FW364 Ecological Problem Solving

Class 4: Quantitative Tools

Sept. 11, 2013

Objectives for Today:

Survey how and why models are usedSurvey 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)

Outline for Today

Help you to understand what’s so special about this monkey

Why quantitative tools (math / models) are useful

Quantitative tools:• 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)

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 meetingsthat 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 = OutputsPredation: No predator saturation (satiation)Harvest: No reduction in angler effort with reduced bag limit

Value:Other researchers / managers / stakeholders know how results were obtainedOthers can evaluate whether the results are valid given knowledge of process and assumptions

Quantitative Tools

Quantitative tools help us to understand natural systems

Quantitative Tools

Help us to handle complexity (work with or just deal with)

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Help us to understand how aspects of the natural world are related

Equation allowed us to see how plant turnover time affects the amount of

herbivore biomass that can be supported

Quantitative tools allow us to do virtual experiments

Quantitative Tools

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 limitationsMesocosms - lose the complexity of food webExperimental additions to lake - unethical for invasive species

Both take a lot of resources (time and money)

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Quantitative tools make predictions that can be tested in the real world

Quantitative Tools

Made predictions for 2008-2012 from those data

Built model of wolf population growth using 1999-2007 data

Predictions can now be evaluated in 2013Models can be refined as needed

would know now if linear or non-linear was best model

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Quantitative tools strengthen adaptive management

Quantitative Tools

Say our goal was 900 wolves by 2012

Hypothetically, say the2008-2012 data suggest population growth was linear

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

Components of Models

Variables: the quantities that change in a modelDependent 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 variablesE.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 functionsMediate the relationship between independent and dependent variablesTypically numbers that we can hopefully estimate with real dataE.g., Assimilation efficiency, per capita birth rate, survival probability

a and b (above) are parameters

Model Complexity

Model complexity range

In general, different types of models fall somewhere along asimple-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)

Model Complexity

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 understandthe 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 factorsThe art is figuring out how to simplify

what to leave out and still get at important processes

Which level of complexity do we use?

MonkeyModel 1

MonkeyReality

DESIRED COMPLEXITYmaybe human?

Pictorial Monkey Model Complexity

Monkey Monkey Monkey

MonkeyModel 2

Model Break!

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

Model Categories:

• Static vs. Dynamic• Discrete vs. Continuous• Deterministic vs. Stochastic• Analytical vs. Numerical Simulation

Types of Models

Static vs. Dynamic

Static models assume system is at steady stateE.g., mass balance; predator and prey populations at carrying capacitiesOften 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 timeCan be used to predict both trajectories and equilibriaMore powerful, but more complex; e.g., population size over time

Time

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Carrying capacityEquilibrium

Static

Dynamic

Discrete vs. Continuous

Discrete models useful for predicting quantities over fixed intervalsTime is modeled in discrete steps; Intervening time is not modeledGood for populations that reproduce seasonally, like moose, salmon(don't use calculus) Extreme example: 13-year cicadas

Continuous models useful for continuous processesCan predict quantities at any time; time is a smooth curveGood for populations that breed continuously, like humans(apply calculus to solve for a point in time)

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Time, t

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Nt Some continuous functiondN/dt Derivative of that function (differential equation)ΔN/Δt Population growth rate

Derivative is the instantaneous slope of the N versus t functionChange in N over time [ΔN/Δt]

(like zooming in at a point on curve until straight line appears)

Discrete vs. Continuous

How we use calculus in continuous models

Deterministic vs. Stochastic

Deterministic models useful for making exact predictions (no uncertainty)

Stochastic models have uncertainty or error built in

Deterministic vs. Stochastic

Deterministic models useful for making exact predictions (no uncertainty)E.g., population will be 5,564 in 3 years

y = a + bxVery 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

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 equationsOur models never fit perfectly

The scatter around the model we usually attribute to “random error”,but this really means “unexplained error”

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Scatter:Points do not fall perfectly along line

Deterministic vs. Stochastic

y = a + bx + errorStochastic 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

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 equationsOur models never fit perfectly

Analytical vs. Numerical Simulation

Analytical models can be solved using algebra and calculusThese are functions we are familiar with from math coursesMore 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 calculusEither because they have discontinuous functionsOr because there are too many variablesE.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

Analytical vs. Numerical Simulation

Example: Equation series without an analytical solution

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

Fluid dynamics around a boat hull

Model Categories:

• Static vs. Dynamic• Discrete vs. Continuous … why?• Deterministic vs. Stochastic• Analytical vs. Numerical Simulation

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 SimulationSolved using algebra

Types of Models

Where do our mass-balance models fit into these categories?

Wrap-Up

Monday: Starting population growth

Chapter 1 in Text (if you want to read)Nt+1 = Nt