1. Models

2. The world of models:
Formal descriptions of natural (ecological) systems
Usually, but not always quantitative (conceptual & physical models)
“Throughout the history of ecology, models have
played a fundamentally important role in the development of insight, providing statements about nature with an economy that only mathematics can provide.”
Hobbs and Ogle (2011)

3. Conceptual Models
River continuum concept

4. A little history:
Darwin: "Mathematics seems to endow one with something like a new sense"

5. Types of Models? Predator-prey Matrix models Metapopulation
Individual-based
Stock-recruit
Food web
Ecosystem
Density dependent
Disease (S-I-R)
Exponential/Geometri c
Logistic
Size spectra
Island biogeography
Mark-recapture
State-variable
Statistical

6. Dichotomies abound...

7. How are classes of models organized?
What’s the question ?
Theoretical
Applied
Level of interest?
Individual
Population
Meta-population
Food web
Ecosystem
How is time considered?
Discrete
Continuous
Do state variables change through time/space?
Static (no)
Dynamic (yes)
Is uncertainty incorporated?
Deterministic (no)
Stochastic (yes)
Is there a mathematical solution?
Analytical (yes)
Simulation (no)
Is the model structured?
Unstructured (no)
Structured (yes)

8. What’s the question?
Theoretical
Applied
Increase general insight into ecological processes
Often mathematically difficult (cutting edge)
But usually ecologically simplified
Describe & predict how systems function
Often mathematically simpler
Usually captures more of the complexity needed for prediction

9. What’s the question? Predator-prey (stable limit-cycles)
Theoretical
Applied
Predator-prey (stable limit-cycles)
Stock-recruit (MSY)

10. Population Models we might explore:
Theoretical
Applied
Euler model
Predator-prey
Individual-based (agent-based)
Stochastic dynamic programming
Metapopulation
Stock-recruit (MSY)
Population viability analysis (PVA)
S-I-R disease
MAXENT
*Occupancy
*Capture-Mark- Recapture

11. Types of Models: Forgoes attempts to explain mechanism
Statistical (pattern)
Mathematical/Scientific (process)
Forgoes attempts to explain mechanism
Data-driven (fit models to data, evaluate)
Begins with description/prediction of how nature might work (hypothesis)

12. Statistical models:
Frequentist (traditional stats classes)-tests assume normality, Ho Ha paradigm
Likelihood (modern analysis classes)- accounts for observation & process error, compete multiple hypotheses simultaneously
Bayesian (v. modern courses)- incorporates prior information about probabilities of competing hypotheses, computationally complex

13. Is uncertainty considered?
Deterministic (no)
Stochastic (yes)
Parameters fixed (ave.)
Multiple runs of same conditions = same outcome
Estimate variation in parameter values & env.
Usually simulation-based, draw from distributions

14. Stochastic models
Observation/measurement error
Process error
Uncertainty in values (counted, measured, analyzed, etc.)
Assumed to be normal and random...
Affects future dynamics of system
Demographic stochasticity: random processes where multiple trials produce diff. outcomes (ex. predation)
Environmental stochasticity: unpredictable variation imposed from “outside” of the system (ex. climate)

15. Is there a mathematical solution?
Computational (no)
Analytical (yes)
Run for range of values (simulations) to observe behavior
More complex models
Equations solved with algebra and calculus (equilibria)
Only the simplest models

16. From Hall (1988) Ecol. Mod.

17. How is time considered?
Discrete
Continuous
Predicts next time step as a function of current param values
Ex. Geometric pop growth
Predicts any future time based on current params
Ex. Exponential pop growth
Nt+1 = Nt

18. What’s the level of interest?
Population
Metapopulation
Directly modeled as numbers of individuals (all equal)
Culmination of individual behaviors
Dynamics of multiple interconnected populations

19. Individual-based models (agent-based)

20. What’s the level of interest?
Food web
Ecosystem
Linked differential equations for each level
Dynamics integrative ecosystem processes (biogeochem, ecosim)

21. Species distribution models (MaxEnt, ecological niche, climate envelope)
Now
Doubling CO2
Lozier et al. (2009)

22. Global Circulation Models
Climate Scenarios
*simulations also called forecasts

23. Is there structure? All individuals the same Simple models
Un-structured
Structured
All individuals the same
Simple models
Individuals assigned to biologically important groups (age/size/stage/geno type)

24. Predictive modeling cycle
(iterative process)

25. General Utility of Models
Formal (usually quantitative) descriptions of ecological systems
Often generate testable hypotheses about systems (iterative tools for learning)
“Experiments” are cheap and easy to run
Essentially impossible to validate for anything but the simplest models
Useful for answering “What if…” scenarios