Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.1/44

Transcription

1 Physics tricks for fun and profit: A physicist s adventures in theoretical ecology Robin E. Snyder robin.snyder@cwru.edu Department of Biology, Case Western Reserve University Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.1/44

2 What is theoretical ecology? Use mathematical models to help understand ecological dynamics Detailed simulations or toy, caricature models Different scales Physiology of individuals Dynamics of a single population Dynamics of communities (collections of species) Energy and nutrient cycling through ecosystems Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.2/44

3 Sample questions How do toxicants affect organism growth? How rapidly will an invasive species spread? What determines the size and frequency of insect outbreaks? When do species that compete for the same resources coexist? How might rising arctic temperatures affect the release of carbon into the atmosphere? Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.3/44

4 Connections with physics Real dynamics: many interacting individuals Observed: macroscopic properties (e.g. population density) Spatially localized interactions which affect dynamics. E.g.: spatial correlations Birth rates may decline with crowding (depends on local pop. density). Seeds land near parent plant population clusters Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.4/44

5 Some of the same tools Mean field theories Lattice models (like the Ising model) Reaction-diffusion equations And others (Fourier transforms, Markov chain models, Langevin equations,...) Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.5/44

6 Differences Classical methods not applicable No Hamiltonian No detailed balance Different interests (e.g. less interest in critical phenomena) No overarching theoretical framework (but do have classic models and results) Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.6/44

7 Overlapping disciplines Physics Applied probability theory Large number of interacting units Chemistry Sociology Theoretical ecology Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.7/44

8 Multidisciplinary tools Physics: Lattice models SDE s Fourier Chemistry: Reaction diffusion eq. Applied prob. theory: Interacting particle systems Markov chains Prob. generating functions Sociology: Network theory Theoretical ecology Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.8/44

9 Why are there so many kinds of plants? Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.9/44

10 Why so many plants? All need same resources. One species best at garnering all resources Champion Plant monoculture (Tilman s rule.) Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.10/44

11 Classic hypotheses Need different relative amounts of resources a few species of plants (Tilman s resource partitioning) Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.11/44

12 Classic hypotheses Need different relative amounts of resources a few species of plants (Tilman s resource partitioning) Tradeoffs in ability to get resources a few more species (e.g. light-nitrogen tradeoff) Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.11/44

13 Classic hypotheses Need different relative amounts of resources a few species of plants (Tilman s resource partitioning) Tradeoffs in ability to get resources a few more species (e.g. light-nitrogen tradeoff) Is that all? Is it enough? Probably not... Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.11/44

14 Current hypotheses Strong candidate: spatial coexistence mechanisms Most theoretical studies make severe simplifications: Usually: spatially homogeneous environment If include env. heterogeneity, then usually Global dispersal Very short-range competition Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.12/44

15 Central question Real dynamics: complex interplay between... Local competition Local dispersal Environmental heterogeneity These have different spatial scales. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.13/44

16 Central question How do these different spatial scales affect spatial coexistence mechanisms? Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.14/44

17 Modeling approach Approach to studying coexistence Criterion for coexistence Regional scale growth Partitioning of spatial coexistence mechanisms Local dynamics Environmental response and competition My example model Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.15/44

18 Modeling approach Approach to studying coexistence Criterion for coexistence Regional scale growth Partitioning of spatial coexistence mechanisms Local dynamics Environmental response and competition My example model Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.15/44

19 Coexistence criterion mutual invasibility Can blue invade the red residents? Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.16/44

20 Coexistence criterion mutual invasibility Can blue invade the red residents? Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.16/44

21 Coexistence criterion mutual invasibility Can blue invade the red residents? Can red invade the blue residents? Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.16/44

22 Coexistence criterion mutual invasibility Can blue invade the red residents? Can red invade the blue residents? Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.16/44

23 Coexistence criterion mutual invasibility Can blue invade the red residents? Can red invade the blue residents? Each can invade the other coexist. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.16/44

24 Coexistence criterion mutual invasibility Note: coexistence is regional on regional scale growth rate. ability to invade depends Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.17/44

25 Notation guide : time ( = now, subscripts resident) : locations, distances : population size = next generation) : species labels ( = invader, : spatial average of population size (regional population) : environmental response : competition : local growth rate : regional growth rate = Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.18/44

26 Regional scale growth Dynamics defined at local scale. Spatial average dynamics. regional (pre-dispersal) Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.19/44

27 Regional scale growth Next year s pop. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.20/44

28 Regional scale growth Next year s pop. This year s pop. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.20/44

29 Regional scale growth Next year s pop. Regional growth rate This year s pop. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.20/44

30 Regional scale growth Next year s pop. This year s pop. Regional growth rate Mutual invasibility. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.20/44

31 Partitioning : Competitive differences and nonspatial coexistence mechanisms : Relative nonlinearity rare opportunities or dangerous extremes : Storage effect covariance between environment and competition : Growth-density covariance populations build up in favorable locations Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.21/44

32 Modeling approach Approach to studying coexistence Criterion for coexistence Regional scale growth Partitioning of spatial coexistence mechanisms Local dynamics Environmental response and competition My example model Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.22/44

33 Environment and competition Local growth rate depends on and. Environmental response E.g. germination rate : Usually different for different species Varies in space Competition : depends on... how far away neighbors are what species they are what their environment is like Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.23/44

34 Competition kernels U jk U jk y y of species distance = competition kernel. Measures how much a plant affects a plant of species when separated by Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.24/44

35 Competition We assume species Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.25/44

36 Competition We assume species Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.25/44

37 Competition We assume species Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.25/44

38 Competition We assume species = resident Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.25/44

39 My example the lottery model E C At least one seed arrives at every location. At each location, a single seed succeeds in establishing itself. Prob. of establishment: Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.26/44

40 My example the lottery model F j Species fecundity = (fecundity)(prob. of establishment) Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.27/44

41 My example the lottery model Seeds disperse, parents die. z k j (z) z prob. of species j dispersing a distance Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.28/44

42 My example the lottery model Local dynamics: where and Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.29/44

43 Local dynamics Several competing spatial scales: Environment E U k (z) j Competition z Dispersal z k j (z) z Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.30/44

44 Resident population density Perturbative approach: Resident density drives competition. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.31/44

45 Resident density response function Fourier transform of local resident dynamics density response function: resident + + X = + + environment response res. density Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.32/44

46 Resident density response function Short-range resident dispersal and long-range residentresident competition strong resonance in resident density response function. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.33/44

47 Figure format E Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.34/44

48 Resident response to env. variation E E Competition is driven by resident density, which tracks environmental variation better when Resident dispersal short-range Resident-resident competition long-range Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.35/44

49 Partitioning : Competitive differences and nonspatial coexistence mechanisms : Relative nonlinearity rare opportunities or dangerous extremes : Storage effect covariance between environment and competition : Growth-density covariance populations build up in favorable locations Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.36/44

50 Partitioning : Competitive differences and nonspatial coexistence mechanisms : Relative nonlinearity rare opportunities or dangerous extremes : Storage effect covariance between environment and competition : Growth-density covariance populations build up in favorable locations Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.36/44

51 Storage effect ( ) E Self-competition limits resident s growth in favorable areas. (Cov large) Not much limits invader s growth in favorable areas. (Cov small) Compete primarily with residents; peaks of resident density may not coincide with favorable. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.37/44

52 Storage effect ( ) Cov Cov Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.38/44

53 Storage effect ( ) Cov Cov For sinusoidal environmental variation Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.38/44

54 Storage effect ( ) Cov Cov For sinusoidal environmental variation Cov Cov Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.38/44

55 Storage effect ( ) E Maximize Cov : Short-range resident dispersal Short-range resident-resident competition Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.39/44

56 Storage effect ( ) E E Make Cov small and pos. or large and neg. (let invaders take full advantage of good spots): Invader and resident good spots largely offset: short-range resident-invader competition Invader and resident good spots largely the same: long-range resident-invader competition Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.40/44

57 Storage effect ( ) The storage effect helps the invader (boosts ) when Short-range resident dispersal Invader and resident good spots largely offset: short-range res.-inv. competition and res.-res. competition Invader and resident good spots largely the same: long-range res.-inv. competition, (res.-res. competition inconsistent) Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.41/44

58 Storage effect Storage effect is strongest for intermediate scales of environmental variation. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.42/44

59 Conclusions Spatial variation in env. conditions additional coexistence mechanisms. Can quantify their strengths. Strengths depend on relative spatial scales of dispersal kernels, competition kernels, and environmental variation Can use Fourier methods to make general statements about the effects of relative spatial scales without specifying the form of environmental variation or the forms of the kernels. Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.43/44

60 Advertisement Case biology is becoming more quantitative. Hiring theorists and mathematically inclined experimentalists. Developing biology courses for physical science majors Improved computing infrastructure, incl. a cluster for parallel programming. New Bachelor of Science major in systems biology We welcome collaborations with physics faculty and grad. students. Intro. to ecological modeling class next fall or spring Physics tricks for fun and profit: A physicist s adventures in theoretical ecology p.44/44