}{\lambda}=\frac{1}{R^{*}\times
f_{s}}=\frac{L}{N}$ (4)

Therefore, if $L$ is large compared to $N$ there is likely a large proportion
of communicating lifetime of difference between CETIs. Likewise if $N$ is
relatively large compared to $L$ then there are likely many civilizations
whose broadcast epochs are of similar length and they may share some
similarity in technological development. This could be considered a
superficial metric of differential technological advancement though these
assumptions beyond the scope of this paper.

###  2.1 Existence versus contact

In [8], a model was demonstrated showing that the maximum distance a broadcast
from a CETI is expected to have appreciable signal-to-noise, $D$, combines
with $L$ to determine requirements for civilization density to hope to achieve
any contact. In [2, 4] a more detailed analysis about the density and
possibility of contact between ETIs was made using assumptions of average
distances between CETIs assuming they are isotropically distributed throughout
the galaxy and using the mathematics of packing to determine the overall
density. Here we will explore a slightly different problem of the probability
that two co-existing civilizations in a region of space can receive each
other’s messages or communicate.

Assume a volume normalized version of the Drake Equation where $N$ is
normalized per volume of space as $n$ and $R^{*}$ is normalized per volume of
space as $r^{*}$.

$n=r^{*}\times f_{s}\times L$ (5)

A key question is that given a volume of space how probable is it that first,
at least two CETI exist concurrently for any period of time and second, how
probable is it that they both survive long enough for one to send at least one
message and receive a response. For a Poisson field where the Poisson variable
for emergence of points is volume normalized as $\lambda_{V}$ and the expected
number of points to emerge per unit time in a volume $V$ is $\lambda_{V}V$,
the probability for a point to have another point a distance $l$ away is given
by [12] (p. 46, though derived earlier by [13, 14]).

Define $P$ as the probability that a CETI is neighbors with at least one other
CETI within a defined volume of space. Assuming an isotropic model of
homogenous space and viability similar to [2, 4], the probability that a CETI
emerging at a point _does not_ have other neighboring CETI within a sphere of
radius $l$ is [12]

$1-P=\exp{\Bigg{(}-\frac{4}{3}\pi\lambda_{V}l^{3}\Bigg{)}}=\exp{\Bigg{(}-\frac{4}{3}\pi
l^{3}(r^{*}\times f_{s})\Bigg{)}}$ (6)

So likewise, the probability of at least one other CETI at a distance $l$ is
given by

$P=1-\exp{\Bigg{(}-\frac{4}{3}\pi\lambda_{V}l^{3}\Bigg{)}}=1-\exp{\Bigg{(}-\frac{4}{3}\pi
l^{3}(r^{*}\times f_{s})\Bigg{)}}$ (7)

For two CETI to feasibly communicate with each other at least once, they
cannot be more than $L/2$ light-years distant from each other. Therefore,
setting $l=L/2$, the probability of communication between at least two CETI
can be given by

$P=1-\exp{\Bigg{(}-\frac{4}{3}\pi(r^{*}\times
f_{s})\frac{L^{3}}{8}\Bigg{)}}=1-\exp{\Bigg{(}-\frac{\pi(r^{*}\times
f_{s})L^{3}}{6}\Bigg{)}}$ (8)

The probability here should be viewed as a maximum probability given the
estimates of CETI emergence rates and lifetime. In short, it is the
probability that two CETI can feasibly communicate given the emergence rate
for CETI per $L/2$ years per light years cubed. This of course is assuming
they will communicate if possible but just because two CETI can communicate
doesn’t mean they will and it also doesn’t mean they find each other even when
looking. However, this can provide us with interesting estimates for a lower
bound rate for the emergence of CETI.

Given a spherical volume of space with radius $d$ and volume $(4/3)\pi d^{3}$,
for a given $P$ we can calculate the emergence rate per volume per $L/2$ years
and then translate this into $d$, the radius for this volume of space. For
example for a $L=1,000$ and feasible contact probabilities of 0.01, 0.25, 0.5,
and 0.99 the CETI emergence rates per light-year cubed ($r^{*}\times f_{s}$)
per 500 years would have to be $2.0\times 10^{-11}$, $5.5\times 10^{-10}$,
$1.3\times 10^{-9}$, and $9.0\times 10^{-9}$ respectively. This translates to
the emergence of a CETI once per 500 years with radii of 2,285 light-years,
757 light-years, 568 light-years, or 298 light-years. This equation does,
however, assume that both CETI remain fixed at their places of origin.
Exceptions to this definitely adjust the outcome but will not be treated in
this paper.

##  3 Interstellar civilization interaction – a definition

The interaction, or at least awareness, between at least two interstellar
technological civilizations is the ostensible goal of all types of SETI. How
this could possibly affect the dynamics of a postulated wider interstellar
community is the subject of countless works of fiction as well as
philosophical and scientific essays.

In a basic sense, interaction over long time frames can be defined in many
ways, but in terms of the Drake equation, we will define the effects of
interaction (or lack of effects) by the statistical dependence between the
average communicating lifetime $L$ based on the number of coexisting
civilizations $N$. Therefore, if $I_{\Delta t}(N,L)$ is defined as the mutual
information between the number of co-existing civilizations and their average
lifetime over a measured period $\Delta t$, a basic quantification of the
presence of interaction effects on the Drake Equation is

$I_{\Delta t}(N,L)>0$ (9)

The mutual information is chosen as a metric given it accounts for all types
of statistical dependence, linear and nonlinear, and thus is a comprehensive
description. If it is zero, it does not rule out interaction but shows that
such interaction has almost no impact on the trajectory of the lifetime of
interstellar civilizations. However, in the case most or all of the
statistical dependence between $N$ and $L$ is linear, we can instead define
the presence of interaction as the covariance.

$\mbox{Cov}_{\Delta t}(N,L)\neq 0$ (10)

The covariance interpretation has the added benefit of being positive or
negative depending on the beneficial or malign effects on the lifetime of
civilizations based on the presence of others. While difficult to answer in
the abstract, especially with sparse known data about other life in the
cosmos, the answer to this question would be one of the most important
regarding the dynamics of advanced intelligent life in our universe.

###  3.1 Interaction of CETI - the Allee Effect

Under the conditions of the previous section where any one CETI can influence
another, it is an open question how the dynamics of that influence can work
out. We will assume the results of CETI interaction manifest in changes to
their expected lifetime. An interaction parameter, $r$, is defined that
measures the increase (positive values) or decrease (negative values) in the
lifetime of a civilization proportional to $r$ and the number of other
civilizations it can contact. In other words, if an isolated CETI is expected
to have a lifetime of $L_{0}$ then interaction effects change this to
$L_{0}+rN$, where $L_{0}$ is $L$ when $r=0$.

This condition, where $N$ and $L$ are interdependent, the change in $N$ from a
change in $L$ can be modeled as

$\displaystyle\Delta N=R^{*}\times f_{s}\times\Delta L$ (11)
$\displaystyle\Delta N=R^{*}\times f_{s}\times rN$ $\displaystyle
N(t)=N_{0}e^{(R^{*}\times f_{s}\times r)t}$

Clearly, when $r>0$, the number of CETI will increase exponential while when
$r<0$, the stable point is reached when $N$ declines to zero. For both of
these cases the covariances between $N$ and $L$ are positive and negative
respectively.

This model is admittedly simple and shows a simple cooperative/competitive
based change in average lifetime based on the parameter $r$ and $N$. A more
comprehensive model of interaction among CETIs that can be borrowed from
ecology is known as the Allee Effect. First discovered by W.C. Allee in
observations the of how the growth rates of the flour beetle _Tribolium
confusum_ were greatest at intermediate population densities, the Allee Effect
now encompasses a wide range of effects where fitness, usually measured as the
rate of population growth, varies according to the population density.

Typically, Allee Effect models have population density thresholds. Below the
threshold population growth rates decrease, possibly becoming negative. Above
the threshold the population growth rates increase until they reach a peak,
and they then decline until the population reaches carrying capacity and the
population growth rate goes to zero. The model for the Allee Effect amongst
CETIs here is borrowed from the model elucidated in [15, 16] and incorporates
not only $r$, but the carrying capacity, $K$, as well as the Allee threshold
$A$. In this model $L$ changes to

$L=L_{0}+rN\Bigg{(}1-\frac{N}{K}\Bigg{)}\Bigg{(}\frac{N}{A}-1\Bigg{)}$ (12)

Therefore, the rate of change for $N$ can be shown as

$\Delta N=R^{*}\times f_{s}\times
rN\Bigg{(}1-\frac{N}{K}\Bigg{)}\Bigg{(}\frac{N}{A}-1\Bigg{)}$ (13)

Of note in Equation 13 there are three fixed points. $N=0$ which is unstable,
$N=K$ which is stable at carrying capacity and $N=A$ which is also unstable.
At these fixed points, the formula for $N$ reduces to the traditional Drake
Equation with $L=L_{0}$. For $N<A$, the overall rate of growth is negative
leading to a declining count of CETI while at $N>A$ the growth rate increases
until it hits a maximum rate and thereafter declines, still remaining positive
until carrying capacity for CETIs is reached. An example of the expected
behavior is given in Figure 1.

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Figure 1: Plots of $\Delta L$ based on different values of $A$. Based on
$L_{0}=1,000$, $r=1$, and $K=50$.

One aspect of this expression of the Allee Effect may be unrealistic for CETI
lifetimes, however. Namely, that lower densities, below the threshold $A$,
would cause a decrease in the expected communicating lifetime. A more sparse
universe should likely reflect the traditional Drake Equation, not represent a
decline. Therefore a modified version of the Allee Effect can be used as
follows

$\begin{cases}L=L_{0},&\text{if}\ N<=A\\\
L=L_{0}+rN\Bigg{(}1-\frac{N}{K}\Bigg{)}\Bigg{(}\frac{N}{A}-1\Bigg{)},&\text{otherwise}\end{cases}$
(14)

The results of the dynamics assuming Equation 14 are shown in Figure 2. Where
$A$ is low enough for $N$ to commonly pass the threshold such as $A=5$ in this
example, the number of CETI can easily exceed the expectations of the
traditional Drake Equation and reach the “carrying capacity”. Otherwise, the
results basically fluctuate around the level of $N$ the traditional equation
expects. Likewise, if $r<0$ (not shown), then $N=A$ is still an unstable fixed
point but cannot be exceeded and the CETI population cannot exceed it for any
sustained period of time and just keeps to the traditional Drake Equation
expectations. In this Allee Effect model, there is no driver for sustained
complete extinction of all CETI.

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---|---
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Figure 2: Plots of $N$ over time assuming CETI emergence is based on a Poisson
process with $\lambda$=0.01, $L_{0}=1,000$, and $N_{0}=10$ for different
values of $A$. Other model constants are the same such as $r=1$ and $K=50$.

##  4 Conclusion

The Drake Equation continues to provide a fertile ground for research. Despite
justified criticisms [17], it remains a useful guide to theorize about the
possible conditions for the presence of ETIs. Here we have used the simple
assumptions of the traditional Drake Equation and a Poisson process underlying
CETI emergence to build basic models of CETI interaction as well as look at
relative likelihoods of communications between CETI. In short, interaction
between CETI can be beneficial or detrimental with the former almost always
allowing more CETI to co-exist than implied in the simple Drake Equation. On
the other hand, if CETI have an overall detrimental effect on one another the
Drake Equation expectations are the best case and a relatively quiet universe
the worst.

The relationship between CETI was defined using changes in the average
lifetime based on CETI density since it is a clear and simple way that
feedback between CETI can be incorporated into the macro-trends guiding the
presence of technological civilizations in space. Many other models of
interaction can be postulated and developed but it is likely that all must
address the question of how the overall density of CETI and their longevity
are affected by their interactions.

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