Mathematical Ecology: A Century of Progress, and Challenges for the Next Century

Simon Levin (September 21, 2016)

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Slides

2:26 Mathematical Ecology: A Century of Progress and Challenges for the Next Century
2:29 With thanks to
3:00 Mathematics has been used to address ecological problems for centuries
3:47 Population Sizes from Fibonacci Sequence
4:22 This is a Trivial Example of a Deep Result
5:20 The (stable) age distribution carries much information about births and deaths
5:51 Later theory extended these considerations to continuous age/time
6:34 Mathematics has been used to address ecological problems for centuries
7:03 Mathematics has been used to address ecological problems for centuries (cont.)
7:23 Theoretical Ecology Has A Long and Rich History
7:51 Volterra's Contributions to Ecology
8:06 Volterra's Contributions to Ecology (cont.)
8:24 Volterra brilliantly examined the properties of multi-species communities and their statistics
8:37 But this spawned an increasingly sterile mathematical tradition, ignoring ecological reality
9:05 Nonetheless, the challenges remain unabated
9:19 Mathematics has been used to address ecological problems for centuries
9:44 EPIDEMICS - Classical Theory (Kermack - McKendrick)
10:16 Kermack - McKendrick Model - 2 Competing Strains
10:49 Models of the dynamics of iinfectious disease have played a fundamental role in their management, for example in vaccination strategies
11:10 Mathematics has been used to address ecological problems for centuries
11:22 All of these examples lead to some grand challenges, to which I will return
11:29 Mathematics stimulated by ecological problems
11:55 History of Statistics & Probability
11:59 Other Areas of Math Stimulated by Biology
12:45 Other Areas of Math Stimulated by Biology (cont.)
14:04 Other Areas of Math Stimulated by Biology (cont.)
15:02 Mathematical and systems thinking is increasing in influence
15:34 Ecology for Bankers
16:26 Ecology for Bankers (cont.)
17:09 Financial Regulation
17:33 World Politics
18:09 Grand Challenges of Mathematical Systems Theory
20:21 Ecosystems and the Biosphere are Complex Adaptive Systems
22:02 So too are the socio-economic systems with which they are interlinked
22:18 Stock markets crash... as collective consequence of individual actions
22:26 There may be Critical Biosphere Thresholds
22:42 Many... but not all... such transitions have characteristic early warning signals
23:48 When do these work? Avoid excessive claims
24:07 Caution is needed... Mechanisms need to be identified
25:13 Getting Mechanisms right crucial
25:30 Challenges of Systems Theory
25:46 Sustainability must focus on macroscopic features, while recognizing that control of those rests at lower levels of organization
26:31 Forest growth models can scale from individual to ecosystem
27:40 Vegetation models have been sucessful in explaining global patterns, though not individual species abundances
28:21 And similar approaches can examine collective decision-making
28:33 Agent-Based Models Help
29:20 But reduced dimensional descriptions will be essential for robustness of conclusions
29:50 Time out for questions?
34:04 Systems models can inform the potential for critical transitions
35:59 Staver et al. 2011 (Ecology) and Staver & Levin (Amer. Natur.)
37:56 Such phenomena are widespread
38:16 Individual-based models relate the small-scale to the large scale
38:51 Diffusion models inadequate: Organisms don't move randomly, but follow gradients or actively aggregate
39:19 Lagrangian-Eulerian Connections
40:37 But real aggregations are heterogeneous assemblages of individuals
40:45 The dynamics of collective phenomena and collective decision-making
41:02 Role of leadership and collective decision-making
41:26 Two-Target Model
42:55 1 Informed Individuals in Group of 100
43:34 10 Informed Individuals in Group of 100
44:02 Cometitioin and Consensus
44:42 Theoretically and empirically, unopinionated individuals are crucial to nature of consensus.
45:26 Challenges of Systems Theory
45:42 Public goods problems are widespread in socio-economic and ecological contexs
47:00 The Tragedy of the Commons
47:12 The Commons solution (Hardin, Ostrom)
47:41 Social norms can sustain and enhance prosocial behavior
49:02 Fairness norms can provide "Mutual coercion, mutually agreed upon"
49:24 Equity-Driven Ostracism: Nash Equilibria
50:36 Equity-Driven Ostracism
51:06 Tavoni, Schluter, Levin: Coordination Game
51:45 Lessons Learned
52:38 Avinash Dixit-Simon Levin: Prosociality and Multiple Groups
52:57 Ostrom: Climate Change
53:12 Climate Clubs: Overcoming Free-riding in International Climate Policy
53:20 Incomplete Cooperation and Co-benefits: Deepening Climate Cooperation with a Proliferation of Small Agreements
53:36 Club Approach
53:47 Summary so far
54:42 Voting Theory and Models of Collective Action
54:55 Modeling of Collective Decision-Making Represents a New Frontier
55:20 Attitudinal shifts affect action on issues like climate change
56:17 Conclusion: Ecological systems and socio-economic systems alike are complex adaptive systems
56:30 Systems Challenges
57:31 These are challenges old, yet new
57:48 Galileo said that
58:25 Thus, we have reason to continue to anticipate and enjoy what Wigner call
59:12 Question: You were talking about anticipating critical transitions on a global socio-economic scale. Do you think there is already sufficient data out there to determine whether or not we are close to such a transition, or is more data needed?
63:16 Question: Modeling ecological systems, we are challenged by requiring increasing complexity to reflect complexity of nature, but at the same time are mindful that as we increase the number of parameters in a model, we increase the inherent variability. How do we deal with this paradox?
67:15 Question: This is a very broad question, but could you name some areas of biology where there is potential for a role of mathematics, but hasn't gotten enough attention yet? Also, which subfields of mathematical biology are you personally most excited about in the future?
73:02 Question: What rules for local behavior have been successful in agent-based modeling where individual people are the agents, for example in voting patterns? Also, do you have any suggested reading?
75:34 Question: There has been a lot of excitement over the past few years about the predictive capabilities of empirical dynamic modeling. I was just curious to hear your opinion about the role of equation based model moving forward in an era where we can do great predictions from data alone.
78:17 Question: My question pertains to international cooperation with respect to an effort to eradicate a disease (e.g. polio). What happens if some nations choose not to vaccinate (thereby halting eradication efforts?) Is there anything the global community can do in this case?

Abstract

Mathematical ecology is one of the oldest and most successful branches of mathematical biology, and one that has profited both ecology and mathematics. The great mathematician Volterra was a pioneer a century ago, and the subject has built on the dynamical systems approaches he introduced. As attention has turned to the ecological challenges of the present- climate change, biodiversity loss, critical transitions and the management of the global commons, new methods have entered from stochastic processes to game theory, from statistical physics to topological data analysis, and with a heavy emphasis on high-speed computation. In this talk, I will trace out some of the historic successes, and introduce modern challenges.