Matthew Faldyn
Community Ecology
April 22, 2014
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“One way of emphasizing the singularity of the recent past is [..]
to observe that the total number of humans ever to have lived is
estimated at around (a bit less than) 100 billion. One of Walt
Whitman's poems has a memorable image—thinking of all past
people lined up in orderly columns behind those living—‘row upon
row rise the phantoms behind us’. Actually, looking over our
shoulder, we would see only around 15 rows.”
- Robert M. May
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Biographical Information
Awards and Recognition
Research Overview
Seminal Books:
• Theoretical Ecology: Principles and Applications
• Stability and Complexity in Model Ecosystems
• Seminal Papers:
• Biological Populations with Non-Overlapping Generations
• Simple Mathematical-Models with Very Complicated Dynamics
• Evolutionary Games and Spatial Chaos
• References
• Born in Sydney, New South Wales – Australia
on Jan. 8, 1936 (78)
• B.Sc. (1956) from University of Sydney in
Chemical Engr. & Theoretical Physics
• Ph. D. (1959) in Theoretical Physics
• Post-doc, Division of Engineering and Applied
Physics at Harvard as Gordon MacKay
Lecturer
• Met his wife (Judith) here!
• Currently holds a Professorship at Oxford
University and is a Fellow of Merton College,
Oxford
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• Royal Society Fellowship – 1979
• Corresponding Fellow of Australian
Academy of Science – 1991
• Foreign Member of USNAS – 1992
• Knight Bachelor – 1996
• Chief Scientific Advisor to UK Gov’t (19952000)
• President of the Royal Society (2000-2005)
• Life Peer- House of Lords Commission (Baron
May of Oxford) – 2001
• Honorary University Degrees from: Uppsala,
Yale, Sydney, Princeton, and ETH Zurich
• Weldon Memorial Prize, MacArthur
Foundation Award, Linnean Society of
London Medal, Balzan Prize for Biodiversity,
and many others…
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• Used knowledge and skills as a theoretical physicist
to advance the fields of:
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Population and Community Biology
Infectious Disease Dynamics
Theoretical Ecology
General Theories of Biodiversity
Did most of this through rigorous quantitative analysis!
• Overall Point: Describe how interacting
populations of plants and animals
change over time and space, in response
to natural or human-created disturbance
in two parts
• Initial chapters give an account of the
basic principles governing:
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Structure
Function
Temporal dynamics
Spatial dynamics
• of populations and communities of
plants and animals
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• Later chapters outline applications of these ideas to
practical issues:
• fisheries, infectious diseases, future food supplies, climate
change, and conservation biology
• May ends with a brief synopsis, and poses
philosophical questions
• Throughout, emphasis is placed on questions which still
remain unanswered
• May addresses:
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What makes populations stabilize?
What makes them fluctuate?
Are populations in complex
ecosystems more stable than
populations in simple ecosystems?
• May analyzed the mathematical
roots of population dynamics
• Concluded (opposite to most) that
complex ecosystems themselves do
not lead to population stability
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• Played key role in introducing nonlinear math-models
and deterministic chaos into Ecology
• Most recently, it has:
• Lead nonlinear models to forefront of ecological thinking
• Influenced current thinking about the role of ecosystem
complexity as it relates to threats toward biodiversity
Cited: 869
Cited: 2,631
Cited: 1,232
Google Scholar total citations: 96,618 !
• Two Kinds of Populations:
• Biological pop 1) Man:
• Continuous growth and generations overlap
• nonlinear differential equations
• Biological pop 2) 13-year periodical cicadas:
• population growth takes place at discrete intervals of time
• Nonoverlapping generations
• terms of nonlinear difference equations
• So for single species, simplest differential equation with simple dynamics:
𝑑𝑁
𝑁
= 𝑟𝑁(1 − )
𝑑𝑡
𝐾
• globally stable equilibrium point at N = K for all r > 0.
• May’s Purpose:
• illustrate that many of the differential equations used in population biology have
been discussed inadequately
• May shows that simplest nonlinear equations have large spectrums of behavior
specifically when (r) increases
• The behavior:
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goes from stable equilibrium point
stable cyclic oscillations between 2 population points
stable cycles with 4 points
then 8 points
and so on…
to a chaotic regime
• Eq 1) the difference equation
analog of the logistic differential
equation
• Eq 2) quadratic form of simple,
nonlinear equation
• These simple, purely deterministic, single species models give
arbitrary dynamical behavior once r is big enough (r > 2.692 for
Eq. 1, r > 2.570 for Eq. 2)
Reflects the dynamic behavior of
the population density (Nt/K) as a
function of time (t) through the
differences in the values of (r) for
Eq. 1.
• A) r = 1.8, stable equilibrium
point
• B) r = 2.3, stable 2-point cycle
• C) r = 2.6, stable 4-point cycle
• D) r = 3.3,*  N0/K = 0.075
• E) r = 3.3,*  N0/K = 1.5
• F) r = 5.0, *  N0/K = 0.02
*chaotic regime depends on initial
population value
Multispecies:
• Uses difference equations modeling competition between two
species:
• 𝑁1 𝑡 + 1 = 𝑁1 𝑡 exp{
𝑟1 𝐾1 − α11 𝑁2 𝑡 − α12 𝑁2 𝑡
𝐾1
} (5a)
𝑟2 𝐾2 − α21 𝑁1 𝑡 − α22 𝑁2 𝑡
𝐾2
} (5b)
• 𝑁2 𝑡 + 1 = 𝑁2 𝑡 exp{
• Eq. 5a and 5b are the analog of Lotka-Volterra differential
equation model for 2-species competition
• Important: Even for multispecies, these equations yield stable cyclical
dynamics, up until chaos with increasing (r)
Reflects the stability
characteristics of the
difference equation model
for two-species competition*
• A) r = 1.1
• B) r = 1.5
• C) r = 2.5
• D) r = 4.0
*Conditions (which allow for a stability): r1 = r, r2
= 2r, K1 = K2 = K, α11 = α22 = 1, α12 = α21 = α
What does this all mean?
• Remember: Eq. 1 and 2 are two of the simplest nonlinear (DD insects) difference equations
• Rich dynamical structure
• Arbitrary cycles
• Exhibit chaotic regimes (even for multispecies!)
• Considerable mathematical/ ecological interest of which May
worked to elucidate
• Without understanding this behavior, prove difficult to make
sense of computer simulations or time series analyses
• Looked at first order difference equations
• Found these equations to be simple and deterministic, but
exhibit wide array of dynamics
• Alternate between stable points, bifurcated cycles, random
fluctuations, and then chaos
• Lead to problems and questions regarding finer
mathematical approaches to understanding these
trajectories
• a = 2.707
• b = 3.414
• Base dynamics of
previous equations
• Eq.1 and 2 iterations of
eq. 3, with (a). (Stable)
• Eq. 1, and 2 iterations of
eq. 3, with (b).
(Unstable)
• Figures Illustrate that as the nonlinear functions F(X)
becomes more steeply humped, the basic fixed point of
X* may become unstable
• Alternates to stable fixed period of 2
• Beyond (λ = +1, past 0, to λ = -1) hump steepens
• 2-point cycle become unstable, bifurcates to stable
period of 4
• Leads to 8, 16, 32, 64… 2n
1
• Tables describing:
• the potential parameter values for the F(X) that indicate
stability/ period shifts/ instability
• potential cycles for (k) number of periods under different
parameter scenarios
• Functional Point:
• As the parameter values are varied for F(X), the fundamental
and stable dynamical units become basic periods (k), arising to
bifurcations of stability/ period shifts/ instability
• Simple, deterministic equations possessing chaos have
implications:
1. Fluctuations in census data for animal population
may not be due to environment or sampling error,
but rather the dynamics of population growth model
2. Over time, trajectories diverge into chaos, indicating
long-term model prediction may be impossible
3. When applied, May found that natural populations
remain stable while lab-reared insect populations
show chaotic behavior
4. But…
• The biggest application is in pedagogy:
• May argues traditional mathematics & physics education
poorly prepares students
• Said these equations should be emphasized in elementary
calculus to enrich understanding of non linear systems
• Overall: Simple nonlinear systems do not necessarily
possess simple dynamical properties (and more
people need to know this!)
• Biological attention is given to the prisoners dilemma
• describe the evolution of cooperative behavior
• Previous work dealt with strategies (i.e. tit-for-tat)
• Here, May neglects:
• all strategy
• memories of past encounters
• considers only two types of players: those who always
cooperate (C) or who always defect (D)
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A
• Game played by
two players, each
may choose to
cooperate (C) or
defect (D) in an
encounter
• If both choose C,
both get pay-off (R)
• If one chooses D and the other C, D gets the biggest payoff (T) while C gets smallest pay-off (S)
• If both choose D, both get (P)
• So T > R > P > S
• Here, May labels the pay-offs as:
• T = b (b > 1)
•R=1
•P=S=0
• So mutual C each score 1, mutual D each score 0, and D
score (b) against C (who get 0 here)
• Explored this with substantial computing power, various
values for (b), and various initial proportions of C and D
• The dynamics of the system are dependent on (b)
• The most interesting regime is 2 > (b) > 1.8, allows for C to
grow clusters in D (and vice versa)
• Spatial games can generate what May called an “Evolutionary
Kaleidoscope”
• This simulation is started with a single D at the center of a 99 x
99 square lattice world of C with fixed boundary conditions
(1.8 < (b) < 2)
• Generates nearly infinite patterns
• Symmetry is always maintained because the rules of the game
are symmetrical
• Compared it to Conway’s “Game of Life”
• Prisoners dilemma generates a transition rule based on
the amount of neighbors around a cell
• Patterns shown have complexity and regularity unlike any
other cellular automata
• Zoo: “rotators, eaters, gliders, and blinkers”
• May explored other evolutionary games:
• Other games display similar spatial polymorphisms
• Hawk-dove game gives beautiful patterns
• Prior, Prisoners Dilemma was confined to:
• individuals/ groups who remember past encounters
• make “strategies”
• May’s models display no memory or strategy, the players are
pure C and D
• May argues that deterministically generated spatial structure
within populations is crucial for the evolution of cooperative
behavior
• Ultimately, the motive for this was biological
• Ended up generating patterns of ‘spatial dilemmas’ with
extreme richness and beauty
1. May, R.M. (1973) Stability and Complexity in Model
Ecosystems. In Press. Princeton University Press
2. May, R.M. (1974) Biological populations with nonoverlapping
generations: stable points, stable cycles, and chaos. Science.
186: 645-647
3. May, R.M. (1976) Simple mathematical models with very
complicated dynamics. Nature. 261: 459- 467
4. May, R.M. and McLean, A. (1976) Theoretical Ecology:
Principles and Applications. In Press. Oxford University Press
5. Nowak, M. and May, R.M. (1992) Evolutionary games and
spatial chaos. Letters to Nature. 359: 826-829
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http://mathforum.org/mathimages/index.php/Logistic_Bifurcation
http://www.beyondintractability.org/essay/prisoners-dilemma
http://scholar.google.com/citations?user=ScXat-4AAAAJ
http://www.britannica.com/EBchecked/media/113032/Aerialview-of-the-University-of-Oxford-Oxfordshire-England
http://www.topbritishinnovations.org/~/media/Voting/Images/Biol
ogical_Chaos_detail_2.jpg
http://cacm.acm.org/opinion/articles/74667-scientists-should-beon-tap-not-on-top/fulltext
http://www.sciencearchive.org.au/scientists/interviews/m/may.html
http://churchandstate.org.uk/2011/02/theoretical-ecologyprinciples-and-applications/
http://press.princeton.edu/titles/7050.html
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