The 1990 book *Discordant Harmonies *by Daniel B Botkin, professor of Biology and Environmental Studies, is a wonderful treatment of Ecology, the subject, and Ecology, the policy, as it should be seen. Professor Botkin is first and foremost a teacher, and in doing so he brings the reader to see how ecosystems work, how interactions among coexisting species is complicated and nonlinear, and how human interventions based upon ironclad policies unresponsive to feedback and monitoring data inevitably fail.

Professor Botkin also gives the reader an education about various quasi-biological and quasi-botanical ideas, like those which attend notions of *invasive species*. That oughtn’t be a strange critique to readers of this blog, because I’ve noted it elsewhere. (I’ve noted it twice in fact.)

I promised to read it, and I did, and I took copious notes. This, however, is not a review, for any such review would need to be detailed and would have overwhelmingly positive things to say. This, instead, is a comment about a point which I believe Professor Botkin got wrong in the book, and I’m only qualified to say something because it concerns something squarely within my wheelhouse: The world of quantitative modeling. Note I have done a minimal review elsewhere.

I also need to note that Professor Botkin wrote an update to this book, one called *The Moon in the Nautilus Shell*, written a decade later, which may have other views about the opinions expressed in *Discordant Harmonies*. I cannot say yet. I am reading *The Moon*. I have not yet finished. I may update this comment if I find something which revises or clarifies.

The point I believe Professor Botkin got wrong in *Discordant Harmonies* was his interpretation of what the Lotka-Volterra differential equations model means, how it is to be used, and its implications. This is a major concern of the third chapter of Part I (“The Current Dilemma”). It’s possible for anyone to get something wrong. I do, often, but then I learn and correct. In the case of Professor Botkin, unfortunately, while, per *The Moon* he might eventually get it right, in *Discordant Harmonies* the critique of the Lotka-Volterra model is seized as a paradigm for what is wrong about a computationally based, “machine”-oriented approach to managing ecosystems. Such approaches to natural management may well have severe imperfections, but Professor Botkin’s interpretation of Lotka-Volterra cannot be their basis. Because his interpretation is, well, just wrong.

I should say my own prejudices come from an interest in sessile, botanical communities, most recently, mosses, and to the degree that may or may not have a bearing upon my view, I feel readers should know. My own first introduction to Lotka-Volterra came in the text:

Peter Yodzis, *Competition for Space and the Structure of Ecological Communities*, **Lecture Notes in Biomathematics**, 25, Springer-Verlag, 1978.

These equations are a set of coupled linear differential equations which offer a simple model to describe *aspects* of real ecological communities. They are not a complete description, nor were they ever proposed to be a complete description. But to the degree to which they successfully capture features of actual biological systems, notably ecological systems, their presence gives *powerful ideas* for biologists and people to think about such systems. Professor Raymond Pierrehumbert, a geophysicist, has a dedication in his textbook *Principles of Planetary Climate* which captures this idea:

R. Pierrehumbert,

who taught us to think deeply of simple thingsFor Arnold E Ross,Principles of Planetary Climate(2010), Cambridge University Press, frontspiece.

Lotka-Volterra equations have been used in many successful ways to describe many such systems, not all biological. They are an idea, that *powerful* idea. They are not a complete vision, nor are they, in themselves, the basis for policy.

The critical flaw in Botkin’s treatment is he took from Lotka-Volterra the limit or extreme cases as being the only contribution they had, and he has, in his text, no understanding that these are dynamical systems with complexity and nuance, describing a whole range of behaviors and interactions beyond and besides these canonical limit cases. Indeed, Lotka-Volterra systems come in many orders. The standard presentation and introduction to them is the predator-prey model with a single predator and a single prey, but these can be made endlessly more complicated, introducing a forage for the prey which grows at a fixed rate per unit time, or a forage which grows at such a rate and then is impacted by a drought, or a predator which preys on the previous predator as well as the prey, or many other complications, all within a quantitative framework.

Botkin’s key failure is a statement

Lacking the understanding to analyze and thereby criticize these equations, they accepted them on the basis of authority.

D. Botkin,Discordant Harmonies, page 41

Botkin then goes on to recapitulate what “they” took away from “these equations”, notions of stability. In fact, it’s clear the lack “of understanding” he cites not only failed “field ecologists” in their criticism of the equations, if Botkin is correct, it also failed them in the conclusions which the equations presented. For Lotka-Volterra systems, like many dynamical systems, are *anything* but stable, and anyone who concludes otherwise has grossly misunderstood not only Lotka-Volterra equations, but the whole significance of dynamical systems theory of which they are a small part.

This might be a small piece of a dark corner of biological, but names like Lorenz and Mandelbrot and Smale were out there, investigating *chaos theory* which is all about these kinds of systems. Indeed, they appear in a major way in the ** 1974** textbook by Hirsch and Smale

*Differential Equations, Dynamical Systems, and Linear Algebra*as Chapter 12.

My conclusion is that both for Botkin and his “field ecologists” the problem is and was not the Lotka-Volterra paradigm, but their failure to learn enough mathematics to appreciate what Lotka-Volterra and differential equations meant. And that’s not the fault of Lotka, Volterra, differential equations, or mathematics. That’s Botkin’s fault. That’s the ecologists’ fault.

And that is another reason why I, a retired statistician and quantitative engineer, am trying with the kind help of a few erudite and experienced bryologists and ecologists to bring mathematics back into biology in a practical way, if only in bryology.