Classic two-species predator-prey system: $$\frac{dx}{dt} = \alpha \, x - \beta \, x y$$ $$\frac{dy}{dt} = \delta \, x y - \mu \, y$$
Arguments
- init
Named numeric vector, e.g.
c(x = 40, y = 9).- params
Named numeric vector with four parameters:
- alpha
Prey intrinsic growth rate.
- beta
Predation rate (prey killed per predator per unit time).
- delta
Predator reproduction rate per prey consumed.
- mu
Predator mortality rate. Named
mu(notgamma) to avoid confusion with the recovery rate used in epidemic models.
- times
Numeric vector of output times.
- ...
Additional arguments passed to
ode_solver.
Examples
model_lv(
init = c(x = 40, y = 9),
params = c(alpha = 0.4, beta = 0.04, delta = 0.02, mu = 0.5),
times = seq(0, 100, by = 0.1)
)
#> time x y
#> 1 0.0 40.00000 9.000000
#> 2 0.1 40.13832 9.275438
#> 3 0.2 40.23195 9.561528
#> 4 0.3 40.27884 9.857829
#> 5 0.4 40.27723 10.163773
#> 6 0.5 40.22563 10.478661
#> 7 0.6 40.12286 10.801642
#> 8 0.7 39.96811 11.131711
#> 9 0.8 39.76093 11.467715
#> 10 0.9 39.50131 11.808347
#> 11 1.0 39.18969 12.152152
#> 12 1.1 38.82695 12.497535
#> 13 1.2 38.41445 12.842770
#> 14 1.3 37.95401 13.186022
#> 15 1.4 37.44790 13.525363
#> 16 1.5 36.89884 13.858799
#> 17 1.6 36.30991 14.184296
#> 18 1.7 35.68459 14.499812
#> 19 1.8 35.02662 14.803328
#> 20 1.9 34.34000 15.092880
#> 21 2.0 33.62892 15.366591
#> 22 2.1 32.89767 15.622705
#> 23 2.2 32.15058 15.859613
#> 24 2.3 31.39198 16.075879
#> 25 2.4 30.62610 16.270260
#> 26 2.5 29.85704 16.441723
#> 27 2.6 29.08872 16.589460
#> 28 2.7 28.32480 16.712886
#> 29 2.8 27.56868 16.811647
#> 30 2.9 26.82348 16.885614
#> 31 3.0 26.09199 16.934873
#> 32 3.1 25.37669 16.959714
#> 33 3.2 24.67974 16.960616
#> 34 3.3 24.00296 16.938230
#> 35 3.4 23.34789 16.893354
#> 36 3.5 22.71576 16.826921
#> 37 3.6 22.10754 16.739970
#> 38 3.7 21.52393 16.633631
#> 39 3.8 20.96541 16.509100
#> 40 3.9 20.43226 16.367623
#> 41 4.0 19.92457 16.210477
#> 42 4.1 19.44228 16.038954
#> 43 4.2 18.98520 15.854346
#> 44 4.3 18.55300 15.657934
#> 45 4.4 18.14527 15.450973
#> 46 4.5 17.76155 15.234687
#> 47 4.6 17.40128 15.010258
#> 48 4.7 17.06387 14.778822
#> 49 4.8 16.74870 14.541464
#> 50 4.9 16.45514 14.299212
#> 51 5.0 16.18253 14.053039
#> 52 5.1 15.93022 13.803857
#> 53 5.2 15.69755 13.552522
#> 54 5.3 15.48388 13.299826
#> 55 5.4 15.28859 13.046506
#> 56 5.5 15.11106 12.793243
#> 57 5.6 14.95072 12.540659
#> 58 5.7 14.80700 12.289326
#> 59 5.8 14.67937 12.039764
#> 60 5.9 14.56730 11.792443
#> 61 6.0 14.47032 11.547790
#> 62 6.1 14.38798 11.306186
#> 63 6.2 14.31984 11.067971
#> 64 6.3 14.26550 10.833450
#> 65 6.4 14.22460 10.602888
#> 66 6.5 14.19678 10.376521
#> 67 6.6 14.18173 10.154552
#> 68 6.7 14.17915 9.937156
#> 69 6.8 14.18878 9.724484
#> 70 6.9 14.21035 9.516660
#> 71 7.0 14.24365 9.313789
#> 72 7.1 14.28848 9.115956
#> 73 7.2 14.34465 8.923226
#> 74 7.3 14.41200 8.735650
#> 75 7.4 14.49038 8.553263
#> 76 7.5 14.57967 8.376089
#> 77 7.6 14.67975 8.204138
#> 78 7.7 14.79054 8.037413
#> 79 7.8 14.91195 7.875904
#> 80 7.9 15.04391 7.719596
#> 81 8.0 15.18638 7.568467
#> 82 8.1 15.33931 7.422490
#> 83 8.2 15.50267 7.281630
#> 84 8.3 15.67644 7.145853
#> 85 8.4 15.86061 7.015118
#> 86 8.5 16.05518 6.889383
#> 87 8.6 16.26015 6.768606
#> 88 8.7 16.47553 6.652742
#> 89 8.8 16.70134 6.541747
#> 90 8.9 16.93760 6.435576
#> 91 9.0 17.18433 6.334188
#> 92 9.1 17.44156 6.237539
#> 93 9.2 17.70932 6.145591
#> 94 9.3 17.98764 6.058305
#> 95 9.4 18.27655 5.975648
#> 96 9.5 18.57606 5.897587
#> 97 9.6 18.88621 5.824096
#> 98 9.7 19.20700 5.755151
#> 99 9.8 19.53845 5.690732
#> 100 9.9 19.88057 5.630826
#> 101 10.0 20.23334 5.575424
#> 102 10.1 20.59676 5.524521
#> 103 10.2 20.97077 5.478122
#> 104 10.3 21.35536 5.436235
#> 105 10.4 21.75044 5.398876
#> 106 10.5 22.15594 5.366068
#> 107 10.6 22.57175 5.337842
#> 108 10.7 22.99775 5.314236
#> 109 10.8 23.43379 5.295297
#> 110 10.9 23.87968 5.281082
#> 111 11.0 24.33520 5.271655
#> 112 11.1 24.80011 5.267090
#> 113 11.2 25.27411 5.267473
#> 114 11.3 25.75688 5.272899
#> 115 11.4 26.24802 5.283474
#> 116 11.5 26.74712 5.299316
#> 117 11.6 27.25369 5.320554
#> 118 11.7 27.76719 5.347329
#> 119 11.8 28.28701 5.379795
#> 120 11.9 28.81249 5.418120
#> 121 12.0 29.34287 5.462484
#> 122 12.1 29.87735 5.513079
#> 123 12.2 30.41502 5.570112
#> 124 12.3 30.95489 5.633803
#> 125 12.4 31.49588 5.704386
#> 126 12.5 32.03683 5.782108
#> 127 12.6 32.57647 5.867226
#> 128 12.7 33.11341 5.960011
#> 129 12.8 33.64618 6.060745
#> 130 12.9 34.17319 6.169718
#> 131 13.0 34.69274 6.287229
#> 132 13.1 35.20301 6.413582
#> 133 13.2 35.70210 6.549083
#> 134 13.3 36.18796 6.694040
#> 135 13.4 36.65846 6.848754
#> 136 13.5 37.11136 7.013520
#> 137 13.6 37.54432 7.188617
#> 138 13.7 37.95495 7.374308
#> 139 13.8 38.34075 7.570825
#> 140 13.9 38.69919 7.778370
#> 141 14.0 39.02769 7.997099
#> 142 14.1 39.32368 8.227119
#> 143 14.2 39.58457 8.468473
#> 144 14.3 39.80784 8.721132
#> 145 14.4 39.99104 8.984985
#> 146 14.5 40.13181 9.259823
#> 147 14.6 40.22797 9.545333
#> 148 14.7 40.27750 9.841083
#> 149 14.8 40.27863 10.146513
#> 150 14.9 40.22984 10.460926
#> 151 15.0 40.12994 10.783480
#> 152 15.1 39.97809 11.113184
#> 153 15.2 39.77383 11.448890
#> 154 15.3 39.51713 11.789302
#> 155 15.4 39.20838 12.132971
#> 156 15.5 38.84845 12.478309
#> 157 15.6 38.43867 12.823599
#> 158 15.7 37.98084 13.167011
#> 159 15.8 37.47721 13.506621
#> 160 15.9 36.93046 13.840438
#> 161 16.0 36.34367 14.166431
#> 162 16.1 35.72027 14.482556
#> 163 16.2 35.06401 14.786791
#> 164 16.3 34.37888 15.077170
#> 165 16.4 33.66905 15.351810
#> 166 16.5 32.93880 15.608947
#> 167 16.6 32.19248 15.846963
#> 168 16.7 31.43441 16.064411
#> 169 16.8 30.66883 16.260039
#> 170 16.9 29.89985 16.432801
#> 171 17.0 29.13138 16.581875
#> 172 17.1 28.36712 16.706665
#> 173 17.2 27.61048 16.806804
#> 174 17.3 26.86460 16.882150
#> 175 17.4 26.13228 16.932777
#> 176 17.5 25.41602 16.958966
#> 177 17.6 24.71800 16.961185
#> 178 17.7 24.04005 16.940074
#> 179 17.8 23.38374 16.896426
#> 180 17.9 22.75031 16.831166
#> 181 18.0 22.14073 16.745328
#> 182 18.1 21.55574 16.640036
#> 183 18.2 20.99582 16.516485
#> 184 18.3 20.46125 16.375917
#> 185 18.4 19.95215 16.219609
#> 186 18.5 19.46845 16.048852
#> 187 18.6 19.00997 15.864939
#> 188 18.7 18.57639 15.669150
#> 189 18.8 18.16732 15.462743
#> 190 18.9 17.78228 15.246944
#> 191 19.0 17.42071 15.022938
#> 192 19.1 17.08205 14.791863
#> 193 19.2 16.76566 14.554806
#> 194 19.3 16.47092 14.312801
#> 195 19.4 16.19716 14.066821
#> 196 19.5 15.94373 13.817784
#> 197 19.6 15.70998 13.566547
#> 198 19.7 15.49528 13.313907
#> 199 19.8 15.29898 13.060604
#> 200 19.9 15.12048 12.807320
#> 201 20.0 14.95920 12.554683
#> 202 20.1 14.81457 12.303267
#> 203 20.2 14.68605 12.053593
#> 204 20.3 14.57313 11.806136
#> 205 20.4 14.47533 11.561324
#> 206 20.5 14.39218 11.319541
#> 207 20.6 14.32326 11.081129
#> 208 20.7 14.26817 10.846395
#> 209 20.8 14.22653 10.615607
#> 210 20.9 14.19799 10.389001
#> 211 21.0 14.18224 10.166782
#> 212 21.1 14.17898 9.949128
#> 213 21.2 14.18792 9.736189
#> 214 21.3 14.20884 9.528093
#> 215 21.4 14.24149 9.324944
#> 216 21.5 14.28569 9.126828
#> 217 21.6 14.34123 8.933813
#> 218 21.7 14.40796 8.745949
#> 219 21.8 14.48573 8.563273
#> 220 21.9 14.57441 8.385808
#> 221 22.0 14.67390 8.213567
#> 222 22.1 14.78410 8.046551
#> 223 22.2 14.90492 7.884752
#> 224 22.3 15.03630 7.728156
#> 225 22.4 15.17818 7.576740
#> 226 22.5 15.33053 7.430476
#> 227 22.6 15.49331 7.289333
#> 228 22.7 15.66650 7.153274
#> 229 22.8 15.85009 7.022259
#> 230 22.9 16.04408 6.896248
#> 231 23.0 16.24847 6.775196
#> 232 23.1 16.46327 6.659060
#> 233 23.2 16.68850 6.547795
#> 234 23.3 16.92418 6.441357
#> 235 23.4 17.17033 6.339703
#> 236 23.5 17.42697 6.242792
#> 237 23.6 17.69415 6.150583
#> 238 23.7 17.97188 6.063039
#> 239 23.8 18.26019 5.980125
#> 240 23.9 18.55912 5.901810
#> 241 24.0 18.86867 5.828065
#> 242 24.1 19.18887 5.758867
#> 243 24.2 19.51973 5.694197
#> 244 24.3 19.86126 5.634040
#> 245 24.4 20.21344 5.578388
#> 246 24.5 20.57626 5.527235
#> 247 24.6 20.94969 5.480585
#> 248 24.7 21.33368 5.438447
#> 249 24.8 21.72818 5.400835
#> 250 24.9 22.13311 5.367773
#> 251 25.0 22.54835 5.339291
#> 252 25.1 22.97379 5.315427
#> 253 25.2 23.40927 5.296227
#> 254 25.3 23.85462 5.281748
#> 255 25.4 24.30961 5.272052
#> 256 25.5 24.77401 5.267215
#> 257 25.6 25.24751 5.267320
#> 258 25.7 25.72980 5.272463
#> 259 25.8 26.22049 5.282749
#> 260 25.9 26.71915 5.298294
#> 261 26.0 27.22532 5.319228
#> 262 26.1 27.73845 5.345691
#> 263 26.2 28.25794 5.377837
#> 264 26.3 28.78312 5.415831
#> 265 26.4 29.31325 5.459854
#> 266 26.5 29.84752 5.510097
#> 267 26.6 30.38504 5.566766
#> 268 26.7 30.92481 5.630081
#> 269 26.8 31.46578 5.700274
#> 270 26.9 32.00676 5.777592
#> 271 27.0 32.54650 5.862291
#> 272 27.1 33.08363 5.954643
#> 273 27.2 33.61668 6.054927
#> 274 27.3 34.14405 6.163434
#> 275 27.4 34.66406 6.280463
#> 276 27.5 35.17490 6.406315
#> 277 27.6 35.67466 6.541300
#> 278 27.7 36.16131 6.685722
#> 279 27.8 36.63272 6.839885
#> 280 27.9 37.08666 7.004084
#> 281 28.0 37.52080 7.178600
#> 282 28.1 37.93274 7.363695
#> 283 28.2 38.31998 7.559605
#> 284 28.3 38.68001 7.766531
#> 285 28.4 39.01025 7.984635
#> 286 28.5 39.30811 8.214025
#> 287 28.6 39.57102 8.454748
#> 288 28.7 39.79645 8.706781
#> 289 28.8 39.98194 8.970015
#> 290 28.9 40.12514 9.244250
#> 291 29.0 40.22383 9.529176
#> 292 29.1 40.27600 9.824370
#> 293 29.2 40.27986 10.129278
#> 294 29.3 40.23389 10.443212
#> 295 29.4 40.13686 10.765338
#> 296 29.5 39.98791 11.094672
#> 297 29.6 39.78658 11.430078
#> 298 29.7 39.53278 11.770264
#> 299 29.8 39.22691 12.113792
#> 300 29.9 38.86980 12.459082
#> 301 30.0 38.46275 12.804422
#> 302 30.1 38.00754 13.147988
#> 303 30.2 37.50639 13.487861
#> 304 30.3 36.96196 13.822054
#> 305 30.4 36.37731 14.148535
#> 306 30.5 35.75585 14.465263
#> 307 30.6 35.10132 14.770212
#> 308 30.7 34.41769 15.061412
#> 309 30.8 33.70912 15.336975
#> 310 30.9 32.97989 15.595130
#> 311 31.0 32.23435 15.834249
#> 312 31.1 31.47682 16.052877
#> 313 31.2 30.71155 16.249748
#> 314 31.3 29.94265 16.423806
#> 315 31.4 29.17405 16.574215
#> 316 31.5 28.40947 16.700368
#> 317 31.6 27.65232 16.801884
#> 318 31.7 26.90576 16.878609
#> 319 31.8 26.17262 16.930606
#> 320 31.9 25.45541 16.958144
#> 321 32.0 24.75632 16.961681
#> 322 32.1 24.07721 16.941848
#> 323 32.2 23.41966 16.899431
#> 324 32.3 22.78493 16.835347
#> 325 32.4 22.17401 16.750625
#> 326 32.5 21.58763 16.646385
#> 327 32.6 21.02630 16.523817
#> 328 32.7 20.49032 16.384162
#> 329 32.8 19.97980 16.228696
#> 330 32.9 19.49470 16.058709
#> 331 33.0 19.03482 15.875494
#> 332 33.1 18.59987 15.680332
#> 333 33.2 18.18944 15.474484
#> 334 33.3 17.80307 15.259176
#> 335 33.4 17.44022 15.035596
#> 336 33.5 17.10030 14.804885
#> 337 33.6 16.78269 14.568134
#> 338 33.7 16.48676 14.326377
#> 339 33.8 16.21185 14.080594
#> 340 33.9 15.95731 13.831704
#> 341 34.0 15.72248 13.580568
#> 342 34.1 15.50673 13.327986
#> 343 34.2 15.30943 13.074701
#> 344 34.3 15.12996 12.821400
#> 345 34.4 14.96773 12.568711
#> 346 34.5 14.82219 12.317212
#> 347 34.6 14.69279 12.067429
#> 348 34.7 14.57901 11.819837
#> 349 34.8 14.48038 11.574867
#> 350 34.9 14.39643 11.332906
#> 351 35.0 14.32673 11.094299
#> 352 35.1 14.27088 10.859352
#> 353 35.2 14.22850 10.628338
#> 354 35.3 14.19924 10.401494
#> 355 35.4 14.18279 10.179026
#> 356 35.5 14.17883 9.961114
#> 357 35.6 14.18711 9.747909
#> 358 35.7 14.20737 9.539541
#> 359 35.8 14.23937 9.336114
#> 360 35.9 14.28293 9.137717
#> 361 36.0 14.33784 8.944416
#> 362 36.1 14.40395 8.756264
#> 363 36.2 14.48111 8.573299
#> 364 36.3 14.56920 8.395544
#> 365 36.4 14.66808 8.223012
#> 366 36.5 14.77769 8.055705
#> 367 36.6 14.89792 7.893617
#> 368 36.7 15.02871 7.736731
#> 369 36.8 15.17001 7.585028
#> 370 36.9 15.32178 7.438478
#> 371 37.0 15.48398 7.297051
#> 372 37.1 15.65659 7.160710
#> 373 37.2 15.83961 7.029416
#> 374 37.3 16.03302 6.903127
#> 375 37.4 16.23683 6.781801
#> 376 37.5 16.45105 6.665392
#> 377 37.6 16.67570 6.553857
#> 378 37.7 16.91079 6.447152
#> 379 37.8 17.15636 6.345234
#> 380 37.9 17.41242 6.248060
#> 381 38.0 17.67901 6.155590
#> 382 38.1 17.95615 6.067787
#> 383 38.2 18.24388 5.984617
#> 384 38.3 18.54221 5.906046
#> 385 38.4 18.85117 5.832048
#> 386 38.5 19.17078 5.762598
#> 387 38.6 19.50105 5.697676
#> 388 38.7 19.84198 5.637269
#> 389 38.8 20.19357 5.581365
#> 390 38.9 20.55580 5.529963
#> 391 39.0 20.92864 5.483062
#> 392 39.1 21.31204 5.440672
#> 393 39.2 21.70596 5.402808
#> 394 39.3 22.11031 5.369493
#> 395 39.4 22.52498 5.340755
#> 396 39.5 22.94986 5.316633
#> 397 39.6 23.38479 5.297172
#> 398 39.7 23.82959 5.282428
#> 399 39.8 24.28405 5.272465
#> 400 39.9 24.74793 5.267355
#> 401 40.0 25.22093 5.267183
#> 402 40.1 25.70274 5.272043
#> 403 40.2 26.19297 5.282040
#> 404 40.3 26.69121 5.297289
#> 405 40.4 27.19697 5.317920
#> 406 40.5 27.70973 5.344071
#> 407 40.6 28.22888 5.375897
#> 408 40.7 28.75376 5.413561
#> 409 40.8 29.28364 5.457243
#> 410 40.9 29.81771 5.507135
#> 411 41.0 30.35507 5.563441
#> 412 41.1 30.89474 5.626380
#> 413 41.2 31.43567 5.696185
#> 414 41.3 31.97669 5.773099
#> 415 41.4 32.51653 5.857381
#> 416 41.5 33.05384 5.949299
#> 417 41.6 33.58715 6.049135
#> 418 41.7 34.11489 6.157177
#> 419 41.8 34.63535 6.273723
#> 420 41.9 35.14676 6.399077
#> 421 42.0 35.64718 6.533545
#> 422 42.1 36.13462 6.677434
#> 423 42.2 36.60694 6.831048
#> 424 42.3 37.06191 6.994681
#> 425 42.4 37.49721 7.168616
#> 426 42.5 37.91044 7.353116
#> 427 42.6 38.29913 7.548418
#> 428 42.7 38.66074 7.754727
#> 429 42.8 38.99270 7.972206
#> 430 42.9 39.29243 8.200966
#> 431 43.0 39.55736 8.441059
#> 432 43.1 39.78494 8.692464
#> 433 43.2 39.97271 8.955080
#> 434 43.3 40.11832 9.228709
#> 435 43.4 40.21954 9.513051
#> 436 43.5 40.27435 9.807686
#> 437 43.6 40.28095 10.112071
#> 438 43.7 40.23778 10.425524
#> 439 43.8 40.14361 10.747218
#> 440 43.9 39.99757 11.076180
#> 441 44.0 39.79916 11.411280
#> 442 44.1 39.54828 11.751237
#> 443 44.2 39.24528 12.094620
#> 444 44.3 38.89099 12.439855
#> 445 44.4 38.48668 12.785239
#> 446 44.5 38.03409 13.128953
#> 447 44.6 37.53543 13.469083
#> 448 44.7 36.99333 13.803645
#> 449 44.8 36.41083 14.130610
#> 450 44.9 35.79133 14.447933
#> 451 45.0 35.13854 14.753590
#> 452 45.1 34.45641 15.045605
#> 453 45.2 33.74912 15.322086
#> 454 45.3 33.02093 15.581253
#> 455 45.4 32.27618 15.821472
#> 456 45.5 31.51920 16.041275
#> 457 45.6 30.75425 16.239386
#> 458 45.7 29.98545 16.414737
#> 459 45.8 29.21674 16.566480
#> 460 45.9 28.45184 16.693994
#> 461 46.0 27.69419 16.796887
#> 462 46.1 26.94697 16.874992
#> 463 46.2 26.21301 16.928359
#> 464 46.3 25.49485 16.957248
#> 465 46.4 24.79470 16.962104
#> 466 46.5 24.11444 16.943553
#> 467 46.6 23.45566 16.902369
#> 468 46.7 22.81962 16.839464
#> 469 46.8 22.20735 16.755862
#> 470 46.9 21.61960 16.652677
#> 471 47.0 21.05687 16.531096
#> 472 47.1 20.51947 16.392359
#> 473 47.2 20.00754 16.237739
#> 474 47.3 19.52102 16.068525
#> 475 47.4 19.05974 15.886012
#> 476 47.5 18.62342 15.691482
#> 477 47.6 18.21164 15.486196
#> 478 47.7 17.82395 15.271382
#> 479 47.8 17.45980 15.048232
#> 480 47.9 17.11862 14.817889
#> 481 48.0 16.79978 14.581446
#> 482 48.1 16.50266 14.339941
#> 483 48.2 16.22660 14.094358
#> 484 48.3 15.97094 13.845618
#> 485 48.4 15.73504 13.594584
#> 486 48.5 15.51824 13.342063
#> 487 48.6 15.31993 13.088799
#> 488 48.7 15.13948 12.835481
#> 489 48.8 14.97631 12.582743
#> 490 48.9 14.82986 12.331164
#> 491 49.0 14.69957 12.081272
#> 492 49.1 14.58494 11.833546
#> 493 49.2 14.48548 11.588420
#> 494 49.3 14.40072 11.346282
#> 495 49.4 14.33024 11.107480
#> 496 49.5 14.27363 10.872322
#> 497 49.6 14.23051 10.641083
#> 498 49.7 14.20053 10.414001
#> 499 49.8 14.18337 10.191285
#> 500 49.9 14.17873 9.973115
#> 501 50.0 14.18633 9.759645
#> 502 50.1 14.20593 9.551004
#> 503 50.2 14.23729 9.347300
#> 504 50.3 14.28020 9.148621
#> 505 50.4 14.33449 8.955035
#> 506 50.5 14.39998 8.766596
#> 507 50.6 14.47653 8.583341
#> 508 50.7 14.56401 8.405296
#> 509 50.8 14.66230 8.232473
#> 510 50.9 14.77131 8.064876
#> 511 51.0 14.89095 7.902497
#> 512 51.1 15.02116 7.745323
#> 513 51.2 15.16188 7.593332
#> 514 51.3 15.31306 7.446496
#> 515 51.4 15.47468 7.304785
#> 516 51.5 15.64672 7.168162
#> 517 51.6 15.82915 7.036588
#> 518 51.7 16.02198 6.910022
#> 519 51.8 16.22522 6.788421
#> 520 51.9 16.43886 6.671740
#> 521 52.0 16.66292 6.559935
#> 522 52.1 16.89744 6.452962
#> 523 52.2 17.14242 6.350779
#> 524 52.3 17.39790 6.253342
#> 525 52.4 17.66390 6.160611
#> 526 52.5 17.94046 6.072550
#> 527 52.6 18.22759 5.989123
#> 528 52.7 18.52533 5.910297
#> 529 52.8 18.83370 5.836045
#> 530 52.9 19.15272 5.766342
#> 531 53.0 19.48239 5.701169
#> 532 53.1 19.82273 5.640511
#> 533 53.2 20.17373 5.584357
#> 534 53.3 20.53536 5.532704
#> 535 53.4 20.90762 5.485553
#> 536 53.5 21.29044 5.442912
#> 537 53.6 21.68377 5.404796
#> 538 53.7 22.08754 5.371226
#> 539 53.8 22.50164 5.342233
#> 540 53.9 22.92595 5.317853
#> 541 54.0 23.36033 5.298131
#> 542 54.1 23.80459 5.283123
#> 543 54.2 24.25852 5.272892
#> 544 54.3 24.72188 5.267510
#> 545 54.4 25.19439 5.267062
#> 546 54.5 25.67571 5.271639
#> 547 54.6 26.16549 5.281347
#> 548 54.7 26.66329 5.296301
#> 549 54.8 27.16865 5.316628
#> 550 54.9 27.68103 5.342469
#> 551 55.0 28.19984 5.373974
#> 552 55.1 28.72442 5.411309
#> 553 55.2 29.25405 5.454652
#> 554 55.3 29.78790 5.504192
#> 555 55.4 30.32510 5.560136
#> 556 55.5 30.86468 5.622700
#> 557 55.6 31.40556 5.692117
#> 558 55.7 31.94661 5.768629
#> 559 55.8 32.48655 5.852494
#> 560 55.9 33.02404 5.943980
#> 561 56.0 33.55762 6.043368
#> 562 56.1 34.08570 6.150945
#> 563 56.2 34.60662 6.267011
#> 564 56.3 35.11858 6.391866
#> 565 56.4 35.61967 6.525819
#> 566 56.5 36.10788 6.669177
#> 567 56.6 36.58109 6.822241
#> 568 56.7 37.03709 6.985309
#> 569 56.8 37.47355 7.158664
#> 570 56.9 37.88808 7.342570
#> 571 57.0 38.27820 7.537265
#> 572 57.1 38.64138 7.742957
#> 573 57.2 38.97506 7.959811
#> 574 57.3 39.27665 8.187942
#> 575 57.4 39.54357 8.427404
#> 576 57.5 39.77330 8.678182
#> 577 57.6 39.96335 8.940178
#> 578 57.7 40.11136 9.213202
#> 579 57.8 40.21511 9.496957
#> 580 57.9 40.27256 9.791033
#> 581 58.0 40.28187 10.094892
#> 582 58.1 40.24151 10.407861
#> 583 58.2 40.15021 10.729121
#> 584 58.3 40.00707 11.057707
#> 585 58.4 39.81157 11.392497
#> 586 58.5 39.56361 11.732220
#> 587 58.6 39.26350 12.075452
#> 588 58.7 38.91203 12.420628
#> 589 58.8 38.51046 12.766051
#> 590 58.9 38.06050 13.109907
#> 591 59.0 37.56434 13.450289
#> 592 59.1 37.02458 13.785213
#> 593 59.2 36.44424 14.112654
#> 594 59.3 35.82670 14.430567
#> 595 59.4 35.17566 14.736926
#> 596 59.5 34.49506 15.029751
#> 597 59.6 33.78906 15.307144
#> 598 59.7 33.06192 15.567319
#> 599 59.8 32.31797 15.808632
#> 600 59.9 31.56157 16.029605
#> 601 60.0 30.79695 16.228953
#> 602 60.1 30.02825 16.405595
#> 603 60.2 29.25944 16.558670
#> 604 60.3 28.49423 16.687544
#> 605 60.4 27.73610 16.791813
#> 606 60.5 26.98821 16.871298
#> 607 60.6 26.25345 16.926037
#> 608 60.7 25.53435 16.956277
#> 609 60.8 24.83314 16.962456
#> 610 60.9 24.15174 16.945187
#> 611 61.0 23.49172 16.905240
#> 612 61.1 22.85439 16.843517
#> 613 61.2 22.24078 16.761038
#> 614 61.3 21.65164 16.658911
#> 615 61.4 21.08751 16.538322
#> 616 61.5 20.54870 16.400506
#> 617 61.6 20.03535 16.246736
#> 618 61.7 19.54742 16.078300
#> 619 61.8 19.08475 15.896493
#> 620 61.9 18.64704 15.702598
#> 621 62.0 18.23391 15.497878
#> 622 62.1 17.84489 15.283563
#> 623 62.2 17.47945 15.060846
#> 624 62.3 17.13700 14.830873
#> 625 62.4 16.81695 14.594742
#> 626 62.5 16.51863 14.353493
#> 627 62.6 16.24142 14.108111
#> 628 62.7 15.98464 13.859523
#> 629 62.8 15.74766 13.608596
#> 630 62.9 15.52981 13.356137
#> 631 63.0 15.33049 13.102896
#> 632 63.1 15.14906 12.849564
#> 633 63.2 14.98495 12.596778
#> 634 63.3 14.83758 12.345120
#> 635 63.4 14.70640 12.095121
#> 636 63.5 14.59092 11.847263
#> 637 63.6 14.49062 11.601982
#> 638 63.7 14.40506 11.359668
#> 639 63.8 14.33379 11.120672
#> 640 63.9 14.27642 10.885304
#> 641 64.0 14.23256 10.653840
#> 642 64.1 14.20186 10.426521
#> 643 64.2 14.18399 10.203557
#> 644 64.3 14.17866 9.985130
#> 645 64.4 14.18559 9.771395
#> 646 64.5 14.20453 9.562482
#> 647 64.6 14.23524 9.358501
#> 648 64.7 14.27752 9.159540
#> 649 64.8 14.33118 8.965670
#> 650 64.9 14.39605 8.776943
#> 651 65.0 14.47199 8.593399
#> 652 65.1 14.55886 8.415064
#> 653 65.2 14.65655 8.241951
#> 654 65.3 14.76496 8.074063
#> 655 65.4 14.88402 7.911394
#> 656 65.5 15.01364 7.753930
#> 657 65.6 15.15377 7.601652
#> 658 65.7 15.30438 7.454530
#> 659 65.8 15.46542 7.312535
#> 660 65.9 15.63687 7.175630
#> 661 66.0 15.81873 7.043776
#> 662 66.1 16.01098 6.916933
#> 663 66.2 16.21364 6.795056
#> 664 66.3 16.42670 6.678103
#> 665 66.4 16.65018 6.566028
#> 666 66.5 16.88412 6.458787
#> 667 66.6 17.12851 6.356338
#> 668 66.7 17.38341 6.258638
#> 669 66.8 17.64883 6.165647
#> 670 66.9 17.92479 6.077327
#> 671 67.0 18.21134 5.993643
#> 672 67.1 18.50849 5.914562
#> 673 67.2 18.81627 5.840056
#> 674 67.3 19.13469 5.770101
#> 675 67.4 19.46377 5.704676
#> 676 67.5 19.80352 5.643767
#> 677 67.6 20.15392 5.587363
#> 678 67.7 20.51497 5.535459
#> 679 67.8 20.88663 5.488058
#> 680 67.9 21.26886 5.445165
#> 681 68.0 21.66161 5.406797
#> 682 68.1 22.06480 5.372974
#> 683 68.2 22.47833 5.343725
#> 684 68.3 22.90208 5.319087
#> 685 68.4 23.33590 5.299105
#> 686 68.5 23.77962 5.283834
#> 687 68.6 24.23302 5.273335
#> 688 68.7 24.69586 5.267681
#> 689 68.8 25.16786 5.266955
#> 690 68.9 25.64871 5.271250
#> 691 69.0 26.13802 5.280670
#> 692 69.1 26.63539 5.295329
#> 693 69.2 27.14034 5.315354
#> 694 69.3 27.65235 5.340884
#> 695 69.4 28.17082 5.372070
#> 696 69.5 28.69510 5.409076
#> 697 69.6 29.22446 5.452079
#> 698 69.7 29.75810 5.501270
#> 699 69.8 30.29514 5.556852
#> 700 69.9 30.83461 5.619042
#> 701 70.0 31.37546 5.688071
#> 702 70.1 31.91652 5.764182
#> 703 70.2 32.45657 5.847631
#> 704 70.3 32.99423 5.938686
#> 705 70.4 33.52806 6.037626
#> 706 70.5 34.05650 6.144741
#> 707 70.6 34.57786 6.260326
#> 708 70.7 35.09036 6.384684
#> 709 70.8 35.59211 6.518123
#> 710 70.9 36.08109 6.660949
#> 711 71.0 36.55520 6.813466
#> 712 71.1 37.01221 6.975970
#> 713 71.2 37.44982 7.148745
#> 714 71.3 37.86563 7.332057
#> 715 71.4 38.25718 7.526147
#> 716 71.5 38.62192 7.731222
#> 717 71.6 38.95731 7.947452
#> 718 71.7 39.26076 8.174953
#> 719 71.8 39.52968 8.413784
#> 720 71.9 39.76154 8.663934
#> 721 72.0 39.95386 8.925311
#> 722 72.1 40.10427 9.197727
#> 723 72.2 40.21054 9.480896
#> 724 72.3 40.27061 9.774410
#> 725 72.4 40.28265 10.077742
#> 726 72.5 40.24508 10.390223
#> 727 72.6 40.15664 10.711046
#> 728 72.7 40.01641 11.039252
#> 729 72.8 39.82383 11.373730
#> 730 72.9 39.57878 11.713215
#> 731 73.0 39.28155 12.056291
#> 732 73.1 38.93291 12.401403
#> 733 73.2 38.53409 12.746858
#> 734 73.3 38.08677 13.090851
#> 735 73.4 37.59312 13.431477
#> 736 73.5 37.05571 13.766758
#> 737 73.6 36.47754 14.094669
#> 738 73.7 35.86198 14.413166
#> 739 73.8 35.21269 14.720220
#> 740 73.9 34.53363 15.013849
#> 741 74.0 33.82893 15.292148
#> 742 74.1 33.10285 15.553325
#> 743 74.2 32.35973 15.795728
#> 744 74.3 31.60390 16.017869
#> 745 74.4 30.83963 16.218450
#> 746 74.5 30.07106 16.396380
#> 747 74.6 29.30215 16.550785
#> 748 74.7 28.53665 16.681018
#> 749 74.8 27.77803 16.786663
#> 750 74.9 27.02950 16.867527
#> 751 75.0 26.29394 16.923638
#> 752 75.1 25.57390 16.955231
#> 753 75.2 24.87165 16.962734
#> 754 75.3 24.18910 16.946751
#> 755 75.4 23.52785 16.908043
#> 756 75.5 22.88923 16.847506
#> 757 75.6 22.27428 16.766153
#> 758 75.7 21.68376 16.665089
#> 759 75.8 21.11823 16.545495
#> 760 75.9 20.57801 16.408605
#> 761 76.0 20.06324 16.255688
#> 762 76.1 19.57390 16.088034
#> 763 76.2 19.10983 15.906937
#> 764 76.3 18.67074 15.713681
#> 765 76.4 18.25626 15.509530
#> 766 76.5 17.86591 15.295717
#> 767 76.6 17.49917 15.073437
#> 768 76.7 17.15546 14.843839
#> 769 76.8 16.83417 14.608022
#> 770 76.9 16.53467 14.367032
#> 771 77.0 16.25630 14.121855
#> 772 77.1 15.99840 13.873422
#> 773 77.2 15.76033 13.622603
#> 774 77.3 15.54144 13.370210
#> 775 77.4 15.34110 13.116993
#> 776 77.5 15.15870 12.863649
#> 777 77.6 14.99363 12.610817
#> 778 77.7 14.84534 12.359081
#> 779 77.8 14.71329 12.108977
#> 780 77.9 14.59694 11.860989
#> 781 78.0 14.49581 11.615553
#> 782 78.1 14.40944 11.373065
#> 783 78.2 14.33739 11.133875
#> 784 78.3 14.27925 10.898298
#> 785 78.4 14.23465 10.666610
#> 786 78.5 14.20323 10.439055
#> 787 78.6 14.18466 10.215843
#> 788 78.7 14.17864 9.997160
#> 789 78.8 14.18489 9.783160
#> 790 78.9 14.20316 9.573976
#> 791 79.0 14.23322 9.369718
#> 792 79.1 14.27486 9.170476
#> 793 79.2 14.32789 8.976321
#> 794 79.3 14.39215 8.787307
#> 795 79.4 14.46747 8.603474
#> 796 79.5 14.55374 8.424848
#> 797 79.6 14.65083 8.251444
#> 798 79.7 14.75865 8.083265
#> 799 79.8 14.87711 7.920307
#> 800 79.9 15.00615 7.762554
#> 801 80.0 15.14570 7.609988
#> 802 80.1 15.29572 7.462580
#> 803 80.2 15.45619 7.320301
#> 804 80.3 15.62706 7.183113
#> 805 80.4 15.80834 7.050980
#> 806 80.5 16.00001 6.923858
#> 807 80.6 16.20209 6.801707
#> 808 80.7 16.41457 6.684480
#> 809 80.8 16.63748 6.572135
#> 810 80.9 16.87083 6.464627
#> 811 81.0 17.11464 6.361913
#> 812 81.1 17.36895 6.263950
#> 813 81.2 17.63378 6.170698
#> 814 81.3 17.90916 6.082119
#> 815 81.4 18.19512 5.998177
#> 816 81.5 18.49168 5.918841
#> 817 81.6 18.79887 5.844081
#> 818 81.7 19.11670 5.773874
#> 819 81.8 19.44519 5.708197
#> 820 81.9 19.78434 5.647037
#> 821 82.0 20.13415 5.590383
#> 822 82.1 20.49460 5.538229
#> 823 82.2 20.86567 5.490576
#> 824 82.3 21.24732 5.447433
#> 825 82.4 21.63949 5.408812
#> 826 82.5 22.04210 5.374736
#> 827 82.6 22.45506 5.345231
#> 828 82.7 22.87824 5.320336
#> 829 82.8 23.31150 5.300094
#> 830 82.9 23.75467 5.284558
#> 831 83.0 24.20754 5.273792
#> 832 83.1 24.66987 5.267867
#> 833 83.2 25.14137 5.266865
#> 834 83.3 25.62173 5.270878
#> 835 83.4 26.11059 5.280009
#> 836 83.5 26.60752 5.294373
#> 837 83.6 27.11206 5.314096
#> 838 83.7 27.62369 5.339316
#> 839 83.8 28.14182 5.370184
#> 840 83.9 28.66579 5.406861
#> 841 84.0 29.19489 5.449526
#> 842 84.1 29.72832 5.498368
#> 843 84.2 30.26519 5.553588
#> 844 84.3 30.80456 5.615405
#> 845 84.4 31.34535 5.684047
#> 846 84.5 31.88644 5.759757
#> 847 84.6 32.42657 5.842791
#> 848 84.7 32.96440 5.933416
#> 849 84.8 33.49849 6.031910
#> 850 84.9 34.02727 6.138562
#> 851 85.0 34.54907 6.253668
#> 852 85.1 35.06211 6.377530
#> 853 85.2 35.56451 6.510456
#> 854 85.3 36.05426 6.652751
#> 855 85.4 36.52925 6.804721
#> 856 85.5 36.98728 6.966662
#> 857 85.6 37.42603 7.138859
#> 858 85.7 37.84311 7.321578
#> 859 85.8 38.23607 7.515062
#> 860 85.9 38.60238 7.719522
#> 861 86.0 38.93947 7.935127
#> 862 86.1 39.24476 8.161999
#> 863 86.2 39.51567 8.400199
#> 864 86.3 39.74966 8.649722
#> 865 86.4 39.94424 8.910477
#> 866 86.5 40.09704 9.182286
#> 867 86.6 40.20582 9.464866
#> 868 86.7 40.26851 9.757818
#> 869 86.8 40.28327 10.060619
#> 870 86.9 40.24850 10.372612
#> 871 87.0 40.16292 10.692995
#> 872 87.1 40.02558 11.020818
#> 873 87.2 39.83592 11.354978
#> 874 87.3 39.59378 11.694220
#> 875 87.4 39.29945 12.037137
#> 876 87.5 38.95364 12.382178
#> 877 87.6 38.55757 12.727661
#> 878 87.7 38.11290 13.071784
#> 879 87.8 37.62175 13.412649
#> 880 87.9 37.08670 13.748281
#> 881 88.0 36.51072 14.076655
#> 882 88.1 35.89714 14.395729
#> 883 88.2 35.24964 14.703472
#> 884 88.3 34.57212 14.997899
#> 885 88.4 33.86873 15.277099
#> 886 88.5 33.14373 15.539274
#> 887 88.6 32.40145 15.782762
#> 888 88.7 31.64621 16.006066
#> 889 88.8 30.88230 16.207877
#> 890 88.9 30.11386 16.387092
#> 891 89.0 29.34487 16.542825
#> 892 89.1 28.57908 16.674416
#> 893 89.2 27.82000 16.781436
#> 894 89.3 27.07083 16.863680
#> 895 89.4 26.33447 16.921164
#> 896 89.5 25.61351 16.954111
#> 897 89.6 24.91022 16.962940
#> 898 89.7 24.22652 16.948245
#> 899 89.8 23.56405 16.910779
#> 900 89.9 22.92415 16.851431
#> 901 90.0 22.30785 16.771207
#> 902 90.1 21.71596 16.671210
#> 903 90.2 21.14903 16.552614
#> 904 90.3 20.60739 16.416653
#> 905 90.4 20.09121 16.264594
#> 906 90.5 19.60046 16.097726
#> 907 90.6 19.13499 15.917343
#> 908 90.7 18.69452 15.724730
#> 909 90.8 18.27868 15.521152
#> 910 90.9 17.88700 15.307845
#> 911 91.0 17.51896 15.086006
#> 912 91.1 17.17399 14.856785
#> 913 91.2 16.85147 14.621287
#> 914 91.3 16.55077 14.380557
#> 915 91.4 16.27124 14.135589
#> 916 91.5 16.01222 13.887314
#> 917 91.6 15.77307 13.636606
#> 918 91.7 15.55313 13.384279
#> 919 91.8 15.35177 13.131090
#> 920 91.9 15.16838 12.877735
#> 921 92.0 15.00237 12.624859
#> 922 92.1 14.85316 12.373048
#> 923 92.2 14.72021 12.122840
#> 924 92.3 14.60301 11.874722
#> 925 92.4 14.50104 11.629133
#> 926 92.5 14.41386 11.386471
#> 927 92.6 14.34103 11.147090
#> 928 92.7 14.28213 10.911304
#> 929 92.8 14.23678 10.679393
#> 930 92.9 14.20463 10.451602
#> 931 93.0 14.18536 10.228144
#> 932 93.1 14.17865 10.009204
#> 933 93.2 14.18422 9.794940
#> 934 93.3 14.20183 9.585485
#> 935 93.4 14.23124 9.380951
#> 936 93.5 14.27225 9.181427
#> 937 93.6 14.32465 8.986987
#> 938 93.7 14.38828 8.797686
#> 939 93.8 14.46299 8.613564
#> 940 93.9 14.54866 8.434648
#> 941 94.0 14.64515 8.260954
#> 942 94.1 14.75237 8.092484
#> 943 94.2 14.87025 7.929236
#> 944 94.3 14.99869 7.771194
#> 945 94.4 15.13766 7.618339
#> 946 94.5 15.28710 7.470646
#> 947 94.6 15.44699 7.328082
#> 948 94.7 15.61728 7.190612
#> 949 94.8 15.79798 7.058199
#> 950 94.9 15.98908 6.930800
#> 951 95.0 16.19057 6.808372
#> 952 95.1 16.40248 6.690873
#> 953 95.2 16.62480 6.578258
#> 954 95.3 16.85757 6.470482
#> 955 95.4 17.10080 6.367502
#> 956 95.5 17.35453 6.269275
#> 957 95.6 17.61877 6.175762
#> 958 95.7 17.89356 6.086924
#> 959 95.8 18.17893 6.002726
#> 960 95.9 18.47490 5.923134
#> 961 96.0 18.78150 5.848121
#> 962 96.1 19.09874 5.777660
#> 963 96.2 19.42663 5.711732
#> 964 96.3 19.76519 5.650321
#> 965 96.4 20.11441 5.593416
#> 966 96.5 20.47427 5.541012
#> 967 96.6 20.84475 5.493109
#> 968 96.7 21.22581 5.449714
#> 969 96.8 21.61740 5.410842
#> 970 96.9 22.01943 5.376512
#> 971 97.0 22.43181 5.346752
#> 972 97.1 22.85443 5.321599
#> 973 97.2 23.28714 5.301097
#> 974 97.3 23.72976 5.285298
#> 975 97.4 24.18210 5.274265
#> 976 97.5 24.64390 5.268068
#> 977 97.6 25.11490 5.266790
#> 978 97.7 25.59478 5.270521
#> 979 97.8 26.08317 5.279365
#> 980 97.9 26.57967 5.293435
#> 981 98.0 27.08380 5.312856
#> 982 98.1 27.59505 5.337766
#> 983 98.2 28.11283 5.368315
#> 984 98.3 28.63650 5.404665
#> 985 98.4 29.16534 5.446992
#> 986 98.5 29.69854 5.495485
#> 987 98.6 30.23525 5.550345
#> 988 98.7 30.77450 5.611789
#> 989 98.8 31.31525 5.680045
#> 990 98.9 31.85635 5.755355
#> 991 99.0 32.39657 5.837975
#> 992 99.1 32.93457 5.928170
#> 993 99.2 33.46890 6.026219
#> 994 99.3 33.99802 6.132409
#> 995 99.4 34.52025 6.247037
#> 996 99.5 35.03383 6.370404
#> 997 99.6 35.53687 6.502817
#> 998 99.7 36.02738 6.644584
#> 999 99.8 36.50325 6.796008
#> 1000 99.9 36.96228 6.957386
#> 1001 100.0 37.40217 7.129005