Existence versus contact

In Smith (Reference Smith2009), 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 Prantzos (Reference Prantzos2013, Reference Prantzos2020) 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.

For two CETIs to feasibly communicate with each other at least once, the following condition must apply. Namely, that a signal emitted by one CETI should have a reasonable probability of reaching another CETI at most L/2 years after emission which should allow a bi-directional communication if there is a prompt reply. A corollary is that the volume of space whose points are L/2 light years or less distant from a CETI must have a reasonable probability of N ≥ 2 given the Drake equation parameters. Since the Drake equation is defined over a the entire Milky Way this could cause issues for short-lived civilizations if N is small.

To further address the key assumption, a pertinent question is that if a CETI were to send out a signal, given the expanding area of space that signal encounters, how probable is that the signal reaches another CETI and then how probable is that the receiving CETI is able to send a response that can be received by the first before its quiescence. From our earlier definition, using the Poisson assumption the emergence rate for CETI is λ = N/L for random points emerging in the plane of a circle of radius, R, the probability distribution for the shortest distance between any two random points is given by (Kendall and Moran Reference Kendall and Moran1963, p. 38)

(5)$$\matrix{\;\;\,f( R) \,=\!\!\!\! & 2{\lambda\over \pi R_{\rm G}^2} \pi R \exp{\Bigg(-{\lambda\over \pi R_{\rm G}^2} \pi R^2 \Bigg)} \cr f( R)=\!\!\!\!\!\!\!\!\!\!\! & 2{N\over L}{R\over R_{\rm G}^2} \exp{\Bigg(-{N\over L}\Bigg({R\over R_{\rm G}}\Bigg)^2 \Bigg)} }$$

The expression ${N}/{L\pi R_{\rm G}^2}$ is the emergence rate normalized by the surface area of the Milky Way galactic disc simplified as a circle with galactic radius R _G = 50 000 light-years. Thus the probability of joint communication between CETIs is given by calculating the probability of R from 0 to L/2.

Figure 1 shows a density plot of the probability of two CETIs having bi-directional communication based on values of N and L.

Fig. 1. Contour plot of the probability of bi-directional communication between CETI given N and L in the Milky Way. Based on equation (5).

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 CETIs 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 CETIs can communicate does not mean they will and it also does not 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.

A simple model of CETI interaction

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 operate. We will assume the results of CETI interaction manifest in changes to their expected lifetime. In short, the key variables that will influence the change in lifetime are the density of CETI, given by N and a measure of the strength, and direction, or their interaction given by a new variable, r. The interaction parameter, r, so that the increase (positive values of r) or decrease (negative values of r) in the lifetime of a civilization is proportional to r and N. For this analysis, N assumes all CETIs are within L/2 light years and can interact. Otherwise, interaction cannot be consistently expected.

Assuming 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 relationship, where N and L are interdependent, can be modelled as

(8)$$\matrix{& \Delta N = R^{\ast } \times f_{\rm s} \times \Delta L \cr & \Delta N = R^{\ast } \times f_{\rm s} \times rN \cr & N( t) = N_0{\rm e}^{( R^{\ast } \times f_{\rm s} \times r) t} }$$

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.

A more complex model of CETI interaction – the Allee effect

The previous model is admittedly simple and shows a simple cooperative/competitive-based change in average lifetime based on the parameters r and N. A more realistic model will bound the growth of CETI and will also account for variations in the growth rate. A common model to accomplish this is the logistic growth model and its trademark sigmoidal curve. This varies the effective growth rate based on the relationship between N and a carrying capacity K. One drawback of the logistic model, however, is there are only two stable fixed points for the population of N – either N = 0 or N = K. Additionally, with a logistic model, once the population is N > 0 it will grow until it hits N = K. An alternate model that can account for threshold dynamics, such as a minimum number of co-extant CETIs for a positive impact, and no guarantee of continuous growth in L is preferable.

One longstanding model that incorporates these features 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 dynamics based on different regimes of growth above or below a threshold of population density. First is minimal to negative changes in the growth rates for increasing populations at low densities. These are populations whose density or number are below the threshold A. Once the population density passes the threshold, A, it will start to show an acceleration in its growth rate until opposing forces due to population limit criteria slow it down again until the population threshold K is reached where the growth rate is zero.

The Allee effect is superior over models that only change the growth rate due to carrying capacity since it also allows the modelling of synergistic effects that can enhance growth rates in addition to carrying capacity factors that limit the growth rate. In our model, the CETI lifetime instead of the growth rate of a population will be modelled. At low densities below A, the lifetimes of CETIs will be similar to or lower than what is expected depending on the value of A in the model. Based on the value of A this models a scenario where minimal or no interaction has little effect on CETI lifetime or where CETI interaction can be negative if the threshold to synergistic positive effects is high. Once the population reaches A, however, the effects are modelled as mostly synergistic until the carrying capacity ‘K’ is approached. In all models, the threshold A and beyond is crucial for positive effects on the expected lifetime L to take place. The model for the Allee effect amongst CETIs here is borrowed from the model elucidated in Allee et al. (Reference Allee, Park, Emerson, Park and Schmidt1949) and Courchamp et al. (Reference Courchamp, Clutton-Brock and Grenfell1999), also sometimes called the strong Allee effect.

The Allee threshold is a number that dictates when the density of CETI will lead to mutually positive or negative effects on L based on the value of r. If r > 0, the case we will investigate, between N = 0 and N = A, the change in L is minimal negative or even strongly negative depending on the value of A and r. However once N ≥ A, increases in L occur due to synergistic effects until they decrease again due to the effects of N approaching K. This explicitly models a system where small numbers of co-extant CETIs have little or even a negative association with each other while a greater number of co-extant CETIs leads to synergistic effects within limits.

In this model L changes to

(9)$$L = L_0 + rN\Bigg(1-{N\over K} \Bigg)\Bigg({N\over A}-1 \Bigg)$$

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

(10)$$\Delta N = R^{\ast } \times f_{\rm s} \times rN\Bigg(1-{N\over K} \Bigg)\Bigg({N\over A}-1 \Bigg)$$

Of note in equation (10), there are three fixed points for r > 0, N ∈ {0,  A,  K} but only N ∈ {0,  K} are stable. 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 behaviour is shown in Figure 2.

Fig. 2. Plots of ΔL based on different values of A. Based on L ₀ = 1000, 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:

(11)$$\left\{\matrix{ L = L_0,\; \hfill & {\rm if}\ N \leq A \cr L = L_0 + rN\displaystyle\bigg(1-{N\over K} \bigg)\bigg({N\over A}-1 \bigg),\; & \rm{otherwise} }\right.$$

The results of the dynamics assuming equation (11) are shown in Figure 3. Where A is low enough for N to commonly pass the threshold such as A = 5 in this example, the number of CETIs 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 CETIs.

Fig. 3. Plots of N over time assuming CETI emergence is based on a Poisson process with λ = 0.01, L ₀ = 1000 and N ₀ = 10 for different values of A. Other model constants are the same such as r = 1 and K = 50.