Report

Matthew Faldyn Community Ecology April 22, 2014 6 “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 • • • • 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 5 7 • 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… 7 7 7 7 7 • Used knowledge and skills as a theoretical physicist to advance the fields of: 1. 2. 3. 4. 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: • • • • Structure Function Temporal dynamics Spatial dynamics • of populations and communities of plants and animals 8 • 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: 1. 2. 3. 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 9 • 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: = (1 − ) • 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: • • • • • • 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: • 1 + 1 = 1 exp{ 1 1 − α11 2 − α12 2 1 } (5a) 2 2 − α21 1 − α22 2 2 } (5b) • 2 + 1 = 2 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) B 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 1. 2. 3. 4. 5. 6. 7. 8. 9. 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 7