Why is biodiversity ‘good’? To what extent is this sort of goodness even relevant to ecosystems—as opposed to us humans? I’d like to study this mathematically.
To do this, we’d need to extract some answerable questions out of the morass of subtlety and complexity. For example: what role does biodiversity play in the ability of ecosystems to be robust under sudden changes of external conditions? This is already plenty hard to study mathematically, since it requires understanding ‘biodiversity’ and ‘robustness’.
Luckily there has already been a lot of work on the mathematics of biodiversity and its connection to entropy. For example:
• Tom Leinster, Measuring biodiversity, Azimuth, 7 November 2011.
But how does biodiversity help robustness?
There’s been a lot of work on this. This paper has some inspiring passages:
• Robert E. Ulanowicz,, Sally J. Goerner, Bernard Lietaer and Rocio Gomez, Quantifying sustainability: Resilience, efficiency and the return of information theory, Ecological Complexity 6 (2009), 27–36.
I’m not sure the math lives up to their claims, but I like these lines:
In other words, (14) says that the capacity for a system to undergo evolutionary change or self-organization consists of two aspects: It must be capable of exercising sufficient directed power (ascendancy) to maintain its integrity over time. Simultaneously, it must possess a reserve of flexible actions that can be used to meet the exigencies of novel disturbances. According to (14) these two aspects are literally complementary.
The two aspects are ‘ascendancy’, which is something like efficiency or being optimized, and ‘reserve capacity’, which is something like random junk that might come in handy if something unexpected comes up.
You know those gadgets you kept in the back of your kitchen drawer and never needed… until you did? If you’re aiming for ‘ascendancy’ you’d clear out those drawers. But if you keep that stuff, you’ve got more ‘reserve capacity’. They both have their good points. Ideally you want to strike a wise balance. You’ve probably sensed this every time you clean out your house: should I keep this thing because I might need it, or should I get rid of it?
I think it would be great to make these concepts precise. The paper at hand attempts this by taking a matrix of nonnegative numbers 
If you want to learn more about the underlying math, click on this picture:
The new idea of these authors is that in the context of an ecological network, the mutual information can be understood as ‘ascendancy’, while the conditional entropy can be understood as ‘reserve capacity’.
I don’t know if I believe this! But I like the general idea of a balance between ascendancy and reserve capacity.
They write:
While the dynamics of this dialectic interaction can be quite subtle and highly complex, one thing is boldly clear—systems with either vanishingly small ascendancy or insignificant reserves are destined to perish before long. A system lacking ascendancy has neither the extent of activity nor the internal organization needed to survive. By contrast, systems that are so tightly constrained and honed to a particular environment appear ‘‘brittle’’ in the sense of Holling (1986) or ‘‘senescent’’ in the sense of Salthe (1993) and are prone to collapse in the face of even minor novel disturbances. Systems that endure—that is, are sustainable—lie somewhere between these extremes. But, where?
Azimuth 
I just happen to be reading “Picturing Quantum Processes”, and it occurs to me that when species are dependent on other species then that is a lot like entanglement/decoherence. However conversely, nature puts a lot of effort into destruction and copying which is at the opposite extreme from quantum.
Someone i knew once loaned me this book by Ulanowicz which had all these formulas–i think they are likely or somewhat correct but i found his presentation in that book ‘nonoptimal’.
He was quite popular in some theoretical biology circles. Note the 2nd author in the cited paper –B Lietaer https://sustainable-world.ch
he’s sort of famous in economics.
(the person i knew also was into Lietaer –and was into a UBI in part –universal basic income. Many people are as well but his view of UBI was it all was the same .
He ran a group house in which everyone had the same income (most of which was paid for by the ‘state’ or govt–most people’s incomes were forms of welfare, disability, etc.While the person who promoted the equal incomes sort of operated that way, he was also appointed as the ‘executor’ and hence determined who could get acess to their equal income.
-.
(Richard Levins formerly of Harvard had a slightly different analyses of bio and other diversity. biologicaldiversity.org or https:.//biologicaldiversity.org
I also sort of use the ‘reserve’ and ‘ascendency’ concepts (as noted some sort of scientific/mathematical/rigorous characterization of these might be difficult and contentious–who in their right mind would use their reserves to attempt to climb mount improbable?
as a side note there is a major metropolis or some say town called Reserve, New Mexico –last of the wild wild west. )
I got a nice note from my bank account today that my reserves (which i am saving for retirement) are now -44$. My ascendency might be whatever bills i have paid, and any other ‘capital’ (social, human or otherwise) i have accumulated.
‘we are walking on the streets of chance, where the chances are always next to none, and the intentions unjust’ (iggy pop, the song ‘baby’ on the album idiot, produced by david bowie).
If I recall correctly (it’s been a while :-) ), for a couple of billion years or whatever the most complex form of life on Earth was algae. Not much diversity, but apparently very stable.
I know it is a little off-topic, but since you mentioned the interplay between mutual information and conditional entropy, it reminded me about a question about the combination of entropies, in particular using the property of strong additivity (I think also sometimes known under different names, glomming rule, chain rule,…). This includes, in the general case, several conditional entropies as summands, and it is related to the question if the subsystems, which these conditional entropies represent, and which have to be combined, are independent (simple additivity case) or not (strong additivity case). And I wonder, if this occurs in a classical thermodynamical context ever, since I always only have encountered it in the context of information theory works, where in classical thermodynamics entropies always just simply add?
Maybe more related to your posting, one could imagine the more general case with several conditional entropies taken to be into account. Would that mean ascendancy (mutual information) has to be always a unique, single quantifier, while reserve capacities (conditional entropies) could be split into, in fact, arbitrary number of cases (but according to which classification)? Would that make sense?
It’s surely worth exploring these ideas in a variety of ways. But it just makes sense to take into consideration work that’s already been done and summarized.
In particular, in theory-based ecology these measures are based upon a number of special cases. (See the section on “Finiteness and Diversity”.) These fit into a much bigger framework, and “diversity” is addressed in terms of genetic diversity, not numbers of individuals. There is also a notion that species come and go all the time, due to genetic drift.
So reduction in genetic diversity is due to homogenization of populations, not primarily extinction rates. Clearly the latter can contribute to homogenization, but simply encouraging certain preferred species can do it, too, even if that “encouragement” is selection.
There are phenomena in genetic drift which can reduce diversity spontaneously, like coalescence and genetic bottlenecks.
Many other scholars address this with mathematical theory, including Hubbell. Google Scholar gives 7900 citations to articles about that theory from 2015 to 2020 alone.
Other references include M.A.Nowak’s Evolutionary Dynamics (2006) which is mathematical and uses diversity of genomes in SIV/HIV to explain an extreme benefit to that virus, and uses it to touch on cancer proliferation, something which Durrett has examined, too.
Finally, Kokko (2007, Modelling for Field Biologists) recommends in her chapter 3:
D. A. Roff (1997), Evolutionary Quantitative Genetics.
S. H. Rice (2004), Evolutionary Theory: Mathematical and Conceptual Foundations.
R. McElreath and R. Boyd (2007), Mathematical Models of Social Evolution: A Guide for the Perplexed.
as richer tutorials on modeling and understanding quantitative genetics. Check out her chapter, too, though.
Thanks for the references! I know I have a lot of reading to do. I’ve decided I’ll blog about it as I go along. There’s a particular line of thought I’m trying to clarify, and Ulanowicz’s work seems helpful toward clarifying it, but I don’t really buy his particular formulas for ascendancy and reserve.
I doubt I’ll want to actually work on genetics and genomes. Since I’m a mathematician of a certain rather abstract ilk, I’m aiming for something a bit more general, and simple. But of course it’s important for me to learn more about genetics, population biology, evolutionary biology and the like, because generalities not based on knowledge of specifics tend to go astray.
Great.
Actually genetics and genomes are abstract. After all, they are coded versions of biochemical constructs, even if they express themselves biochemically:
The thing of it is I don’t know how to do diversity without addressing fitness. A definition?
That’s from Kokko (2007). And, to me, it kind of ties hands trying to address diversity without fitness, and, transitively, genetics.
Stay tuned and you’ll see what I’ll do. I don’t think it will look much like genetics—it’ll probably look more like thermodynamics or information theory combined with network theory. In other words: stuff I’ve already studied and written about.
The economists have a concept of diversity (vs. monopoly).
Clearly the authors have never been in a woman’s shoe closet, or they would understand that “reserve capacity “ is a) infinite and b) never , ever “random junk”.