# Remembering Don Ludwig

(November 14, 1933-March 30, 2018)

By Leah Edelstein-Keshet

I had not seen him for several years when I found out that Don Ludwig was ill, and then months later, and just before a planned visit, that he had passed away. It was a shock. The distance to his White Rock abode, his active retirement elsewhere, and finally his ill health had largely kept him away from UBC as an emeritus professor. And yet, his wisdom, mentorship, quick wit, and quiet, understated presence are a vivid memory that I will always carry with me. Don was, in many respects, my mentor: a key influence in my career, and a distinguished colleague whom I was privileged to know. He was a senior and respected luminary, and I was a “youngster” with my first real appointment when we overlapped at UBC, and that time was all too short. But it left me with a life-long appreciation of Don as a scientist, a mentor, and a true scholar.

From the Department of Mathematics archives, I learned that Don Ludwig studied at NYU (BA in 1954), and completed a PhD there with Richard Courant on Propagation of Singularities in Wave Motion in 1959. His early work in classical applied mathematics and hyperbolic partial differential equations (PDE's) meant that he had an influence on the relevant material in the classic Courant-Hilbert text, Methods of Mathematical Physics (In the preface to Volume 2, Courant wrote “Among younger assistants I must particularly mention Donald Ludwig whose active and spontaneous participation has led to a number of significant contributions” p ix). After positions as a Fine Instructor in Princeton (1960-61), and an Assistant professor at UC Berkeley (1961-64), Don returned to NYU as an Associate Professor (1964-69), and was then promoted to Professor (1969-1974).

Dating back to the 1960’s, Don’s published work includes Singularities of the Riemann Function, Solutions of Hyperbolic Equations, the Radon Transform, Caustics, and Propagation of Singularities, to name just a few. According to colleagues, “Don’s knowledge of mathematics was encyclopedic”, in both pure and applied mathematics. “.. there was a lot to learn from him. One thing I learned from him was the care and feeding of asymptotic expansions, of which he was a master,” said John Walsh, an emeritus UBC Mathematics professor. “He helped me understand projective geometry surfaces mapping in the context of anisotropic wave propagation”, recalled Matt Yedlin, a UBC professor in Electrical Engineering. Former PhD student Marc Mangel (now Distinguished Research Professor of Mathematical Biology at the University of California, Santa Cruz) recounts a typical anecdote: “I got to the fourth power in the exponent [extending the Airy function], and thought I had discovered a new special function. Don’s reaction was ‘Oh, that's the Pearcey function’, immediately finding the reprint from 1946 in his filing cabinet.” Mangel also recalls that Don taught him the importance of first considering a basic canonical problem before diving into a harder problem.

In the early 1970’s Don’s interests gradually shifted to ecology. Frank Hoppensteadt recalls that the mathematical biology seminar at the NYU Courant Institute of Mathematical Sciences had been initiated around that time. According to him, “One of the spin-offs of the seminar was a project with Wasily Leontief (Nobel prize in input-output economic analysis) funded by a NOAA Sea Grant to study east-coast US fisheries… this stimulated Don's interest in ecology ... Don and I also started working on a problem in polio inoculation strategies”. The 1970’s were a heady time in mathematical ecology, with novel research of lasting significance that had a profound impact. “Our seminar was very lively and timely, presenting some of the early work on chaos theory”, said Hoppenstaedt. “There was Ricker’s work on fisheries, the discovery that period three implies chaos by Jim Yorke, Ed Lorenz’s paper on chaotic convection.” At about that time Robert May wrote his influential paper, showing that the simple discrete logistic equation has extremely complex (chaotic) dynamics.

A fateful meeting between Donald Ludwig and Colin Clark took place at a conference on Mathematical Biology organized by a University of Victoria professor, Pauline van den Driessche, around 1973. “I had reviewed Don’s paper on the wave equation for Mathematical Reviews,” said Colin, “but this was our only contact prior to 1970. Don and I both switched from PDEs to Math Biology in the early 1970s”, he continued. “When Don expressed delight in the BC environment, I told him that the UBC Math Department was looking for good people [to hire].. ”. As it happens, in 1973, UBC was advertising for the position of Director of the young Institute of Applied Mathematics (IAM). “Don was never seriously interested in an administrative position”, says Brian Seymour, an emeritus UBC Mathematics professor and former IAM Director (1986-1993), “but the committee eventually recommended that we hire two full professors, Don Ludwig and Fred Wan- Fred as Director.”

In this way, Don Ludwig was appointed as Professor at UBC in 1974. As Clark recalls, “Don soon began collaborating closely with professional biologists, such as Buzz Holling, Carl Walters, and Ray Hilborn. We both published widely in Biology journals (plus Economics in my case), mostly jointly with biologists.” Shortly after his move to UBC, there followed a series of papers on stochasticity in ecological topics such as epidemiology, and species competition; Don also began to work on resource management and insect outbreak systems.

In fact, Don had started working on stochastic models of population phenomena even before leaving NYU. His work on Stochastic Approximation for the General Epidemic was published in 1973 in Mathematical Biosciences, followed by a monograph, Stochastic Population Models, which appeared in the Lecture Notes in Biomathematics, a classic series of the time in 1974. About Don’s work in the area, Marc Mangel, tells the following anecdote: Don spent a sabbatical at the Weizmann Institute of Science (WIS) with Prof. Lee A. Segel. (LAS was my former PhD supervisor, but Don’s visit did not overlap with my time there.). Ludwig’s seminar on stochastic epidemics was attended by Sam Karlin. As Don recounted “Sam tore into it with all sorts of criticisms. Then”, Don said with a twinkle “he ended the attack with ‘you've given me a good idea’.” Mangel added another story: In the 1980’s, Don had a T-shirts emblazoned with the words: “What about the stochastic effects?”. He would often wear this discretely under his dress shirt. After a seminar, Don, would occasionally stand up, tear open his shirt, like superman, and expose the relevant question.

By the time Don Ludwig had arrived at UBC, Colin Clark was already well-known for founding the field of economics of fishing (see review by Munro and Sumaila). Don turned his attention to resource management and optimal harvesting in the late 1970’s and eventually followed Colin with his own “fishing expedition”.

As it turned out, Don too made a profound impact on this area, though in a distinct direction. To understand the timeliness of his work it is helpful to recall the history of fisheries in Canada. The colonization of the east coast of Canada and Newfoundland by Europeans since the 1700’s was driven by abundant fisheries, among other factors. The social fabric of the Maritime Provinces had been woven through with the tradition of cod fishing, then considered as a limitless resource. In fact, in the 1960’s, cod populations rose to dramatic heights not previously seen for at least a century. In the mid 1970’s, a precipitous decline was followed by a mild recovery, resulting in optimistic projections. But, by 1992, the fisheries had collapsed to the point that a moratorium on fishing was declared, with devastating effects on the communities that had relied on fishing for a livelihood.

In 1993, Don Ludwig (with Ray Hilborn and Carl Walters), wrote an influential paper that explained some of the policy errors that had contributed to the catastrophe. A numerical indicator denoted “maximum sustainable yield” (MSY) had been used as a guideline to policy makers for how much fish could be “safely harvested”. The MSY is a deterministic quantity. But nature is variable, fickle, and noisy. Conditions change over the years, adding stochasticity into the mix. The naïve assumption of determinism could, in some circumstances, overpredict population size, or under-predict mortality and environmental pressure. Furthermore, human greed, and tendency to overlook long-term consequences had not been considered previously in forecasts. Ludwig’s work with Hilborn and Walters put paid to the naïve assumptions and explained the reasons for the collapse. The work was featured in a full-length article by William K Stevens in the New York Times (April 20, 1993).

Don continued to work on resource management well into his retirement (in the mid 1990’s), including publishing an article on ecology, conservation and public policy in The Annual Review on Ecology and Systematics in 2001. Former PhD student Lynn van Coller recalls “On a visit 5 years after my PhD, I discovered that Don was writing papers to challenge government policy on fishing in the local waters which was severely depleting fish stocks. He was very outspoken and confrontational in his writing. I asked him about it and he said that he had already made his name, and now it was time to give back.” Don was not afraid to ruffle some feathers (or, perhaps “fish-scales,” for a more appropriate metaphor). In Don Ludwig’s own words (preserved in email to van Coller), “[My worry] is that humans seem to be unable to restrain themselves from devastating the planet. A particular component of that is the absurd notion that economic growth can and should be unrestrained and perpetual. The so-called miracle of compound interest is a corollary. I have been pondering the implications of this weird notion since 1972, when Colin Clark showed that an economically rational thing to do with many whale species is to harvest them all and invest the proceeds.” On the other hand, in his typical humor, he also wrote to her “It helps to think about geology and geological time scales, because then our present environmental debacle doesn't seem that important. I'm sure that a few thousand, or maybe a few million years will straighten things out nicely.”

Aside from their work in resource management, Don Ludwig and Colin Clark have shared many interests during their overlapping careers (occasionally to Colin’s irritation). In 1995, at a Pacific Northwest Workshop in Mathematical Biology (PNWWMB) dedicated to honoring Don and Colin, both recently-retired, Colin said something to the effect: “Whatever I was working on, whether it was fisheries, or resource management, there was Don, copying me and getting into my area. So I’d switch to a new area, but there he was again, following me..” Interestingly, Clark and Ludwig published only one joint paper on conservation biology (with Tony Sinclair, in 2000), several years after they had both retired.

Walsh paraphrases some of the essence of the theories that Clark developed and that Ludwig explained to him: “Basically, money grows much faster than resources, so you make less by properly managing a resource than by just turning it all into money right away, and investing that. Not only do you get a greater return, you also have the windfall profit for selling it in the first place… Don had the ability to get right down to the essentials, and explain other people's work, as well as his own.”

Personally, my all-time favorite paper by Don Ludwig is his seminal work on the spruce bud worm outbreaks, published in 1978 with DD Jones and Buzz Holling. This paper is a true “classic”, cited over 800 times in subsequent scientific literature, and adopted as a set-piece in some of the most popular books on Mathematical Biology, including those of JD Murray, of Steve Strogatz and many others. The reason is clear: it is a key illustrative example of excellence in modeling.

What makes this paper outstanding? First, like the work on fisheries collapse, it addresses an ecological problem of serious economic importance. Spruce budworm outbreaks cause significant damage to forestry, notoriously hard to control. Second, the paper is an elegant minimal model that manages to capture the essence of the interactions (logistic growth of the budworm and mortality due to avian predation, quantification of forest by branch surface area and energy reserves). Third, the paper exploits distinct timescales of forest and insect growth to study the fast (budworm) and slow (forest) variables separately, reducing the analysis to an elegant sequence of 2D phase-plane diagrams. The geometric analysis is both intuitive and informative. Fourth, steps in the process of model derivation and analysis are presented in pedagogically, in a very readable exposition. Finally, aside from the simplified qualitative description, the model is tested with estimates of actual growth rates, maximal densities, and key dimensionless quantities, as well as other parameter values taken from the biology. Numerical solutions demonstrate the presence of typical cycle of outbreaks. In short, this paper, coming ahead of most modern math-biology texts was visionary in its geometric and qualitative ideas, and in the way it links mathematical reasoning with biological facts and hypotheses. It is as current today as on its date of publication 4 decades ago. It was a perfect example of bistability for a lecture I delivered recently, shortly after Don Ludwig passed away.

Don Ludwig made a huge impact on ecology, on science, and on applied mathematics. For this he earned prestigious awards (Guggenheim Fellow l970-7l, UBC Killam Fellow l978-80, Fellow of the Royal Society of Canada l983). He also held influential positions on editorial boards of many journals (Mathematical Methods in Applied Science, Bulletin of Mathematical Biology, Journal of Mathematical Biology, Theoretical Population Biology, and Ecology). He was Chief Editor of Acta Biotheoretica 1986 and Managing Editor of SIAM Review l977-81.

But aside from these broadly influential roles, Don also made a direct impact on people who knew him. Don’s former departmental colleague John Walsh said “Don made a big contribution to our department, and not just for the mathematics he published, which was impressive. He was always happy to welcome you into his office when you wanted to learn something from him. Or tell him something. He was always interesting, both for his math and for his humor. He knew a lot, and was happy to share it.” Carl Walters recalls that “Don was an amazing character, who was always gentle and pleasant, even when showing us how silly and overly complicated most of the mathematical models were that we were trying to use for ecological data. He would look at a math problem that we identified, tip his head to the side, and write down the answer; he then became a bit upset when asked to show his proof -- he was very much an intuitive thinker.” While the list of students who studied under Don is not long, several of his former students are internationally-known scientists. Prof John Rinzel (former head of the Mathematical Research Branch of NIDDK at National Institutes of Health (NIH), and Professor of Neural Science and Mathematics NYU since 1997) studied with Don Ludwig at NYU. John writes: “I became at risk for getting drafted to Vietnam midway through the first year of my PhD (1967/1968). But a career in the US Public Health Service at the NIH would be a way to avoid the draft. Within 6 months, with Don’s help, I got through a thesis-based MS and headed off to NIH. Don’s guidance and understanding were significant.” Later on, in 1973, Rinzel was considering offers for a post-PhD trajectory. After he described negotiations with one university, Don commented: "If that's how they treat you when they are trying to recruit you, imagine how they'll treat you if you agree to come." John recalls that notable quote and passes it on as a life-lesson to many others.

Thinking back to student days, Marc Mangel also recalls that “Don used to say that they build little anxiety sensors into a keyboard, to make things work poorly when you are anxious about getting them done.”

Another one of Don’s PhD students, Lynn van Coller, eventually made a career in banking, quantitative portfolio construction, and consulting. She recalls that Don taught her to think well. “He never gave me an answer, just explained where he thought there was something that could be done better, and why. And how to have the most impact when writing for an audience.” When Lynn asked Don Ludwig how he decided that it was time for her to write up her thesis, he replied “It looks as though you’ve had enough and want to move on.” And as it turns out, he was correct, according to Lynn.

I did not know Don Ludwig well outside the university setting, but I gather that, among many avocations, outdoor activities played a major role in his life. Neil Fenichel (member of Electrical Engineering department at UBC until 1984) had known Don from NYU, and arrived at UBC at the same time, in 1974. He was part of Don's cross-country skiing group. According to Neil, “Don's specialty was skiing into creek beds. We used to take math department visitors on "memorable" ski trips.” John Rinzel was one such visitor in the late 1970’s. “I was a novice, and we were unable to rent cross-country skiis on short notice,” he recalls. The solution was that John would follow Don and Neil on snowshoes. “We went to Whistler, [and there were] a lot of ups/downs and through the woods, deep snow .. 11 miles, that seemed like 111 miles. I thought my legs were going to collapse.” John remembers that “it was an unforgettable experience.” Frank Hoppenstaedt also recalled that “Don's family and mine camped and skied together numerous times in Utah and BC. Cross-country skiing with Don was usually a thrill packed adventure.”

My inbox still contains an email from Don, sent on 6/3/16. He wrote “I’m happy and proud to see that you have won an award from CAIMS...I’m glad that I was able to help bring you to UBC.” Thank you, Don, for believing in me, for your mentorship and for being such an important influence in my scientific career at UBC.

References:

D. Ludwig (1973) Stochastic Approximation for the General Epidemic, J. Appl. Prob. l0: 263-276.

D. Ludwig (l975) Final Size Distributions for Epidemics, Math. Biosci 23: 33-46.

D. Ludwig (1974) Stochastic Population Theories, Vol. 3 of Lecture Notes in Biomathematics, Springer Verlag.

D. Ludwig, D.D. Jones, C.S. Holling (1978) Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and the Forest, J. Animal Ecology 47: 3l5-332

D. Ludwig, H. Ray Hilborn, C. Walters (1993) Uncertainty, resource exploitation, and conservation: lessons from history, Ecological Applications 3 (4): 548-549.

D. Ludwig, M. Mangel, B. Haddad (2001) Ecology, conservation and public policy. Annual Review of Ecology and Systematics. 32:481-517

S. Strogatz (2018) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press.

J.D. Murray (1989) Mathematical Biology, Springer.

A.R. E. Sinclair, D. Ludwig, C.W. Clark (2000) Conservation in the real world, Science 289 (5486): 1875-1875.

G.R. Munro, U. R. Sumaila (2015). On the Contributions of Colin Clark to Fisheries Economics.

Environmental and Resource Economics, 61(1), 1-17.

W. K. Stevens (1993) Biologists Fear Sustainable Yield Is Unsustainable Idea, New York Times, April 20, 1993, C00004.