Run model of mutualism with saturating functional response
run_mutualism.RdGiven a vector of time, initial values, and parameters, this function runs a phenomenological model of direct mutualism between two species as described by Holland (2012). This model features a saturating functional response for interspecific density dependence, which produces a stable equilibrium where both mutualist species can coexist above their respective population carrying capacities.
Arguments
- time
Vector of time units over which to run the model, starting at 0.
timecan also be supplied as the total length of the simulation.- init
Named vector of initial population sizes.
N1: Initial population size of species 1N2: Initial population size of species 2
- params
Named vector of model parameters.
r1: Per capita growth rate of species 1r2: Per capita growth rate of species 1a12: Effect of species 2 on species 1a21: Effect of species 1 on species 2b1: Half-saturation constant for species 1b2: Half-saturation constant for species 2d1: Rate of self-limitation for species 1d2: Rate of self-limitation for species 2
Value
A data frame of population sizes (N1, N2) simulated through time,
produced by solving mutualism() with deSolve::ode() given time, init,
and params. This data frame has additional attributes recording the model
run (model), input parameter values (params) and population carrying
capacities calculated from the parameter values (K).
Note
See the following for further context and details regarding this model: Holland, N.J. (2012). Population dynamics of mutualism. Nature Education Knowledge, 3, 2.
Examples
# Define full time series and run model with default parameter values
run_mutualism(time = 0:10)
#> time N1 N2
#> 1 0 20.00000 40.00000
#> 2 1 36.39801 54.98436
#> 3 2 49.85543 67.71351
#> 4 3 56.65409 75.96815
#> 5 4 59.29797 80.41556
#> 6 5 60.23751 82.57074
#> 7 6 60.56841 83.55962
#> 8 7 60.68756 84.00130
#> 9 8 60.73187 84.19599
#> 10 9 60.74890 84.28124
#> 11 10 60.75564 84.31844
# Define length of simulation and run model with default parameter values
run_mutualism(time = 10)
#> time N1 N2
#> 1 0.0 20.00000 40.00000
#> 2 0.1 21.50882 41.47669
#> 3 0.2 23.07089 42.97066
#> 4 0.3 24.67861 44.47747
#> 5 0.4 26.32334 45.99249
#> 6 0.5 27.99550 47.51093
#> 7 0.6 29.68482 49.02790
#> 8 0.7 31.38056 50.53851
#> 9 0.8 33.07181 52.03790
#> 10 0.9 34.74775 53.52136
#> 11 1.0 36.39797 54.98435
#> 12 1.1 38.01267 56.42255
#> 13 1.2 39.58294 57.83199
#> 14 1.3 41.10089 59.20898
#> 15 1.4 42.55982 60.55023
#> 16 1.5 43.95428 61.85284
#> 17 1.6 45.28008 63.11432
#> 18 1.7 46.53430 64.33258
#> 19 1.8 47.71520 65.50598
#> 20 1.9 48.82214 66.63325
#> 21 2.0 49.85547 67.71353
#> 22 2.1 50.81637 68.74631
#> 23 2.2 51.70674 69.73144
#> 24 2.3 52.52907 70.66909
#> 25 2.4 53.28626 71.55969
#> 26 2.5 53.98157 72.40394
#> 27 2.6 54.61847 73.20276
#> 28 2.7 55.20055 73.95727
#> 29 2.8 55.73145 74.66872
#> 30 2.9 56.21478 75.33851
#> 31 3.0 56.65409 75.96815
#> 32 3.1 57.05280 76.55921
#> 33 3.2 57.41421 77.11331
#> 34 3.3 57.74142 77.63214
#> 35 3.4 58.03739 78.11736
#> 36 3.5 58.30486 78.57066
#> 37 3.6 58.54639 78.99370
#> 38 3.7 58.76436 79.38812
#> 39 3.8 58.96097 79.75554
#> 40 3.9 59.13823 80.09752
#> 41 4.0 59.29797 80.41556
#> 42 4.1 59.44190 80.71112
#> 43 4.2 59.57154 80.98562
#> 44 4.3 59.68829 81.24038
#> 45 4.4 59.79343 81.47669
#> 46 4.5 59.88810 81.69576
#> 47 4.6 59.97334 81.89875
#> 48 4.7 60.05009 82.08674
#> 49 4.8 60.11921 82.26076
#> 50 4.9 60.18146 82.42180
#> 51 5.0 60.23752 82.57075
#> 52 5.1 60.28802 82.70848
#> 53 5.2 60.33352 82.83578
#> 54 5.3 60.37453 82.95342
#> 55 5.4 60.41149 83.06209
#> 56 5.5 60.44481 83.16245
#> 57 5.6 60.47486 83.25512
#> 58 5.7 60.50196 83.34065
#> 59 5.8 60.52642 83.41959
#> 60 5.9 60.54849 83.49243
#> 61 6.0 60.56841 83.55962
#> 62 6.1 60.58641 83.62160
#> 63 6.2 60.60266 83.67876
#> 64 6.3 60.61735 83.73146
#> 65 6.4 60.63063 83.78005
#> 66 6.5 60.64263 83.82484
#> 67 6.6 60.65349 83.86612
#> 68 6.7 60.66332 83.90417
#> 69 6.8 60.67222 83.93923
#> 70 6.9 60.68027 83.97154
#> 71 7.0 60.68756 84.00130
#> 72 7.1 60.69417 84.02872
#> 73 7.2 60.70016 84.05398
#> 74 7.3 60.70559 84.07724
#> 75 7.4 60.71052 84.09867
#> 76 7.5 60.71499 84.11841
#> 77 7.6 60.71904 84.13658
#> 78 7.7 60.72273 84.15332
#> 79 7.8 60.72607 84.16873
#> 80 7.9 60.72911 84.18292
#> 81 8.0 60.73187 84.19599
#> 82 8.1 60.73438 84.20801
#> 83 8.2 60.73666 84.21909
#> 84 8.3 60.73874 84.22928
#> 85 8.4 60.74063 84.23867
#> 86 8.5 60.74235 84.24731
#> 87 8.6 60.74391 84.25526
#> 88 8.7 60.74534 84.26258
#> 89 8.8 60.74664 84.26932
#> 90 8.9 60.74782 84.27553
#> 91 9.0 60.74890 84.28123
#> 92 9.1 60.74988 84.28649
#> 93 9.2 60.75078 84.29133
#> 94 9.3 60.75160 84.29578
#> 95 9.4 60.75235 84.29987
#> 96 9.5 60.75303 84.30364
#> 97 9.6 60.75365 84.30711
#> 98 9.7 60.75422 84.31031
#> 99 9.8 60.75474 84.31324
#> 100 9.9 60.75521 84.31595
#> 101 10.0 60.75564 84.31844
# Run model with custom input values
tmax <- 10
start <- c(N1 = 100, N2 = 50)
pars <- c(r1 = 1, r2 = 1, a12 = 1.2, a21 = 0.8, b1 = 20, b2 = 20, d1 = 0.04, d2 = 0.02)
run_mutualism(time = tmax, init = start, params = pars)
#> time N1 N2
#> 1 0.0 100.00000 50.00000
#> 2 0.1 83.70023 53.21507
#> 3 0.2 73.81523 56.18362
#> 4 0.3 67.30379 58.89673
#> 5 0.4 62.77894 61.35311
#> 6 0.5 59.51722 63.55802
#> 7 0.6 57.10395 65.52199
#> 8 0.7 55.28422 67.25939
#> 9 0.8 53.89267 68.78710
#> 10 0.9 52.81738 70.12340
#> 11 1.0 51.97998 71.28697
#> 12 1.1 51.32411 72.29621
#> 13 1.2 50.80825 73.16871
#> 14 1.3 50.40135 73.92090
#> 15 1.4 50.07978 74.56786
#> 16 1.5 49.82538 75.12324
#> 17 1.6 49.62405 75.59926
#> 18 1.7 49.46475 76.00674
#> 19 1.8 49.33883 76.35519
#> 20 1.9 49.23941 76.65292
#> 21 2.0 49.16106 76.90717
#> 22 2.1 49.09947 77.12418
#> 23 2.2 49.05119 77.30936
#> 24 2.3 49.01347 77.46733
#> 25 2.4 48.98413 77.60209
#> 26 2.5 48.96141 77.71703
#> 27 2.6 48.94392 77.81508
#> 28 2.7 48.93056 77.89873
#> 29 2.8 48.92043 77.97010
#> 30 2.9 48.91284 78.03100
#> 31 3.0 48.90722 78.08298
#> 32 3.1 48.90313 78.12735
#> 33 3.2 48.90022 78.16524
#> 34 3.3 48.89821 78.19761
#> 35 3.4 48.89689 78.22526
#> 36 3.5 48.89609 78.24888
#> 37 3.6 48.89567 78.26908
#> 38 3.7 48.89553 78.28634
#> 39 3.8 48.89559 78.30111
#> 40 3.9 48.89579 78.31374
#> 41 4.0 48.89608 78.32455
#> 42 4.1 48.89644 78.33380
#> 43 4.2 48.89682 78.34172
#> 44 4.3 48.89721 78.34850
#> 45 4.4 48.89761 78.35431
#> 46 4.5 48.89799 78.35928
#> 47 4.6 48.89836 78.36354
#> 48 4.7 48.89870 78.36720
#> 49 4.8 48.89902 78.37033
#> 50 4.9 48.89932 78.37301
#> 51 5.0 48.89959 78.37531
#> 52 5.1 48.89984 78.37729
#> 53 5.2 48.90006 78.37898
#> 54 5.3 48.90026 78.38043
#> 55 5.4 48.90044 78.38168
#> 56 5.5 48.90060 78.38275
#> 57 5.6 48.90074 78.38367
#> 58 5.7 48.90087 78.38446
#> 59 5.8 48.90098 78.38514
#> 60 5.9 48.90108 78.38572
#> 61 6.0 48.90117 78.38622
#> 62 6.1 48.90124 78.38665
#> 63 6.2 48.90131 78.38701
#> 64 6.3 48.90137 78.38733
#> 65 6.4 48.90142 78.38760
#> 66 6.5 48.90147 78.38784
#> 67 6.6 48.90151 78.38804
#> 68 6.7 48.90154 78.38821
#> 69 6.8 48.90157 78.38836
#> 70 6.9 48.90160 78.38849
#> 71 7.0 48.90162 78.38860
#> 72 7.1 48.90164 78.38869
#> 73 7.2 48.90166 78.38877
#> 74 7.3 48.90168 78.38884
#> 75 7.4 48.90169 78.38890
#> 76 7.5 48.90170 78.38895
#> 77 7.6 48.90171 78.38900
#> 78 7.7 48.90172 78.38903
#> 79 7.8 48.90173 78.38907
#> 80 7.9 48.90173 78.38910
#> 81 8.0 48.90174 78.38912
#> 82 8.1 48.90174 78.38914
#> 83 8.2 48.90175 78.38916
#> 84 8.3 48.90175 78.38917
#> 85 8.4 48.90175 78.38919
#> 86 8.5 48.90176 78.38920
#> 87 8.6 48.90176 78.38921
#> 88 8.7 48.90176 78.38922
#> 89 8.8 48.90176 78.38922
#> 90 8.9 48.90176 78.38923
#> 91 9.0 48.90177 78.38924
#> 92 9.1 48.90177 78.38924
#> 93 9.2 48.90177 78.38924
#> 94 9.3 48.90177 78.38925
#> 95 9.4 48.90177 78.38925
#> 96 9.5 48.90177 78.38925
#> 97 9.6 48.90177 78.38926
#> 98 9.7 48.90177 78.38926
#> 99 9.8 48.90177 78.38926
#> 100 9.9 48.90177 78.38926
#> 101 10.0 48.90177 78.38926