Modeling as a Tool for Fish

Ecology and Management

Daniel E. Shoup, Ph.D.

Assistant Professor

Department of Natural Resource Ecology & Management

Oklahoma State University

http://nrem.okstate.edu/shouplab/modeling_workshop.htm

Outline for Talk

• What is a model and why would I want to use one?

• Basic info needed to predict changes to fish populations

• FAST modeling software

• Excel options

Introduction

What is a Model?

• Models are a simplified representation of real thing or concept:

� Physical models – scaled representations of a real thing.

Army Corps of Engineers physical model of Mississippi River (at St. Louis, MO)

Introduction

What is a Model?

• Models are a simplified representation of real thing or concept:

� Physical models – scaled representations of a real thing.

� Conceptual models – flow charts or other representations of “ideas” or other abstract concepts.

Introduction

What is a Model?

• Models are a simplified representation of real thing or concept:

� Physical models – scaled representations of a real thing.

� Conceptual models – flow charts or other representations of ideas.

� Mathematical models – equations used to predict/describe something of interest.

� A simple example:

– Predicting fish weight by knowing its length

– Weight = a * lengthb

Introduction

� Mathematical models – equations used to predict/describe something of interest.

� A simple example:

– Predicting fish weight by knowing its length

– Weight = a * lengthb

same as: log(weight)=log(a)+b*(length)

y = 4.8306*10-6 x3.191

0

1

2

3

4

5

250 350 450 550 650 750

Weig

ht

(kg

)

Total Length (mm)

So what does a 450mm fish weigh?0.0000048306 * 4503.191 = 1.414

or about 3 lb

� Another example: What makes a fish population get larger or smaller?

– Birth/Immigration (coming into population)

– Death/Emigration (leaving population)

� We can describe this with a math equation:

– Nt+1 = Nt + B – D

Where:Nt+1 = pop size in next time step (e.g., next yr)

Nt = current pop size

B = number of births/time step

D = number of deaths/time step

Introduction

Introduction

• How can I use this in Fish Ecology/Management?

� Can I improve the population size or size structure of a sportfish?

� Effect of length regulations

� Harvest quota/bag limit

� Test for growth overfishing = harvesting fish at a rate that reduces maximum yield (i.e., no fish reach growth potential)

� Is the current bag limit sustainable?

� Test for recruitment overfishing = excessive harvest of adults does not allow adequate reproduction to sustain population

� Play “what if” for parameters you cannot easily/accurately measure – determine range of possible outcomes:

� Mortality estimates (harvest or natural)

� Reproductive rates

– Explicitly consider variable recruitment

Introduction

• Some common objections to using models:

� They are too complicated (I hate math/I’m no good at math).

� …Let the software handle the complexity

� They aren’t real…so their answer may not be right.

� …But is better than no information (more correct than other “guesses” you may come up with)

� …Can overcome by adding random variability to model = range of outcomes

• Some amusing, but insightful quotes:

� "For every complex question there is a simple and wrong solution." A. Einstein.

� "Make your theory as simple as possible, but no simpler." A. Einstein

� "All models are wrong but some are useful." George E.P. Box

Outline for Talk

• What is a model and why would I want to use one?

• Basic info needed to predict changes to fish populations

• FAST modeling software

• Excel options

Data types for modeling

• Basic info needed to predict changes to fish populations

� Population size

� Growth rate (mean length at each age)

� Length-weight relationships

� Mortality rate

� Reproductive or recruitment rate

• If we are missing one or more of these:

� Go measure it.

� Try several hypothetical values to see the range of possible outcomes.

• Let’s look in some detail at how we can measure each of these.

Data types for modeling

� Population size:

� Mark-recapture or multi-sample depletion methods

� Quantitative gear (hydroacoustics, trawl, purse seine, etc.)

� CPUE and catchability coefficient

� Pick a number…just remember it is not real

– Many models provide yield per recruit = population size is not critical

Data types for modeling

� Growth rate

� von Bertalanffy growth curve or mean length at age

( )( )( )0*1

ttk

t eLl−−

∞−=

050

100150200250300350400450

0 1 2 3 4 5 6

Len

gth

(m

m)

Age (yr)

lt = fish size at time “t”

L∞

= Maximum fish size

(average)

e = base of natural log

(2.718281828…)

k = constant describing growth

rate

t0 = time at which fish was

0mm long

Data types for modeling

� Length-weight relationships

� Derived for population of interest using equation

– Weight = a * lengthb

– Fit using field data as linear equation

log(weight)=b*(length) + log(a)

� Use standard weight equation with known or estimated relative weight (Wr)

– Wr = 100 represents 75th percentile of weights

– Wr ≈ 91 is usually “average” (see Standard Methods book for mean values of several species).

Data types for modeling

� Mortality rate

� Instantaneous vs. natural mortality

– Instantaneous mortality = use with any time step, but has no intuitive meaning.

– Annualized mortality = % dying per year (but cannot be used to give number dying in < 1/yr time step)

Annualized rate = 1 – (e - instantaneous rate)

where e = inverse natural log symbol

Data types for modeling

� Mortality rate

� Instantaneous vs. natural mortality

� Catch curve

ln(Nt) = ln(N0) – Z*t

Y = b – mX

Where:

Nt = population size at age t

N0 = population size at age 0

Z = instantaneous mortality rate (positive value despite fact that slope is negative)

t = age class

S = annual survivorship = e-Z

A = annual mortality = 1 - S

Theoretical maximum age = ln(1) – b/mWhere: b = y-intercept and m = slope from catch curve

y = -0.4256x + 6.2087R2 = 0.9989

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 1 2 3 4 5 6 7 8 9 10

Log

eC

PU

E

t (time) = Age (yr)

Data types for modeling

� Mortality rate

� Instantaneous vs. natural mortality

� Catch curve

� Separate fishing mortality and natural mortality

Z = F + M

Where: Z = total mortality rate (instantaneous)

F = fishing mortality rate (instantaneous)

M = natural mortality rate (instantaneous)

– The same relationship exists with annualized mortality rates (% of population dying each year)

A = u + v

Where: A = total mortality rate (A = 1 - e-Z)

u = fishing mortality

v = natural mortality

Data types for modeling

� Mortality rate

� Instantaneous vs. natural mortality

� Catch curve

� Separate fishing mortality and natural mortality

� Conditional fishing and natural mortality

– cf = annualized fishing mortality (u) if there were no natural mortality

» In reality, u < cf because some fish died of natural causes before they could be harvested.

– cm = annualized natural mortality (v) if there were no fishing.

» In reality, v < cm because some fish were caught and harvested before they could die of natural causes.

Data types for modeling

� Reproduction

� Spawning potential ratio (SPR)

– Is a measure of the effect of harvest on reproduction by entire population.

SPR =

– If SPR < 35%, most populations will decline (recruitment overfishing).

# Eggs prodcued when fish are removed by fishing# Eggs produced when no fishing mortality occurs

Outline for Talk

• What is a model and why would I want to use one?

• Basic info needed to predict changes to fish populations

• FAST modeling software

• Excel options

Using FAST software

• FAST software (Fisheries Analysis and Simulation Tool)

� Does 2 things:

1. Calculate statistics from fisheries data (note: only one species/site at a time…not best option for routine analysis)

– Mean length at age

– Length-weight relationships & relative weights

– Calculate PSD categories

– Fit von Bertalanffy growth curves (L∞, k, t0)

– Mortality estimation

» Catch curve using your data (total mortality, survival, theoretical max age)

» Estimate natural mortality (“guesses” based on typical populations…no data required)

– Length-fecundity relationship (requires data)

Using FAST software

• FAST software (Fisheries Analysis and Simulation Tool)

� Does 2 things:

1. Calculate statistics from fisheries data

2. Model the population under different conditions using 2 kinds of models:

1) Yield-per-recruit model (Beverton-Holt, Jones 1957

modification)

» Can systematically vary mortality (harvest or natural mortality).

» Can systematically vary minimum length regulation

» Can also model fixed minimum length and/or slot limits

Using FAST software

• FAST software (Fisheries Analysis and Simulation Tool)

� Does 2 things:

1. Calculate statistics from fisheries data

2. Model the population under different conditions using 2 kinds of models:

1) Yield-per-recruit model (Beverton-Holt, Jones 1957

modification)

2) Dynamic Pool model (Ricker 1975)

» Allows mortality (harvest and natural mortality) to vary with age class

� Will not systematically vary as YPR model did

» Allows variable or fixed recruitment to be modeled

» Software is available from AFS for $75/license

» Windows vista/7 must use “compatibility mode”

Parameters from log(TL) vs log(weight) regression

Theoretical max age from catch curve

Parameters from von Bertalanffy curve

Don’t need…FAST will do for you

Minimum length regulation (Min TL)Start population size (No)

Set lowest/highest mortality rates to test and “step”, FAST will step through all of these

Specifying lengths of interest (mm) will allow you to track abundance of up to 3 length classes

Allows size-based reproductive output (SPR = Spawning Potential Ratio)…opens new dialog box

Model by varying Min Length or slot limit moves the Min TL from model parameters to a new box that you fill out for minimum and maximum values to test (like mortality).

Model by varying Min Length or slot limit moves the Min TL from model parameters to a new box that you fill out for minimum and maximum values to test (like mortality).

Model by varying Min Length or slot limit moves the Min TL from model parameters to a new box that you fill out for minimum and maximum values to test (like mortality).

Length when fish first are long enough that anglers harvest (recruited to creel)

Once everything is parameterized, click on the “L” icon to load the parameters and “R” to run the model, then, click “View Output” and select variables for a table or figure.

Use these buttons to switch between Yield per recruit (YP) and Dynamic Pool (DP) models.

Let’s go over the DP model settings

Set parameters up the same way you did for YP model.

Num Years = how many years to simulate…do at least 20 to reach equilibrium

Age of interest = specify an age class for which you may want additional output

Select species being modeled…FAST will then list stock-trophy sizes

Can specify different natural (cm) and fishing (cf) mortality rates for up to 4 age groups

Several options exist that allow you to introduce variable year-class strength based on random variation in recruitment.

Conditional fishing mortality (% of population that is harvested/yr)

Size at harvest (minimum length regulation)

Total weight harvested at a given mortality and minimum length

Yield contour plot

Conditional fishing mortality (% of population that is harvested/yr)

Size at harvest (minimum length regulation)

Number harvested

Catch contour plot

FAST versus Spreadsheets

• Some limitations of FAST:

� Cannot change the equations it uses…so the assumptions are fixed.

� Example, slot limits fail to produce better trophy potential because there is no effect of harvest on growth rate (i.e., growth is constant…but in the real world competition usually slows growth of larger populations)

� You cannot get output for factors the program was not designed to produce.

• Excel offers reasonably easy way to produce models…and the sky is the limit on functionality (assuming you know excel well enough)

� I am providing a modeling spreadsheet that handles basically everything FAST does and more. Feel free to modify/use for whatever you may want (Acknowledgement…this spreadsheet was based on spreadsheet produced during Mike Allen’s modeling workshop at SDAFS 2011.

• If you have questions, feel free to contact me (dshoup@okstate.edu or 405-744-9671) and I can help you with FAST or the spreadsheet.