This vignette exemplifies the use of the devRate package to fit a development rate model to an empirical dataset (laboratory experiments at air constant temperatures) and to build a simple phenological model.
The dataset for T. solanivora was retrieved from Crespo-Perez et al. 20111, using Web Plot Digitizer2. The dataset is also included in the package in the exTropicalMoth object which is a list of the raw data presented below (a list of three elements for eggs, larva, and pupa), and a list of the NLS fit to the Taylor model (taylor_81; a list of three elements for eggs, larva, and pupa).
rawDevEggs <- matrix(c(10, 0.031, 10, 0.039, 15, 0.047, 15, 0.059, 15.5, 0.066,
13, 0.072, 16, 0.083, 16, 0.100, 17, 0.100, 20, 0.100, 20, 0.143, 25, 0.171,
25, 0.200, 30, 0.200, 30, 0.180, 35, 0.001), ncol = 2, byrow = TRUE)
rawDevLarva <- matrix(c(10, 0.010, 10, 0.014, 10, 0.019, 13, 0.034, 15, 0.024,
15.5, 0.029, 15.5, 0.034, 15.5, 0.039, 17, 0.067, 20, 0.050, 25, 0.076,
25, 0.056, 30, 0.0003, 35, 0.0002), ncol = 2, byrow = TRUE)
rawDevPupa <- matrix(c(10, 0.001, 10, 0.008, 10, 0.012, 13, 0.044, 15, 0.017,
15, 0.044, 15.5, 0.039, 16, 0.034, 15.5, 0.037, 16, 0.051, 17, 0.051,
20, 0.080, 20, 0.092, 25, 0.102, 25, 0.073, 30, 0.005,
35, 0.0002), ncol = 2, byrow = TRUE)
data(exTropicalMoth)
str(exTropicalMoth)
R> List of 2
R> $ raw :List of 3
R> ..$ eggs : num [1:16, 1:2] 10 10 13 15 15 15.5 16 16 17 20 ...
R> ..$ larva: num [1:14, 1:2] 10 10 10 13 15 15.5 15.5 15.5 17 20 ...
R> ..$ pupa : num [1:17, 1:2] 10 10 10 13 15 15 15.5 15.5 16 16 ...
R> $ NLSmodel:List of 3
R> ..$ eggs :List of 6
R> ...
R> ..$ larva:List of 6
R> ...
R> ..$ pupa :List of 6
R> ...Before attempting to fit any model to the empirical data, the devRate function “devRateFind” search the database for previous articles fitting which models to the organism, either by Order, Family, or species. The most used models are presented at the top of the data.frame.
devRateFind(orderSP = "Lepidoptera")
R> equation occu
R> 2 campbell_74 108
R> 19 lactin2_95 32
R> 21 briere1_99 23
R> 10 taylor_81 22
R> 4 logan6_76 19
R> 22 briere2_99 18
R> 1 davidson_44 15
R> 13 harcourtYee_82 13
R> 5 logan10_76 12
R> 16 hilbertLogan_83 10
R> 11 wang_82 9
R> 6 sharpeDeMichele_77 8
R> 18 lactin1_95 8
R> 7 analytis_77 7
R> 14 poly4 5
R> 3 stinner_74 4
R> 12 poly2 4
R> 17 lamb_92 4
R> 23 kontodimas_04 4
R> 24 damos_08 4
R> 9 schoolfieldHigh_81 3
R> 8 schoolfield_81 1
R> 15 rootsq_82 1
R> 20 wangengel_98 1
R> 25 regniere_12 1Here we can see that at the time of thie vignette, campbell_74 model was used 108 times to characterize the relatiohship between development rate and temperature for the Lepidoptera Order, and the taylor_81 model 22 times.
devRateFind(familySP = "Gelechiidae")
R> equation occu
R> 1 campbell_74 12
R> 10 lactin2_95 7
R> 2 logan6_76 4
R> 9 lactin1_95 4
R> 11 briere1_99 4
R> 12 damos_08 4
R> 4 analytis_77 3
R> 5 taylor_81 2
R> 3 logan10_76 1
R> 6 harcourtYee_82 1
R> 7 poly4 1
R> 8 rootsq_82 1If we focus on the Family (Gelechiidae), campbell_74 has been used 12 times, and taylor_81 2 times.
devRateFind(species = "Tecia solanivora")
R> [1] equation occu
R> <0 rows> (or 0-length row.names)Unfortunately, the species Tecia solanivora is not in the package database, so that we have to find a closely-related species.
The “devRateInfo” function provides additional information on these models and on parameter estimations.
devRateInfo(eq = taylor_81)
R> ----------------------------------------
R> Gauss
R> ----------------------------------------
R> Taylor, F. (1981) Ecology and evolution of physiological time in insects.
R> American Naturalist, 1-23. Lamb, RJ. (1992) Developmental rate of
R> Acyrthosiphon pisum (Homoptera: Aphididae) at low temperatures: implications
R> for estimating rate parameters for insects. Environmental Entomology 21(1):
R> 10-19.
R>
R> rT ~ Rm * exp(-1/2 * ((T - Tm)/To)^2)
R> <environment: namespace:devRate>
R>
R> Parameter estimates from the literature (eq$startVal):
R>
R> ... genSp stage param.Rm param.Tm param.To ref
R> 1 ... Geocoris articolor all 0.0550 37.2000 8.8000 Taylor 1981
R> 2 ... Geocoris pallens all 0.0630 37.0000 8.6000 Taylor 1981
R> 3 ... Geocoris punctipes all 0.0440 33.8000 7.7000 Taylor 1981
R> 4 ... Lygus desetinus all 0.0560 32.1000 9.8000 Taylor 1981
R> 5 ... Lygus hesperus all 0.0620 36.2000 12.8000 Taylor 1981
R> 6 ... Acyrthosiphon pisum all 0.1650 26.2000 9.0000 Taylor 1981
R> 7 ... Acyrthosiphon pisum all 0.1510 27.5000 11.0000 Taylor 1981
R> 8 ... Brevicoryne brassicae all 0.0970 26.4000 10.5000 Taylor 1981
R> 9 ... Brevicoryne brassicae all 0.1000 26.6000 8.4000 Taylor 1981
R> 10 ... Hyadaphis pseudobrassicae all 0.1440 26.0000 9.1000 Taylor 1981
R> 11 ... Macrosiphum euphorbiae all 0.1400 30.6000 14.6000 Taylor 1981
R> 12 ... Myzus persicae all 0.1290 24.7000 9.3000 Taylor 1981
R> 13 ... Myzus persicae all 0.1550 26.3000 10.4000 Taylor 1981
R> 14 ... Circulifer tenelus all 0.0520 35.2000 9.2000 Taylor 1981
R> 15 ... Empoasca fabae all 0.0710 30.7000 8.9000 Taylor 1981
R> 16 ... Callosobruchus rhodesianus all 0.0390 31.2000 8.0000 Taylor 1981
R> 17 ... Crioceris asparagi all 0.0710 37.6000 13.3000 Taylor 1981
R> ...
R> 62 ... Acyrthosiphon pisum all 0.1990 26.1000 9.9000 Lamb 1992
R> 63 ... Acyrthosiphon pisum all 0.1900 26.3000 10.3000 Lamb 1992
R> 64 ... Symmetrischema tangolias larva 0.0476 28.5800 10.5000 Sporleder et al. 2016
R> 65 ... Symmetrischema tangolias pupa 0.0990 31.8000 11.1000 Sporleder et al. 2016
R> 66 ... Nephus includens all 0.0441 34.9999 11.0220 Kontodimas et al. 2004
R> 67 ... Nephus bisignatus all 0.0338 32.4999 10.7097 Kontodimas et al. 2004
R> 68 ... Cydia pomonella egg 0.2468 30.9122 9.0574 Aghdam et al. 2009
R> 69 ... Cydia pomonella larva 0.0637 29.6918 8.7660 Aghdam et al. 2009
R> 70 ... Cydia pomonella pupa 0.0836 30.6414 8.8934 Aghdam et al. 2009
R> 71 ... Cydia pomonella all 0.0316 29.9478 8.5990 Aghdam et al. 2009
R> ...
R>
R> Comments:
R> "The curve must be truncated to the right of Tm because of lethal effects of
R> short exposures to high temperatures. The rate at which development rate falls
R> away from Tm is measured by To." Taylor 1981Here we can see that Symmetrischema tangolias study by Sporleder et al. 2016 has used the taylor_81 model.
Information from the database can be plotted using the “devRatePlotInfo” function if we want to have a first look on the taylor_81 model.
devRatePlotInfo (eq = taylor_81, sortBy = "ordersp",
ylim = c(0, 0.20), xlim = c(0, 50))Now that we have a model candidate and starting parameter estimates from a closely-related species, we can start the fitting process with our empirical data (NLS fit). The empirical data can be fitted to any model in the database with the “devRateModel” function, which returns an object of class “nls”.
mEggs <- devRateModel(eq = taylor_81,
temp = exTropicalMoth$raw$eggs[,1],
devRate = exTropicalMoth$raw$eggs[,2],
startValues = list(Rm = 0.05, Tm = 30, To = 5))
mLarva <- devRateModel(eq = taylor_81,
temp = exTropicalMoth$raw$larva[,1],
devRate = exTropicalMoth$raw$larva[,2],
startValues = list(Rm = 0.05, Tm = 25, To = 10))
mPupa <- devRateModel(eq = taylor_81,
temp = exTropicalMoth$raw$pupa[,1],
devRate = exTropicalMoth$raw$pupa[,2],
startValues = list(Rm = 0.1, Tm = 30, To = 10))
The summary of the “devRateModel” can be optained with the “devRatePrint” function.
devRatePrint(myNLS = mLarva,
temp = exTropicalMoth$raw$larva[,1],
devRate = exTropicalMoth$raw$larva[,2])
R> ##################################################
R> ### Parameter estimates and overall model fit
R> ##################################################
R>
R> Formula: rT ~ Rm * exp(-1/2 * ((T - Tm)/To)^2)
R>
R> Parameters:
R> Estimate Std. Error t value Pr(>|t|)
R> Rm 0.068383 0.009473 7.219 1.71e-05 ***
R> Tm 21.408732 0.729062 29.365 8.41e-12 ***
R> To 5.544986 0.865264 6.408 5.02e-05 ***
R> ---
R> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
R>
R> Residual standard error: 0.0133 on 11 degrees of freedom
R>
R> Number of iterations to convergence: 13
R> Achieved convergence tolerance: 6.653e-06
R>
R> ##################################################
R> ### Confidence intervals for parameters
R> ##################################################
R> 2.5 % 97.5 %
R> Rm 0.04981547 0.08694955
R> Tm 19.97979621 22.83766817
R> To 3.84909961 7.24087239
R>
R> ##################################################
R> ### Residuals distribution and independence
R> ##################################################
R> ### Normality of the residual distribution
R>
R> Shapiro-Wilk normality test
R>
R> data: stats::residuals(myNLS)
R> W = 0.97915, p-value = 0.9696
R>
R> ### Regression of the residuals against a lagged version of themselves
R> ### and testing if the slope of the resulting relationship is significantly
R> ### different from 0:
R>
R> Call:
R> stats::lm(formula = stats::residuals(myNLS)[-N] ~ stats::residuals(myNLS)[-1])
R>
R> Residuals:
R> Min 1Q Median 3Q Max
R> -0.021388 -0.011116 0.002149 0.009943 0.020078
R>
R> Coefficients:
R> Estimate Std. Error t value Pr(>|t|)
R> (Intercept) 0.0005691 0.0036255 0.157 0.878
R> stats::residuals(myNLS)[-1] -0.1654787 0.2965227 -0.558 0.588
R>
R> Residual standard error: 0.01307 on 11 degrees of freedom
R> Multiple R-squared: 0.02753, Adjusted R-squared: -0.06087
R> F-statistic: 0.3114 on 1 and 11 DF, p-value: 0.588
R>
R> ##################################################
R> ### Comparing models
R> ##################################################
R> ### Using AIC and BIC
R> Akaike Information Criterion (AIC): -76.6001352945075
R> Bayesian Information Criterion (BIC): -74.0439059760465Empirical data can be plotted against the model using the “devRatePlot” function.
devRatePlot(eq = taylor_81,
nlsDR = mEggs,
temp = exTropicalMoth$raw$eggs[,1],
devRate = exTropicalMoth$raw$eggs[,2],
pch = 16, ylim = c(0, 0.2))
devRatePlot(eq = taylor_81,
nlsDR = mLarva,
temp = exTropicalMoth$raw$larva[,1],
devRate = exTropicalMoth$raw$larva[,2],
pch = 16, ylim = c(0, 0.2))
devRatePlot(eq = taylor_81,
nlsDR = mPupa,
temp = exTropicalMoth$raw$pupa[,1],
devRate = exTropicalMoth$raw$pupa[,2],
pch = 16, ylim = c(0, 0.2))From the “nls” object obtained above, we can build a simple phenology model using temperature time series (e.g., to forecast the organism outbreaks). In this example the temperature dataset is built from a Normal distribution of mean 15 and a standard deviation of 1, with a time step of one day. The development models used are those previously fitted with the Taylor model for the three stages. We assumed that the average time for female adults to lay eggs was of 1 day. We simulated 500 individuals, with a stochasticity in development rate centered on the development rate, with a standard deviation of 0.015 (Normal distribution).
forecastTsolanivora <- devRateIBM(
tempTS = rnorm(n = 100, mean = 15, sd = 1),
timeStepTS = 1,
models = list(mEggs, mLarva, mPupa),
numInd = 500,
stocha = 0.015,
timeLayEggs = 1)
print(forecastTsolanivora)
R> [[1]]
R> g1s1 g1s2 g1s3 g2s1
R> [1,] 14 43 68 83
R> [2,] 14 41 67 83
R> [3,] 14 44 79 96
R> [4,] 13 43 75 92
R> [5,] 15 43 68 84
R> [6,] 13 37 65 80
R> [7,] 14 42 68 85
R> [8,] 14 40 67 84
R> [9,] 15 44 78 94
R> [10,] 14 45 76 91
R> ...
R> [500,] 14 40 70 87
R>
R> [[2]]
R> [[2]][[1]]
R> Nonlinear regression model
R> model: rT ~ Rm * exp(-1/2 * ((T - Tm)/To)^2)
R> data: data.frame(rT = devRate, T = temp)
R> Rm Tm To
R> 0.1934 25.3418 -6.8939
R> residual sum-of-squares: 0.01303
R>
R> Number of iterations to convergence: 13
R> Achieved convergence tolerance: 8.264e-06
R>
R> [[2]][[2]]
R> Nonlinear regression model
R> model: rT ~ Rm * exp(-1/2 * ((T - Tm)/To)^2)
R> data: data.frame(rT = devRate, T = temp)
R> Rm Tm To
R> 0.06838 21.40873 5.54499
R> residual sum-of-squares: 0.001947
R>
R> Number of iterations to convergence: 13
R> Achieved convergence tolerance: 6.653e-06
R>
R> [[2]][[3]]
R> Nonlinear regression model
R> model: rT ~ Rm * exp(-1/2 * ((T - Tm)/To)^2)
R> data: data.frame(rT = devRate, T = temp)
R> Rm Tm To
R> 0.0978 22.0114 4.7515
R> residual sum-of-squares: 0.002394
R>
R> Number of iterations to convergence: 14
R> Achieved convergence tolerance: 3.032e-06
R>
R>
R> [[3]]
R> [1] 13.73130 16.34298 15.14231 17.88713 13.84732 14.35198 16.15267 ...
R> [21] 14.88865 15.13143 14.41983 15.66492 14.32055 16.48751 16.47430 ...
R> [41] 13.86618 15.10432 15.81162 15.54097 13.39031 16.13360 15.25043 ...
R> [61] 14.62629 15.47062 14.06978 15.24159 14.68912 14.90124 16.97001 ...
R> [81] 15.80832 15.33636 16.05875 16.28860 14.14241 13.81280 16.15399 ...Results can be plotted using the “devRateIBMPlot” function. From this simple model we can observe that if eggs of the first generation are layed at time t = 0, adults should be seen at t = [65:85], if the temperature time series correspond to the temperatures experienced by the organism and in the absence of other limiting factors.
devRateIBMPlot(ibm = forecastTsolanivora, typeG = "density")Crespo-Pérez, V., Rebaudo, F., Silvain, J.-F. and Dangles, O. (2011) Modeling invasive species spread in complex landscapes: the case of potato moth in Ecuador. Landscape Ecology, 26, 1447–1461.↩
Rohatgi, A. (2015) WebPlotDigitalizer: HTML5 Based Online Tool to Extract Numerical Data from Plot Images.↩