How many spiders are in your home (Part II)

18 Apr, 2026

This is part II, providing a justification for my previous post: How many spiders are in your home?.

I will now attempt to prove there is some very, very small realm of legitimacy to this function. For reference, the function:

$$\begin{array}{c}f(x)=0.20\sqrt{m(x+t)}+0.35(w)\\ \\ \mathrm{W}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{x}\mathrm{\ge }\mathrm{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{\ge }0.\\ \\ \mathrm{L}\mathrm{e}\mathrm{t}\mathrm{m}\mathrm{=}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{y}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{e}(\mathrm{i}\mathrm{f}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{t}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{s})\mathrm{,}\\ \mathrm{t}\mathrm{=}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{b}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{,}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}(\mathrm{n}\mathrm{o}\mathrm{.}\mathrm{o}\mathrm{f}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s})\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{,}\\ \mathrm{w}\mathrm{=}\mathrm{n}\mathrm{o}\mathrm{.}\mathrm{o}\mathrm{f}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{y}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{e}.\end{array}$$

General assumptions to start with...

• The function assumes that no spiders inhabit the home prior to the end of construction. Spiders only inhabit the home immediately following the completion of the home's construction. We have to start at a zero point somewhere ¹.
• The function assumes space and time is bounded by reasonable home-sized and temporal values. For example no home is 1m^2 or 1,000,000m^2. It's less accurate at the extremes. Similarly for time, the function works best at the start and the middle so to speak. The extreme end of the function is less accurate (technically it never ends reaching infinity). I am assuming most homes are <100 years old. Anything beyond 100 years is rare ² with homes usually knocked down and rebuilt before then. I also anticipate the vast majority of people using this function are in a home >6 months old and <100 years old. Shout out to anyone in a 1000+ year old castle.

A square root function

Arguably the most important decision made was to model the spider population over time with a square root function.

My function has a rapid increase in the spiders population at the start. However characteristic of the square root function is an ever decreasing growth rate (that importantly remains positive, increasing the total population indefinitely). Remember it is not just one species of spider, it is multiple, a community of spiders that I am modelling. Therefore the growth rate can be fairly rapid at the start. I will call this phase 1. Most, if not all the spiders in phase 1 (at the start) will be from outside spiders moving into the newly built home. It is only later, when the inside population is able to reproduce offspring, the inside population can continue to grow, although at a less rapid rate, and is when phase 2 starts. Phase 2 is represented by the tail of the square root function. In my opinion, Phases 2 also occurs at an ever decreasing growth rate. This is as habitat availability reduces over time as more and more spiders exploit and populate new areas within the home. Eventually the spider population in your home will stabilise and reach carrying capacity. In an urban home setting, the idea of carrying capacity I consider to be mostly true, however there are some caveats (as I discuss later).

So then, if I acknowledge there is a carrying capacity, why not just use a logistic function? This is a very fair point. Especially given the long history of the logistic function being used for population dynamics in ecology. I decided against this for a couple reasons.

Firstly, to my knowledge, the logistic function has only been used to describe population dynamics in a single specific species - and not multiple species (such as a community of spiders, which is what I need). Consequently, this makes it hard to determine the main requirements needed for the logistic function - or even to modify such a function. As a minimum, I need to know what the initial population is, the intrinsic rate of natural increase (again for the entire community) and the carrying capacity. Instead I had to create my own constants (pretty much out of thin air - although not entirely) 0.20 and 0.35 (refer above). I will explain how I got these values later. This may seem hypocritical, as I was happy adding co-efficients to variables (such as no. of windows), so why can I not have a stab at assigning a value to the carrying capacity?

The true reason was I wanted the function to be easier for me to understand and reliant on less estimations. When adding 0.20 or 0.35 to the function, I am at least able to understand how these co-efficients affect the function. To properly calculate the carrying capacity or intrinsic rate of natural increase, you need to be able to estimate and combine many more values (e.g. birth rate, death rate, survivorship, migration, immigration etc). There is an amalgamation difficult to untangle - and honestly I don't have enough experience to understand how logistic functions work properly, to also be able to modify such a function! At least with a square root function, I felt like I understood it more, so I have more confidence in modifying it.

Finally, even if I did know with certainty all these values, knowing such things doesn't guarantee that your function is any more accurate. At the end of the day it is still a vibes based function. Without empirical evidence what is to say a logistic function is better than any other type of function (and this includes my square root function too). So why should I bother making a function at all? - because it's cool! It's better to make a wrong function than to cower away and make no function at all.

I mulled over other types of functions. I could have used a log function, but I considered the growth rate to be too rapid (at the start) and not growing fast enough towards the tail (in fact it almost asymptotes, a problem I explain next). In comparison to logs, it is easier dilating a square root function to taper the rapid growth rate at the start.

Another main reason for choosing the square root function was that in the real world asymptotes are very hard to place, and put a hard limit on the population. This was also why I didn't choose a reciprocal -1/x function either. So many variables can increase or decrease the total spider population in a home. I ended up preferring to keep the population of spiders ever increasing, over longtime spans, even if this becomes more unrealistic (refer to my second dot point above).

In a nutshell this was why I selected a square root function.³

I will now go over in detail on how I decided specific variables and constants in my function....

Variables and constants

In my mind, habitat availability and prey abundance are the two most important factors in determining the total number of spiders in your home (aka the spider abundance).

Therefore my function in someway or another needs to, at the very least, be able to model these two factors over space and time. To transform the function into three dimentional space and time, I am using three variables:

• Size of home
• Time since construction
• No. of windows

In will now summarise...

Size of home (m)

The bigger the house the bigger the potential for spiders to inhabit it.

I am using the size of home to act as a proxy for habitat availability. Pretty simple, I believe that total habitat availability increases as the size of the home increases. The bigger the home, the more space available for spiders to inhabit. As the spider population grows, less and less habitat remains. There is only a limited amount of space. Resulting in habitat availability reducing over time.

Time since construction (t)

Time is very important variable. The age of home, or time since construction (t) moves the curve horizontally along the x-axis. Most importantly it determines what stage the spider population is in (phase 1 or 2). Time since construction provides crucial detail on how long the spider population in a home has existed - and therefore have had time to increase. The longer the time, the more potential for spiders to enter, reproduce and populate the home.

More technically, I mean the time since the home has been built in its' entirety. Or since complete renovation. After the previous home was knocked down and rebuilt again. As small renovations or extensions are not likely to cause the spider population to plummet to 0. So I am referring to the age of the home since completed construction.

More mathematically, it is important both of these variables t and m (representing time and space), are inside the square root. For m, this determines the rate at which the population increases and decreases over time (as habitat availability reduces). Larger homes will have a faster growing population than smaller homes, and vice versa. Whilst for t, this gives the function the ability to adjust time, shifting time to the present day. As my original question is asking how many spiders are in your home now, not in the past or future.

Upon reflection, I think both variables seem logical enough. Justifiable maybe.

It is my next variable I will have to explain in more detail...

Total no. of windows (w)

I do not think that that prey abundance necessarily reduces with time (like habitat availability). Therefore I placed it outside of the square root. In my view, it fluctuates throughout the year. At times there may be more prey than the spider population can deal with. Such as in the spring and summer, where insects are plentiful and more active. Other times in the year there is less. Various species will have different strategies to deal this change in supply over time.⁴ At the community level though, In my function I am ignoring the impact fluctuating prey abundance may have on the spider abundance (and focusing more on the net increase over time).

Hang on a second... wouldn't the spider community need a steady supply of prey? Why am I ignoring this?

It is because spiders are ambush predators, they don't expend energy hunting down their prey. It's more feast or famine. I believe there is a surplus of prey in the spring and summer, and they wait out during the colder months. They make the most of summer, when every fly, midge and moth becomes trapped in the home.

But that being said, there are still some insect prey inhabiting homes all throughout the year, and spiders would still be opportunistic with stochastic encounters with their prey. Also other urban dwelling insects do return, and start populating newly built home - just as spider do. So inside homes, prey does increase over time, supporting the growing spider community.

Further, spiders can go very long periods without food. In fact it has been reported their standard metaolic rate is on average ~50% less than all other arthropods - but this may be an over-generalisation, as metabolism vary greatly between species, also onset of starvation is determined by initial lipid reserves and size of spider (and other factors).

But how can I estimate prey abundance?

Windows are key. They not only allow a method for entry/exit when open. They are awesome at capturing prey. When trapped inside during the day, flies, midges and other insects will seek the natural outside light. I often watch flies tying to escape and buzzing into the windows repeatedly. Where lurking in the corners of these windows the patient spider is waiting with a web ready to capture them. Windows attract all sorts of insects at night too. In Nephila plumipes, an urban dwelling species, urban lighting (and I assume this includes the internal lighting of homes) can increase prey capture. Additionally, this increased prey abundance resulted in increased fecundity in this species. Thus this links prey abundance to spider abundance.

However linking artificial lighting to being beneficial for all spiders is a little difficult. In fact I discovered a counter example. For Eriophora biapicata artificial light reduced the number of eggs produced. So the benefits of this artificial light may be species-specific.

Regardless, I don't think it is controversial to state that in general, prey being abundant is good for spiders, irrespective of night or day, and windows are excellent locations for spiders.

I have documented evidence of this myself:

Spiders on the edges of windows (inside)	Spiders on the edges of the same windows (outside)

Overall I am using total no. of windows as a proxy for prey abundance. So how on earth did I get the 0.35 coefficient?

Well in my unit 12 out of the 17 total windows had at least one spider (on the window or surrounding window frame). Despite the obvious problem of having a very small sample size, this suggested the probability of encountering at least one spider on any given window is 70.5%. In fact you might find multiple. Now I admit my windows are by far from the cleanest, and I am quite lazy/ don't ever really move spiders out of my unit. So I do consider this to be an overestimation. So I lowered this. The lowest value I was prepared to go down to was 0.35, or for every window there is 35% chance of finding one spider. Thus the coefficient for total number of windows is 0.35.

There is one final constant I have left to explain... 0.2

The intrinsic rate of natural increase in a community of spiders

I am taking some liberties now. It is really hard to give an accurate co-efficient to determine the natural growth rate of a community of house spiders. Every spider is different, species can have drastically different life cycles. Fecundity, gestation period, survival and longevity all changes dependent on the species, with there is a wide range possible values. While some spiders live for 1-2 years, trapdoor spiders can live up to 40 years. Not just this, the climate and temperature of the house all impact spider populations. Interestingly, increased temperatures provided via the urban heat island effect, may also benefit spiders and is correlated to with increased size and fecundity.

All in all to get some sort of consensus is very difficult. Further, the research hasn't been done on many species, so a lot is currently unknown. The total spider abundance can vary greatly even between houses as highlighted by this study.

But the answer to this is 0.2. The value seemed to be a reasonable co-efficient to serve as a proxy for the intrinsic rate of natural increase in a community of spiders. I had to choose something.

Finally, there are a couple other factors that would be remiss not to write about...

Human inhabitants

Human inhabitants can have great influence over the habitat availability. Regions with high human disturbance. For example, if a spider was to start making a web on the fridge door handle, this would be quickly removed by humans. Humans will move spiders outside/kill any spider that makes the unfortunate mistake of being in an exposed, noticeable locations in areas humans occupy. Whereas if that same spider was to make a web behind the fridge, it is far less likely to be removed.

As such I believe a home unoccupied for long periods of time could result in a small surge in the total spider population. Without humans, spiders can suddenly occupy new areas in optimal locations humans would have usually occupied. However this is not guaranteed. In this circumstance, we are assuming habitat availability is the limiting factor. However if prey abundance is a limiting factor, the presence or absence of humans likely won't make much difference.

For this reason I did not include human inhabitants as a variable in the function. Further, whilst not strictly assumed mathematically, in reality almost all readers of this blog will have some sort of home with internet connection. Therefore in almost all cases, we can assume human inhabitation of any given home. Shout out to any non-human species reading my blog.

Interactions between variables

I believe time since construction (t) also interacts, very slowly, with habitat availability (whereupon size of home is being used as a proxy). There is a small interaction between these two variables, (t) and (m). I will now explain.

Early on, immediately following construction, there is lots of available habitat. The more the populations establish, and occupy optimal locations in your house (such as the window frames), the less habitat is available and thus reducing potential for population growth. Soon the only places left are unattractive areas, that make survival more difficult (either by having less prey or being in a hostile, exposed location). And this is partly my justification for the square root graph, and the ever decreasing population growth rate. Although importantly it never asymptotes. My evidence for this is the following:

I believe as the home ages, the more nooks and crannies start being exposed. Think about cracks in wood as it becomes rotten, leaks in roofs. This will increase the total habitat availability, albeit at a slow rate - and can occur in optimal locations too (such as in window frames). Therefore the total population will still increase very slowly over time just due to the fact the house is aging.

This is evidence against the logistic function or any function where there is an x-axis asymptote. It further supported my decision to use a square root function.

Conclusion

I know I may have made a mountain out of a mole hill here. It is quite a long explanation for a very simple equation. But I would feel wrong if I just presented my function without any explanation. Not showing the working out behind my function, in my view, makes it have no legitimacy at all! And it is a completely worthless function. At least now I can argue there is a tiny realm of legitimacy. You could almost argue it is more of a theory than a function if anything.

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Footnotes

#o.g. obs