Exploitation of natural populations – Food etc

Exploitation of natural populations – Food etc

http://www.mmcta.org/images/whaling_whale1.jpg

http://www.lakeviewcottage.com/logging-2.jpg

http://www.treehugger.com/files/nz-trawling-02.jpg

Harvesting

Exploitation of natural populations – Entertainment and “Sport”http://www.cairnsfishing.com/images/photos/photo61.jpghttp://www.accuratereloading.com/z200110.jpg

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Imagine an earthworm* – 7 segments long

This earthworm produces two segments per day

t0, N0 = 7

t1, N1 = 9

t2, N2 = 11

t3, N3 = 13

* - Dr Anesh Govendar’s “Earth-worm model”

http://io.uwinnipeg.ca/~simmons/images/lb7fg1a.gifhttp://www.bu.edu/lernet/GK12/eric/earthworm.jpg

SUSTAINABILITY

What is the maximum sustainable number of segments that can be harvested each day?

If you harvest three segments per day:

t0, N0 = 7

t1, N1 = 6

t2, N2 = 5

t3, N3 = 4

t4, N4 = 3

t5, N5 = 2

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Over-exploitation

7 + 2 = 9

9 – 3 = 6

t0, N0 = 7

If you harvest one segment per day:

t1, N1 = 8

t2, N2 = 9

t3, N3 = 10

t4, N4 = 11

t5, N5 = 12

7 + 2 = 9

9 – 1 = 8

Under-exploitation

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t0, N0 = 7

If you harvest two segments per day:

t1, N1 = 7

t2, N2 = 7

t3, N3 = 7

t4, N4 = 7

t5, N5 = 7

t6, N6 = 7

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7 + 2 = 9

9 – 2 = 7

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Total over 12 day projection:

3 per day = 151 per day = 122 per day = 24

Not Sustainable

Sustainable

Maximum

Building harvesting into a model

Model applies for a species showing discrete breeding – use R not r

Assume intra-specific competition occurs

Nt+1 = (Nt R) / {1 + [Nt.(R-1)/K]}

Step 1

Project population into future for x time units without competition

Calculate R and recruitment (additions) for each time step

Plot population size against time (line-graph)

Plot recruitment against population size (X-Y graph)

Determine what the maximum recruitment is and when it occurs

If N0 = 12, R = 1.63, K = 368 974

Project 100 years

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t N R Additions0 12 1.6300 81 20 1.6299 122 32 1.6299 203 52 1.6299 334 85 1.6298 535 138 1.6296 876 225 1.6294 1427 367 1.6290 2318 597 1.6283 3759 972 1.6273 61010 1582 1.6256 99011 2572 1.6229 160212 4174 1.6185 258113 6755 1.6114 413014 10885 1.6003 653415 17418 1.5829 1015416 27572 1.5567 1535017 42922 1.5187 2226318 65185 1.4668 3042519 95610 1.4012 3836320 133974 1.3265 4374921 177723 1.2505 4452422 222247 1.1816 4036223 262610 1.1254 3292824 295538 1.0833 2462925 320167 1.0539 1725126 337417 1.0342 1153527 348952 1.0214 747528 356428 1.0133 474729 361174 1.0082 297530 364150 1.0051 185031 365999 1.0031 114432 367143 1.0019 70533 367849 1.0012 43434 368283 1.0007 26735 368550 1.0004 16436 368714 1.0003 10137 368814 1.0002 6238 368876 1.0001 3839 368914 1.0001 2340 368937 1.0000 1441 368951 1.0000 942 368960 1.0000 5

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]}

Nt+1 = ((Nt R) / {1 + [Nt.(R-1)/K]}) – ht+1

Constant Yield Model

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R 1.63 Fixed Annual Yield 1K 368974 Total Harvest 100

t N R Additions Harvest0 12 1.4941 6 01 18 1.5390 10 12 28 1.5709 16 13 43 1.5923 26 14 69 1.6062 42 15 111 1.6150 68 16 179 1.6204 111 17 290 1.6236 181 18 471 1.6252 295 19 766 1.6257 479 1

10 1245 1.6252 778 111 2023 1.6236 1261 112 3284 1.6204 2037 113 5322 1.6150 3273 114 8594 1.6062 5210 115 13805 1.5924 8177 116 21982 1.5710 12551 117 34533 1.5392 18620 118 53153 1.4944 26276 119 79429 1.4353 34577 120 114007 1.3644 41544 121 155550 1.2879 44787 122 200338 1.2145 42981 123 243319 1.1516 36881 124 280199 1.1025 28727 125 308926 1.0671 20735 126 329661 1.0429 14158 127 343819 1.0271 9304 128 353123 1.0169 5961 129 359085 1.0105 3758 130 362843 1.0065 2345 131 365188 1.0040 1454 132 366641 1.0024 898 133 367539 1.0015 553 134 368092 1.0009 340 135 368431 1.0006 209 136 368640 1.0003 128 137 368769 1.0002 79 1

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10 1375 1.6255 860 111 2235 1.6234 1393 112 3628 1.6197 2248 113 5877 1.6136 3606 114 9483 1.6039 5727 115 15210 1.5887 8954 116 24163 1.5654 13661 117 37824 1.5311 20088 118 57913 1.4833 27990 119 85902 1.4215 36207 120 122109 1.3488 42589 121 164698 1.2722 44835 122 209534 1.2005 42011 123 251545 1.1403 35281 124 286826 1.0941 27004 125 313830 1.0613 19238 126 333068 1.0391 13016 127 346084 1.0246 8501 128 354585 1.0153 5425 129 360010 1.0095 3411 130 363422 1.0058 2125 131 365547 1.0036 1316 132 366863 1.0022 812 133 367675 1.0014 500 134 368175 1.0008 307 135 368482 1.0005 189 136 368671 1.0003 116 137 368787 1.0002 71 1

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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?

t N R Additions0 12 1.6300 81 20 1.6299 122 32 1.6299 203 52 1.6299 334 85 1.6298 535 138 1.6296 876 225 1.6294 1427 367 1.6290 2318 597 1.6283 3759 972 1.6273 61010 1582 1.6256 99011 2572 1.6229 160212 4174 1.6185 258113 6755 1.6114 413014 10885 1.6003 653415 17418 1.5829 1015416 27572 1.5567 1535017 42922 1.5187 2226318 65185 1.4668 3042519 95610 1.4012 3836320 133974 1.3265 4374921 177723 1.2505 4452422 222247 1.1816 4036223 262610 1.1254 3292824 295538 1.0833 2462925 320167 1.0539 1725126 337417 1.0342 1153527 348952 1.0214 747528 356428 1.0133 474729 361174 1.0082 297530 364150 1.0051 185031 365999 1.0031 114432 367143 1.0019 70533 367849 1.0012 43434 368283 1.0007 26735 368550 1.0004 16436 368714 1.0003 10137 368814 1.0002 6238 368876 1.0001 3839 368914 1.0001 2340 368937 1.0000 1441 368951 1.0000 942 368960 1.0000 5

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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

t N R Additions0 12 1.6300 81 20 1.6299 122 32 1.6299 203 52 1.6299 334 85 1.6298 535 138 1.6296 876 225 1.6294 1427 367 1.6290 2318 597 1.6283 3759 972 1.6273 61010 1582 1.6256 99011 2572 1.6229 160212 4174 1.6185 258113 6755 1.6114 413014 10885 1.6003 653415 17418 1.5829 1015416 27572 1.5567 1535017 42922 1.5187 2226318 65185 1.4668 3042519 95610 1.4012 3836320 133974 1.3265 4374921 177723 1.2505 4452422 222247 1.1816 4036223 262610 1.1254 3292824 295538 1.0833 2462925 320167 1.0539 1725126 337417 1.0342 1153527 348952 1.0214 747528 356428 1.0133 474729 361174 1.0082 297530 364150 1.0051 185031 365999 1.0031 114432 367143 1.0019 70533 367849 1.0012 43434 368283 1.0007 26735 368550 1.0004 16436 368714 1.0003 10137 368814 1.0002 6238 368876 1.0001 3839 368914 1.0001 2340 368937 1.0000 1441 368951 1.0000 942 368960 1.0000 5

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Population Size Recruitment

Look at the relationship between recruitment and time.

In this case we should start harvesting 44 524 individuals from time 22

How do we know when to start harvesting?

t N R Additions0 12 1.6300 81 20 1.6299 122 32 1.6299 203 52 1.6299 334 85 1.6298 535 138 1.6296 876 225 1.6294 1427 367 1.6290 2318 597 1.6283 3759 972 1.6273 61010 1582 1.6256 99011 2572 1.6229 160212 4174 1.6185 258113 6755 1.6114 413014 10885 1.6003 653415 17418 1.5829 1015416 27572 1.5567 1535017 42922 1.5187 2226318 65185 1.4668 3042519 95610 1.4012 3836320 133974 1.3265 4374921 177723 1.2505 4452422 222247 1.1816 4036223 262610 1.1254 3292824 295538 1.0833 2462925 320167 1.0539 1725126 337417 1.0342 1153527 348952 1.0214 747528 356428 1.0133 474729 361174 1.0082 297530 364150 1.0051 185031 365999 1.0031 114432 367143 1.0019 70533 367849 1.0012 43434 368283 1.0007 26735 368550 1.0004 16436 368714 1.0003 10137 368814 1.0002 6238 368876 1.0001 3839 368914 1.0001 2340 368937 1.0000 1441 368951 1.0000 942 368960 1.0000 5

If you remove this number of individuals (starting at time 22), the population will remain at a constant size.

In other words these individuals are surplus to the population and we refer to this type of model as a Surplus Production Model.


t N R Additions Harvest0 12 1.6300 8 01 20 1.6299 12 02 32 1.6299 20 03 52 1.6299 33 04 85 1.6298 53 05 138 1.6296 87 06 225 1.6294 142 07 367 1.6290 231 08 597 1.6283 375 09 972 1.6273 610 010 1582 1.6256 990 011 2572 1.6229 1602 012 4174 1.6185 2581 013 6755 1.6114 4130 014 10885 1.6003 6534 015 17418 1.5829 10154 016 27572 1.5567 15350 017 42922 1.5187 22263 018 65185 1.4668 30425 019 95610 1.4012 38363 020 133974 1.3265 43749 021 177723 1.0000 0 022 177723 1.0000 0 4452423 177724 1.0000 0 4452424 177724 1.0000 0 4452425 177724 1.0000 0 4452426 177724 1.0000 0 4452427 177725 1.0000 0 4452428 177725 1.0000 0 4452429 177725 1.0000 0 4452430 177725 1.0000 0 4452431 177726 1.0000 0 4452432 177726 1.0000 0 4452433 177726 1.0000 0 4452434 177726 1.0000 0 4452435 177727 1.0000 0 4452436 177727 1.0000 0 4452437 177727 1.0000 0 4452438 177727 1.0000 0 4452439 177727 1.0000 0 4452440 177727 1.0000 0 4452441 177728 1.0000 0 4452442 177728 1.0000 0 44524

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R 1.63 Fixed Annual Yield 48976.4K 368974 Total Harvest 3869136

t N R Additions Harvest0 12 1.6300 8 01 20 1.6299 12 02 32 1.6299 20 03 52 1.6299 33 04 85 1.6298 53 05 138 1.6296 87 06 225 1.6294 142 07 367 1.6290 231 08 597 1.6283 375 09 972 1.6273 610 010 1582 1.6256 990 011 2572 1.6229 1602 012 4174 1.6185 2581 013 6755 1.6114 4130 014 10885 1.6003 6534 015 17418 1.5829 10154 016 27572 1.5567 15350 017 42922 1.5187 22263 018 65185 1.4668 30425 019 95610 1.4012 38363 020 133974 1.3265 43749 021 177723 0.9749 -4452 022 173271 0.9752 -4296 48976.423 168974 0.9752 -4194 48976.424 164780 0.9749 -4141 48976.425 160640 0.9743 -4134 48976.426 156506 0.9733 -4172 48976.427 152334 0.9720 -4259 48976.428 148074 0.9703 -4398 48976.429 143677 0.9680 -4594 48976.430 139082 0.9651 -4860 48976.431 134223 0.9612 -5208 48976.432 129015 0.9561 -5659 48976.433 123356 0.9494 -6244 48976.434 117112 0.9402 -7006 48976.435 110106 0.9272 -8011 48976.436 102095 0.9083 -9360 48976.437 92735 0.8791 -11216 48976.438 81519 0.8300 -13855 48976.439 67664 0.7374 -17771 48976.440 49894 0.5204 -23928 48976.441 25966 -0.3254 -34415 48976.442 -8449 7.4507 -54501 48976.443 -62949 2.6043 -100991 48976.4

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R 1.63 Fixed Annual Yield 40071.6K 368974 Total Harvest 3165656

t N R Additions Harvest0 12 1.6300 8 01 20 1.6299 12 02 32 1.6299 20 03 52 1.6299 33 04 85 1.6298 53 05 138 1.6296 87 06 225 1.6294 142 07 367 1.6290 231 08 597 1.6283 375 09 972 1.6273 610 010 1582 1.6256 990 011 2572 1.6229 1602 012 4174 1.6185 2581 013 6755 1.6114 4130 014 10885 1.6003 6534 015 17418 1.5829 10154 016 27572 1.5567 15350 017 42922 1.5187 22263 018 65185 1.4668 30425 019 95610 1.4012 38363 020 133974 1.3265 43749 021 177723 1.0251 4453 022 182176 1.0233 4247 40071.623 186423 1.0215 4005 40071.624 190428 1.0196 3737 40071.625 194166 1.0178 3453 40071.626 197618 1.0160 3160 40071.627 200778 1.0143 2868 40071.628 203646 1.0127 2583 40071.629 206230 1.0112 2311 40071.630 208540 1.0099 2054 40071.631 210595 1.0086 1816 40071.632 212411 1.0075 1598 40071.633 214009 1.0065 1400 40071.634 215408 1.0057 1222 40071.635 216630 1.0049 1063 40071.636 217693 1.0042 922 40071.637 218615 1.0036 798 40071.638 219413 1.0031 689 40071.639 220102 1.0027 594 40071.640 220695 1.0023 511 40071.641 221206 1.0020 439 40071.642 221645 1.0017 377 40071.643 222022 1.0015 323 40071.6

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Small errors in MSY can have

BIG consequences

GarbageOVER - Exploitation

UNDER - Exploitation

Some Stock Recruit Curves

http://www.pmel.noaa.gov/foci/sebscc/results/megrey/spawner-recruit.gif

Walleye-PollockE Beiring Sea

http://oregonstate.edu/instruct/fw465/sampson/anchovy/anchov18.jpg

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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

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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

Which is larger: aa or bb?

WHY?Efficiency Catch

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NOW –

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

WHY?

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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]}

Nt+1 = ((Nt R) / {1 + [Nt.(R-1)/K]}) – ht+1

R 1.63 Efficiency 1K 368974 Effort 0.1

EE 0.1Total Harvest 2258460


10 553 1.4656 257 6111 810 1.4650 377 9012 1187 1.4640 551 13213 1738 1.4627 804 19314 2542 1.4607 1171 28215 3713 1.4578 1700 41316 5413 1.4536 2455 60117 7867 1.4476 3521 87418 11389 1.4390 5000 126519 16388 1.4271 6999 182120 23387 1.4107 9604 259921 32992 1.3888 12826 366622 45818 1.3606 16520 509123 62338 1.3259 20315 692624 82653 1.2856 23603 918425 106256 1.2417 25684 1180626 131941 1.1973 26029 1466027 157969 1.1554 24544 1755228 182513 1.1185 21620 2027929 204133 1.0878 17931 2268130 222064 1.0637 14143 2467431 236207 1.0454 10721 2624532 246928 1.0319 7883 2743633 254811 1.0222 5669 2831234 260480 1.0154 4011 2894235 264491 1.0106 2806 2938836 267297 1.0073 1947 2970037 269243 1.0050 1343 2991638 270587 1.0034 923 3006539 271510 1.0023 633 3016840 272144 1.0016 433 30238

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Harvesting after reproduction

The shape of both lines should be similar – one is just 10% of the other (EE = 0.1)

Total Harvest = 2 258 460




10 553 1.4656 257 6111 810 1.4650 377 9012 1187 1.4640 551 13213 1738 1.4627 804 19314 2542 1.4607 1171 28215 3713 1.4578 1700 41316 5413 1.4536 2455 60117 7867 1.4476 3521 87418 11389 1.4390 5000 126519 16388 1.4271 6999 182120 23387 1.4107 9604 259921 32992 1.3888 12826 366622 45818 1.3606 16520 509123 62338 1.3259 20315 692624 82653 1.2856 23603 918425 106256 1.2417 25684 1180626 131941 1.1973 26029 1466027 157969 1.1554 24544 1755228 182513 1.1185 21620 2027929 204133 1.0878 17931 2268130 222064 1.0637 14143 2467431 236207 1.0454 10721 2624532 246928 1.0319 7883 2743633 254811 1.0222 5669 2831234 260480 1.0154 4011 2894235 264491 1.0106 2806 2938836 267297 1.0073 1947 2970037 269243 1.0050 1343 2991638 270587 1.0034 923 3006539 271510 1.0023 633 3016840 272144 1.0016 433 30238




10 75 1.2012 15 1011 90 1.2012 18 1212 108 1.2012 22 1513 130 1.2012 26 1814 156 1.2011 31 2115 188 1.2011 38 2516 226 1.2010 45 3117 271 1.2010 54 3718 325 1.2009 65 4419 391 1.2008 78 5320 469 1.2007 94 6421 563 1.2006 113 7622 676 1.2005 136 9223 812 1.2003 163 11024 975 1.2001 195 13225 1170 1.1999 234 15926 1403 1.1996 280 19027 1683 1.1993 335 22828 2019 1.1989 401 27429 2420 1.1984 480 32830 2900 1.1978 574 39331 3474 1.1971 685 47032 4159 1.1962 816 56333 4975 1.1953 971 67334 5946 1.1941 1154 80435 7100 1.1927 1368 96036 8468 1.1910 1618 114337 10086 1.1891 1907 136138 11993 1.1867 2240 161639 14232 1.1840 2619 191640 16852 1.1809 3048 2265

BE

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Before After

Harvesting BEFORE or AFTER the

population has had a chance to

reproduce can have profound impacts

on population size!

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

http://www.cmhenderson.com/images/jsrnclds.jpg

http://ww

w.lew

es-flood-action.org.uk/lfa-images/spenceslane.jpg

http://i1.trekearth.com/photos/16779/drought-victim

.jpg

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





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



Good FactorGood 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



Good FactorGood 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


0

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Time (years)

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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!

The models built to date assume that all individuals in the population are equal

How can we build harvesting into an age-specific model?

We know that this is not reasonable

Not easily…………………..certainly not in this course

The age structured models we have constructed ignore intra-specific competition and are based on a constant R – i.e. exponential models

0

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49

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63

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Time

Nu

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ers R = 1.63

Age Structured Harvesting

That said – when you start to harvest from a population (inevitably during its growth phase), the population is maintained in a constant state of growth

Population (growing under intra-specific competition)

Harvest (MSY)

Population (growing without intra-specific competition)

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As a consequence, using Fixed R Models may not be unreasonable……….

So…………………

Start off with the basic life table, calculate p and m, and project the population into the future (e.g.) 100 time units.

Plot total population size against time

X a F p m v* v l0 89756 0 0.7 0 1.0000 1.0000 11 62829 81678 0.3 1.3 0.4473 1.7473 0.72 18849 28273 0.2 1.5 0.3237 1.8237 0.213 3770 6786 0.1 1.8 0.1799 1.9799 0.0424 377 829 0 2.2 0 2.2000 0.00425 0 0 0 0 0 0

X a F p m v* v l0 89756 0 0.7 0 1.0000 1.0000 11 62829 81678 0.3 1.3 0.4473 1.7473 0.72 18849 28273 0.2 1.5 0.3237 1.8237 0.213 3770 6786 0.1 1.8 0.1799 1.9799 0.0424 377 829 0 2.2 0 2.2000 0.00425 0 0 0 0 0 0

p 0.7 0.3 0.2 0.1 0 0m 0 1.3 1.5 1.8 2.2 0X 0 1 2 3 4 5 TOTAL R0 89756 62829 18849 3770 377 0 175581 1.1583891 117566 62829 18849 3770 377 0 203391 1.2201382 142873 82296 18849 3770 377 0 248165 1.2230163 174663 100011 24689 3770 377 0 303509 1.2231824 213665 122264 30003 4938 377 0 371247 1.2231235 261342 149566 36679 6001 494 0 454081 1.2231196 319650 182939 44870 7336 600 0 555395 1.2231227 390971 223755 54882 8974 734 0 679316 1.2231218 478206 273680 67127 10976 897 0 830886 1.2231219 584903 334744 82104 13425 1098 0 1016274 1.223121

10 715408 409432 100423 16421 1343 0 1243027 1.22312111 875031 500786 122830 20085 1642 0 1520373 1.223121

POPULATION

X a F p m v* v l0 89756 0 0.7 0 1.0000 1.0000 11 62829 81678 0.3 1.3 0.4473 1.7473 0.72 18849 28273 0.2 1.5 0.3237 1.8237 0.213 3770 6786 0.1 1.8 0.1799 1.9799 0.0424 377 829 0 2.2 0 2.2000 0.00425 0 0 0 0 0 0

Calculate Reproductive Value

Set up a parallel table with harvest information – i.e. the number of individuals of each age class you will harvest each year

Use a fixed yield model – i.e. harvest a fixed number (h) of individuals of each age class at each time interval and make each value in the harvest column for a particular age class equal to the h value for that age class

at t0, all h values = 0: harvest 1 0 year old from t1: keep a running total

p 0.7 0.3 0.2 0.1 0 0 TOTALS 100 0 0 0 0 0 100m 0 1.3 1.5 1.8 2.2 0 h 1 0 0 0 0 0X 0 1 2 3 4 5 TOTAL R X 0 1 2 3 4 5 TOTAL0 89756 62829 18849 3770 377 0 175581 1.158389 0 0 0 0 0 0 0 01 117566 62829 18849 3770 377 0 203391 1.220138 1 1 0 0 0 0 0 12 142873 82296 18849 3770 377 0 248165 1.223016 2 1 0 0 0 0 0 13 174663 100011 24689 3770 377 0 303509 1.223182 3 1 0 0 0 0 0 14 213665 122264 30003 4938 377 0 371247 1.223123 4 1 0 0 0 0 0 15 261342 149566 36679 6001 494 0 454081 1.223119 5 1 0 0 0 0 0 16 319650 182939 44870 7336 600 0 555395 1.223122 6 1 0 0 0 0 0 17 390971 223755 54882 8974 734 0 679316 1.223121 7 1 0 0 0 0 0 18 478206 273680 67127 10976 897 0 830886 1.223121 8 1 0 0 0 0 0 19 584903 334744 82104 13425 1098 0 1016274 1.223121 9 1 0 0 0 0 0 1

10 715408 409432 100423 16421 1343 0 1243027 1.223121 10 1 0 0 0 0 0 1

POPULATION HARVEST

Subtract harvest from the population table: BEFORE OR AFTER REPRODUCTION?

p 0.7 0.3 0.2 0.1 0 0 TOTALS 100 0 0 0 0 0 100m 0 1.3 1.5 1.8 2.2 0 h 1 0 0 0 0 0X 0 1 2 3 4 5 TOTAL R X 0 1 2 3 4 5 TOTAL0 89756 62829 18849 3770 377 0 175581 1.158383 0 0 0 0 0 0 0 01 117565 62829 18849 3770 377 0 203390 1.220132 1 1 0 0 0 0 0 12 142871 82295 18849 3770 377 0 248162 1.22301 2 1 0 0 0 0 0 13 174660 100010 24689 3770 377 0 303505 1.223178 3 1 0 0 0 0 0 14 213661 122262 30003 4938 377 0 371240 1.223119 4 1 0 0 0 0 0 15 261336 149563 36679 6001 494 0 454071 1.223116 5 1 0 0 0 0 0 16 319642 182935 44869 7336 600 0 555382 1.223119 6 1 0 0 0 0 0 17 390961 223749 54880 8974 734 0 679298 1.223119 7 1 0 0 0 0 0 18 478192 273672 67125 10976 897 0 830862 1.22312 8 1 0 0 0 0 0 19 584886 334734 82102 13425 1098 0 1016244 1.22312 9 1 0 0 0 0 0 1

POPULATION HARVEST

p 0.7 0.3 0.2 0.1 0 0 TOTALS 790000 948000 790000 790000 1185000 0 4503000m 0 1.3 1.5 1.8 2.2 0 h 10000 12000 10000 10000 15000 0X 0 1 2 3 4 5 TOTAL R X 0 1 2 3 4 5 TOTAL0 89756 62829 18849 3770 377 0 175581 1.158389 0 0 0 0 0 0 0 01 117566 62829 18849 3770 377 0 203391 1.220138 1 0 0 0 0 0 0 02 142873 82296 18849 3770 377 0 248165 1.223016 2 0 0 0 0 0 0 03 174663 100011 24689 3770 377 0 303509 1.223182 3 0 0 0 0 0 0 04 213665 122264 30003 4938 377 0 371247 1.223123 4 0 0 0 0 0 0 05 261342 149566 36679 6001 494 0 454081 1.223119 5 0 0 0 0 0 0 06 319650 182939 44870 7336 600 0 555395 1.223122 6 0 0 0 0 0 0 07 390971 223755 54882 8974 734 0 679316 1.223121 7 0 0 0 0 0 0 08 478206 273680 67127 10976 897 0 830886 1.223121 8 0 0 0 0 0 0 09 584903 334744 82104 13425 1098 0 1016274 1.223121 9 0 0 0 0 0 0 010 715408 409432 100423 16421 1343 0 1243027 1.223121 10 0 0 0 0 0 0 011 875031 500786 122830 20085 1642 0 1520373 1.223121 11 0 0 0 0 0 0 012 1070269 612521 150236 24566 2008 0 1859600 1.223121 12 0 0 0 0 0 0 013 1309069 749188 183756 30047 2457 0 2274517 1.223121 13 0 0 0 0 0 0 014 1601150 916348 224756 36751 3005 0 2782010 1.223121 14 0 0 0 0 0 0 015 1958400 1120805 274904 44951 3675 0 3402736 1.223121 15 0 0 0 0 0 0 016 2395361 1370880 336241 54981 4495 0 4161959 1.223121 16 0 0 0 0 0 0 017 2929818 1676753 411264 67248 5498 0 5090581 1.223121 17 0 0 0 0 0 0 018 3583523 2050872 503026 82253 6725 0 6226398 1.223121 18 0 0 0 0 0 0 019 4383083 2508466 615262 100605 8225 0 7615641 1.223121 19 0 0 0 0 0 0 020 5361043 3068158 752540 123052 10061 0 9314853 1.223121 20 0 0 0 0 0 0 021 6557206 3752730 920447 150508 12305 0 11393196 1.210956 21 0 0 0 0 0 0 022 7928658 4578044 1115819 174089 51 0 13796662 1.213383 22 10000 12000 10000 10000 15000 0 5700023 9623593 5538061 1363413 213164 2409 0 16740640 1.215046 23 10000 12000 10000 10000 15000 0 57000

POPULATION HARVEST

PLAYAdjust harvest of each age class, adjust time of first harvest in order to maximize total harvest. REMEMBER – Population must not crash!

0

1E+13

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0 1 2 3 4

Optimising harvest in an age-structured model can be time-consuming!

Better yields were (in any case) obtained from harvesting under a constant effort scenario in our previous models and there is no reason to suppose that this will be different here! It is also quicker!!!

Start off as before with a population table and a parallel harvest table.

Set the harvest table up with an Effort row, an Efficiency row and an EE row (Effort x Efficiency). All values in the Efficiency row should equal 1 (i.e. the harvesting method is 100% efficient). In the first instance make all values of Effort = 0, except E0 (=0.00000001). That means all values of EE will equal 0, except EE0, which will equal 0.00000001

TOTALS 0 0 0 0 0 0 0Effort 0.00000001 0 0 0 0 0

p 0.7 0.3 0.2 0.1 0 0 Efficiency 1 1 1 1 1 1m 0 1.3 1.5 1.8 2.2 0 EE 0.00000001 0 0 0 0 0X 0 1 2 3 4 5 TOTAL R X 0 1 2 3 4 5 TOTAL0 89756 62829 18849 3770 377 0 175581 1.158389 0 0 0 0 0 0 0 01 117566 62829 18849 3770 377 0 203391 1.220138 1 0 0 0 0 0 0 02 142873 82296 18849 3770 377 0 248165 1.223016 2 0 0 0 0 0 0 03 174663 100011 24689 3770 377 0 303509 1.223182 3 0 0 0 0 0 0 04 213665 122264 30003 4938 377 0 371247 1.223123 4 0 0 0 0 0 0 05 261342 149566 36679 6001 494 0 454081 1.223119 5 0 0 0 0 0 0 06 319650 182939 44870 7336 600 0 555395 1.223122 6 0 0 0 0 0 0 07 390971 223755 54882 8974 734 0 679316 1.223121 7 0 0 0 0 0 0 08 478206 273680 67127 10976 897 0 830886 1.223121 8 0 0 0 0 0 0 09 584903 334744 82104 13425 1098 0 1016274 1.223121 9 0 0 0 0 0 0 010 715408 409432 100423 16421 1343 0 1243027 1.223121 10 0 0 0 0 0 0 011 875031 500785 122830 20085 1642 0 1520372 1.223121 11 0 0 0 0 0 0 012 1070269 612521 150236 24566 2008 0 1859600 1.223121 12 0 0 0 0 0 0 013 1309068 749188 183756 30047 2457 0 2274517 1.223121 13 0 0 0 0 0 0 0

POPULATION

HARVEST

You now need to make the harvest of a particular age class at a particular time equal to EE multiplied by the number of individuals of that age class at that time. Having done that, you will need to subtract this number from the number of individuals of that age at that time from the population table. This will inevitably result in Circular Argument errors in MSExcel….which means that you need to build the formulae projecting populations into the future into the harvest table too!

Ensure that harvest at t0 = 0

TOTALS 2922413 0 0 0 0 0 2922413Effort 0.00000001 0 0 0 0 0

p 0.7 0.3 0.2 0.1 0 0 Efficiency 1 1 1 1 1 1m 0 1.3 1.5 1.8 2.2 0 EE 0.00000001 0 0 0 0 0X 0 1 2 3 4 5 TOTAL R X 0 1 2 3 4 5 TOTAL0 89756 62829 18849 3770 377 0 175581 1.158389 0 0 0 0 0 0 0 01 117566 62829 18849 3770 377 0 203391 1.220138 1 0 0 0 0 0 0 02 142873 82296 18849 3770 377 0 248165 1.223016 2 0 0 0 0 0 0 03 174663 100011 24689 3770 377 0 303509 1.223182 3 0 0 0 0 0 0 04 213665 122264 30003 4938 377 0 371247 1.223123 4 0 0 0 0 0 0 05 261342 149566 36679 6001 494 0 454081 1.223119 5 0 0 0 0 0 0 06 319650 182939 44870 7336 600 0 555395 1.223122 6 0 0 0 0 0 0 07 390971 223755 54882 8974 734 0 679316 1.223121 7 0 0 0 0 0 0 08 478206 273680 67127 10976 897 0 830886 1.223121 8 0 0 0 0 0 0 09 584903 334744 82104 13425 1098 0 1016274 1.223121 9 0 0 0 0 0 0 010 715408 409432 100423 16421 1343 0 1243027 1.223121 10 0 0 0 0 0 0 011 875031 500785 122830 20085 1642 0 1520372 1.223121 11 0 0 0 0 0 0 012 1070269 612521 150236 24566 2008 0 1859600 1.223121 12 0 0 0 0 0 0 013 1309068 749188 183756 30047 2457 0 2274517 1.223121 13 0 0 0 0 0 0 0

POPULATION

HARVEST

With such a low level of Effort, you should only start to harvest any whole organisms at time 32

TOTALS 0 0 0 377 525 118 533 279 585 803 749 0 657 110 922 282Effort 0 0 0 0.1 1 0

p 0.7 0.3 0.2 0.1 0 0 Efficiency 1 1 1 1 1 1m 0 1.3 1.5 1.8 2.2 0 EE 0 0 0 0.1 1 0X 0 1 2 3 4 5 TOTAL R X 0 1 2 3 4 5 TOTAL0 89756 62829 18849 3770 377 0 175581 1.151518 0 0 0 0 0 0 0 01 116737 62829 18849 3770 0 0 202184 1.214848 1 0 0 0 0 377 0 3772 141289 81716 18849 3770 0 0 245623 1.218604 2 0 0 0 0 377 0 3773 172131 98902 24515 3770 0 0 299317 1.219559 3 0 0 0 0 377 0 3774 209970 120491 29671 4903 0 0 365035 1.219159 4 0 0 0 0 377 0 3775 255975 146979 36147 5934 0 0 445036 1.219221 5 0 0 0 0 490 0 4906 312091 179183 44094 7229 0 0 542597 1.219222 6 0 0 0 0 593 0 5937 380509 218464 53755 8819 0 0 661547 1.219219 7 0 0 0 0 723 0 7238 463924 266356 65539 10751 0 0 806571 1.214669 8 0 0 0 0 882 0 8829 563266 324747 79907 11797 0 0 979717 1.215342 9 0 0 0 1311 1075 0 238610 684598 394286 97424 14383 0 0 1190691 1.215276 10 0 0 0 1598 1180 0 277811 831978 479218 118286 17536 0 0 1447019 1.215267 11 0 0 0 1948 1438 0 338712 1011073 582385 143766 21291 0 0 1758515 1.215272 12 0 0 0 2366 1754 0 411913 1228730 707751 174715 25878 0 0 2137074 1.215271 13 0 0 0 2875 2129 0 500414 1493240 860111 212325 31449 0 0 2597125 1.215271 14 0 0 0 3494 2588 0 608215 1814691 1045268 258033 38219 0 0 3156211 1.215271 15 0 0 0 4247 3145 0 739116 2205342 1270284 313580 46446 0 0 3835653 1.215271 16 0 0 0 5161 3822 0 898317 2680089 1543740 381085 56444 0 0 4661359 1.215271 17 0 0 0 6272 4645 0 1091618 3257036 1876063 463122 68595 0 0 5664816 1.215271 18 0 0 0 7622 5644 0 1326619 3958182 2279925 562819 83362 0 0 6884288 1.215271 19 0 0 0 9262 6860 0 1612220 4810265 2770728 683978 101307 0 0 8366278 1.215271 20 0 0 0 11256 8336 0 1959321 5845778 3367186 831218 123116 0 0 10167298 1.215271 21 0 0 0 13680 10131 0 2381022 7104206 4092044 1010156 149619 0 0 12356026 1.215271 22 0 0 0 16624 12312 0 28936

POPULATION

HARVEST

PLAYAdjust Effort of each age class, adjust time of first harvest in order to maximize total harvest. REMEMBER – Population must not crash!

Documents

Exploitation of natural populations – Food etc