Stochastic logistic model with environmental noise. Brian Dennis Dept Fish and Wildlife Resources and Dept Statistical Science University of Idaho

Transcription

1 Stochastic logistic model with environmental noise Brian Dennis Dept Fish and Wildlife Resources and Dept Statistical Science University of Idaho

2 Questions I. What plausible biological mechanisms, if any, lead to logistic growth? Derivations of & mechanisms behind logistic II. How can the logistic be made more useful? Accounting for stochastic forces Í Fitting to data and evaluating

3 œ<r R œ+r,r.r _ <.> R # # R : population abundance at time > <: maximum per-unit-abundance growth rate R : equilibrium abundance " R transformation: ] œ yields linear DE for ] solution trajectory: R œ R R R "! / R! <>

4 time population abundance

5 I. Derivations/mechanisms A. Verhulst (1838) "logistique # α added,r and,r to the exponential growth DE as an adjustment B. Lotka (1925).R.> œ7r +R,R # Handwaves about the sign of, Taylor series expands around a particular point what point does he have in mind? Lotka's derivation is parroted in many ecology textbooks and websites (Hutchinson 1978) C. Chapman (1928).R R.> O œ<r environmental resistance œ<r "

6 D. Dennis and Patil (1984), Dennis (1989a) Re-formulation of Lotka.R.> œr1r 1R is per-unit-abundance growth rate, presumed decreasing Near R, 1R w œ!, 1 R! w 1 R 1 R R R 1 R œ +,R w w + œ R1 R, œ 1 R

7 population abundance (1/N)dN/dt

8 E. Substrate-product model: "something replacing something else" Invasions: cheat grass replacing native grassland. Epidemics: infecteds replacing susceptibles. Commerce: Wal-Mart replacing K-Mart. Innovation: DVDs replacing VHSs. Dissemination of ideas: Like an autocatalytic chemical reaction (Pearl and Reed 1920, Reed and Berkson 1929):.R.> " œ5ro R

9

10

11 F. (Williams 1972) Explicit limiting nutrient (or substrate) 1. Closed system.g.> œ α " 5 GR α: conversion (in)efficiency.r.> œ5gr " mass balance: G αrœg αr Ê G œ G αr αr!!!! œ<r R <œ5 G αr, Rœ G αr.r < ".> R # "!! α!!

12 2. Open system (like a laboratory microbial chemostat).g.> œ5! # α 5GR 5G "! #: nutrient concentration in inflow 5! : dilution rate.r.> œ5gr 5R "! mass balance: G α Rœ# G α R # /!! 5 >! Ê.R.> is a logistic with time-dependent coefficients (Dennis 1978), is exactly logistic if # œg αr, and becomes logistic as >p!!

13 3. Closed system, Michaelis-Menton nutrient uptake C + V Þ C~V Ä V + N V: uptake bottleneck (gut volume, cell membrane sites, filter surface, etc.) Ê rate of product formation per unit abundance JG (analogous to Michaelis-Menten enzyme kinetics).g JGR.> œ α O G.R JGR.> œ O G O G

14 population abundance growth rate

15 time population abundance

16 II. Stochastic forces and data analysis A. Problems with using the logistic as is. 1. Life is stochastic (data depart from model in noisy ways) 2. How to do inferences (estimation, evaluation) 3. Modeling objectives might require more realistic modeling

17

18

19

20 B. Off-the-shelf solution: least squares (Leach 1981) Data: time series population abundances 0 > " > # â > ; 8! 8" 8# â 8; Model parameters: <, R, 5 # Minimize departures of observations from logistic trajectory: ; WW <, R, œ R 83 R 8! 5 # 3œ" " / 8! <> 3 This procedure implicitly assumes a stochastic logistic model in which the underlying growth is deterministic and the departures from the trajectory are due to observation or measurement error. Ecological populations themselves are stochastic (process noise). #

21 C. Models of process noise Seek: a useful, general stochastic version of.r œ 7ÐRÑ.> Answer: a diffusion process is in the form.r> œ 7ÐR> Ñ.[ >, where: R > is population abundance at time >,.[ > µ normal(!,.>) 7Ð8Ñ is the infinitesimal mean ( skeleton is the infinitesimal variance

22 D. Stochastic logistic with environmental noise : 7Ð8Ñ œ <8 < œ 5 # 8 # 5 # (I'll use " 5" instead of R from here on... "equilibrium" of the stochastic model will be seen to be a probability distribution) this form Êln R has a constant infinitesimal variance > Ê substantial fluctuations at all population sizes, not just at low population sizes

23

24 Transition probability density function: :8>l8,, > +! Pr + R Ÿ, œ :8, >l8.8! :8>l8,.8œ1 # `: 1 ` 7: `> œ 2 `8# `8! (Fokker-Planck equation) The FP for the stochastic logistic has been solved! :8>l8, œ: /! - -! -% E -> 9 - is related to a confluent hypergeometric function!

25 Stationary pdf of a diffusion process: :8>l8, Ò :8! :8 œ / as >p #.8 stationary pdf exists if :8.8œ! 1

26 Stationary pdf for stochastic logistic: " >α α " " 8 :8 œ 8 / α (gamma distribution) 2< 2< 5 55 α œ 1, " œ # # stationary pdf exists if α!

27

28 Moments of R > :? 8, > œ E R lr œ 8 / /! >!! #! `? `? `> œ `8! `8 / / /! #! (backward equation)? 8,0 œ 8 / /!!

29 Approximation for the transition pdf (Dennis 1989b) 1. Find approximations for? 8, > and? 8, > "! #! 2. Use a gamma pdf with those moments

30 1. Singular perturbation (small noise expansion) of backward equation: write?/ 8!, > œ A! 8!, > # 5A 8, > 5 A 8, > â "! #! substitute into the PDE, equate coefficients of like powers of 5, solve resulting PDEs for A, A,..., subject to initial condition (laborious)! " Wieszak (1988) proved that this perturbation is asymptotically correct

31 !? 8, > 5Î " / "! 5 8 8! <> 55 #5 #<> #5 <> 5 8 <> #< 8! 8! 8! $ 5 8! <> 8! " / / " #< >/ " /! # # 5 8 <> #! 8! 5 # 5 # %5 & #<> %5 <> " # < 8! # 8! # % 5 8! <> 5 8! <> 5 8! <> 8 8 8!? 8, > 5 Î " / #< / # / < >/ / # " /!!! 2. Use a gamma distribution w/ matching moments as transition pdf: " > α > " " > 8 :8>l8, 8 / α! >α?" α 8, > œ # >! >?? # " # " " " 8, > œ >!??? # " #

32 E. Simulations: does the approximation work?.r œ <R < R.> 5R.[ > > 5 > # > > 1000 realizations ending at each time > ( > œ 1, 2, 3,...until stationarity) Density probability plot: excellent graphical tool for assessing goodness of fit for a continuous distribution (related to the probability plot)

33 ], r.v. with continuous distribution J C, 0 C : cdf & pdf models to be assessed C, C,..., C data (ordered) " # ; usual probability plot : " 3 ; 3 plot J vs. C " # straight line indicates good fit, but the interpretations of departures from fit are not transparent

34 density probability plot (Jones and Daly 1995, Jones 2004): " 3 ; 3 # plot 0 J vs. C superimposed with plot of 0 C "

35

36

37

38

39

40

41

42

43 Uses of the stochastic logistic: data analysis! <, 5, 5 # unknown parameters 0 > " > # â > ; 8! 8" 8# â 8; time series data (pop. abundances) 7 œ>!, 7 œ> >,..., 7 œ> > intervals " " # # " ; ; ; " Likelihood function: # P<5,, 5 œ:8, 7 l8 :8, 7 l8 â:8, 7 l8 " "! # # " ; ; ; " # # s<, s 5, 5s maximum likelihood estimates (values of <, 5, 5 that jointly maximize P<5,, 5 # ) requires numerical maximization: optim() function in R, etc.

44 Residuals: " αs 3 * αs3 3 *s D œ " s "Î$ are approximately normal(0, 1) distributed, where α 3 αs œ α 8, s<, s " 5, 5s " s œ " 8, s <, s 5, 5 s # " " #

45

46

47

48

49

50 References: Chapman 1928 Ecology, Dennis 1978 Mathematical Biosciences, Dennis 1989a Natural Resource Modeling, Dennis 1989b in MacDonald et al. eds. Estimation and analysis of insect populations, Dennis and Patil 1984 Mathematical Biosciences, Hutchinson 1978 An introduction to population ecology, Jones 2004 The American Statistician, Jones and Daly 1995 Communications in Statistics Simulation and Computation, Leach 1981 JRSS A, Lotka 1925 Elements of physical biology, Pearl and Reed 1920 PNAS, Reed and Berkson 1929 J Physical Chemistry, Verhulst 1838 Correspondance Mathématique et Physique, Wieszak 1988 UI PhD Thesis, Williams 1972 in Deevey, ed. Growth by intussusception: ecological essays in honor of G. E. Hutchinson.