Chances are good Modeling stochastic forces in ecology

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Harry Neal Cooper
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1 Chances are good Modeling stochastic forces in ecology Ecologists have a love-hate relationship with mathematical models The current popularity in population ecology of models so abstract that it is difficult to see how a testable hypothesis might be deduced from them is a more serious matter. How much weight and epistemological status do the authors and supporters of such models attach to the conclusions that might be drawn from them? Does the popularity of such models mark the first step toward Scholasticism? Andrewartha and Birch (1954) Part of the problem was that mathematical models in ecology were deterministic, consisting of differential equations, difference equations, etc. Ecological data, by contrast, typically revealed wild fluctuations (see Figures):

2 Solid circles: female grizzly bears with cubs, unduplicated sightings, Open circles: one-step expected values of fitted stochastic Ricker model.

3 The claim (May 1974) that such wild fluctuations might in fact be deterministic and explained with some low-dimensional mathematical model (if only we knew the form of the model) was the ultimate provocation. (In the '70s, in practically the same breath, the mathematical ecologists claimed that dynamic models, linearized around equilbrium, explained population abundance patterns in ecological communities.) What were ecologists to make of mathematical models? Were these differential equations in their textbooks intended to be taken seriously as scientific hypotheses? In all of the biomathematics literature, was there a single mathematical model that stuck as a succesful explanation of a real system?

4 Stochasticity population processes are fundamentally births, deaths, migrations stochasticity is the random variability in such processes random : unexplained, high-dimensional variability that is conveniently modeled by probability distributions

5 Importance of stochasticity noise is pervasive in ecological data, and satisfactory explanations (ecological theory) must accomodate such variability connecting deterministic models with data requires explicit modeling of stochastic forces (departures of data from model) deterministic concepts, such as point equilibria, cycles, and chaos must be rethought in stochastic contexts accurate modeling and estimation of stochastic forces crucial to species preservation efforts in conservation biology

6 Deterministic model: 8 œ:8 "! 8! : number of individuals in a cohort at year >œ0 :: survival fraction 8 " : number of individuals in cohort alive 1 year later Demographic stochasticity each individual in cohort experiences an independent random experiment to determine a life table outcome e.g. for each of 8! individuals, roll the dice of life to determine whether the individual lives or dies :: probability of survival R " : number of survivors is a random variable

7 Pr ÒR œ8 Óœ Š : " Ð" :Ñ! ", " " 8 8! " 8 œ " 0, 1, 2, notation: R " µ binomial( 8!, :) expected value: variance: 8 EaR b œ! 8 PrcR œ8 d " " 8œ0 8 "! œ:8! " " VaR b œ! a8 EaR bb TcR œ8 d " " " 8œ0 "! œ:ð1 :Ñ8 ( œ! 8 Ñ #!! " " coefficient of variation: CVaR b œ œ " ÈVaR" b EaR b " " Ð" :Ñ É È 8! : note: CVaR " bp0 as 8! becomes large (model becomes essentially deterministic)

8 Environmental stochasticity good years and bad years: survival probability varies model: T taken to be a random variable with continuous distribution on a0, 1b R œt8 "! expected value: EaR b œ EaTb8 "! variance: VaR b œ V atb8 # ( œ " 8 # Ñ "!! coefficient of variation: CVaR b œ CVaTb " note: CV ar bdoes not go to zero as 8 increases "!

9 Both demo and enviro stochasticity 0 T a: b: probability density function for T ' 0 a:.: b œ1! " T first draw value of T (say, :); then draw value of R " R Tœ:µ binomial a8, : b PrÒR "! œ8 Ó œ " " " ! 8 T ' Š! : " Ð" :Ñ! " 0 Ð:Ñ.: " from conditional probability: EÐR" Ñœ EE Ò ÐR" Tœ: ÑÓ VÐR" Ñœ EV Ò ÐR" Tœ: ÑÓ VE Ò ÐR" Tœ: ÑÓ expected value: EÐR Ñœ EÐ8 TÑœ8 EÐTÑ "!!

10 variance: VÐR Ñœ EÒ8 TÐ" TÑÓ VÐ8 TÑ "!! œ8 EÒTÐ" TÑÓ 8 VÐTÑ!! # ( œ! 8 " 8 )!! # Variance stabilizing transformations V ÐR" Ñœ@Ð8! Ñ (conditional on 8! ) want to find transformation \" œ1r a " b such that the variance of \ " is a constant (i.e. does not depend on ) 8! such a transformation represents the scale on which the stochasticity in R " can be modeled as additive noise \ œ1ðr Ñ 1Ð8 Ñ ÐR 8 Ñ1w " "! "! Ð8! Ñ w "! "!! VÐ\ Ñ Vc1Ð8 Ñ ÐR 8 Ñ1 Ð8 Ñd

11 w w # "!!! œ VaR 1Ð8 Ñb œ c1ð8 Ñ Ñ w # Set c1ð8ñ œ- and solve for 1a8b! (a positive constant) "'.8 # 1Ð8Ñ œ- - where -" ( œ È-! ) and -# are arbitrary constants demographic: a8b œ! 8 or just - # È! È # 1Ð8Ñ œ " 8-1Ð8Ñ œ È8

12 environmental: a8b œ " 8 # or just - 1Ð8Ñ œ " " ln 8-18 a b œ ln8 # both d & e: a8b œ! 8 " 8 # - # # 1Ð8Ñ œ " È" ln ˆ È" È"! reparameterize a b œ! 8 " 8 # œ 9 Ð" < Ñ%8 < 8 # <: relative weight of enviro stochasticity component

13 1Ð8Ñ œ # ln " È< 8 " È< 8 %Ð" < Ñ È< # # ( - œ È9 and - œ Ð#ÎÈ< Ñln Ð# È< Ñ) " # if < œ 1, 18 a b œ ln8 if < p0, 18 a bpè8

14 Stochastic dynamic models deterministic survival as a dynamic model: 8 œ:8 > > " where >œ0, 1, 2, Chains of conditional distributions (Markov, etc.) PrcR œ8 R œ8 d œ:8 a 8 b > > > " > " > > " : a b: probability mass (or density) function for R > given the fixed size of the previous population demographic stochastic model of survival :8 a > 8> " b œ Š : > Ð" :Ñ > " > " > > Start with, pick, then pick, R R! " #

15 ex. demo plus enviro :8 a > 8> " b= ' Š! : " Ð" :Ñ! " 0 Ð:Ñ.: " ! 8 T " Nonlinear autoregressive (NLAR) processes deterministic population model: ex. Ricker model: 8 œ08 a > > " 8 œ8 exp a+,8 > > " > " stochstic version: transform and add noise : B œ18 a b œ108 a a bb œ2ðb Ñ > > > " > " b b

16 \ œ2\ a b I > > " > where I > has a normal distribution with mean 0 and variance 5 # ( I > µ normalˆ 0, 5 #, with I", I#,..., uncorrelated. The above implies that the conditional distribution of \ > given \ > " œb> " is normal ˆ 2B a > " b, 5# Þ\ > is called a nonlinear autoregressive (NLAR) process. Conditional pdf for \ > is: :B a B b œ > > " # " a B 2 a B # 5 bb ˆ # > > " 5 # 1 exp # #

17 ex. environmental stochastic Ricker: lnr œ ln R +,R I > > " > " > \ > > " > \ œ\ +, e > " I The conditional pdf for :B a B b œ > > " \ > is: " # a B B +, e # 5 B > " # ˆ # > > " 5 # 1 exp # Multivariate NLAR process Deterministic model now is a system of difference equations 8 œ08 a > > " where 8 > is a vector of state variables, and 0 a b is a vector of functions b b

18 ex. LPA model of laboratory flour beetle populations P> œ,e> " expð -/6 P> " -/+ E> " Ñ T> œ P> " Ð". 6Ñ E> œ T> " exp Ð -:+ E> " Ñ E> " Ð". + Ñ P> œ feeding larvae at time >, T> œ nonfeeding larvae, pupae, & callow adults at time >E, > œsexually mature adults at time >

19 Stochastic version: \ œ2\ a b I > > " > where \ > is the vector of transformed variables, 2 a bis the map on the transformed scale, and I > is a random vector with a multivariate normal distribution having mean vector 0 and variance-covariance matrix D. ex. LPA model, demographic stochasticity ÈP> œ È,E expð - P - E Ñ > " /6 > " /+ > " ÈT œ ÈP Ð" Ñ I > > ". 6 #> I "> ÈE> œ ÈT exp Ð - E Ñ E Ð". Ñ > " :+ > " > " + I $>

20 Joint distribution of \, \,..., \ starting from B! :B a, B,..., B B " # ;! b " # ; œ:b a B b:b a B bâ: ˆ B B "! # " ; ; " joint pdf of the trajectory B, B,..., B starting at fixed size B! " # ; Advantages of chain models These types of chain models are excellent for analyzing time series data Many discrete time ecological processes

21 2. Diffusion processes (stochastic differential equations) deterministic model: stochastic version:.8>.> œ78 a > b.8 œ78 a b.> > >.R œ7r a b.> È@R a b.[ > > > > here.[ > µ normala0,.> b, 7Ð8Ñ is the infinitesimal mean function, a bis the infinitesimal variance function..r > is defined rigorously in terms of an Ito stochastic a b scales the noise, a b œ! 8 a b œ " 8 # environmental, etc.

22 ex. Stochastic logistic equation with environmental noise > > < 5 > # > # >.R œ ˆ <R R.> É" R.[ Conditional pdf ( transition distribution ) is solution to a PDE: Prc+ R, d œ ' :8ß a > 8 b.8 > +,! `: `8 œ " # # ` c@: d `7: c d `8# `8 Note: in chain models, it is usually not possible to write down the form of :8ß a > 8! b. But in diffusion processes, :8 a 8 b œ:8ß a 1 8 b > > " > > "

23 Advantages of diffusion processes: Many closed formulas for statistical properties of diffusion processes have been derived Many other stochastic processes can be approximated by diffusion processes

24 Stationary distributions In some stochastic models, the transition distribution converges to a stationary distribution (independent of time and initial condition) :8ß a > 8! bp:8 a b as >p_ In some other stochastic models, such as in models where extinction is possible, the transition distribution can become quasistationary for a long time In diffusion processes, if a stationary distribution exists, it is given by G 78 a a a b :8 a b œ exp 2'.8

25 ex. stochastic logistic, enviro noise = - > ab = = " -8 :8 a b œ 8 e, 0 8 _ 2< 2< " 5 " - œ, =œ 1 (gamma distribution; special case is the chisquare distribution)

26 First passage properties 0a8+, ;, b: probability of reaching + before,, starting from 8, where +Ÿ8Ÿ, 0a8+, ;, b œ, ' 8exp c AaB bd.b ', exp c AaB bd.b + ' 7B a a b where AB a b œ 2.B Interestingly, stable and unstable equilibria in 7Ð8Ñ correspond to inflection points in 0 taken as a function of 8

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It's Only Fitting. Fitting model to data parameterizing model estimating unknown parameters in the model

It's Only Fitting. Fitting model to data parameterizing model estimating unknown parameters in the model It's Only Fitting Fitting model to data parameterizing model estimating unknown parameters in the model Likelihood: an example Cohort of 8! individuals observe survivors at times >œ 1, 2, 3,..., : 8",