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Exploitation of natural populations – Entertainment and “Sport”http://www.cairnsfishing.com/images/photos/photo61.jpghttp://www.accuratereloading.com/z200110.jpg
Step 2We will start by using a constant yield model (i.e. a fixed number of individuals are harvested from the population each year)
Label a cell Fixed Annual Yield and, in the adjacent cell, put an arbitrary starting value – say 1
Create a harvest (h) column next to the “additions” column. This is the number of individuals that you are going to harvest in each interval. Make each cell address equal to the value of your Fixed Annual Yield, with the exception of t0, which make equal to 0
Calculate Total harvest over the 100-year projection by summing the harvest column (should equal 100, at this stage)
You must now subtract the harvest each year from the numbers in the population …….BUT…….do you subtract the harvest from the population BEFORE or AFTER it has reproduced?
What is the difference?Nt+1 = ((Nt – ht+1).R) / {1 + [(Nt – ht+1).(R-1)/K]}
STEP 3Adjust your value of Fixed Annual Yield and adjust the time of first harvesting in order to maximise the total harvest over the 100 year projection
BUT remember – it is important that the final R (R99) value is greater than or equal to 1.0000 (i.e. the population is sustainable)
PLAYAdvantages
Fixed Yield Models are liked by industry because they can plan plant and workforce in advanceCommunities like Fixed Yield Models because they know how much money will be coming in – in advance
Disadvantages
Data-hungry: small errors in Yield can result in population crashes
Management implications of MSY models
1. Set a Total Allowable Catch (TAC) each year
2. Apportion TAC amongst rights’ holders
3. Open the resource to exploitation
4. Keep a cumulative log of harvest and close access to right’s holder when
TAC reached
How do you calculate MSY without playing around?
Remember the earthworm model – the MSY was achieved by harvesting the number of “recruiting” segments to the worm (population): two per day.
How do we find the equivalent of the “two segments per day” MSY in our population?
Maximum recruitment to the population occurs not at the carrying capacity (when the population size is at its maximum), but at some intermediate density.If you allow the population to increase beyond this intermediate density you are decrease the number of recruits.
Look at the Stock : Recruit curve
In our example here – the maximum number of individuals that recruit to the population is 44 524 over the period 21- 22
Harvesting from a population (in a sustainable way) does not harm the population.
By preventing a population from reaching the carrying capacity, you maintain it in a constant state of growth and ensure that the negative effects of intra-specific competition are reduced.
WHY?
Constant Effort ModelLet us imagine a population, size N.
You go out today and spend 2 hours harvesting from the population (with efficiency e) and come back with aa individuals
But you went out yesterday and spent 4 hours harvesting from the population (with efficiency e), and came back with bb individuals
Which is larger: aa or bb?
Number of Hours Catch2 123 194 225 346 367 388 489 5810 6411 6212 72
0
10
20
30
40
50
60
70
80
0 2 4 6 8 10 12 14
Effort (hours)
To
tal
Cat
ch
WHY?
Let us imagine a fish population, size N.
You go out today and spend 2 hours harvesting from the population using a motor-powered vessel and a trawl net and come back with aa individuals
But you went out yesterday and spent 2 hours harvesting from the population using a canoe and a throw net, and came back with bb individuals
You go out today and spend 2 hours harvesting from the population using a motor-powered vessel and a net and come back with 120 546 individuals
But you went out yesterday and spent 2 hours harvesting from the population using a motor-powered vessel and a net and came back with 98 113 individuals
Catch is proportional to effort, efficiency and population size.
If we fix efficiency (assume hereafter that it is equal to 1), then catch will reflect population size and effort.
If we fix effort, then catch will reflect population size: in other words the numbers caught will reflect some fixed proportion of the population.
Step 2Label a cell Efficiency and, in the adjacent cell, enter a value of 1 (100% efficient)
Label a cell Effort and, in the adjacent cell, put an arbitrary starting value – say 0.1. You are going to play around with this number in just a minute.
Label a cell EE (Effort x Efficiency) and make it equal to the product of the aforementioned Efficiency and Effort cells (it should equal 0.1).
Create a harvest (h) column next to the “additions” column. This is the number of individuals that you are going to harvest in each interval. Make this number in each cell equal to the product of the EE cell and the population size: with the exception of t0, which make equal to 0
In order to avoid circular arguments that will arise when you subtract the harvest from the population size, you need to re-enter the formula to calculate population size at t+1 into the harvest calculations.
Thus – the harvest at (e.g.) t5 is calculated as
Building a Constant Effort Model
h5 = ((N4 R) / {1 + [N4.(R-1)/K]}) * EE
Step 2Calculate Total harvest over the 100-year projection by summing the harvest column
You must now subtract the harvest each year from the numbers in the population …….BUT…….do you subtract the harvest from the population BEFORE or AFTER it has reproduced?
What is the difference? Nt+1 = ((Nt – ht+1).R) / {1 + [(Nt – ht+1).(R-1)/K]}
STEP 3Adjust your value of EFFORT and adjust the time of first harvesting in order to maximise the total harvest over the 100 year projection
BUT remember – it is important that the final R (R99) value is greater than or equal to 1.0000 (i.e. the population is sustainable)
PLAYAdvantages
Fixed Effort Models are liked by management authorities because they are not as sensitive as Fixed Yield models to mistakes. A 10% change in effort will not necessarily crash the population, whilst a 10% increase in a Fixed Yield probably will!
Disadvantages
Because the numbers harvested each year will vary (with population size), industry and communities have problems planning in advance.
Management implications of MSY models
1. Set a Total Effort each year
2. Apportion Effort amongst rights’ holders
3. Open the resource to exploitation
4. Keep a cumulative log of effort and close access to rights’ holder when Total
Effort reached
Effort can be limited by (in the case of fishing)
•Closed seasons•Closed areas•Fleet size, vessel type, engine power•Gear used: number of hooks or lines, mesh size of nets•Time at sea•etc
Building environmental variability into your modelsAll the models you have developed so far are deterministic (essentially fixed), but we know that populations change in size all the time due to extrinsic factors such as the weather. Weather conditions have an impact on the amount of resources available to a population, which in turn influences the carrying capacity.
We need to build some sort of environmental variability into our models if they are to more “accurately” reflect patterns in the real world, and to minimise the chance of over-exploiting the population when we start to harvest.
Before we start this exercise, we need to know how often “bad” or “good” weather conditions occur, and we need to know how these affect the carrying capacity. Whilst the first set of information can be readily obtained from long-term weather sets, the latter is difficult to pin down. That does not matter in a modeling scenario – because we are exploring the processes rather than the actual numbers.
In our models, we are going to use random numbers to indicate the state of the weather each year, and we are going to ask MSExcel to look at these numbers and see if they are greater or less than the numbers we propose to indicate “good” or “bad” weather, and then to assign a carrying capacity accordingly. This modified k value will then be used in our equations to model population size
Weather calculations
If “bad” weather happens, on average, once every 15 years, we can say that the probability of bad weather is 1 / 15 = 0.0667
If “good” weather happens, on average, once every 9 years, we can say that the probability of good weather is 1 / 9 = 0.111
In your spreadsheet, you should set up a new column labeled “Weather A”. Weather is a random number in our model, so ask MSExcel to generate a random number each year =RAND()
The next thing we need to do is to ask MSExcel to identify the weather each year as Good, Bad or Normal, and we do this using the IF function.
Incorporating weather into an unexploited population with:
N0 = 12, R = 1.63, K = 368 974
At this point, no two of us are going to have the same weather conditions. Every time you do something to the worksheet, your weather conditions will change (as they will too if you press the F9 key). You can convert the constantly changing numbers into values using the edit, copy, paste special, values function – BUT DON’T yet
The logic of the IF function is as follows….
We ask MSExcel to look at the contents of a particular cell address and if the contents conform to some pre-established condition, then it will return one answer and if it doesn’t then it will return another answer
For example – Set up a dataset spanning four columns (A-D) and 10 rows. Put titles to each column in row 1 as indicated, and fill column A with random numbers.
In column B, we are going to ask MSExcel to look at each cell in column A and, if it is smaller than 0.20, to return a value of “YES” in the corresponding cell of Column B. Otherwise to return a value of “NO”
The “ ” signs are important when dealing with text in formulae, but should be ignored if using numbers =IF(A2<0.20,”YES”,”NO”)
Repeat the exercise for column c
You can combine IF arguments. For example, in column D we are asking MSExcel to look at the contents of cells in column A and then IF they are larger than 0.9 to assign an answer of “BIG”, IF they are smaller than 0.2 to assign an answer of “SMALL”, otherwise to return an answer of “Average”
RowColumn
A B C D
1 Random Number <0.20 >0.90 Big or Small?
2 0.60 NO NO Average
3 0.65 NO NO Average
4 0.60 NO NO Average
5 0.77 NO NO Average
6 0.39 NO NO Average
7 0.04 YES NO SMALL
8 0.99 NO YES BIG
9 0.96 NO YES BIG
10 0.96 NO YES BIG
=IF(A2<0.20,”SMALL”,IF(A2>0.90,”BIG”,”Average”))
Okay – having set up our weather (in Weather A), define it as GOOD, BAD or NORMAL in a Weather B column using the =IF Function
R 1.63 0.066667K 368974 0.111111
0.25922441.69
623566
t N R Additions Weather A Weather B New K0 12 1.6300 8 0.54 NORMAL 3689741 20 1.6299 12 0.50 NORMAL 3689742 32 1.6299 20 0.28 NORMAL 3689743 52 1.6299 33 0.30 NORMAL 3689744 85 1.6298 53 0.05 GOOD 6235665 138 1.6296 87 0.01 GOOD 6235666 225 1.6294 142 0.20 NORMAL 3689747 367 1.6290 231 0.90 NORMAL 3689748 597 1.6283 375 0.34 NORMAL 3689749 972 1.6273 610 0.45 NORMAL 368974
10 1582 1.6256 990 0.50 NORMAL 36897411 2572 1.6229 1602 0.76 NORMAL 36897412 4174 1.6185 2581 0.26 NORMAL 36897413 6755 1.6114 4130 0.28 NORMAL 36897414 10885 1.6003 6534 0.59 NORMAL 36897415 17418 1.5829 10154 0.55 NORMAL 36897416 27572 1.5567 15350 0.45 NORMAL 36897417 42922 1.5187 22263 0.44 NORMAL 36897418 65185 1.4668 30425 0.25 NORMAL 36897419 95610 1.4012 38363 0.32 NORMAL 36897420 133974 1.3265 43749 0.75 NORMAL 36897421 177723 1.2505 44524 0.54 NORMAL 36897422 222247 1.1816 40362 0.32 NORMAL 36897423 262610 1.1254 32928 0.04 GOOD 62356624 295538 1.0833 24629 0.20 NORMAL 36897425 320167 1.0539 17251 0.84 NORMAL 36897426 337417 1.0342 11535 0.17 NORMAL 36897427 348952 1.0214 7475 0.47 NORMAL 36897428 356428 1.0133 4747 0.95 BAD 9224429 361174 1.0082 2975 0.62 NORMAL 36897430 364150 1.0051 1850 0.34 NORMAL 36897431 365999 1.0031 1144 0.75 NORMAL 36897432 367143 1.0019 705 0.01 GOOD 623566
Good FactorGood k
Probability of Bad WeatherProbability of Good WeatherBad FactorBad k
How?
1.0000
Random Number
Weather is a random number in our model and varies from 0.0000 – 1.0000
In other words, if the random number is less than 0.0667, then it must be bad weather
0.0667
Bad Weather
The probability of bad weather is 0.0667: the probability of good weather is 0.111
0.1111
Good Weather
On the other hand, if the random number is more than 0.8889 (1.0000 – 0.1111) , then it must be good weather
R 1.63 0.066667K 368974 0.111111
0.25922441.69
623566
t N R Additions Weather A Weather B New K0 12 1.6300 8 0.54 NORMAL 3689741 20 1.6299 12 0.50 NORMAL 3689742 32 1.6299 20 0.28 NORMAL 3689743 52 1.6299 33 0.30 NORMAL 3689744 85 1.6298 53 0.05 GOOD 6235665 138 1.6296 87 0.01 GOOD 6235666 225 1.6294 142 0.20 NORMAL 3689747 367 1.6290 231 0.90 NORMAL 3689748 597 1.6283 375 0.34 NORMAL 3689749 972 1.6273 610 0.45 NORMAL 368974
10 1582 1.6256 990 0.50 NORMAL 36897411 2572 1.6229 1602 0.76 NORMAL 36897412 4174 1.6185 2581 0.26 NORMAL 36897413 6755 1.6114 4130 0.28 NORMAL 36897414 10885 1.6003 6534 0.59 NORMAL 36897415 17418 1.5829 10154 0.55 NORMAL 36897416 27572 1.5567 15350 0.45 NORMAL 36897417 42922 1.5187 22263 0.44 NORMAL 36897418 65185 1.4668 30425 0.25 NORMAL 36897419 95610 1.4012 38363 0.32 NORMAL 36897420 133974 1.3265 43749 0.75 NORMAL 36897421 177723 1.2505 44524 0.54 NORMAL 36897422 222247 1.1816 40362 0.32 NORMAL 36897423 262610 1.1254 32928 0.04 GOOD 62356624 295538 1.0833 24629 0.20 NORMAL 36897425 320167 1.0539 17251 0.84 NORMAL 36897426 337417 1.0342 11535 0.17 NORMAL 36897427 348952 1.0214 7475 0.47 NORMAL 36897428 356428 1.0133 4747 0.95 BAD 9224429 361174 1.0082 2975 0.62 NORMAL 36897430 364150 1.0051 1850 0.34 NORMAL 36897431 365999 1.0031 1144 0.75 NORMAL 36897432 367143 1.0019 705 0.01 GOOD 623566
Good FactorGood k
Probability of Bad WeatherProbability of Good WeatherBad FactorBad k
Next you must set new K values based upon the effect that weather has on K
Under Good weather conditions New K = 1.69K Under Bad weather conditions, New K = 0.25K
Under Normal weather conditions New K = K
Use another =IF function
R 1.63 0.066667K 368974 0.111111
0.25922441.69
623566
t N R Additions Weather A Weather B New K0 12 1.6300 8 0.54 NORMAL 3689741 20 1.6299 12 0.50 NORMAL 3689742 32 1.6299 20 0.28 NORMAL 3689743 52 1.6299 33 0.30 NORMAL 3689744 85 1.6298 53 0.05 GOOD 6235665 138 1.6296 87 0.01 GOOD 6235666 225 1.6294 142 0.20 NORMAL 3689747 367 1.6290 231 0.90 NORMAL 3689748 597 1.6283 375 0.34 NORMAL 3689749 972 1.6273 610 0.45 NORMAL 368974
10 1582 1.6256 990 0.50 NORMAL 36897411 2572 1.6229 1602 0.76 NORMAL 36897412 4174 1.6185 2581 0.26 NORMAL 36897413 6755 1.6114 4130 0.28 NORMAL 36897414 10885 1.6003 6534 0.59 NORMAL 36897415 17418 1.5829 10154 0.55 NORMAL 36897416 27572 1.5567 15350 0.45 NORMAL 36897417 42922 1.5187 22263 0.44 NORMAL 36897418 65185 1.4668 30425 0.25 NORMAL 36897419 95610 1.4012 38363 0.32 NORMAL 36897420 133974 1.3265 43749 0.75 NORMAL 36897421 177723 1.2505 44524 0.54 NORMAL 36897422 222247 1.1816 40362 0.32 NORMAL 36897423 262610 1.1254 32928 0.04 GOOD 62356624 295538 1.0833 24629 0.20 NORMAL 36897425 320167 1.0539 17251 0.84 NORMAL 36897426 337417 1.0342 11535 0.17 NORMAL 36897427 348952 1.0214 7475 0.47 NORMAL 36897428 356428 1.0133 4747 0.95 BAD 9224429 361174 1.0082 2975 0.62 NORMAL 36897430 364150 1.0051 1850 0.34 NORMAL 36897431 365999 1.0031 1144 0.75 NORMAL 36897432 367143 1.0019 705 0.01 GOOD 623566
Good FactorGood k
Probability of Bad WeatherProbability of Good WeatherBad FactorBad k
=IF(F9=“GOOD”,G$6,IF(F9=“BAD”,G$4,B$2))
A B C D E F G
123456789
You must now make your population numbers reflect this new K value
R 1.63 0.066667K 368974 0.111111
0.25922441.69
623566
t N R Additions Weather A Weather B New K New N0 12 1.6300 8 0.48 NORMAL 368974 121 20 1.6299 12 0.70 NORMAL 368974 202 32 1.6299 20 0.24 NORMAL 368974 323 52 1.6299 33 0.40 NORMAL 368974 524 85 1.6298 53 0.28 NORMAL 368974 855 138 1.6296 87 0.06 GOOD 623566 1386 225 1.6294 142 0.60 NORMAL 368974 2257 367 1.6290 231 0.48 NORMAL 368974 3678 597 1.6283 375 0.09 GOOD 623566 5979 972 1.6273 610 0.57 NORMAL 368974 972
10 1582 1.6256 990 0.99 BAD 92244 157511 2572 1.6229 1602 0.68 NORMAL 368974 256012 4174 1.6185 2581 0.25 NORMAL 368974 415413 6755 1.6114 4130 0.58 NORMAL 368974 672414 10885 1.6003 6534 0.92 NORMAL 368974 1083615 17418 1.5829 10154 0.61 NORMAL 368974 1734116 27572 1.5567 15350 0.46 NORMAL 368974 2745317 42922 1.5187 22263 0.52 NORMAL 368974 4274518 65185 1.4668 30425 0.45 NORMAL 368974 6493519 95610 1.4012 38363 0.27 NORMAL 368974 9528020 133974 1.3265 43749 0.08 GOOD 623566 14166921 177723 1.2505 44524 0.76 NORMAL 368974 18594322 222247 1.1816 40362 0.61 NORMAL 368974 23004923 262610 1.1254 32928 0.12 NORMAL 368974 26922924 295538 1.0833 24629 0.54 NORMAL 368974 30064125 320167 1.0539 17251 0.18 NORMAL 368974 32382026 337417 1.0342 11535 0.98 BAD 92244 16435027 348952 1.0214 7475 0.80 NORMAL 368974 20918828 356428 1.0133 4747 0.36 NORMAL 368974 25124029 361174 1.0082 2975 0.61 NORMAL 368974 286583
Good FactorGood k
Probability of Bad WeatherProbability of Good WeatherBad FactorBad k
0
100000
200000
300000
400000
500000
6000000 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Time (years)
Nu
mb
ers
REMEMBER – NO TWO will have the same results
Okay – having now built weather into your model for an unexploited population – you must now build it into your Fixed Yield and Fixed Effort harvesting
models
Remember – the population must not crash in your simulations, the Final R (R99) should be greater than or equal to 1.0000, and you should aim to maximize the harvest over the 100 year period
PLAYYour model run is from a single simulation – based on the particular set of random numbers generated in that single run.
Ideally you need to repeat the simulation (looking at final R, total harvest, and whether the population crashes or not) thousands of times for each starting time, fixed yield and fixed effort model that you use in order to come up with appropriate values!
You can do this using macros………………but that is another story!