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How many types of dinosaurs were there? And how can a theoretical examination of the question be applied to RPG games?

This is going to be one of the more unusual posts here at Campaign Mastery. For a start, it’s full of maths, and for another it’s about real science (at least at first). But the content is relevant to everything from Sci-Fi to D&D/PFRPG campaigns, and that makes it worth sharing.

I’m going to put the maths in bold so that readers can skip over my working and go straight to the results on the last bolded line – or to the next bit of explanation – simply by looking for the next bit that isn’t jumping up and screaming “Different! Different! Look at me!!” In addition, I’ll be putting occasional side comments and other irrelevancies in boxes to distinguish them from meaningful content.

How did I come to write this article?

A whole heap of elements had to gel to get me thinking the right thoughts at the right moment to get inspired to write this. The most proximate cause was watching the fourth and final part of a documentary series on Dinosaurs; not only was this about dinosaur breeding patterns, but (like most of the rest of the series) there was enough content recycled from earlier episodes that it encouraged drifting thoughts.

Throw in the occasional passing conversation with a twitter gaming bud, John Kahane, about Dinosaurs. I last mentioned John in connection with Stormy Weather – making unpleasant conditions player-palatable, about making climate-oriented encounters interesting for players in RPGs, who is heavily into dinosaurs for his Primeval the RPG campaign (or maybe that’s the other way around, and he’s into that RPG because he’s into dinosaurs).

Include various other minor stimuli and passing fancies lurking around in my subconscious and this is the result.
 

A brief warning to both Creationists and Science proponants

I totally respect my readers rights to believe as they wish, even if I disagree with those beliefs, and hope that they similarly afford me and the other readers of this site our own opinions.

I never want to be told what to think or what to believe, and that totally means that I refuse to tell others those things. I make my own assessments and judgments, and am happy to permit everyone else to do the same.

This article may be upsetting to those of a creationist doctrine or faith, because it assumes that the evidence that has been discovered, and the interpretation of that evidence by traditional science, is correct. If that is likely to anger or offend you, I would prefer it if you stop reading this article now. Upsetting readers – any readers – is not something I do willingly.

If you are able to treat what follows as an intellectual exercise, you should feel free to disregard that warning. If, on the other hand, you feel that a discussion of these matters will offend, save both of us the grief and move on to the next article, which I hope will be more acceptable. There are some good ones listed to the right of this page.

I will be monitoring comments very carefully; flame wars will not be tolerated. Theology- or ideology-based comments attacking the article or anyone else’s opinions, and any comment treating any faith in a derogatory manner, will not be accepted. Let’s all be civil to each other, Okay?

Mistakes can happen and things sometimes slip through cracks in the system (including the first draft of these words). If and when I become aware of such, they will be corrected ASAP.

The time available

According to the Geologic record, Dinosaurs ruled the Earth for 165 million years. There would almost certainly have been a prior period where they were around but had not yet become top dog of the evolutionary pile, though we have no idea how long that was. If it was as little as 1% of their reign, it’s 1,650,000 years – an impressive number in and of itself. If there was one first “true dinosaur species” at the beginning of their history, can we get some clue as to the number of different species of Dinosaur that ultimately came into existence? And how fast did evolution have to take place? What is the relationship between these two numbers?

Initially, I was very much feeling and fumbling my way through to a coherent answer to those questions, purely for my own intellectual curiosity. And I was very aware as I proceeded that I was making a number of rather dodgy assumptions. I’ll signpost those when we get to them.

To start with, we need some estimate of the number of generations, because it’s not time so much as the birth-maturity-breed-die cycle that dictates the pace of evolution.

To actually relate timespan to number of generations, you need to know the average lifespan of a dinosaur. Based purely on an intuitive value derived from modern animal lifespans, I picked an entirely arbitrary 15 years – remembering that this includes all dinosaurs from the very big to the very small, weighted according to relative populations, and counts every egg that was laid whether or not the young survived. I presumed that there were a large number of beasts that had quite lengthy lifespans (30-40 years), a few even more long-lived types (especially in the oceans), and an equally large number with shorter lifespans (1-2 years), with an even spread in between of a median value.

165 million divided by 15 years average lifespan over all species gives approximately 11 million generations.

The TV documentary in question, Clash Of The Dinosaurs, then provided a reality check by stating that more than 10 million generations of dinosaurs came and went. (I think they may have already suggested 10.6 million, but this was a factoid that I didn’t consciously absorb at the time, so I didn’t want to rely on it). But, between the vagueness and universality of the specific question, and the potential of a pre-rule time period of unknown length, I accepted 11 million as being a good round approximation.
622792_88155825

Nevertheless, there are all sorts of assumptions implicit in this number that might not be all that accurate. It’s a working number that can be adjusted as necessary, or as better information comes to light.

If the average is closer to 10 years, we get a number that’s closer to 17 million generations. If the average is closer to 20 years, we get a generation count of about 9 or 10 million generations. The first number is much higher, but the second is not all that far removed from the 11 million value employed. In fact, to get the same ratio of number-of-generations – 17 million to 11 million – we need an average closer to 22-23 years to a generation, which gives 7.33 million generations.

To me, that means that the most likely direction of any error in the estimate is in the direction of vastly more generations; it takes a much more significant extension of average lifespan to have a drastic effect on the estimate being approximately right.

The Percentage Of Progress

The next significant decision was to define the process of evolution from one species into a new one. Because I was interested in how big a percentage change per generation was required, it was natural for me to define this as 100%. In other words, evolution was defined as an average percentage change per generation of unknown scale, but when enough of these changes accumulated, the total would reach 100 and one species would have become another.

I didn’t realize it at the time, though I came to appreciate as work on the problem continued, but this permitted me to sidestep all sorts of thorny issues, like a functional definition of the difference between a new species and its immediate evolutionary ancestors. I had all sorts of thoughts about traits that bred true, and an inability to breed with the precursor species, (except possibly to produce sterile offspring) and meaningful change vs natural variations between individuals; I didn’t end up needing any of it, because I stated at the outset that whatever the functional definition was,, the species had achieved it when the total reached 100% accumulated difference from the average member of the precursor species. A bullet dodged!

The effectiveness of change

As a general rule of thumb, there are three obvious outcomes of any evolutionary change experienced by an individual:

  • The change can better fit the individual creature to the needs of its environment or ecological niche
  • The change can make no immediate difference, though it may provide a platform for an advantage at a later date after other evolutionary changes; or
  • The change can make the individual a worse match to the needs of its environment or ecological niche.

I assumed that these three outcomes were equally likely, and that members of group 3 would not survive long enough to make a meaningful contribution to the development of a new species.

That means that whatever the rate of evolution was, only 2/3 of it would actually contribute to the accumulated % of evolution to the new species.

I now realize that this is makes a number of really dubious assumptions, and is a massive oversimplification, to boot.

You could argue that the existing mechanisms of biology are so evolved already toward success – the legacy of many generations of prior existence through the evolutionary chain – that almost all changes would be negative or neutral ones.

You could counter-ague that negative changes are more likely not to survive to breed, at least, not as often as more successful and neutral variations, but that the whole negative evolution should not be thrown away as non-contributing. For example, being a worse fit for the current environment or ecological niche makes the creature potentially better equipped to move into a new ecological niche or penetrate a new environment.

At the same time, this doesn’t account for the potential for beneficial genes to be lost through accident, or negative ones to be conserved by random chance – or that these two chances balance each other out, at the very least.

You can also point out that one generation may move in direction X – longer legs, for example – while the next generation moves in direction anti-X, i.e. toward shorter legs.

What’s more, this employs an extremely simple, even Mendelian, standard of genetics that I know to be a vast oversimplification. Some changes can be beneficial in one respect and negative in another at the same time, for example the propensity for sickle-cell anemia amongst African Americans – a potentially lethal impairment in blood chemistry that increases resistance to Malaria.

So this value is riddled with uncertainty. But it’s hard to argue that there shouldn’t be some allowance for the distribution of changes into a pattern of more likely to be conserved and less-likely to be conserved, which in turn reduces the actual rate of evolutionary change to a smaller “effective” rate, and which is defined as the “efficiency” of random evolutionary change. In the absence of a more meaningful number, I’ve stuck with the 2/3 assessment rate.

Generations to a new species

I’ve used logarithms to make the math easier to handle.

Defining
R = % rate of evolutionary change per generation (average),
T = % of accumulated change to produce a new species (defined as 100%),
E = Efficiency ratio, defined above as 2/3, and
n = Number of Generations,

we get:

formula 1

formula 1

taking the logarithm of both sides and substituting 100 for T, we get:

formula 2

formula 2

Solving for n gives:

formula 3

formula 3

This is as good a time as any to point out that this all assumes that there is a consistent overall evolutionary rate, an assumption that is almost certainly not true to at least some extent.

The level to which this is the case is subject to considerable debate. So the results at any given moment in time are probably more-or-less in the ballpark but may be wildly and radically wrong.

Statistics can predict a general outcome – and make no mistake, this is a statistical analysis – but it fails miserably at predicting actual concrete outcomes.

Species count

Using the resulting n, it becomes possible to calculate how many species result in a given time frame, defined as a number of generations:

G = number of generations
S = number of species at the end of G generations, starting with 1
n = number of generations to a new species

formula 4

formula 4

It was at about this point, having achieved my original goal and plugged various numbers into the formulae – something that we’ll do in a minute – that I started catching a glimpse of how useful this could be to GMs, and expanding my scope beyond simply satisfying my original intellectual curiosity.

The +1 and -1 within the calculation exclude the original parent species and then add it back at the end. This means that you get the right result if G=n, i.e. there is only long enough to produce one new species.

Of course, this throws in a whole other unwarranted assumption: that if you start with one species and wait the required number of generations, that you will only get one new species. Call this the Boom factor. I’ve set it to 2, so that at the end of n generations, one species has become two.

The reality could be two, three, four, or fifty, depending on all sorts of external factors, like the number of empty ecological niches to be populated – when the Dinosaurs were wiped out, this is what enabled Mammals to take over as the dominant land species. Instead of 2, the exponential base would have been something much higher. Five, ten, twenty? I don’t know, and I doubt anyone has ever seriously asked the question before, at least not in this exact way.

If those niches are already filled, however, it’s likely that the result is going to be fairly low and close to the 2 shown above simply as a result of competition.

Something else that needs to be pointed out: This calculation is all about how many species arise, not how many survive for an appreciable length of time. To factor that in, you would need to determine a survival rate for new species and then allow for that in the exponent – because a species that doesn’t survive doesn’t serve as the source for a new species.

You can simplify this problem somewhat by defining “an appreciable length of time” as long enough for a species to evolve into a new one. But this factor is too complex for my simple mathematical logic, so I have chosen to ignore it.

Generations to reach a species count

Redefining S to mean a target species count, and rearranging that last formula to solve for G gives the number of generations needed to reach that target:

formula 5

formula 5

This isn’t quite as simple as it could be – there’s a function performed on a constant value – the Boom Factor of 2. I mention this so that if anyone wants to experiment with different Boom Factors, they know they need to use this version of the formula, and replace the 2 with their new Boom Factor, as shown below:

formula 5, simplified version

formula 5, simplified version

If, however, you stay with the assumed B factor of 2, doing that calculation (anti-log of reciprocal log 2) and moving the result out of the exponent makes life a little easier:

formula 6

formula 6 – that’s 2099 multiplied by the result of 10 to the power of (S-1)

Actual calculations: R=0.1%

So, let’s plug in an actual number and see what happens in 165 million years, or 11 million generations. I’ll do the calculations step-by-step:


Formula 3:
1. R x E / 100 = 0.1 x 2/3 / 100 = 2/3000 = 0.000666666666667
2. +1 = 1.000666666666667
3. log of this = log (1.000667) = 0.000289433187589
4. reciprocal (1/x) of this = 3,455.02880416
5. times 2 = n = 6,910.05760832 = approx 6910 generations.
Conversion:
6. x 15 (assumed average generation length) = 103,650 years.
Formula 4:
7. G / n = 11,000,000 / 6910 = 1,591.89580318
8. -1 = 1590.89580318
9. 2 to the power of 1590.89580318 = overload on my calculator.
10. Logarithms to the rescue:
calc7
10 (cont): 0.30103 x 1590.89580318 = 478.907363631275
11: anti-log (0.907363631275) = 8.08
12: so 2^1590.89580318 = 8.08 x 10^478.

Jaws may drop at will.

That’s 80, 800, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000 species.

Actual calculations: R=0.01%

Okay, so maybe 0.1% is too much. Let’s try 0.01%.


Formula 3:
1. R x E / 100 = 0.01 x 2/3 / 100 = 2/30,000 = 6.66666666667e-05
2. +1 = 1.00006666667
3. log of this = log (1.00006666667) = 2.89520004043e-05
4. reciprocal (1/x) of this = 34,539.9276747
5. times 2 = n = 69,079.8553493 = approx 69,080 generations.

Remarkably, slowing the evolution rate 10-fold has increased the generation count 10-fold. Exactly as it should have done!

Conversion:
6. x 15 (assumed average generation length) = 1,036,200 years.
Formula 4:
7. G / n = 11,000,000 / 69,080 = 159.23566879
8. -1 = 158.23566879
9. 2 to the power of 158.23566879 = 4.30212168052e+47. I don’t need the bigger powers-of-ten workaround this time.

Only in comparison to the earlier number does this result look anything other than completely preposterous.

430, 212, 168, 052, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000. Repeat: over 430 thousand million, million million million, million million million. This is less than the number of hydrogen atoms in the earth. In fact, it’s one for every 309 such atoms, or thereabouts.

Actual calculations: R=0.001%

More than anything else, this is putting into perspective exactly what eleven million generations actually means. It’s a huge number. So, 0.01% is still too much. Let’s try 0.001%.


Formula 3:
1. R x E / 100 = 0.001 x 2/3 / 100 = 2/300,000 = 6.66666666667e-06
2. +1 = 1.00000666667
3. log of this = log (1.00000666667) = 2.89528834264e-06
4. reciprocal (1/x) of this = 345,388.742556
5. times 2 = n = 690,777.485112 = approx 690,777 generations.
Conversion:
6. x 15 (assumed average generation length) = 10,361,662 years.
Formula 4:
7. G / n = 11,000,000 / 690,777.485112 = 15.9240858845
8. -1 = 14.9240858845
9. 2 to the power of 14.9240858845 = 31,088.338291.

From the ridiculous to the unlikely, but we’re in the ballpark all of a sudden, or so it seems.
622789_20652227

Estimating the real species count

31,100 species seems too low, by quite a margin. Or too high. There are 4,800 species of lizard in the world right now, and 5,416 species of mammals, and new species are still being discovered all the time. In fact, one reputable estimate is that Eighty-six percent of all land-dwelling species and 91 percent in the water have yet to be discovered and cataloged – most of them insects and fish and other primitive life-forms, of course. In contrast, Wikipedia states that only 1,000 species of dinosaur have been discovered, and according to some sources, up to 1/3 of these are actually misidentified juveniles of other known species.

So, taking the pessimistic view, let’s say that there are actually only 666 known species. And that we have only found 14% of the species – that’s doing as well as we have in terms of the life forms living today. That works out to 4,757 species that are available for us to identify. Now let’s factor in the spottiness of the geologic record, and assume that these are in fact only 14% of all the species that could have ended up leaving fossils for us – and that a large majority therefore did not. We end up with 33,979.6 – call it 33,980. Suddenly that 31,088 result is looking very plausible.

Eleven Million Generations is a lot of time for species to come and go. I would not be at all surprised if the real number turned out to be higher than these 30-odd-thousand, perhaps MUCH higher – by a factor of 1,000. Don’t think that’s plausible? Try this for size:

An alternative approach

This time, let’s start with those 5.416 species of mammals, and add in 1/3 of the lizard species, representing them moving into ecological slots that were left vacant by the deaths of the dinosaurs, That’s 7,016 species. Now, mammals are one of the most closely-examined types of animals out there; I don’t think it’s reasonable that 86% of mammals remain undiscovered. Let’s halve that to 43%. That gets us to 16,316 species.

The primitive biosphere was a bit simpler than the world today – there was no such thing as grass, and a lot of animals live on grass. So let’s say that not all the full modern biodiversity can be accommodated. Let’s play it safe and drop the number of ecological vacancies to 80% of what we have so far. This will also reflect the increased competition that results from more of the world’s landmasses being in one or two super-continents. 13,050 species is the count so far.

Mammals first evolved about 165 million years ago – that’s not a lot different from the length of time that dinosaurs were in charge. Their period of dominance is much shorter, however, and that’s what matters in terms of evolution – how long they have had to reach the current species count from a relatively small beginning. That’s about 66 million years, only 40% as long as the dinosaurs. So we aren’t talking about 11 million generations – if we use the same average generational length of 15 years, we (unsurprisingly) get to about 4.4 million generations.

Taking the R=0.01 value and applying it to only 4.4 million generations:

7. G / n = 4,400,000 / 690,777.485112 = 6.3696343538
8. -1 = 5.3696343538
9. 2 to the power of 5.3696343538 = 41.3448105493

But we don’t want this number to be just 41 species; our R must be wrong, we want it to come out to 13,050. Working backwards, I get R = 0.00230340274661 and n of 14,671. And taking that rate forward for the full 11 million generations, I get:

Formula 3:
1. R x E / 100 = 0.00230340274661 x 2/3 / 100 = 1.53560183107e-05
2. +1 = 1.00001535602
3. log of this = log (1.00001535602) = 6.66898281197e-06
4. reciprocal (1/x) of this = 149,947.90483
5. times 2 = n = 299,895.80966 = approx 300,000 generations.
Conversion:
6. x 15 (assumed average generation length) = 4,500,000 years.
Formula 4:
7. G / n = 11,000,000 / 299,895.80966 = 36.6794054658
8. -1 = 35.6794054658
9. 2 to the power of 35.6794054658 = approx 55, 026, 421, 772

I don’t think the 34 million or so that I proposed as an extreme but possible result at the end of the previous section – an underestimate by 10,000 – is all that unreasonable in comparison to the 55 thousand million that the mammalian evolutionary rate offers? It’s only 0.06% of what it could arguably be!!

Once again, just when you think you have a handle on the size of 11 million generations it turns around and bites you on the dinosaur tail! That, plus the power of compound interest and exponential growth are all it takes. Quite frankly, I think it highly unlikely that humans will ever identify more than a fraction of 1% of the dinosaur species that once lived, for all the reasons listed in this section – and that’s being very generous. The 666 or so that we think we have now is a hair under 2% of even the 33,980 prediction.
522601_84606698

The Sci-fi application

Planet X was colonized 20,000 years ago. How much genetic drift has there been? We very handily have a very accurate R% from the preceding section – which is why I reported it so precisely at the time: 0.00230340274661. Using the calculations given above, we get 300,000 generations; there hasn’t been anywhere near that much time. 20,000 years is only 1,333 generations. And that’s 0.44444% of the required timescale.

Ah, but we’re talking about willful genetic manipulation here. So forget that natural R – let’s take it right up to 1%, and not all of it intentional. And let’s assume that virtually of it is conserved or it never gets out of the test tube. And let’s assume that 5 years is plenty of time for an artificial generation, instead of 15. Now what happens?

Formula 3:
1. R x E / 100 = 1 x 1 / 100 = 0.01
2. +1 = 1.01
3. log of this = log (1.01) = 0.00432137378264
4. reciprocal (1/x) of this = 231.407892559
5. times 2 = n = 462.815785118 = approx 463 generations.
Conversion:
6. G = 20,000 /5 = 4,000 generations.
Formula 4:
7. G / n = 4,000 / 462.815785118 = 8.64274756528
8. -1 = 7.64274756528
9. 2 to the power of 7.64274756528 = 199.846371047.

There are nearly 200 fully-adapted new species. And even a small reduction in that generation length results in a massive increase in the number of generations and hence in the number of new species possible. 4 years instead of 5 is an extra 1000 generations and 694 additional new species. 3 years instead of 5 is an extra 2,667 generations and 10,635 additional new species!

Of course, what this fails to factor in is that it might take a long time to get genetic engineering right. Instead of almost complete conservation, we might be talking about a 99.99% failure rate – which is an E of 0.01, and takes that 1% and turns it back into 0.01%, effectively. The result is going to be only marginally faster than natural evolution. In fact, at an error rate of 99.993333%, it’s exactly as fast as mammalian evolution.

A failure rate of 0.1%, or 1 in 1,000 cases resulting in fatality, is enough to get a drug taken off the market by the in the US. A fatal reaction rate of 1 in 10,000 is marginal, and only used in the most extreme of cases, or where there is some way to mitigate against the possible danger. Only if the patient is already terminally ill is there any likelihood that such a medication’s use might be condoned. Even if the side effects are less serious, that 1 in 1000 cases may still be enough to leave the medication on shaky ground. This puts into perspective the suggested acceptable error rates for genetic engineering, and how low the E has to be.

Of course, if we aren’t doing the experimentation, or aren’t the subjects, we get a very different story. A 1% fatal-error rate in livestock or crops is entirely acceptable if there is a reasonable benefit to the other 99%.

It’s also worth remembering that 463 generations is still a lot, even at only 5 years each – better than 2300 years. At 3 years each, we’re talking 1,389 years to get a new species. 1389 years ago was the year 625. Persians were attacking Constantinople (unsuccessfully), Edinburgh had not yet been founded, and the Middle East was still reeling under an attack by the forces of Mohammad the previous year.

The Fantasy Application

The same sort of calculations – using a more natural evolutionary rate, possibly boosted by Magic – can be used to determine how long it has taken for all the creatures in the various monster sourcebooks have taken to evolve. It’s also worth remembering that herbivores are woefully under-represented, as I pointed out in my series on creating ecology-based encounters.

But things get even more interesting when you stop focusing on the whole population and look at just a single major group of species. Dragons, for example. Long-lived, implying a high generation length – perhaps a century. A known number of species – usually 12, but there can be more. How long have dragons been around?

Similarly, there are only 4 or 5 kinds of Elf.

If you listen to Tolkien, there are 3 races of Halfling – they have not yet diverged sufficiently to be counted as separate species, though they all have traits that breed true most of the time amongst their subpopulations. They are perhaps half-way to a dividing of the ways.

Using the step-by-step approach, it’s easy to set up a spreadsheet and play around with numbers to your heart’s content. And if the numbers you get don’t suit you, you know that you have to introduce some effect beyond the basic evolutionary rate to explain things – and can even quantify it. Is there some tainting quality that takes Metallic Dragon eggs and turns them into Colored Dragons? That would halve the number of species required. Of course, this would then require an adjustment to social practices of Dragons, or few of the colored dragons would survive hatching for very long. Perhaps the traits breed true most of the time after the initial metagenesis? No matter how you adapt and interpret the circumstances, you end up with interesting backstory for your game world.

And what if you end up with numbers that say that there are four times as many species as your sourcebooks present? That’s an open license to innovate and create variations – alternative species that will keep your characters guessing.

That’s the real value of these calculations: they offer insight and spur creativity. Make of them what you will.

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