Environmental science.

Modelling Environmental Systems, ENVSCI

310 Assignment 3: Population Models

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Modelling of Environmental Systems: ENVSCI 310

Assignment 3 – Population Dynamics & Harvesting

In this assignment you will explore the dynamics of single species undergoing logistic growth

and experiencing different types and amounts of harvesting and how, in this context,

environmental stochasticity can alter the dynamics of simple deterministic models. This

assignment relates to material discussed in Lectures VII and XXII. The assignment is

based on material in Donovan and Welden (2002).

The Excel spreadsheet ‘LabThree-HarvestModels.xlsx’ (on Cecil) contains two

worksheets, each of which contains templates for building the models we are focusing on.

The first part of this handout details the models and their background, and the second part

outlines what you need to do for this assignment. You need to read, and make sure you

understand, the background material before you move on to the questions. Gillman

(2001) provides an introduction to the logistic model of population dynamics that

we will explore in this lab.

1. Logistic Population Model

The Logistic model modifies the simple Exponential model (introduced back in Lecture VII)

by including limits to population growth in the form of a carrying capacity (K). The carrying

capacity is the maximum amount of individuals that the environment can support indefinitely.

In other words, the Logistic model removes the Exponential model’s assumption of there

being no limits to population growth:

)1(

Eq. 1

where: N = population size, t = time, r = the instantaneous population growth rate and K =

the environmental carrying capacity.

The first part of Eq. 1 (rN) is the growth component. If r > 0 the population will increase, but

if r < 0 then it will decline. The second part (1-N/K) reduces population growth as a function

of how close the population is to the carrying capacity. When N is much smaller than the

carrying capacity then N/K will be close to zero and population growth will be close to

exponential (rN). We can interpret this as an abundance of available resources in the

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Assignment 3: Population Models

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environment. As N increases, N/K will tend towards one, and population growth will slow as

resource availability begins to become more limiting. Thus, starting from a small size the

population will grow exponentially before growth slows as the population’s abundance

approaches the carrying capacity. Population growth will be fastest when N = K/2, and we

can use this information to work out the amount of harvesting a population can support (see

below). At the carrying capacity the population stops growing and so dN/dt = 0; that is, there

are sufficient resources to sustain the population size, but no more.

In this lab we will use the discrete form of the logistic model, which is given by:

Eq. 2

In Eq. 2, r

d

is the discrete population growth rate. The discrete growth rate is the amount

that the population changes in each time-step; if r

d

> 0, the population is increasing, if r

d

< 0

it is decreasing and if r

d

= 0.0 the population is static.

2. Harvesting

A fundamental goal for managers of harvested populations, such as fisheries or game

species, is to ensure that the amount harvested is sustainable but not excessively limiting.

The amount that can be harvested while guaranteeing population persistence is called the

maximum sustainable yield (MSY), a concept that has been the focus of considerable

debate in fisheries science and elsewhere. Heino and Enberg (2008) review these

concepts, and their apparent failings, in an applied context. For a population following the

continuous logistic growth model (Eq. 1) the population grows fastest when N = K/2 and at

this point the growth rate (N) is (rK) / 4. Nt

is the change in population size from one time-step to the next, so is equal to Nt

– Nt-1

. Thus, assuming the population is initially greater

than K/2, the MSY is (rK)/4 (in other words, the amount that we can safely remove from the

population in a given time period is the same as the amount added to the population over

that time period). We can use this as a reasonable approximation of the MSY for the

discrete form of the model.

Harvesting can take one of two forms: (i) fixed-quota harvest, or (ii) fixed-effort harvest.

Under a fixed-quota system a constant amount of individuals is removed from the

population; this amount is termed Q (0 < Q < K) and results in the discrete logistic model

(Eq. 3) becoming:

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Assignment 3: Population Models

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Eq. 3

Under a fixed-effort harvest a certain fixed effort is expended on harvesting each year (e.g.,

as under a time-based quota, a restricted number of harvest licenses being issued, etc.).

We can represent this as a proportional reduction (E; 0 < E < 1) in the population:

Eq. 4

If either form of harvesting results in greater than (rK)/4 individuals being removed from the

population per unit time then the population will ultimately be over-exploited to the point of

extinction.

3. Adding Stochasticity

A key assumption of all of the models described above is that r

d

and K do not change over

time. If we know N0

, r

d

, K and t, we can calculate the population size at any time in the

future (or the past). Likewise, starting from the same conditions, the population will show

exactly the same behavior: in a deterministic model – the outcome is determined solely by

the inputs.

This depiction of population dynamics is clearly unrealistic; for example, the environmental

carrying capacity (K) might vary over time as a result of environmental factors such as

climate and the same factors may affect the population’s growth rate (r

d

). Variability

associated with ‘good’ and ‘bad’ years for population growth is known as environmental

stochasticity. Thus, in our model we would need to represent this variability in the

parameters. A simple way to include stochastic effects in the model is to vary r

d

from year-to-year by multiplying r

d

by some other variable (D) that varies with time. We also need to

decide how much r

d

varies each year – the larger the variation the more pronounced the

environmental stochasticity. We can modify the discrete logistic (Eq. 2) model as follows:

NDSrrNN

t

t ddtt 1) (

1

Eq. 5

In Eq. 5, D is the random component and we will fix it to be in the range -1 to +1. S

determines the size of this random variation relative to the value of r

d

– in other words, it is a

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Assignment 3: Population Models

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scaling factor. For example, if you set r

d

to 0.5 and S to 0.25, the value of r

d

will vary

between 0.375 (i.e., 0.5 + 0.5 × 0.25 × -1) and 0.625 (i.e., 0.5 + 0.5 × 0.25 × 1). Thus, the

larger S, the wider the range of values r

d

can take on. We would include harvesting in

this model by removing the required number or fraction of individuals, as per Eqs 3

and 4.

The question we’re interested in is how does the stochastic version of the logistic model

differ from the deterministic one? And does it make any difference for predicting how much

of a population can sustainably be harvested?

Excel Instructions

The spreadsheet templates are contained in the file ‘LabThree-HarvestModels.xlsx’. This

file is available on Cecil. The two worksheets are:

Part 1 – Logistic Harvest Models

Part 2 – Stochastic LHM

Each worksheet is set up as a template in which you will need to build each model. Cells

highlighted in yellow are those where parameter values are stored (e.g., r

d

and K) – you will

need to enter appropriate values in these. Grey cells are those where values you calculate

will be entered (i.e., by entering an equation at the top of the column and copying it down).

There are comments in some of the cells to guide you; these cells have red triangles in

them. The template also includes charts that will automatically plot some of the relevant

data once calculated.

The key to efficiently completing the lab is to set the spreadsheet up carefully at the start of

the exercise. You should use cell referencing in the equations such that you refer to the

cells where r

d

and K, etc. are stored (B6 and B7 in the ‘Logistic Harvest Models’ sheet). This

way you can enter the equation once, drag it down the columns and when you change the

values in cells B6 and B7 the entire sheet will update, allowing you to easily assess the

outcomes of changing those parameters. You will need to be careful in how you

reference the cells – see the guide at the end of this handout if you are unfamiliar with

absolute vs. relative cell referencing (and see also the introductory tutorial to Excel on Cecil).

In the final part of the lab you will need to generate random deviates from a uniform

distribution (in essence, you are going to conduct a Monte Carlo-style analysis). You can do

this in Excel using the =randbetween() function, which takes the form:

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Assignment 3: Population Models

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=randbetween(a,b) and generates a random deviate from a uniform distribution between

a and b. This function only returns integer values but we need floating point numbers. To do

this, and imagining we want numbers between -1 and 1 with three decimal places we would

use: =randbetween(-1000,1000) / 1000

The Report

The lab write-up is due in Week Ten – the week starting on Monday October 6th –

before the start of your practical session. Please submit it to the Student Resource

Centre (4

th

floor HSB) before the start of your practical session.

Your report needs to contain:

Introduction

A brief background to the models you will use, with 2-3 references. Some relevant readings

are listed at the end of the handout.

Issue 1 – Logistic population dynamics without harvesting

You will start by looking at the logistic model in the absence of harvesting (on sheet ‘Logistic

Harvest Models’). For this part of the report you need to include (supporting the graphs with

interpretation):

Examples of time series graphs for the discrete-time logistic model (Eq. 2) showing

how the population behaves with r

d

> 0, r

d

< 0 and r

d

= 0 for at least two values of K.

A useful way to show this is to put multiple population trajectories on the same graph.

For each of these analyses, you should also include a plot of Nt

vs. Nt

, and some

brief interpretation.

Suitable values to get underway with are: N0

= 1, r

d

= 0.5 and K = 2000.

Issue 2 – Logistic population dynamics under harvesting

Once you understand how the population will behave in the absence of harvesting we can

assess the effects of harvesting on the population (on sheet ‘Logistic Harvest Models’). For

this part of the report you need to include (again supporting the graphs with interpretation):

Examples of time series graphs for the Logistic model showing how the population

behaves under fixed-quota (Eq. 3) and fixed-effort (Eq. 4) harvesting for at least two

values of Q and E (holding r

d

and K fixed). For each of these graphs, you should

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Assignment 3: Population Models

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also include a plot of Nt

vs. Nt

alongside either the quota or fixed effort line, and

some interpretation.

Suitable starting values are: N0

= 2000, r

d

= 0.5 and K = 2000. In selecting values of

Q and E to use remember that, assuming N > K/2, the MSY is (rK) / 4.

Issue 3 – Logistic population dynamics under harvesting with environmental stochasticity

Once you understand how the population will behave under harvesting in a deterministic

environment we can move onto considering the effects of environmental stochasticity (on

sheet ‘Stochastic LHM’). For this part of the report you need to include (again supporting the

graphs with interpretation):

Examples of time series graphs for the Logistic model showing how the population

behaves under fixed-quota (Eq. 3) and fixed-effort (Eq. 4) harvesting under low (S

around 0.1) and high (S around 0.5) environmental uncertainty when harvesting is

close to the MSY.

Because this form of the model is stochastic, its behaviour needs to be considered in terms

of its ‘typical’ dynamics. You should run the model 10 times under the same conditions (i.e.,

same values of N0, K and r

d

), each time copying and pasting the output to a new sheet

(make sure to use Paste → Paste Values). Note that pressing F9 will recalculate the sheet

generating a new population trajectory. When you’ve done this you can calculate the

average (=AVERAGE() in Excel) population size and its standard deviation (=STDEV() in

Excel) at each time period (t) across the ten model runs you have carried out.

To synthesise this analysis of the stochastic models you need to include:

A plot of the average and standard deviation of the population size over time under

high and low environmental uncertainty (on the same graph). You should also

overlay a plot of the deterministic version of the model with the same values of N0

, r

d

and K for comparison (either use the results from above or use Eq. 3 and 4).

The key question you need to address is how including uncertainty (stochasticity) in the

model affects the sustainability of population harvesting. How do the deterministic and

stochastic versions of the harvest model differ? Does the MSY provide a sustainable harvest

in the form of the model with environmental stochasticity? Is the MSY still a useful guide in a

stochastic system?

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Summary

Finally, you need to include a brief discussion of the limitations of the models you

have explored. In particular, what are the relative advantages and disadvantages of

stochastic as opposed to deterministic models?

References and Reading

Botsford, L.W., Castilla, J.C. & Peterson, C.H. (1997). The management of fisheries and

marine ecosystems. Science, 277, 509 –515.

Donovan, T.M. & Welden, C. (2002). Spreadsheet Exercises in Conservation Biology and

Landscape Ecology. Sinauer Associates, Sunderland, Mass.

[http://www.uvm.edu/rsenr/vtcfwru/spreadsheets/?Page=conbiolandecol/conbio_landecol.ht

m]

Gillman, M. (2001) Population dynamics: introduction. Encyclopedia of Life Sciences

(Online). John Wiley & Sons, Ltd. [http://dx.doi.org/10.1038/npg.els.0003164]

Heino, M. & Enberg, K. (2008) Sustainable use of populations and overexploitation.

Encyclopedia of Life Sciences (Online). John Wiley & Sons, Ltd.

[http://dx.doi.org/10.1002/9780470015902.a0020476]

Pauly, D., Christensen, V., Guenette, S., Pitcher, T.J., Sumaila, U.R., Walters, C.J., et al.

(2002). Towards sustainability in world fisheries. Nature, 418, 689–695.

Worm, B., Hilborn, R., et al. (2009) Rebuilding global fisheries. Science, 325, 578 –585.

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Assignment 3: Population Models

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Absolute and Relative Cell References in Excel

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This Assignment will require you to build some simple models in Excel. This will require

careful referencing of cells, using both Absolute and relative references. The differences

between these types of cell referencing are explained below.

Relative References

By default, Excel adjusts copied formulas so that the cell references are changed relative to

their new location. Most of the time, you will want to adjust the cell references as you copy a

formula. These adjusting references are known as relative cell references. (Excel is copying

the formula relative to where you are and where you are going; Fig. 1).

Figure 1 Example of a formula copied using relative references; note how the references have changed to refer

to column C.

This example shows that the original formula, in cell B6 was =SUM(B2:B5). This formula

was copied one cell to the right, therefore, Excel automatically updated the formula using a

relative reference, making the new formula =SUM(C2:C5).

Absolute References

Often you might want to use a constant in a formula (for example, the value or r in the

Exponential model, or r and K in the Logistic). In this case you would use an absolute cell

reference. When a formula containing an absolute cell reference is copied to a new location,

the cell reference is not adjusted. To create an absolute cell reference, you will need to add

dollar signs ($) in front of the column and row elements for the cell referenced. When you

copy any formula that contains absolute cell references, the absolute cell references will be

maintained (Figure 2).

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See also: http://office.microsoft.com/en-nz/excel-help/switch-between-relative-absolute-and-mixed-references-HP010342940.aspx.

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Assignment 3: Population Models

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Figure 2 Example of a formula copied using absolute references; note how the reference to cell C1 has not

changed when we drag the formula to the right.

This example shows that there is a 5% increase in total costs. The original formula in cell B8

was =B7*$C$1, because the amount of the increase is in cell C1, we want that to remain

absolute. When we copy the formula to column C, the only cell number that changed was B7

to C7, while the $C$1 cell remained constant in the copy process. You can also hold either

only the row or the column constant by just using a single dollar sign in the appropriate place

(e.g. $A1 vs. A$1).

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