layout: true background-color: #fdf6e3 class: center, top --- class: middle # Theory of uncertainty, ecological variation, and the limits of prediction <div class="my-footer"> <a href="https://carlboettiger.info"> <i class="fas fa-user "></i> Carl Boettiger</a> | <a href="https://berkeley.edu"> <i class="fas fa-briefcase "></i> UC Berkeley</a> | <a href="https://twitter.com/cboettig"> <i class="fab fa-twitter "></i> @cboettig</a> </div> --- layout: false background-image: url(../image-library/royalsociety/pensive-polarbear.jpg) background-position: center background-size: 130% class: center, top --- layout: true background-color: #fdf6e3 class: center, top --- # What is "noise" anyway? .pull-left[ ## Stochasticity ### Demographic / intrinsic ### Environmental / extrinisic ### Individual heterogeneity ] .pull-right[ ## Model uncertainty ### Parameter uncertainty ### Structural uncertainty ] ## Measurement uncertainty --- # *Noise the Nusiance* # Stochastic Phenomena # Noise as Information # Decisions Under Uncertainty # Limits of Prediction --- # The canonical model `\begin{equation} \textrm{d} N_t = \underbrace{f(N_t) \textrm{d} t}_{\textrm{det skeleton}} + \underbrace{\sigma_d \sqrt{N_t} \phantom{\cdot}\textrm{d}B_t^{(d)} }_{\textrm{demographic noise}} + \underbrace{\sigma_e N_t \phantom{\cdot}\textrm{d}B_t^{(e)}}_{\textrm{environmental noise}} \label{canonical} \end{equation}` - (e.g. Lande 1993, Foley 1994, etc) - rough jusification in Kurtz 1970, 1973, 1978 - Deterministic skeleton -> expected dynamics --- # Noise the Nusiance # *Stochastic Phenomena* # Noise as Information # Decisions despite Uncertainty # Limits of Prediction --- # Stochastic Phenomena: Quasi-cycles <!-- --> Nisbet & Gurney 1976 --- # Stochastic Phenomena: Quasi-cycles <!-- --> --- # Stochastic Phenomena: Noisy switch `$$X_{t+1} = X_t + \underbrace{X_t r \left(1 -\frac{X_t}{K} \right)}_{\textrm{ growth, } g(X_t)} - \underbrace{\frac{a X_t ^ Q}{X_t^ Q + H ^ Q}}_{\textrm{consumption, } c(X_t)} + \xi_t,$$` <!-- --> May 1977 --- # Stochastic Phenomena: Noisy switch `$$X_{t+1} = X_t + \underbrace{X_t r \left(1 -\frac{X_t}{K} \right)}_{\textrm{ growth, } g(X_t)} - \underbrace{\frac{a X_t ^ Q}{X_t^ Q + H ^ Q}}_{\textrm{consumption, } c(X_t)} + \xi_t,$$`ß <!-- --> --- # Stochastic Phenomena: Noisy switch <!-- --> --- <!-- --> Rubella prevelance in Copenhagen illustrating potential stochastic switching dynamics between two regimes, reproduced from Keeling et al 2001 --- # Noise the Nusiance # Stochastic Phenomena # *Noise as Information* # Decisions Under Uncertainty # Limits of Prediction --- # Knowledge from Noise? ## Mechanistic understanding .pull-left[ Levins 1969: `$$\frac{d n}{d t} = \underbrace{c n \left(1 - \frac{n}{N}\right)}_{\textrm{birth}} - \underbrace{e n}_{\textrm{death}}$$` Logistic: `$$\frac{d n}{d t} = \underbrace{c n}_{\textrm{birth}} \underbrace{\left(1 - \frac{n}{N}\right)}_{\textrm{death}}$$` ] .pull-right[ `$$\sigma^2 = \frac{b(n) + d(n)}{2 \partial_n b(n) - 2\partial_n d(n)}$$` Levins: `$$\sigma^2 = N\frac{e}{c}$$` Logistic: `$$\sigma^2 = K$$` ] --- # Knowledge from Noise? ## Early Warning Signals <!-- --> --- <!-- --> --- ## Tipping points: empirical examples <!-- --> Dai et al (2015) --- ## Tipping points: empirical examples <!-- --> Dai et al (2015) --- ## Critical-speeding up? `$$dX_t = X_t \frac{r}{\beta} \left( \frac{X_t}{\beta A} -1 \right) \left(1 - \frac{X_t}{\beta C} \right) dt + \sigma dB_t$$` - Carrying capacity `\(C\)`, Allee threshold `\(A\)`, available territory `\(\beta\)` <!-- --> [Titus et al (2019)](https://arxiv.org/abs/1901.08084) --- # Noise the Nusiance # Stochastic Phenomena # Noise as Information # *Decisions Under Uncertainty* # Limits of Prediction --- layout: false background-image: url(../image-library/noaa/noaa_fisheries.jpg) background-position: center background-size: 190% class: center, top --- layout: true background-color: #fdf6e3 class: center, top --- # MSY & Constant Escapement <!-- --> -- `$$\frac{dN}{dt} = \underbrace{r N \left(1 - \frac{N}{K} \right)}_{f(N)} - H$$` `$$H_{\text{MSY}} = f(N) = r K / 4, \phantom S B_{\text{MSY}} = K/2$$` Schaefer 1954; Clark 1973 --- ## Decisions Under Uncertainty: does it matter? <!-- --> Reed 1979 --- ## Paradox of Measurement Uncertainty <!-- --> Clark & Kirkwood, 1987 --- ## Paradox of Measurement Uncertainty <!-- --> Sethi et al (2005) --- <!-- --> --- <!-- --> Memarzadeh & Boettiger (2019) --- <!-- --> --- # Noise the Nusiance # Stochastic Phenomena # Noise as Information # Decisions Under Uncertainty # *Limits of Prediction* --- # Limits of Prediction <!-- --> --- # Limits of Prediction <!-- --> --- # Limits of Prediction <!-- --> --- ## Perfect information: <!-- --> --- ## Perfect knowledge of structure (MCMC estimate of parameters) <!-- -->