1. FW364 Ecological Problem Solving Class 4: Quantitative Tools
Sept. 11, 2013

2. Outline for Today
Objectives for Today:
Survey how and why models are used
Survey different categories of models
Goal for Today:
Help you to “get the gist” of modeling in general
(we will go into more detail later about most models discussed today)
Help you to understand what’s so special about this monkey

3. Quantitative Tools Why quantitative tools (math / models) are useful
make our process and assumptions transparent
help us to understand natural systems (can often be not intuitive)
allow us to do virtual experiments (cheaper than real experiments)
make predictions that can be tested in the real world
strengthen adaptive management (predictions, understand outcome)

4. Quantitative Tools
Quantitative tools make our process and assumptions transparent
Process examples:
The DNR suggests a 25% reduction for walleye bag limit
Models can be presented in reports and at public meetings
that show exactly how the 25% reduction was calculated
 Models are helpful for showing that management
decisions are not arbitrary
Assumption examples:
Mass balance: Steady-state assumption: Inputs = Outputs
Predation: No predator saturation (satiation)
Harvest: No reduction in angler effort with reduced bag limit
Value:
Other researchers / managers / stakeholders know how results were obtained
Others can evaluate whether the results are valid given knowledge of process and assumptions

5. Quantitative Tools SH SP TH * cFP TP * FP =
Quantitative tools help us to understand natural systems SH SP
TH * cFP
TP * FP =
Help us to understand how aspects of the natural world are related
c is fraction of net plant production incorporated by herbivores
Help us to handle complexity (work with or just deal with)
Equation allowed us to see how plant turnover time affects the amount of herbivore biomass that can be supported

6. Quantitative Tools
Quantitative tools allow us to do virtual experiments
Virtual experiment example:
What happens to salmon biomass if zebra mussel biomass doubles?
What then happens to prey of salmon?
What happens if another mussel (e.g., quagga mussels) invades?
We can answer these questions by altering model variables / parameters
Could also use “real” experiments, but there are limitations
Mesocosms - lose the complexity of food web
Experimental additions to lake - unethical for invasive species
Both take a lot of resources (time and money)

7. Quantitative Tools
Quantitative tools make predictions that can be tested in the real world
Built model of wolf population growth using data
Made predictions for from those data
Predictions can now be evaluated in 2013
Models can be refined as needed
would know now if linear or non-linear was best model

8. Quantitative Tools Quantitative tools strengthen adaptive management
Hypothetically, say the
data suggest population growth was linear
Say our goal was 900 wolves by 2012
and we assumed in 2007 that population growth would be non-linear
We adjust our model to include linear growth
Perhaps introduce more wolves to account for slower population growth

9. Components of Models Variables: the quantities that change in a model
Dependent variable: The quantity that we want to estimate / predict (y)
E.g., The amount of pollutant in the lake; population size
Independent variables: variable being manipulated or followed (x)
E.g., Time
Functions: describe relationships between state variables
E.g., Lynx abundance is a function of hare abundance
Lynx abundance = f (hare abundance)
Could be linear function (y = a + bx)
lynx abundance = a + b * hare abundance
Parameters: constants that specify functions
Mediate the relationship between independent and dependent variables
Typically numbers that we can hopefully estimate with real data
E.g., Assimilation efficiency, per capita birth rate, survival probability
a and b (above) are parameters

10. Model Complexity Model complexity range
In general, different types of models fall somewhere along a
simple-complex continuum
Simple
Complex
More general, behavior easy to understand (why the model
predicts what it does), unrealistic
More specific, realistic
Example: Salmon stocking models
Simple: Total # salmon in lake = f(# naturally reproduced, # stocked salmon)
Complex: Total # salmon in lake = f(# naturally reproduced, # stocked salmon,
competition, harvest, # prey, # predators)

11. Model Complexity Which level of complexity do we use?
Sometimes simple is best, some times complex, sometimes use both
“Make everything as simple as possible, but not simpler”
~ Albert Einstein
When the model is too complex, it can get very hard to understand
the model results and connect them to assumptions
 There is no point in constructing a model that is an exact representation of nature…
…would be as hard to understand as the system we're trying to model!
But there is a tendency to want to consider all the factors
The art is figuring out how to simplify
 what to leave out and still get at important processes

12. Pictorial Monkey Model Complexity
Monkey Reality
Monkey Model 1
Monkey Model 2
Monkey pictorial model
Monkey
maybe human?
DESIRED COMPLEXITY

13. Model Break! http://mvhs1.mbhs.edu/mvhsproj/deer/deer.html
Let’s think about deer…

14. Types of Models Model Categories: Static vs. Dynamic
Discrete vs. Continuous
Deterministic vs. Stochastic
Analytical vs. Numerical Simulation

15. Static vs. Dynamic Static models assume system is at steady state
E.g., mass balance; predator and prey populations at carrying capacities
Often much easier to use: can build an equation for steady state as function of different parameters & see how parameters affect equilibrium
e.g., how attack rate of predator affects carrying capacity
Dynamic models provide a trajectory of some variable over time
Can be used to predict both trajectories and equilibria
More powerful, but more complex; e.g., population size over time
Dynamic
Static
Carrying capacity
Equilibrium
Population size
Time

16. Discrete vs. Continuous
Discrete models useful for predicting quantities over fixed intervals
Time is modeled in discrete steps; Intervening time is not modeled
Good for populations that reproduce seasonally, like moose, salmon
(don't use calculus) Extreme example: 13-year cicadas
Continuous models useful for continuous processes
Can predict quantities at any time; time is a smooth curve
Good for populations that breed continuously, like humans
(apply calculus to solve for a point in time)
Discrete
Continuous
13-year Cicadas vs. Humans
Population Size
Population Size

17. Discrete vs. Continuous
How we use calculus in continuous models
Population size, N
Time, t
Nt Some continuous function
dN/dt Derivative of that function (differential equation)
ΔN/Δt Population growth rate
Derivative is the instantaneous slope of the N versus t function
Change in N over time [ΔN/Δt]
(like zooming in at a point on curve until straight line appears)
13-year Cicadas vs. Humans

18. Deterministic vs. Stochastic
Deterministic models useful for making exact predictions (no uncertainty)
Stochastic models have uncertainty or error built in
13-year Cicadas vs. Humans

19. Deterministic vs. Stochastic Very simple deterministic model
Deterministic models useful for making exact predictions (no uncertainty)
E.g., population will be 5,564 in 3 years
y = a + bx
Very simple deterministic model
if we want to know y (dependent variable), we simply plug in values for a, b (parameters or constants) and then vary x (x will often be time)
Advantage of Deterministic Models:
Great as general tools for understanding ecological problems because they are simpler and easier to understand than stochastic models
Drawback of Deterministic Models:
There are no real-life situations in ecology where we can make exact predictions and expect them to be right
13-year Cicadas vs. Humans

20. Deterministic vs. Stochastic
Stochastic models have uncertainty or error built in
Advantage of Stochastic Models:
More realistic: the ecological world is messy
I.e., not fully describable by sets of deterministic equations
Our models never fit perfectly
The scatter around the model we usually attribute to “random error”,
but this really means “unexplained error”
13-year Cicadas vs. Humans
Scatter:
Points do not fall perfectly along line
Population Size

21. Deterministic vs. Stochastic Stochastic model example
Stochastic models have uncertainty or error built in
Advantage of Stochastic Models:
More realistic: the ecological world is messy
I.e., not fully describable by sets of deterministic equations
Our models never fit perfectly
y = a + bx + error
Stochastic model example
Prediction (y) is based on a, b, x and error (stochasticity)
Model predicts a range (cloud of points) for y (not a single value for each x)
Using statistics we can put bounds on the likely values of y
E.g., in 5 years we predict there will be between 1000 and 1500 wolves in the UP with 95% confidence
13-year Cicadas vs. Humans

22. Analytical vs. Numerical Simulation
Analytical models can be solved using algebra and calculus
These are functions we are familiar with from math courses
More general (can be applied in many contexts)
e.g., All our mass balance problems (T = S/F)
Some dynamic models (e.g., exponential population growth)
Numerical simulation models cannot be solved using algebra and calculus
Either because they have discontinuous functions
Or because there are too many variables
E.g., Stochastic models (have “randomness” involved)
 Need a computer to solve iteratively:
Plug in starting values (real numbers)
Computer calculates output for each time step
Advantage of Numerical Simulation Models:
Greater realism, easier to use with available software (don't need to be a math whiz)
Drawback of Numerical Simulation Models:
Harder to understand behavior
13-year Cicadas vs. Humans

23. Analytical vs. Numerical Simulation Fluid dynamics around a boat hull
Example: Equation series without an analytical solution
Fluid dynamics around a boat hull
13-year Cicadas vs. Humans
Computers are used to plug in different combinations of numbers for the variables to determine what combinations work to make the equations balance
We’ll create complex conceptual models in Stella and let Stella do numerical simulations for us to solve them

24. Where do our mass-balance models fit into these categories?
Types of Models
Where do our mass-balance models fit into these categories?
Model Categories:
Static vs. Dynamic
Discrete vs. Continuous … why?
Deterministic vs. Stochastic
Analytical vs. Numerical Simulation

25. Where do our mass-balance models fit into these categories?
Types of Models
Where do our mass-balance models fit into these categories?
Model Categories:
Static vs. Dynamic
 Steady state (no time component)
Discrete vs. Continuous …why?
 No time component
Deterministic vs. Stochastic
 No uncertainty
Analytical vs. Numerical Simulation
Solved using algebra

26. Nt+1 = Nt Wrap-Up Monday: Starting population growth
Chapter 1 in Text (if you want to read)
Nt+1 = Nt