Density Dependence and Harvest (Weeks 5-6)
Chase Stevens
05/15/2023
Background:
Although we expect that populations can produce more offspring than needed for replacement, they cannot grow exponentially for long. A typical population trajectory through time is “S” shaped (also called a “sigmoid curve”), with growth in percent per year slowing as the population reaches a “carrying capacity” (K). The most common equation to represent this pattern is the logistic equation:
dN/dt (change in N with change in time) = rN (1 - N/K)
The discrete form of this equation is: N(t + 1) = N(t) + r * N(t) * (1 - N(t)/K)
The logistic equation (and minor modifications of it) is used regularly in harvest management. The equation predicts that the largest number of individuals will be added to the population in the next year when the population size is 1/2 of K. Note that this is different from proportional population growth rate - the number added (production) is N (t+1) - N(t).
Importantly, for these models, r is considered an “intrinsic property” of the population, rather than a result like lambda. r itself does not change, and is the maximum rate of growth that the population can reach. r is a continuous rate often called “r max”. The logistic model equation assumes that a population will grow at a rate of “r” when it is at its lowest density because animals are not competing for resources (but see lecture notes on depensation to see why this may sometimes be a bad assumption). As soon as the population starts to increase, the “realized” growth rate (which we can express as lambda, N(t+1)/N(t)) will be less than exp^r due to density dependence caused by intraspecific competition. You can see how this works by writing out the equation on a piece of paper, choosing an r and a K value, and calculating N(t+1) when N(t) = 1, 2, 10, one half of K, K-1, K, and K+10. This is a good exercise to help understand how the equation actually works.
Exercise 1. Logistic equation. Start at an initial population size and project a population through time using parameter values for r and K. To find N(t + 1), use the discrete form of the logistic equation (15 points total):
N(t + 1) = N(t) + r * N(t)*(1 - N(t)/K)
We need to set our r, K and N0 values with the following code:
#maximum intrinsic growth rate
r <- 0.3
# carrying capacity
K <- 500
# initial population size
N0 <- 5We can now fill in the values of our N column using for-loops (we learned about these during our extinction analysis) and the logistic equation given above. We first have to create a vector to store the values of N and then set our N value at time 0 to the N0 value we established initially using bracket notation
A. Plot N(t) through time (3 points)
numyears <- 50
N <- rep(NA, numyears) #makes an empty vector (NA) with length numyears. This is an easy way to make an object to fill
N[1] <- N0 #initialize the vector for the first year
for(t in 1:(numyears-1)){ #Note the indexing from 1 to (numyears-1) so that the last element in the vector is N[numyears] and not N[numyears+1], which is bigger than the vector we made
#N[t+1] <- #Use the appropriate equation
}
numyears<-50
N<-rep(NA, numyears)
N[1]<-N0
for(t in 1:(numyears-1)){N[t+1]<-N[t] + r*N[t]*(1- N[t]/K)
}
plot(N, xlab ="Years",ylab ="Initial Population",main ="Initial Population through Time",pch =20)
B. Plot N(t+1) vs. N(t) (3 points)
Use appropriate indexing to plot an x-axis with N(t) and y-axis with N(t+1). This graph shows an approach toward equilibrium, which occurs when N(t) = N(t+1). This will not look that interesting for the value of r that we have chosen, but if you increase r to 3 or more, the dynamics get very interesting and there is no equilibirium but rather the population enters cycles or chaos. For our purposes, this exercise is a stepping stone to get to part C. I want you to see how to access elements 2 through 50 on the y-axis, N(t+1) and 1 through 49 on the x axis, N(t), which is necessary to make the plot.
plot(N[1:49], N[2:50],
main ="N(t+1) vs N(t)",
xlab ="N(t+1)",
ylab ="N(t)",
pch = 20)
C. Plot an approximation of dN/dt vs N (3 points)
You need to plot the change in N divided by the change in t. To do this, make a vector of (N(t+1) - N(t)), which is the change in N from one year to the next. dt = 1, so this column is your dN/dt (dividing by 1 gives you the same number back). Plot N(t) (x-axis) vs change in N (y-axis),
dN<-(N[2:50]-N[1:49])
dN <- rep ( NA , numyears )
for ( t in 1 : ( numyears - 1 ) )
{ dN[t+ 1 ] <- N [t+ 1 ] - N [t ]
}
plot ( N , dN,
main = " Approximation of dN / dt vs N ",
xlab = " Population " ,
ylab = " Change in Population " ,
pch = 19 )
D. PLOT N(t) vs. lambda (3 points)
You will have to find lambda for every time step
lam<-rep(NA, numyears)
for(t in 1:(numyears-1)){
lam[t+1]<-N[t+1] / N[t]
}
plot(N, lam,
main ="N(t) vs Lambda",
xlab ="Population",
ylab ="Lambda",
pch =19)
E. At what population size do you see maximum growth? (1 point)
##N=K/2 ##N=500/2 ##250
F. What could your maximum sustainable harvest rate per year be if r = 0.6 and K = 500? (This is what we call Maximum Sustainable Yield) Hint: It’s whatever the maximum production is because you can harvest that production without changing the population, which is how “sustainable” is defined - it can be sustained. (2 points)
r<-0.6
K<-500
MSY<-(r*K)/4
MSY## [1] 75Exercise 2. California Condor population dynamics. (3 points total)
Six condor breeding pairs (12 condors) are re-introduced into Pinnacles National Park from a captive breeding program. The suitable habitat is relatively small and biologists estimate it can only support about 250 condors (its carrying capacity). The condor population initially grows with an maximum intrinsic growth rate of r = 0.08.
A. What is the maximum possible growth rate for the population (in terms of change in # of condors year, dN/dt)? (3 points)
Hint: You need to find the dN/dt for this population when it is at half of its carrying capacity, following the logistic growth equation, dN/dt - rN(1-(N/K)).
r<-.08
K<-250/2
N<- 12
dN_dt <- r*N*( 1 -( N/K ) )
dN_dt## [1] 0.86784Exercise 3. Maximum sustained yield of Lake Trout under logistic population growth. (8 points total)
A fisheries biologist wants to maximize the yield to a fishery for lake trout by maintaining a population of lake trout at ~11,500 individuals. Another biologist wants to increase yield by stocking more fish into the lake. The first fish biologist disagrees saying that she is maintaining maximum sustained yield (MSY), and stocking will not increase the rate of fish produced (i.e. overstocking could decrease sustained yield).
A. If the biologist thinks MSY will be produced at 11,500 individuals, what is the assumed carrying capacity of the lake? (2 point)
##K=11500*2 ##K=23000
B. Assume that the population growth rate r for trout is 0.11. Find the population growth rate (dN/dt), which is approximated by N(t+1) - N(t) at the population size that yields MSY. (3 points)
r<-0.11
N<-11500
K<-23000
dNdt<-r*N*(1-N/K)
dNdt## [1] 632.5C. Find the population growth rate dN/dt if the second biologist stocks the lake to a total of 14,000 trout assuming that the first fish biologist was correct about the carrying capacity of the lake. How much more or less is the sustainable yield when the population is held at 14,000 trout? (3 points)
r<-0.11
N<-14000
K<-23000
dNdt<-r*N*(1-N/K)
dNdt## [1] 602.6087Exercise 4. Recruitment of game birds. (9 points total)
Imagine that in 2016, you were appointed to run a wildlife management area in Southwestern Oregon At this same time, an upland game bird stocking program transplanted 8 wild turkeys and 14 ruffed grouse into your area. You are in charge of determining the management target population sizes, with the intention of maximizing yield in the long term, as well as the sustainable harvest targets. A graduate student at OSU has recently determined that wild turkey population growth on your management area closely follows a Ricker curve. She estimates \(\alpha\) is 7 and \(\beta\) is 0.003448. She also estimated that the grouse have Beverton-Holt recruitment dynamics. In this case, \(\alpha\) is 0.0012 and \(\beta\) is 0.35 using the form of Beverton-Holt from the PPT not the form used in the video.
A. Produce a plot of adult population size vs. expected recruitment for each species (turkeys follow Ricker dynamics and grouse follow Beverton Holt). The adult popualtion size should range from 1:800. Add the 1:1 replacement line to your plot (i.e when y=x, the abline() function is the easiest way to do this). Label both axes. (4 points)
##The Ricker function: Recruits = alpha*N*exp(-beta*N)
##The Beverton-Holt function has various equivalent forms.
##In this problem use Recruits = 1/(alpha + beta/N)
a<-7
b<-0.003448
x<-1:800
turkey<-a*x*exp(-b*x)
turkey## [1] 6.975906 13.903788 20.783896 27.616475 34.401772 41.140030
## [7] 47.831494 54.476406 61.075007 67.627536 74.134234 80.595337
## [13] 87.011083 93.381708 99.707445 105.988529 112.225192 118.417665
## [19] 124.566178 130.670962 136.732244 142.750250 148.725208 154.657343
## [25] 160.546878 166.394036 172.199040 177.962109 183.683465 189.363326
## [31] 195.001910 200.599434 206.156113 211.672162 217.147796 222.583227
## [37] 227.978668 233.334328 238.650418 243.927147 249.164723 254.363353
## [43] 259.523242 264.644597 269.727621 274.772517 279.779488 284.748735
## [49] 289.680458 294.574857 299.432130 304.252475 309.036089 313.783167
## [55] 318.493904 323.168494 327.807130 332.410004 336.977308 341.509232
## [61] 346.005964 350.467695 354.894611 359.286900 363.644746 367.968336
## [67] 372.257853 376.513482 380.735403 384.923799 389.078851 393.200739
## [73] 397.289641 401.345737 405.369202 409.360215 413.318950 417.245583
## [79] 421.140287 425.003236 428.834603 432.634559 436.403274 440.140920
## [85] 443.847664 447.523677 451.169124 454.784174 458.368992 461.923743
## [91] 465.448592 468.943704 472.409239 475.845362 479.252233 482.630013
## [97] 485.978861 489.298938 492.590401 495.853408 499.088116 502.294681
## [103] 505.473259 508.624004 511.747070 514.842610 517.910778 520.951723
## [109] 523.965599 526.952554 529.912739 532.846303 535.753393 538.634158
## [115] 541.488743 544.317295 547.119960 549.896882 552.648205 555.374073
## [121] 558.074628 560.750011 563.400366 566.025831 568.626547 571.202653
## [127] 573.754288 576.281590 578.784696 581.263742 583.718866 586.150201
## [133] 588.557882 590.942044 593.302821 595.640344 597.954746 600.246158
## [139] 602.514711 604.760536 606.983762 609.184518 611.362932 613.519132
## [145] 615.653245 617.765398 619.855717 621.924326 623.971351 625.996915
## [151] 628.001143 629.984156 631.946078 633.887029 635.807132 637.706506
## [157] 639.585272 641.443549 643.281456 645.099110 646.896630 648.674133
## [163] 650.431735 652.169552 653.887699 655.586292 657.265444 658.925270
## [169] 660.565882 662.187392 663.789914 665.373559 666.938437 668.484659
## [175] 670.012335 671.521574 673.012486 674.485177 675.939757 677.376332
## [181] 678.795009 680.195894 681.579093 682.944711 684.292853 685.623623
## [187] 686.937124 688.233460 689.512733 690.775045 692.020498 693.249193
## [193] 694.461231 695.656712 696.835734 697.998398 699.144803 700.275045
## [199] 701.389224 702.487435 703.569777 704.636345 705.687234 706.722541
## [205] 707.742359 708.746784 709.735909 710.709828 711.668634 712.612418
## [211] 713.541274 714.455293 715.354565 716.239182 717.109234 717.964810
## [217] 718.806000 719.632894 720.445578 721.244142 722.028672 722.799257
## [223] 723.555983 724.298935 725.028200 725.743864 726.446011 727.134726
## [229] 727.810093 728.472195 729.121117 729.756941 730.379749 730.989623
## [235] 731.586646 732.170898 732.742460 733.301414 733.847837 734.381811
## [241] 734.903415 735.412727 735.909826 736.394790 736.867696 737.328623
## [247] 737.777645 738.214841 738.640287 739.054057 739.456227 739.846873
## [253] 740.226069 740.593888 740.950406 741.295694 741.629828 741.952878
## [259] 742.264917 742.566018 742.856252 743.135690 743.404403 743.662462
## [265] 743.909937 744.146897 744.373413 744.589552 744.795384 744.990978
## [271] 745.176401 745.351721 745.517005 745.672321 745.817735 745.953313
## [277] 746.079121 746.195226 746.301692 746.398584 746.485967 746.563905
## [283] 746.632463 746.691704 746.741690 746.782486 746.814154 746.836757
## [289] 746.850355 746.855012 746.850788 746.837744 746.815942 746.785441
## [295] 746.746302 746.698585 746.642349 746.577654 746.504558 746.423119
## [301] 746.333397 746.235450 746.129334 746.015107 745.892826 745.762549
## [307] 745.624332 745.478230 745.324300 745.162598 744.993178 744.816096
## [313] 744.631407 744.439164 744.239423 744.032237 743.817659 743.595744
## [319] 743.366543 743.130111 742.886499 742.635759 742.377944 742.113106
## [325] 741.841294 741.562562 741.276959 740.984537 740.685345 740.379434
## [331] 740.066853 739.747653 739.421881 739.089588 738.750822 738.405631
## [337] 738.054064 737.696169 737.331993 736.961584 736.584990 736.202257
## [343] 735.813431 735.418560 735.017689 734.610865 734.198134 733.779540
## [349] 733.355129 732.924946 732.489036 732.047443 731.600212 731.147386
## [355] 730.689010 730.225127 729.755780 729.281013 728.800868 728.315389
## [361] 727.824616 727.328593 726.827362 726.320964 725.809441 725.292834
## [367] 724.771184 724.244532 723.712919 723.176385 722.634970 722.088714
## [373] 721.537656 720.981838 720.421297 719.856072 719.286204 718.711729
## [379] 718.132688 717.549118 716.961056 716.368542 715.771612 715.170304
## [385] 714.564655 713.954703 713.340483 712.722033 712.099389 711.472587
## [391] 710.841663 710.206653 709.567593 708.924518 708.277463 707.626463
## [397] 706.971554 706.312769 705.650143 704.983710 704.313505 703.639561
## [403] 702.961912 702.280591 701.595633 700.907069 700.214933 699.519257
## [409] 698.820075 698.117418 697.411318 696.701809 695.988921 695.272686
## [415] 694.553136 693.830302 693.104215 692.374907 691.642407 690.906747
## [421] 690.167956 689.426066 688.681106 687.933107 687.182097 686.428107
## [427] 685.671166 684.911303 684.148547 683.382928 682.614473 681.843212
## [433] 681.069173 680.292385 679.512874 678.730670 677.945800 677.158292
## [439] 676.368173 675.575470 674.780211 673.982422 673.182131 672.379364
## [445] 671.574148 670.766509 669.956472 669.144065 668.329314 667.512243
## [451] 666.692879 665.871247 665.047372 664.221280 663.392996 662.562543
## [457] 661.729949 660.895235 660.058428 659.219552 658.378630 657.535687
## [463] 656.690747 655.843833 654.994969 654.144178 653.291484 652.436910
## [469] 651.580479 650.722213 649.862137 649.000271 648.136639 647.271264
## [475] 646.404167 645.535370 644.664896 643.792765 642.919001 642.043625
## [481] 641.166657 640.288120 639.408035 638.526422 637.643302 636.758697
## [487] 635.872628 634.985113 634.096175 633.205833 632.314108 631.421019
## [493] 630.526587 629.630832 628.733772 627.835429 626.935820 626.034966
## [499] 625.132886 624.229599 623.325123 622.419479 621.512684 620.604757
## [505] 619.695717 618.785582 617.874370 616.962101 616.048790 615.134458
## [511] 614.219121 613.302798 612.385505 611.467261 610.548083 609.627987
## [517] 608.706993 607.785116 606.862373 605.938782 605.014359 604.089120
## [523] 603.163083 602.236264 601.308679 600.380344 599.451276 598.521491
## [529] 597.591004 596.659832 595.727990 594.795494 593.862359 592.928601
## [535] 591.994235 591.059276 590.123740 589.187641 588.250996 587.313817
## [541] 586.376121 585.437921 584.499233 583.560071 582.620449 581.680382
## [547] 580.739883 579.798968 578.857649 577.915941 576.973858 576.031413
## [553] 575.088620 574.145493 573.202045 572.258289 571.314239 570.369908
## [559] 569.425309 568.480455 567.535359 566.590034 565.644493 564.698749
## [565] 563.752813 562.806698 561.860418 560.913984 559.967409 559.020704
## [571] 558.073882 557.126955 556.179935 555.232833 554.285662 553.338433
## [577] 552.391158 551.443849 550.496516 549.549171 548.601825 547.654491
## [583] 546.707178 545.759898 544.812662 543.865481 542.918366 541.971327
## [589] 541.024375 540.077521 539.130775 538.184148 537.237650 536.291292
## [595] 535.345083 534.399035 533.453156 532.507458 531.561950 530.616641
## [601] 529.671543 528.726664 527.782014 526.837603 525.893440 524.949536
## [607] 524.005898 523.062537 522.119462 521.176683 520.234207 519.292045
## [613] 518.350205 517.408696 516.467528 515.526708 514.586247 513.646151
## [619] 512.706431 511.767094 510.828149 509.889605 508.951470 508.013752
## [625] 507.076459 506.139600 505.203182 504.267215 503.331705 502.396661
## [631] 501.462090 500.528001 499.594401 498.661298 497.728699 496.796613
## [637] 495.865046 494.934006 494.003501 493.073537 492.144123 491.215265
## [643] 490.286970 489.359246 488.432100 487.505539 486.579570 485.654199
## [649] 484.729433 483.805280 482.881746 481.958837 481.036561 480.114924
## [655] 479.193932 478.273593 477.353911 476.434894 475.516548 474.598880
## [661] 473.681895 472.765600 471.850000 470.935103 470.020913 469.107437
## [667] 468.194681 467.282651 466.371352 465.460790 464.550971 463.641901
## [673] 462.733585 461.826029 460.919238 460.013219 459.107975 458.203513
## [679] 457.299838 456.396955 455.494870 454.593588 453.693114 452.793452
## [685] 451.894609 450.996589 450.099397 449.203038 448.307518 447.412839
## [691] 446.519009 445.626031 444.733910 443.842651 442.952258 442.062736
## [697] 441.174090 440.286324 439.399443 438.513450 437.628351 436.744149
## [703] 435.860850 434.978457 434.096974 433.216406 432.336757 431.458031
## [709] 430.580232 429.703364 428.827431 427.952437 427.078386 426.205281
## [715] 425.333128 424.461928 423.591687 422.722408 421.854095 420.986750
## [721] 420.120379 419.254984 418.390569 417.527137 416.664693 415.803238
## [727] 414.942778 414.083315 413.224852 412.367393 411.510941 410.655500
## [733] 409.801072 408.947660 408.095268 407.243900 406.393557 405.544243
## [739] 404.695960 403.848713 403.002504 402.157335 401.313210 400.470131
## [745] 399.628101 398.787123 397.947200 397.108335 396.270529 395.433786
## [751] 394.598109 393.763499 392.929960 392.097494 391.266103 390.435790
## [757] 389.606557 388.778407 387.951343 387.125365 386.300478 385.476682
## [763] 384.653981 383.832377 383.011871 382.192466 381.374164 380.556967
## [769] 379.740877 378.925897 378.112028 377.299273 376.487632 375.677109
## [775] 374.867706 374.059423 373.252263 372.446228 371.641320 370.837540
## [781] 370.034890 369.233372 368.432988 367.633739 366.835628 366.038655
## [787] 365.242822 364.448131 363.654583 362.862181 362.070925 361.280816
## [793] 360.491858 359.704050 358.917395 358.131893 357.347547 356.564356
## [799] 355.782324 355.001451plot(x, turkey,
main ="Adult Turkey Population Size vs Expected Recruitment",
xlab ="Adult Turkey Population Size",
ylab ="Recruits",
pch =20)
abline(a=0,b=1,col ="red")
a2<-0.0012
b2<-0.35
x<-1:800
grouse<-1/(a2 + b2 / x)
grouse## [1] 2.847380 5.675369 8.484163 11.273957 14.044944 16.797312
## [7] 19.531250 22.246941 24.944568 27.624309 30.286344 32.930845
## [13] 35.557987 38.167939 40.760870 43.336945 45.896328 48.439182
## [19] 50.965665 53.475936 55.970149 58.448459 60.911017 63.357973
## [25] 65.789474 68.205666 70.606695 72.992701 75.363825 77.720207
## [31] 80.061983 82.389289 84.702259 87.001024 89.285714 91.556460
## [37] 93.813387 96.056623 98.286290 100.502513 102.705411 104.895105
## [43] 107.071713 109.235353 111.386139 113.524186 115.649606 117.762512
## [49] 119.863014 121.951220 124.027237 126.091174 128.143133 130.183221
## [55] 132.211538 134.228188 136.233270 138.226883 140.209125 142.180095
## [61] 144.139887 146.088596 148.026316 149.953140 151.869159 153.774464
## [67] 155.669145 157.553290 159.426987 161.290323 163.143382 164.986251
## [73] 166.819013 168.641750 170.454545 172.257480 174.050633 175.834085
## [79] 177.607914 179.372197 181.127013 182.872435 184.608541 186.335404
## [85] 188.053097 189.761695 191.461268 193.151888 194.833625 196.506550
## [91] 198.170732 199.826238 201.473137 203.111495 204.741379 206.362855
## [97] 207.975986 209.580838 211.177474 212.765957 214.346350 215.918713
## [103] 217.483108 219.039596 220.588235 222.129086 223.662207 225.187656
## [109] 226.705491 228.215768 229.718543 231.213873 232.701812 234.182416
## [115] 235.655738 237.121832 238.580750 240.032547 241.477273 242.914980
## [121] 244.345719 245.769541 247.186495 248.596632 250.000000 251.396648
## [127] 252.786624 254.169976 255.546751 256.916996 258.280757 259.638080
## [133] 260.989011 262.333594 263.671875 265.003897 266.329705 267.649341
## [139] 268.962848 270.270270 271.571649 272.867025 274.156442 275.439939
## [145] 276.717557 277.989337 279.255319 280.515542 281.770045 283.018868
## [151] 284.262048 285.499624 286.731634 287.958115 289.179104 290.394639
## [157] 291.604755 292.809489 294.008876 295.202952 296.391753 297.575312
## [163] 298.753666 299.926847 301.094891 302.257830 303.415698 304.568528
## [169] 305.716353 306.859206 307.997118 309.130122 310.258250 311.381532
## [175] 312.500000 313.613685 314.722617 315.826828 316.926346 318.021201
## [181] 319.111425 320.197044 321.278090 322.354590 323.426573 324.494068
## [187] 325.557103 326.615705 327.669903 328.719723 329.765193 330.806340
## [193] 331.843191 332.875772 333.904110 334.928230 335.948158 336.963921
## [199] 337.975543 338.983051 339.986468 340.985820 341.981132 342.972428
## [205] 343.959732 344.943068 345.922460 346.897932 347.869507 348.837209
## [211] 349.801061 350.761085 351.717305 352.669743 353.618421 354.563362
## [217] 355.504587 356.442119 357.375979 358.306189 359.232770 360.155743
## [223] 361.075130 361.990950 362.903226 363.811977 364.717224 365.618987
## [229] 366.517286 367.412141 368.303571 369.191598 370.076239 370.957514
## [235] 371.835443 372.710044 373.581337 374.449339 375.314070 376.175549
## [241] 377.033792 377.888819 378.740648 379.589297 380.434783 381.277123
## [247] 382.116337 382.952440 383.785450 384.615385 385.442260 386.266094
## [253] 387.086903 387.904704 388.719512 389.531345 390.340219 391.146149
## [259] 391.949153 392.749245 393.546441 394.340759 395.132212 395.920816
## [265] 396.706587 397.489540 398.269690 399.047052 399.821641 400.593472
## [271] 401.362559 402.128918 402.892562 403.653506 404.411765 405.167352
## [277] 405.920281 406.670568 407.418224 408.163265 408.905704 409.645555
## [283] 410.382831 411.117545 411.849711 412.579342 413.306452 414.031052
## [289] 414.753157 415.472779 416.189931 416.904626 417.616876 418.326693
## [295] 419.034091 419.739081 420.441676 421.141888 421.839729 422.535211
## [301] 423.228346 423.919147 424.607623 425.293788 425.977654 426.659230
## [307] 427.338530 428.015564 428.690344 429.362881 430.033186 430.701270
## [313] 431.367144 432.030820 432.692308 433.351618 434.008762 434.663751
## [319] 435.316594 435.967302 436.615887 437.262357 437.906725 438.548998
## [325] 439.189189 439.827307 440.463362 441.097364 441.729323 442.359249
## [331] 442.987152 443.613041 444.236926 444.858817 445.478723 446.096654
## [337] 446.712619 447.326628 447.938689 448.548813 449.157007 449.763282
## [343] 450.367647 450.970110 451.570681 452.169367 452.766180 453.361126
## [349] 453.954214 454.545455 455.134855 455.722424 456.308170 456.892101
## [355] 457.474227 458.054555 458.633094 459.209851 459.784836 460.358056
## [361] 460.929520 461.499235 462.067210 462.633452 463.197970 463.760770
## [367] 464.321862 464.881253 465.438951 465.994962 466.549296 467.101959
## [373] 467.652959 468.202303 468.750000 469.296056 469.840479 470.383275
## [379] 470.924453 471.464020 472.001982 472.538347 473.073123 473.606315
## [385] 474.137931 474.667978 475.196464 475.723394 476.248776 476.772616
## [391] 477.294922 477.815700 478.334956 478.852698 479.368932 479.883665
## [397] 480.396902 480.908652 481.418919 481.927711 482.435034 482.940894
## [403] 483.445298 483.948251 484.449761 484.949833 485.448473 485.945688
## [409] 486.441484 486.935867 487.428843 487.920417 488.410596 488.899386
## [415] 489.386792 489.872821 490.357479 490.840770 491.322702 491.803279
## [421] 492.282507 492.760392 493.236940 493.712156 494.186047 494.658616
## [427] 495.129870 495.599815 496.068455 496.535797 497.001845 497.466605
## [433] 497.930083 498.392283 498.853211 499.312872 499.771272 500.228415
## [439] 500.684307 501.138952 501.592357 502.044525 502.495463 502.945174
## [445] 503.393665 503.840940 504.287004 504.731861 505.175518 505.617978
## [451] 506.059246 506.499328 506.938227 507.375950 507.812500 508.247882
## [457] 508.682102 509.115162 509.547069 509.977827 510.407440 510.835913
## [463] 511.263251 511.689457 512.114537 512.538495 512.961336 513.383063
## [469] 513.803681 514.223195 514.641608 515.058926 515.475153 515.890292
## [475] 516.304348 516.717325 517.129228 517.540061 517.949827 518.358531
## [481] 518.766178 519.172770 519.578313 519.982810 520.386266 520.788684
## [487] 521.190068 521.590423 521.989752 522.388060 522.785349 523.181625
## [493] 523.576890 523.971150 524.364407 524.756665 525.147929 525.538202
## [499] 525.927487 526.315789 526.703112 527.089458 527.474832 527.859238
## [505] 528.242678 528.625157 529.006678 529.387245 529.766861 530.145530
## [511] 530.523256 530.900041 531.275891 531.650807 532.024793 532.397854
## [517] 532.769992 533.141210 533.511513 533.880903 534.249385 534.616960
## [523] 534.983633 535.349407 535.714286 536.078272 536.441368 536.803579
## [529] 537.164907 537.525355 537.884927 538.243626 538.601455 538.958417
## [535] 539.314516 539.669754 540.024135 540.377662 540.730337 541.082164
## [541] 541.433147 541.783287 542.132588 542.481053 542.828685 543.175487
## [547] 543.521463 543.866614 544.210944 544.554455 544.897152 545.239036
## [553] 545.580110 545.920378 546.259843 546.598506 546.936371 547.273441
## [559] 547.609718 547.945205 548.279906 548.613823 548.946958 549.279314
## [565] 549.610895 549.941702 550.271739 550.601008 550.929512 551.257253
## [571] 551.584235 551.910459 552.235929 552.560647 552.884615 553.207837
## [577] 553.530315 553.852051 554.173047 554.493308 554.812834 555.131629
## [583] 555.449695 555.767035 556.083650 556.399544 556.714719 557.029178
## [589] 557.342922 557.655955 557.968278 558.279894 558.590806 558.901016
## [595] 559.210526 559.519339 559.827457 560.134882 560.441617 560.747664
## [601] 561.053025 561.357702 561.661699 561.965017 562.267658 562.569625
## [607] 562.870920 563.171545 563.471503 563.770795 564.069424 564.367392
## [613] 564.664702 564.961354 565.257353 565.552699 565.847395 566.141444
## [619] 566.434846 566.727605 567.019722 567.311200 567.602041 567.892246
## [625] 568.181818 568.470759 568.759071 569.046756 569.333816 569.620253
## [631] 569.906069 570.191267 570.475847 570.759813 571.043165 571.325907
## [637] 571.608040 571.889566 572.170487 572.450805 572.730522 573.009639
## [643] 573.288160 573.566085 573.843416 574.120156 574.396307 574.671869
## [649] 574.946846 575.221239 575.495050 575.768280 576.040932 576.313007
## [655] 576.584507 576.855434 577.125791 577.395577 577.664797 577.933450
## [661] 578.201540 578.469067 578.736034 579.002442 579.268293 579.533589
## [667] 579.798331 580.062522 580.326162 580.589255 580.851801 581.113801
## [673] 581.375259 581.636175 581.896552 582.156390 582.415692 582.674459
## [679] 582.932692 583.190395 583.447567 583.704211 583.960328 584.215921
## [685] 584.470990 584.725537 584.979564 585.233072 585.486064 585.738540
## [691] 585.990502 586.241952 586.492891 586.743321 586.993243 587.242659
## [697] 587.491571 587.739980 587.987887 588.235294 588.482203 588.728615
## [703] 588.974531 589.219953 589.464883 589.709322 589.953271 590.196732
## [709] 590.439707 590.682196 590.924202 591.165726 591.406768 591.647332
## [715] 591.887417 592.127026 592.366160 592.604820 592.843008 593.080725
## [721] 593.317972 593.554752 593.791064 594.026912 594.262295 594.497216
## [727] 594.731675 594.965675 595.199216 595.432300 595.664928 595.897102
## [733] 596.128822 596.360091 596.590909 596.821278 597.051199 597.280673
## [739] 597.509702 597.738288 597.966430 598.194131 598.421392 598.648214
## [745] 598.874598 599.100546 599.326059 599.551138 599.775785 600.000000
## [751] 600.223785 600.447141 600.670070 600.892573 601.114650 601.336303
## [757] 601.557533 601.778342 601.998731 602.218700 602.438252 602.657387
## [763] 602.876106 603.094411 603.312303 603.529783 603.746851 603.963511
## [769] 604.179761 604.395604 604.611041 604.826073 605.040701 605.254926
## [775] 605.468750 605.682173 605.895197 606.107822 606.320050 606.531882
## [781] 606.743319 606.954362 607.165012 607.375271 607.585139 607.794618
## [787] 608.003708 608.212411 608.420728 608.628659 608.836207 609.043371
## [793] 609.250154 609.456555 609.662577 609.868219 610.073484 610.278373
## [799] 610.482885 610.687023plot(x, grouse,
main ="Adult Grouse Population Size vs. Expected Recruitment",
xlab ="Adult Grouse Population Size",
ylab ="Recruits",
pch =20)
abline(a=0,b=1,col ="red")
B. Using the graph produced in part A, determine the maximum recruitment of turkeys and grouse. This will later determine the maximum number of individuals of each species that can be harvested in a year, which is the difference between recruitment and your 1:1 line (“surplus production”). Remember to report the units when you give your answer. Hint: R has functions for finding the min, mean, max, etc. of any vector of numbers (2 points)
max(turkey)## [1] 746.855max(grouse)## [1] 610.687C. What is the population size where sustainable yield (harvest) is expected to be maximized for each species? (3 points)
Hint: to figure this out, you will first need to find the largest distance between the replacement line and each of the other lines. For each species, you can subtract the Y values of the replacement line from the recruitment function and save them as a vector. Then use the function which.max() to find the value of N where the difference is maximized. This will be the number of fowl producing the MSY. Write a description of your results in a sentence, including units.
##Turkey max 206 ##grouse max 201
turkeymax<-c((a*x*exp(-b*x)) - x)
turkeymax## [1] 5.9759056 11.9037881 17.7838956 23.6164749 29.4017715
## [6] 35.1400302 40.8314943 46.4764060 52.0750067 57.6275363
## [11] 63.1342339 68.5953374 74.0110834 79.3817079 84.7074452
## [16] 89.9885290 95.2251917 100.4176648 105.5661784 110.6709619
## [21] 115.7322435 120.7502504 125.7252085 130.6573430 135.5468779
## [26] 140.3940361 145.1990397 149.9621094 154.6834653 159.3633262
## [31] 164.0019100 168.5994336 173.1561127 177.6721623 182.1477963
## [36] 186.5832274 190.9786677 195.3343279 199.6504181 203.9271471
## [41] 208.1647230 212.3633527 216.5232424 220.6445971 224.7276210
## [46] 228.7725173 232.7794882 236.7487350 240.6804582 244.5748571
## [51] 248.4321303 252.2524753 256.0360889 259.7831668 263.4939037
## [56] 267.1684938 270.8071299 274.4100043 277.9773081 281.5092316
## [61] 285.0059645 288.4676951 291.8946112 295.2868995 298.6447462
## [66] 301.9683360 305.2578534 308.5134816 311.7354030 314.9237994
## [71] 318.0788515 321.2007391 324.2896415 327.3457368 330.3692024
## [76] 333.3602149 336.3189500 339.2455827 342.1402870 345.0032363
## [81] 347.8346030 350.6345587 353.4032743 356.1409198 358.8476645
## [86] 361.5236768 364.1691243 366.7841740 369.3689918 371.9237432
## [91] 374.4485925 376.9437035 379.4092392 381.8453618 384.2522327
## [96] 386.6300125 388.9788612 391.2989379 393.5904010 395.8534082
## [101] 398.0881163 400.2946815 402.4732592 404.6240040 406.7470701
## [106] 408.8426104 410.9107777 412.9517235 414.9655989 416.9525544
## [111] 418.9127394 420.8463029 422.7533931 424.6341575 426.4887429
## [116] 428.3172953 430.1199601 431.8968821 433.6482052 435.3740728
## [121] 437.0746276 438.7500114 440.4003656 442.0258308 443.6265469
## [126] 445.2026532 446.7542883 448.2815901 449.7846960 451.2637424
## [131] 452.7188655 454.1502005 455.5578822 456.9420444 458.3028207
## [136] 459.6403437 460.9547456 462.2461579 463.5147113 464.7605362
## [141] 465.9837620 467.1845178 468.3629319 469.5191320 470.6532452
## [146] 471.7653982 472.8557166 473.9243259 474.9713507 475.9969151
## [151] 477.0011426 477.9841560 478.9460777 479.8870294 480.8071321
## [156] 481.7065065 482.5852724 483.4435492 484.2814558 485.0991102
## [161] 485.8966303 486.6741330 487.4317348 488.1695518 488.8876992
## [166] 489.5862919 490.2654442 490.9252697 491.5658816 492.1873925
## [171] 492.7899145 493.3735590 493.9384372 494.4846593 495.0123352
## [176] 495.5215744 496.0124856 496.4851771 496.9397568 497.3763318
## [181] 497.7950088 498.1958940 498.5790932 498.9447114 499.2928533
## [186] 499.6236231 499.9371243 500.2334601 500.5127330 500.7750453
## [191] 501.0204984 501.2491935 501.4612312 501.6567115 501.8357343
## [196] 501.9983985 502.1448028 502.2750453 502.3892238 502.4874355
## [201] 502.5697770 502.6363446 502.6872341 502.7225407 502.7423593
## [206] 502.7467843 502.7359095 502.7098283 502.6686337 502.6124183
## [211] 502.5412740 502.4552925 502.3545649 502.2391819 502.1092337
## [216] 501.9648100 501.8060003 501.6328935 501.4455780 501.2441417
## [221] 501.0286724 500.7992571 500.5559826 500.2989351 500.0282004
## [226] 499.7438641 499.4460111 499.1347259 498.8100928 498.4721954
## [231] 498.1211171 497.7569407 497.3797487 496.9896233 496.5866460
## [236] 496.1708981 495.7424605 495.3014135 494.8478373 494.3818114
## [241] 493.9034151 493.4127272 492.9098261 492.3947900 491.8676964
## [246] 491.3286225 490.7776454 490.2148414 489.6402866 489.0540568
## [251] 488.4562274 487.8468731 487.2260687 486.5938883 485.9504057
## [256] 485.2956944 484.6298275 483.9528777 483.2649172 482.5660182
## [261] 481.8562521 481.1356902 480.4044034 479.6624623 478.9099370
## [266] 478.1468972 477.3734125 476.5895519 475.7953843 474.9909779
## [271] 474.1764008 473.3517207 472.5170051 471.6723208 470.8177346
## [276] 469.9533127 469.0791213 468.1952258 467.3016917 466.3985839
## [281] 465.4859670 464.5639054 463.6324630 462.6917036 461.7416904
## [286] 460.7824864 459.8141543 458.8367566 457.8503551 456.8550116
## [291] 455.8507875 454.8377440 453.8159417 452.7854411 451.7463023
## [296] 450.6985853 449.6423494 448.5776539 447.5045577 446.4231195
## [301] 445.3333974 444.2354496 443.1293336 442.0151068 440.8928265
## [306] 439.7625493 438.6243318 437.4782301 436.3243002 435.1625977
## [311] 433.9931780 432.8160960 431.6314065 430.4391641 429.2394228
## [316] 428.0322366 426.8176591 425.5957436 424.3665433 423.1301108
## [321] 421.8864987 420.6357592 419.3779442 418.1131055 416.8412944
## [326] 415.5625621 414.2769595 412.9845371 411.6853452 410.3794341
## [331] 409.0668533 407.7476525 406.4218810 405.0895878 403.7508216
## [336] 402.4056309 401.0540640 399.6961689 398.3319932 396.9615845
## [341] 395.5849900 394.2022566 392.8134311 391.4185599 390.0176894
## [346] 388.6108653 387.1981336 385.7795397 384.3551287 382.9249459
## [351] 381.4890358 380.0474430 378.6002118 377.1473862 375.6890102
## [356] 374.2251271 372.7557804 371.2810133 369.8008684 368.3153886
## [361] 366.8246162 365.3285934 363.8273622 362.3209643 360.8094412
## [366] 359.2928341 357.7711843 356.2445323 354.7129190 353.1763847
## [371] 351.6349696 350.0887136 348.5376564 346.9818377 345.4212966
## [376] 343.8560723 342.2862037 340.7117294 339.1326880 337.5491175
## [381] 335.9610562 334.3685418 332.7716119 331.1703039 329.5646552
## [386] 327.9547025 326.3404829 324.7220329 323.0993888 321.4725869
## [391] 319.8416632 318.2066534 316.5675933 314.9245181 313.2774631
## [396] 311.6264633 309.9715536 308.3127686 306.6501426 304.9837100
## [401] 303.3135048 301.6395609 299.9619119 298.2805914 296.5956326
## [406] 294.9070687 293.2149326 291.5192570 289.8200746 288.1174177
## [411] 286.4113184 284.7018089 282.9889209 281.2726862 279.5531362
## [416] 277.8303022 276.1042154 274.3749067 272.6424069 270.9067466
## [421] 269.1679562 267.4260660 265.6811062 263.9331065 262.1820968
## [426] 260.4281067 258.6711655 256.9113026 255.1485470 253.3829276
## [431] 251.6144731 249.8432122 248.0691733 246.2923846 244.5128743
## [436] 242.7306703 240.9458003 239.1582920 237.3681728 235.5754701
## [441] 233.7802110 231.9824225 230.1821313 228.3793643 226.5741479
## [446] 224.7665085 222.9564724 221.1440655 219.3293138 217.5122431
## [451] 215.6928790 213.8712469 212.0473722 210.2212801 208.3929955
## [456] 206.5625434 204.7299485 202.8952354 201.0584285 199.2195521
## [461] 197.3786305 195.5356875 193.6907471 191.8438331 189.9949690
## [466] 188.1441782 186.2914842 184.4369101 182.5804789 180.7222135
## [471] 178.8621367 177.0002712 175.1366395 173.2712639 171.4041666
## [476] 169.5353699 167.6648955 165.7927655 163.9190014 162.0436249
## [481] 160.1666575 158.2881204 156.4080348 154.5264219 152.6433025
## [486] 150.7586975 148.8726276 146.9851132 145.0961750 143.2058332
## [491] 141.3141079 139.4210193 137.5265873 135.6308317 133.7337723
## [496] 131.8354286 129.9358201 128.0349662 126.1328860 124.2295988
## [501] 122.3251234 120.4194789 118.5126838 116.6047570 114.6957169
## [506] 112.7855819 110.8743704 108.9621005 107.0487904 105.1344580
## [511] 103.2191211 101.3027976 99.3855049 97.4672608 95.5480825
## [516] 93.6279875 91.7069928 89.7851157 87.8623730 85.9387817
## [521] 84.0143585 82.0891201 80.1630831 78.2362640 76.3086790
## [526] 74.3803444 72.4512765 70.5214912 68.5910045 66.6598321
## [531] 64.7279900 62.7954937 60.8623587 58.9286006 56.9942346
## [536] 55.0592760 53.1237399 51.1876414 49.2509955 47.3138170
## [541] 45.3761207 43.4379212 41.4992332 39.5600710 37.6204491
## [546] 35.6803818 33.7398833 31.7989677 29.8576489 27.9159410
## [551] 25.9738577 24.0314128 22.0886200 20.1454927 18.2020445
## [556] 16.2582888 14.3142388 12.3699078 10.4253089 8.4804551
## [561] 6.5353593 4.5900345 2.6444933 0.6987486 -1.2471872
## [566] -3.1933015 -5.1395818 -7.0860159 -9.0325914 -10.9792961
## [571] -12.9261180 -14.8730450 -16.8200651 -18.7671666 -20.7143377
## [576] -22.6615665 -24.6088416 -26.5561514 -28.5034844 -30.4508292
## [581] -32.3981746 -34.3455092 -36.2928221 -38.2401019 -40.1873379
## [586] -42.1345190 -44.0816345 -46.0286735 -47.9756253 -49.9224794
## [591] -51.8692251 -53.8158521 -55.7623499 -57.7087081 -59.6549166
## [596] -61.6009652 -63.5468437 -65.4925421 -67.4380504 -69.3833588
## [601] -71.3284573 -73.2733364 -75.2179862 -77.1623972 -79.1065598
## [606] -81.0504645 -82.9941019 -84.9374627 -86.8805377 -88.8233175
## [611] -90.7657930 -92.7079552 -94.6497951 -96.5913038 -98.5324722
## [616] -100.4732917 -102.4137535 -104.3538488 -106.2935692 -108.2329060
## [621] -110.1718507 -112.1103949 -114.0485302 -115.9862483 -117.9235411
## [626] -119.8604002 -121.7968176 -123.7327853 -125.6682951 -127.6033393
## [631] -129.5379099 -131.4719990 -133.4055990 -135.3387021 -137.2713008
## [636] -139.2033873 -141.1349542 -143.0659940 -144.9964994 -146.9264629
## [641] -148.8558773 -150.7847353 -152.7130297 -154.6407535 -156.5678995
## [646] -158.4944608 -160.4204303 -162.3458013 -164.2705667 -166.1947199
## [651] -168.1182541 -170.0411625 -171.9634387 -173.8850759 -175.8060676
## [656] -177.7264075 -179.6460890 -181.5651058 -183.4834516 -185.4011200
## [661] -187.3181050 -189.2344002 -191.1499996 -193.0648972 -194.9790869
## [666] -196.8925627 -198.8053188 -200.7173492 -202.6286481 -204.5392099
## [671] -206.4490287 -208.3580988 -210.2664147 -212.1739708 -214.0807615
## [676] -215.9867814 -217.8920250 -219.7964870 -221.7001619 -223.6030445
## [681] -225.5051296 -227.4064119 -229.3068862 -231.2065475 -233.1053907
## [686] -235.0034108 -236.9006027 -238.7969615 -240.6924824 -242.5871605
## [691] -244.4809910 -246.3739690 -248.2660900 -250.1573492 -252.0477419
## [696] -253.9372637 -255.8259098 -257.7136759 -259.6005574 -261.4865500
## [701] -263.3716492 -265.2558506 -267.1391501 -269.0215432 -270.9030259
## [706] -272.7835938 -274.6632429 -276.5419690 -278.4197681 -280.2966362
## [711] -282.1725692 -284.0475632 -285.9216143 -287.7947186 -289.6668723
## [716] -291.5380716 -293.4083127 -295.2775919 -297.1459054 -299.0132498
## [721] -300.8796212 -302.7450162 -304.6094313 -306.4728628 -308.3353074
## [726] -310.1967615 -312.0572219 -313.9166851 -315.7751478 -317.6326067
## [731] -319.4890586 -321.3445002 -323.1989283 -325.0523398 -326.9047315
## [736] -328.7561004 -330.6064434 -332.4557575 -334.3040396 -336.1512869
## [741] -337.9974964 -339.8426652 -341.6867904 -343.5298692 -345.3718989
## [746] -347.2128766 -349.0527996 -350.8916652 -352.7294708 -354.5662137
## [751] -356.4018912 -358.2365009 -360.0700401 -361.9025064 -363.7338972
## [756] -365.5642101 -367.3934427 -369.2215925 -371.0486573 -372.8746345
## [761] -374.6995221 -376.5233175 -378.3460187 -380.1676233 -381.9881292
## [766] -383.8075342 -385.6258362 -387.4430330 -389.2591226 -391.0741029
## [771] -392.8879718 -394.7007274 -396.5123677 -398.3228907 -400.1322945
## [776] -401.9405772 -403.7477370 -405.5537719 -407.3586802 -409.1624601
## [781] -410.9651099 -412.7666276 -414.5670117 -416.3662605 -418.1643723
## [786] -419.9613454 -421.7571782 -423.5518692 -425.3454168 -427.1378194
## [791] -428.9290754 -430.7191835 -432.5081422 -434.2959499 -436.0826053
## [796] -437.8681069 -439.6524535 -441.4356436 -443.2176759 -444.9985490which.max(turkeymax)## [1] 206grousemax<-c((1/(a2+b2/x)) - x)
grousemax## [1] 1.8473804 3.6753689 5.4841629 7.2739572 9.0449438
## [6] 10.7973124 12.5312500 14.2469410 15.9445676 17.6243094
## [11] 19.2863436 20.9308452 22.5579869 24.1679389 25.7608696
## [16] 27.3369447 28.8963283 30.4391819 31.9656652 33.4759358
## [21] 34.9701493 36.4484591 37.9110169 39.3579725 40.7894737
## [26] 42.2056663 43.6066946 44.9927007 46.3638254 47.7202073
## [31] 49.0619835 50.3892894 51.7022587 53.0010235 54.2857143
## [36] 55.5564598 56.8133874 58.0566229 59.2862903 60.5025126
## [41] 61.7054108 62.8951049 64.0717131 65.2353525 66.3861386
## [46] 67.5241856 68.6496063 69.7625123 70.8630137 71.9512195
## [51] 73.0272374 74.0911736 75.1431335 76.1832208 77.2115385
## [56] 78.2281879 79.2332696 80.2268827 81.2091255 82.1800948
## [61] 83.1398866 84.0885957 85.0263158 85.9531396 86.8691589
## [66] 87.7744641 88.6691450 89.5532901 90.4269871 91.2903226
## [71] 92.1433824 92.9862511 93.8190128 94.6417502 95.4545455
## [76] 96.2574796 97.0506329 97.8340848 98.6079137 99.3721973
## [81] 100.1270125 100.8724353 101.6085409 102.3354037 103.0530973
## [86] 103.7616946 104.4612676 105.1518876 105.8336252 106.5065502
## [91] 107.1707317 107.8262381 108.4731369 109.1114952 109.7413793
## [96] 110.3628547 110.9759863 111.5808383 112.1774744 112.7659574
## [101] 113.3463497 113.9187130 114.4831081 115.0395956 115.5882353
## [106] 116.1290863 116.6622074 117.1876564 117.7054908 118.2157676
## [111] 118.7185430 119.2138728 119.7018122 120.1824158 120.6557377
## [116] 121.1218316 121.5807504 122.0325468 122.4772727 122.9149798
## [121] 123.3457189 123.7695407 124.1864952 124.5966319 125.0000000
## [126] 125.3966480 125.7866242 126.1699762 126.5467512 126.9169960
## [131] 127.2807571 127.6380803 127.9890110 128.3335944 128.6718750
## [136] 129.0038971 129.3297045 129.6493406 129.9628483 130.2702703
## [141] 130.5716487 130.8670254 131.1564417 131.4399388 131.7175573
## [146] 131.9893374 132.2553191 132.5155421 132.7700454 133.0188679
## [151] 133.2620482 133.4996243 133.7316342 133.9581152 134.1791045
## [156] 134.3946389 134.6047548 134.8094885 135.0088757 135.2029520
## [161] 135.3917526 135.5753123 135.7536657 135.9268471 136.0948905
## [166] 136.2578296 136.4156977 136.5685279 136.7163531 136.8592058
## [171] 136.9971182 137.1301222 137.2582496 137.3815319 137.5000000
## [176] 137.6136850 137.7226174 137.8268275 137.9263456 138.0212014
## [181] 138.1114245 138.1970443 138.2780899 138.3545900 138.4265734
## [186] 138.4940684 138.5571031 138.6157054 138.6699029 138.7197232
## [191] 138.7651934 138.8063405 138.8431912 138.8757721 138.9041096
## [196] 138.9282297 138.9481583 138.9639210 138.9755435 138.9830508
## [201] 138.9864682 138.9858204 138.9811321 138.9724277 138.9597315
## [206] 138.9430676 138.9224599 138.8979320 138.8695073 138.8372093
## [211] 138.8010610 138.7610854 138.7173052 138.6697429 138.6184211
## [216] 138.5633618 138.5045872 138.4421190 138.3759791 138.3061889
## [221] 138.2327698 138.1557430 138.0751295 137.9909502 137.9032258
## [226] 137.8119768 137.7172237 137.6189865 137.5172855 137.4121406
## [231] 137.3035714 137.1915977 137.0762389 136.9575143 136.8354430
## [236] 136.7100442 136.5813367 136.4493392 136.3140704 136.1755486
## [241] 136.0337922 135.8888195 135.7406484 135.5892968 135.4347826
## [246] 135.2771234 135.1163366 134.9524398 134.7854501 134.6153846
## [251] 134.4422604 134.2660944 134.0869033 133.9047037 133.7195122
## [256] 133.5313451 133.3402187 133.1461492 132.9491525 132.7492447
## [261] 132.5464415 132.3407586 132.1322115 131.9208158 131.7065868
## [266] 131.4895397 131.2696897 131.0470518 130.8216409 130.5934718
## [271] 130.3625592 130.1289178 129.8925620 129.6535062 129.4117647
## [276] 129.1673517 128.9202814 128.6705676 128.4182243 128.1632653
## [281] 127.9057043 127.6455549 127.3828306 127.1175449 126.8497110
## [286] 126.5793422 126.3064516 126.0310523 125.7531573 125.4727794
## [291] 125.1899314 124.9046259 124.6168757 124.3266932 124.0340909
## [296] 123.7390811 123.4416761 123.1418881 122.8397291 122.5352113
## [301] 122.2283465 121.9191465 121.6076233 121.2937885 120.9776536
## [306] 120.6592303 120.3385301 120.0155642 119.6903441 119.3628809
## [311] 119.0331858 118.7012700 118.3671444 118.0308200 117.6923077
## [316] 117.3516182 117.0087623 116.6637507 116.3165939 115.9673025
## [321] 115.6158868 115.2623574 114.9067245 114.5489984 114.1891892
## [326] 113.8273071 113.4633621 113.0973642 112.7293233 112.3592493
## [331] 111.9871520 111.6130412 111.2369264 110.8588173 110.4787234
## [336] 110.0966543 109.7126193 109.3266278 108.9386892 108.5488127
## [341] 108.1570074 107.7632825 107.3676471 106.9701101 106.5706806
## [346] 106.1693675 105.7661795 105.3611256 104.9542144 104.5454545
## [351] 104.1348548 103.7224236 103.3081696 102.8921012 102.4742268
## [356] 102.0545548 101.6330935 101.2098512 100.7848361 100.3580563
## [361] 99.9295199 99.4992351 99.0672098 98.6334520 98.1979695
## [366] 97.7607704 97.3218623 96.8812532 96.4389506 95.9949622
## [371] 95.5492958 95.1019588 94.6529589 94.2023035 93.7500000
## [376] 93.2960559 92.8404786 92.3832753 91.9244533 91.4640199
## [381] 91.0019822 90.5383474 90.0731225 89.6063148 89.1379310
## [386] 88.6679784 88.1964637 87.7233938 87.2487757 86.7726161
## [391] 86.2949219 85.8156997 85.3349562 84.8526981 84.3689320
## [396] 83.8836646 83.3969022 82.9086515 82.4189189 81.9277108
## [401] 81.4350337 80.9408938 80.4452975 79.9482511 79.4497608
## [406] 78.9498328 78.4484733 77.9456884 77.4414843 76.9358670
## [411] 76.4288425 75.9204169 75.4105960 74.8993859 74.3867925
## [416] 73.8728215 73.3574788 72.8407703 72.3227017 71.8032787
## [421] 71.2825070 70.7603923 70.2369403 69.7121565 69.1860465
## [426] 68.6586159 68.1298701 67.5998147 67.0684551 66.5357968
## [431] 66.0018450 65.4666053 64.9300828 64.3922830 63.8532110
## [436] 63.3128722 62.7712717 62.2284148 61.6843066 61.1389522
## [441] 60.5923567 60.0445252 59.4954628 58.9451744 58.3936652
## [446] 57.8409399 57.2870036 56.7318612 56.1755176 55.6179775
## [451] 55.0592460 54.4993277 53.9382274 53.3759499 52.8125000
## [456] 52.2478823 51.6821015 51.1151623 50.5470693 49.9778271
## [461] 49.4074402 48.8359133 48.2632509 47.6894574 47.1145374
## [466] 46.5384954 45.9613357 45.3830627 44.8036810 44.2231947
## [471] 43.6416084 43.0589262 42.4751526 41.8902917 41.3043478
## [476] 40.7173252 40.1292281 39.5400606 38.9498270 38.3585313
## [481] 37.7661777 37.1727704 36.5783133 35.9828105 35.3862661
## [486] 34.7886841 34.1900685 33.5904233 32.9897523 32.3880597
## [491] 31.7853492 31.1816248 30.5768904 29.9711498 29.3644068
## [496] 28.7566653 28.1479290 27.5382018 26.9274874 26.3157895
## [501] 25.7031119 25.0894582 24.4748322 23.8592375 23.2426778
## [506] 22.6251567 22.0066778 21.3872447 20.7668609 20.1455301
## [511] 19.5232558 18.9000415 18.2758906 17.6508068 17.0247934
## [516] 16.3978539 15.7699918 15.1412104 14.5115132 13.8809035
## [521] 13.2493847 12.6169603 11.9836334 11.3494074 10.7142857
## [526] 10.0782715 9.4413681 8.8035787 8.1649066 7.5253550
## [531] 6.8849271 6.2436261 5.6014551 4.9584174 4.3145161
## [536] 3.6697543 3.0241352 2.3776617 1.7303371 1.0821643
## [541] 0.4331465 -0.2167133 -0.8674121 -1.5189469 -2.1713147
## [546] -2.8245125 -3.4785374 -4.1333863 -4.7890563 -5.4455446
## [551] -6.1028481 -6.7609640 -7.4198895 -8.0796216 -8.7401575
## [556] -9.4014943 -10.0636292 -10.7265594 -11.3902821 -12.0547945
## [561] -12.7200938 -13.3861773 -14.0530421 -14.7206856 -15.3891051
## [566] -16.0582977 -16.7282609 -17.3989919 -18.0704880 -18.7427466
## [571] -19.4157651 -20.0895407 -20.7640709 -21.4393531 -22.1153846
## [576] -22.7921629 -23.4696853 -24.1479494 -24.8269525 -25.5066922
## [581] -26.1871658 -26.8683709 -27.5503049 -28.2329654 -28.9163498
## [586] -29.6004558 -30.2852807 -30.9708223 -31.6570780 -32.3440454
## [591] -33.0317221 -33.7201056 -34.4091937 -35.0989838 -35.7894737
## [596] -36.4806609 -37.1725431 -37.8651180 -38.5583832 -39.2523364
## [601] -39.9469754 -40.6422977 -41.3383010 -42.0349833 -42.7323420
## [606] -43.4303750 -44.1290801 -44.8284550 -45.5284974 -46.2292052
## [611] -46.9305761 -47.6326079 -48.3352985 -49.0386456 -49.7426471
## [616] -50.4473008 -51.1526045 -51.8585562 -52.5651537 -53.2723949
## [621] -53.9802776 -54.6887997 -55.3979592 -56.1077539 -56.8181818
## [626] -57.5292408 -58.2409289 -58.9532439 -59.6661839 -60.3797468
## [631] -61.0939306 -61.8087333 -62.5241528 -63.2401873 -63.9568345
## [636] -64.6740927 -65.3919598 -66.1104338 -66.8295129 -67.5491950
## [641] -68.2694782 -68.9903606 -69.7118402 -70.4339152 -71.1565836
## [646] -71.8798436 -72.6036932 -73.3281305 -74.0531538 -74.7787611
## [651] -75.5049505 -76.2317202 -76.9590685 -77.6869933 -78.4154930
## [656] -79.1445656 -79.8742094 -80.6044226 -81.3352034 -82.0665499
## [661] -82.7984605 -83.5309332 -84.2639665 -84.9975584 -85.7317073
## [666] -86.4664114 -87.2016690 -87.9374783 -88.6738376 -89.4107452
## [671] -90.1481994 -90.8861985 -91.6247408 -92.3638246 -93.1034483
## [676] -93.8436101 -94.5843083 -95.3255414 -96.0673077 -96.8096055
## [681] -97.5524332 -98.2957891 -99.0396717 -99.7840793 -100.5290102
## [686] -101.2744630 -102.0204360 -102.7669275 -103.5139361 -104.2614601
## [691] -105.0094980 -105.7580481 -106.5071090 -107.2566791 -108.0067568
## [696] -108.7573405 -109.5084289 -110.2600202 -111.0121131 -111.7647059
## [701] -112.5177972 -113.2713854 -114.0254692 -114.7800469 -115.5351171
## [706] -116.2906782 -117.0467290 -117.8032678 -118.5602931 -119.3178037
## [711] -120.0757979 -120.8342743 -121.5932316 -122.3526682 -123.1125828
## [716] -123.8729739 -124.6338401 -125.3951799 -126.1569921 -126.9192751
## [721] -127.6820276 -128.4452483 -129.2089356 -129.9730883 -130.7377049
## [726] -131.5027841 -132.2683246 -133.0343249 -133.8007838 -134.5676998
## [731] -135.3350717 -136.1028981 -136.8711776 -137.6399090 -138.4090909
## [736] -139.1787220 -139.9488010 -140.7193266 -141.4902975 -142.2617124
## [741] -143.0335700 -143.8058691 -144.5786082 -145.3517863 -146.1254019
## [746] -146.8994539 -147.6739409 -148.4488618 -149.2242152 -150.0000000
## [751] -150.7762148 -151.5528585 -152.3299298 -153.1074275 -153.8853503
## [756] -154.6636971 -155.4424666 -156.2216577 -157.0012690 -157.7812995
## [761] -158.5617479 -159.3426131 -160.1238938 -160.9055889 -161.6876972
## [766] -162.4702175 -163.2531486 -164.0364895 -164.8202388 -165.6043956
## [771] -166.3889586 -167.1739267 -167.9592987 -168.7450735 -169.5312500
## [776] -170.3178270 -171.1048035 -171.8921782 -172.6799502 -173.4681182
## [781] -174.2566812 -175.0456380 -175.8349876 -176.6247289 -177.4148607
## [786] -178.2053820 -178.9962917 -179.7875888 -180.5792721 -181.3713405
## [791] -182.1637931 -182.9566287 -183.7498463 -184.5434449 -185.3374233
## [796] -186.1317806 -186.9265156 -187.7216274 -188.5171149 -189.3129771which.max(grousemax)## [1] 201Exercise 5. Constant harvest and constant effort harvest. (15 points total)
Now lets explicity add harvest to the logistic growth equation.
Constant harvest rate H: N(t + 1) = N(t) + r * N(t) * (1 - N(t)/K) - H
Constant effort harvest with fishing mortality F: N(t + 1) = N(t) + r * N(t) * (1 - N(t)/K) - F * N(t)
We can again set our r, K and N0 values. We’ll start the population at carrying capacity
r <- 0.3
K <- 500
N0 <- 500A. Now simulate the population under constant harvest rates of H = 20, 30, 40 and graph the output of N(t) through time using y axis limits from 0 to 600 for all graphs. (5 points)
##Harvest rate of 20
H <-20
N <- rep (NA,numyears)
N[1] <-N0
for ( t in 1 : (numyears-1)){
N[t+1] = N [t]+r*N[t] * (1 -N[t]/K) -H }
plot ( N,
main = " N ( t ) through Time ( H = 20 ) ",
xlab = " Years " ,
ylab = " Population ",
ylim = c ( 0 , 600 ) )
##Harvest rate of 30
H<-30
N <- rep (NA,numyears)
N[1] <-N0
for ( t in 1 : (numyears-1)){
N[t+1] = N [t]+r*N[t] * (1 -N[t]/K) -H }
plot ( N,
main = " N ( t ) through Time ( H = 30 ) ",
xlab = " Years " ,
ylab = " Population ",
ylim = c ( 0 , 600 ) )
##Harvest rate of 40
H<-40
N <- rep (NA,numyears)
N[1] <-N0
for ( t in 1 : (numyears-1)){
N[t+1] = N [t]+r*N[t] * (1 -N[t]/K) -H }
plot ( N,
main = " N ( t ) through Time ( H = 40 ) ",
xlab = " Years " ,
ylab = " Population ",
ylim = c ( 0 , 600 ) )
Describe the results
As the harvest rate increases the population decreases at a greater rate over time.
B. Plot dN/dt and the three H values on the same plot. (4 points)
Hint: 1. Recall that dN/dt is not the same as N(t+1). 2. Use the abline() function to make horizontal lines. You can use the lty option to make the lines dashed or dotted if you so choose and the lwd option to change the line width. Label the axes.
Npop <- 1 : 500
r <- 0.3
K <- 500
t <- 50
dN <- r* Npop *( 1 - Npop /K )
plot (Npop , dN,
xlab = " Population",
ylab = " Change in Population",
ylim = c (0,50),
pch = 19)
abline(h= 20, col= "yellow", lty= "twodash", lwd= 2)
abline(h= 30, col= "blue", lty= "twodash", lwd= 2)
abline(h= 40, col= "green", lty= "twodash", lwd= 2)
Describe the relationship between the graphs in parts A and B. How do the graphs in B tell you what the graphs in A must do?
The graphs in part A show the population with different harvest rates. The harvest rates keep the population below the maximum sustainable yield so the population numbers stay higher. Graph B shows the populations and how they would decline if they exceeded the MSY.
C. Now simulate the population under fishing mortality of F = 0.2, 0.3, 0.4 and graph the output of N(t) through time (3 points)
##fish mortality .2
F <- 0.2
N <- rep ( NA , numyears )
N [ 1 ] <- N0
for ( t in 1 : ( numyears - 1 ) ) { N [t + 1 ] = N [t ]+r* N [ t ] * ( 1 - N [t ]/ K ) - F * N [ t ] }
plot(N,
main ="N(t) through Time (F = 0.2)",
xlab ="Years",
ylab ="Population",
ylim =c(0,600))
##fish mortality .3
F <- 0.3
N <- rep ( NA , numyears )
N [ 1 ] <- N0
for ( t in 1 : ( numyears - 1 ) ) { N [t + 1 ] = N [t ]+r* N [ t ] * ( 1 - N [t ]/ K ) - F * N [ t ] }
plot(N,
main ="N(t) through Time (F = 0.3)",
xlab ="Years",
ylab ="Population",
ylim =c(0,600))
##fish mortality .4
F <- 0.4
N <- rep ( NA , numyears )
N [ 1 ] <- N0
for ( t in 1 : ( numyears - 1 ) ) { N [t + 1 ] = N [t ]+r* N [ t ] * ( 1 - N [t ]/ K ) - F * N [ t ] }
plot(N,
main ="N(t) through Time (F = 0.4)",
xlab ="Years",
ylab ="Population",
ylim =c(0,600))
Describe the results
##Higher mortality rates result in lower populations over time.
D. Plot dN/dt and the three F values on the same plot. (3 points)
Hint: Use the abline() function to make lines with intercept 0 and slope F. You can use the lty option to make the lines dashed or dotted if you so choose and the lwd option to change the line width. Label the axes.
Npop<-1:500
r<-0.3
K<-500
t<-50
dN<-r*Npop*(1- Npop/K)
plot(Npop, dN,
xlab ="Population",
ylab ="Change in Population",
ylim =c(0,45),pch =20)
abline(a=0,b=0.2,col ="yellow",lty ="twodash",lwd =2)
abline(a=0,b=0.3,col ="blue",lty ="twodash",lwd =2)
abline(a=0,b=0.4,col ="green",lty ="twodash",lwd =2)
Describe the relationship between the graphs in parts C and D. How do the graphs in D tell you what the graphs in C must do?
##The graphs in D tell the graphs in C what rates to use to keep the populations at MSY.