| Title: | Functions and Data for the Book, a Primer of Ecology with R |
|---|---|
| Description: | Functions are primarily functions for systems of ordinary differential equations, difference equations, and eigenanalysis and projection of demographic matrices; data are for examples. |
| Authors: | M H H Stevens |
| Maintainer: | Hank Stevens <[email protected]> |
| License: | GPL-3 |
| Version: | 1.2.0 |
| Built: | 2025-03-11 03:45:54 UTC |
| Source: | https://github.com/hankstevens/primer |
Functions are primarily functions for systems of ordinary differential equations, difference equations, and eigenanalysis and projection of demographic matrices; data are for examples.
| Package: | primer |
| Type: | Package |
| Version: | 1.0 |
| Date: | 2012-05-16 |
| License: | GPL version 2 or later |
| LazyLoad: | yes |
Hank Stevens <[email protected]>
Stevens, M.H.H. (2009) A Primer of Ecology with R Springer, 2nd printing.
A function for continuous logistic growth with the Allee effect, for use with ode in the deSolve package.
alogistic(t, y, p)alogistic(t, y, p)
t |
Time points for which N wll be returned. |
y |
N, population size |
p |
a vector of logistic growth (r, alpha) and Allee effect parameters (a, tau); a is the threshold population size. |
The user does not put these directly into this function, but rather uses codeode in the deSolve package.
The function is based on the logistic growth equation
with being the threshold population size.
Hank Stevens
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
clogistic,dlogistic, thetalogistic, levins
library(deSolve) p <- c(r=1,alpha=.01, a=10, tau=.1) time <- 0:10 initialN <- 11 out <- ode(y=initialN, times=time, func=alogistic, parms=p) plot(time, out[,-1], type='l')library(deSolve) p <- c(r=1,alpha=.01, a=10, tau=.1) time <- 0:10 initialN <- 11 out <- ode(y=initialN, times=time, func=alogistic, parms=p) plot(time, out[,-1], type='l')
A function to calculate the several properties of randomly constructed bipartite networks. This function installation of packages bipartite, igraph, and rARPACK.
bip_stability( Spp.a = 10:50, Spp.p = 5:25, C = c(0.05, 0.3), IS = NULL, quant = FALSE, ndd = 1, reps = 10 )bip_stability( Spp.a = 10:50, Spp.p = 5:25, C = c(0.05, 0.3), IS = NULL, quant = FALSE, ndd = 1, reps = 10 )
Spp.a |
numeric vector from which to draw a random number of animal species |
Spp.p |
numeric vector from which to draw a random number of animal species |
C |
numeric two element vector for minimum and maximum possible connectances. |
IS |
mean interaction strength. If NULL, it is calculated as the inverse of the matrix dimension, so that the expected sum of interaction strengths equals one. |
quant |
Logical indicating whether the estimate of modularity should be based on the binary adjacency matrix (FALSE, default), or a quantitative matrix (TRUE). |
ndd |
scalar (1 or 2) determining whether the intraspecific negative density dependence is the same in both networks (1), or is scaled to the interaction sizes in each matrix (2). |
reps |
number of random matrices. |
The interaction strengths are exponentially distributed, with mean of IS (rate = 1/IS). Nestedness is WNODF (Almeida-Neto et al. 2011). It can calculate modularity in two ways, using either the binary adjacency matrix (ones and zeroes, Clauset, Newman and Moore 2004) or a quantitative matrix (Beckett 2016); the latter takes much moore time to calculate, so it uses the binary matrix by default. It also tallies diversity as the total number of species and the connectance as the total number of links divided by the number of possible links.
To calculate resilience, the P by A bipartite matrix is expanded to a (P+A) x (P+A) symmetric matrix so the upper triangular portion are the effects of the animals on the plants, and the lower triangular portion is the effects of the plants on the animals.
The diagonal represents intraspecific density dependence; it is filled with the negative of the largest summed row of interaction strengths in the symmetric matrix. The symmatric matrix is interpreted as a Jacobian matrix. Therefore, resilience is the negative value of the real part of the dominant eigen value of the expanded symmetric matrix.
Returns of list of a matrix with columns for species richness, connectance, nestedness, modularity, and resilience.
Hank Stevens <[email protected]>
M. Almeida-Neto and W. Ulrich. A straightforward computational approach for measuring nestedness using quantitative matrices. Environmental Modelling and Software, 26(2):173–178, 2011.
S. J. Beckett. Improved community detection in weighted bipartite networks. Royal Society Open Science, 3:140536, 2016.
A. Clauset, M. E. J. Newman, and M. Cristopher. Finding community structrue in very large networks. Physical Review E, 70:066111, 2004.
# b <- bip_stability(Spp.a = 16:60, Spp.p = 8:30, C=c(.05, 0.5), reps=10, quant=FALSE) # pairs(b)# b <- bip_stability(Spp.a = 16:60, Spp.p = 8:30, C=c(.05, 0.5), reps=10, quant=FALSE) # pairs(b)
Functional group abundances (herbaceous annual, herbaceous perennial, woody) from one of the fields in the Buell-Small long term succession study (http://www.ecostudies.org/bss/). Data are based on visual estimates of percent cover, using annual means of each species, which are then summed for each functional group.
A data frame with 147 observations on the following 3 variables.
a numeric vector; indicates the age of succession since abandonment from agriculture.
plant functional type; a factor with levels
Annual, Perennial, Woody
a numeric vector
Data graciously provided by Scott Meiners (Eastern Illinois University, http://www.ecostudies.org/bss/).
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
data(BSsucc) #lattice::xyplot(value~ AGE, groups=variable, data=BSsucc, #type='smooth', span=.3, ylab="Percent Cover", #xlab="Age Since Abandonment (y)", #auto.key=list(columns=3, space="top", lines=TRUE, #points=FALSE))data(BSsucc) #lattice::xyplot(value~ AGE, groups=variable, data=BSsucc, #type='smooth', span=.3, ylab="Percent Cover", #xlab="Age Since Abandonment (y)", #auto.key=list(columns=3, space="top", lines=TRUE, #points=FALSE))
Numbers of species which were observed in 1–19 sites.
The format is: num [1:19] 32 16 10 9 8 7 8 6 4 5 ...
Data are approximations of histogram values for a single year of data in:
Collins, S.L. and Glenn, S.M. (1991) Importance of spatial and temporal dynamics in species regional abundance and distribution. Ecology, 72, 654–664.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
data(CandG) barplot(CandG, names=1:19)data(CandG) barplot(CandG, names=1:19)
This function allows you to express your love of cats.
cat_function(love = TRUE)cat_function(love = TRUE)
love |
Do you love cats? Defaults to TRUE. |
Hilary Parker
cat_function()cat_function()
Simulates a fluctuating environment over time, and two species' responses to the environment, after Chesson (1994).
chesson( alpha = c(1.1 * 1e-05, 1e-05), d = 0.1, years = 10, N0 = c(1000, 1e+05), w = c(0.6, 1), env.var = 1, specialization = 1, spread = 0.67, type = 1 )chesson( alpha = c(1.1 * 1e-05, 1e-05), d = 0.1, years = 10, N0 = c(1000, 1e+05), w = c(0.6, 1), env.var = 1, specialization = 1, spread = 0.67, type = 1 )
alpha |
a vector of length 2; the negative effects of all individuals (of both species) on each population – typically different among species. |
d |
disturbance rate; the proportion of all individuals killed at each time step. |
years |
numbers of time steps |
N0 |
vector of length 2; initial abundances. |
w |
vector of length 2; average fitnesses for each species. |
env.var |
degree of environmental variability. |
specialization |
determines the narrowness of each species fitness response. |
spread |
determines how far apart the peak fitness responses are. |
type |
determines the form of C, the negative effect of competition. |
The argument type controls the value of , the effect of
competition on reproduction, where the annual finite rate of increase is
. type = 1 causes , that is, a species-specific fixed fraction
of juveniles that depends on each species response to competition. This is
illustrated in a for-loop in Stevens (2009, Ch. 9, Storage Effect,
Simulating Dynamics). Any other value for type results in the same
negative effect on both species that depends on the number of juveniles and
the disturbance rate.
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variables, y. |
Hank Stevens ([email protected])
Chesson, P.L. (1994) Multispecies competition in variable environments. Theoretical Population Biology, 45, 227–276.
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
out <- chesson(years=50) layout(matrix(1:4, nc=2)) matplot(out[["time"]], out[["Ns"]], type='l', lty=c(1:3), xlab="Time", ylab="N", log="y") plot(out[["time"]][-1], out[["env"]], type='l', xlab="Time", ylab="Environment") matplot(out[["env"]], out[["Es"]], xlab="Environment", ylab="Density-independent reproduction") matplot(out[["env"]], out[["Rs"]], xlab="Environment", ylab="Annual growth rate")out <- chesson(years=50) layout(matrix(1:4, nc=2)) matplot(out[["time"]], out[["Ns"]], type='l', lty=c(1:3), xlab="Time", ylab="N", log="y") plot(out[["time"]][-1], out[["env"]], type='l', xlab="Time", ylab="Environment") matplot(out[["env"]], out[["Es"]], xlab="Environment", ylab="Density-independent reproduction") matplot(out[["env"]], out[["Rs"]], xlab="Environment", ylab="Annual growth rate")
A function for continuous logistic growth, for use with ode in the
deSolve package.
clogistic(times, y, parms)clogistic(times, y, parms)
times |
times points that will return N |
y |
N |
parms |
a vector of logistic growth parameters; the first must be r, and the second must be alpha (1/K). |
The user does not put these directly into this function, but rather uses
ode in the deSolve package.
The function implements the logistic growth equation
or equivalently
Returns of list of one component (required by ode).
Hank Stevens <[email protected]>
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
dlogistic, thetalogistic,
levins, lvcomp2, lvcompg
library(deSolve) p <- c(r=1, alpha=.01) time <- 1:10 initialN <- 10 out <- ode(y=initialN, times=time, func=clogistic, parms=p) plot(time, out[,-1], type='l')library(deSolve) p <- c(r=1, alpha=.01) time <- 1:10 initialN <- 10 out <- ode(y=initialN, times=time, func=clogistic, parms=p) plot(time, out[,-1], type='l')
Data set from an experiment testing whether trophic heterogeneity influences long term population sizes.
A data frame with 144 observations on the following 5 variables.
a factor with levels high
low
a numeric vector
a numeric vector
a factor with levels a b
c d
an ordered factor with levels
a.low < d.low < c.low < b.low < c.high <
a.high < d.high < b.high
Stevens, M. H. H. and Steiner, C. E. (2006) Effects of predation and nutrient enrichment on a food web with edible and inedible prey. Freshwater Biology, 51, 666–671.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
data(ClostExp) #lattice::xyplot(log10(No.per.ml) ~ Day|Nutrients, data=ClostExp, groups=rep, type='l')data(ClostExp) #lattice::xyplot(log10(No.per.ml) ~ Day|Nutrients, data=ClostExp, groups=rep, type='l')
This model implements a Levins-type metapopulation model for two species,
after Hastings (1980). For use with ode in the deSolve
package.
compcol(t, y, params)compcol(t, y, params)
t |
Argument for time |
y |
A vector for population 1 and 2 |
params |
Vector or list of parameters |
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variables, y. |
Hank Stevens <[email protected]>
Hastings, A. (1980) Disturbance, coexistence, history, and competition for space. Theoretical Population Biology, 18, 363–373.
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
library(deSolve) pars <- c(c1 = .3, c2 = 1, m1 = .1, m2 = .1) pops <- c(.1,.1) out <- ode(y=pops, t=0:20, fun=compcol, parms = pars) matplot(out[,1], out[,-1], type='l')library(deSolve) pars <- c(c1 = .3, c2 = 1, m1 = .1, m2 = .1) pops <- c(.1,.1) out <- ode(y=pops, t=0:20, fun=compcol, parms = pars) matplot(out[,1], out[,-1], type='l')
Multi-species competition colonization model, with habitat destruction,
after Nee and May (1992). For use with ode in the deSolve
package.
compcolM(t, y, params)compcolM(t, y, params)
t |
Argument for each time point |
y |
A vector for the populations |
params |
Vector or list of parameters |
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variables, y. |
Hank Stevens <[email protected]>
Nee, S. and May, R.M. (1992) Dynamics of metapopulations: habitat destruction and competitive coexistence. Journal of Animal Ecology, 61, 37–40.
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
library(deSolve) S <- 10 ci <- 2^seq(-5, 5, length=S) m <- rep(.1, S) params <- list(ci=ci, m=m, S=S, D=0) init.N <- rep(0.01, S); t=seq(1, 200, .1) cc.out <- ode(init.N, t, compcolM, params) matplot(t, cc.out[, -1], type="l", ylab="Proportion of Habitat", xlab="Years")library(deSolve) S <- 10 ci <- 2^seq(-5, 5, length=S) m <- rep(.1, S) params <- list(ci=ci, m=m, S=S, D=0) init.N <- rep(0.01, S); t=seq(1, 200, .1) cc.out <- ode(init.N, t, compcolM, params) matplot(t, cc.out[, -1], type="l", ylab="Proportion of Habitat", xlab="Years")
A data set containing size, survival, and flowering in smooth coneflower
data(coneflower)data(coneflower)
A data frame with 136 rows and 5 variables:
natural log of total leaf area in the first year
natural log of total leaf area in the second year (Ap = A prime)
indicator of survival from year one to year 2 (0,1)
indicator of flowering in year 2 (0,1)
Data provided graciously by Rachel Collins (Roanoke College).
data(coneflower) ggplot2::qplot(logA, logAp, data=coneflower)data(coneflower) ggplot2::qplot(logA, logAp, data=coneflower)
A data set containing recruit size and survival.
data(coneflowerrecruits)data(coneflowerrecruits)
A data frame with 36 rows and 3 variables:
total leaf area in the first year
indicator of survival from year one to year 2 (0,1)
total leaf area in the third year
A data set containing size and seed set of individual coneflowers.
data(coneflowerseeds)data(coneflowerseeds)
A data frame with 136 rows and 5 variables:
natural log of total leaf area in the first year
number of seeds produced in the first year
data(coneflowerseeds) ggplot2::qplot(logAs, seeds, data=coneflowerseeds)data(coneflowerseeds) ggplot2::qplot(logAs, seeds, data=coneflowerseeds)
Single species discrete logistic growth – a difference equation.
A function for continuous logistic growth, for use with ode in the
deSolve package, using method = 'euler' and integer time steps.
dlogistic(t, y, p)dlogistic(t, y, p)
t |
times points that will return N |
y |
N |
p |
a vector of labeled logistic growth parameters; the first must be labeled rd, and the second must be labeled alpha (the value of alpha is 1/K). |
Of the form,
Returns of list of one component (required by ode).
Hank Stevens ([email protected])
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
library(deSolve) # MUST use the 'euler' integration method with integer time steps p <- c(rd=1, alpha=.01) time <- 0:10 initialN <- 10 out <- ode(y=initialN, times=time, func=dlogistic, parms=p, method='euler') plot(time, out[,-1], type='l')library(deSolve) # MUST use the 'euler' integration method with integer time steps p <- c(rd=1, alpha=.01) time <- 0:10 initialN <- 10 out <- ode(y=initialN, times=time, func=dlogistic, parms=p, method='euler') plot(time, out[,-1], type='l')
A function to simulate discrete 2 species Lotka-Volterra competition.
dlvcomp2(N, alpha, rd = c(1, 1))dlvcomp2(N, alpha, rd = c(1, 1))
N |
a vector of length two, containing N[t] for both species. |
alpha |
a 2 x 2 matrix of the magnitudes of the per capita (negative) effects (i.e., positive value results in a negative effect). |
rd |
a vector of length 2 containing the discrete growth increments for 2 species. |
Returns a vector of length 2, containing N[t+1] for both species.
Hank Stevens ([email protected])
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
dlogistic, lvcomp2,
lvcomp3, lvcompg
alphs <- matrix(c( .01, .005, .008, .01), ncol=2, byrow=TRUE) t <- 20 N <- matrix(NA, nrow=t+1, ncol=2) N[1,] <- c(10,10) for(i in 1:t) N[i+1,] <- dlvcomp2(N[i,], alphs) matplot(0:t, N, type='l', col=1, ylim=c(0,110)) abline(h=1/alphs[1,1], lty=3) text(0, 1/alphs[1,1], "K", adj=c(0,0)) legend("right", c(expression("Sp.1 "*(alpha[21]==0.008)), expression("Sp.2 "*(alpha[12]==0.005))), lty=1:2, bty='n')alphs <- matrix(c( .01, .005, .008, .01), ncol=2, byrow=TRUE) t <- 20 N <- matrix(NA, nrow=t+1, ncol=2) N[1,] <- c(10,10) for(i in 1:t) N[i+1,] <- dlvcomp2(N[i,], alphs) matplot(0:t, N, type='l', col=1, ylim=c(0,110)) abline(h=1/alphs[1,1], lty=3) text(0, 1/alphs[1,1], "K", adj=c(0,0)) legend("right", c(expression("Sp.1 "*(alpha[21]==0.008)), expression("Sp.2 "*(alpha[12]==0.005))), lty=1:2, bty='n')
A function for the propagule rain or mainland-island metapopulation
dynamics, for use with ode in the deSolve package.
gotelli(t, y, parms)gotelli(t, y, parms)
t |
Argument for time |
y |
A scalar for the population variable |
parms |
Vector or list of parameters |
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variable (a scalar for the proportion of sites occupied). |
Hank Stevens <[email protected]>
Gotelli, N.J. (1991) Metapopulation models: the rescue effect, the propagule rain, and the core-satellite hypothesis. The American Naturalist, 138, 768–776.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
gotelli, hanski, lande,
MetaSim, clogistic
## The function is currently defined as function (t, y, parms) { p <- y[1] with(as.list(parms), { dp <- ce * (1 - p) - e * p return(list(dp)) }) } library(deSolve) p <- c(ce=.1, e=.01) time <- 1:10 initialN <- .3 out <- ode(y=initialN, times=time, func=gotelli, parms=p) plot(time, out[,-1], type='l')## The function is currently defined as function (t, y, parms) { p <- y[1] with(as.list(parms), { dp <- ce * (1 - p) - e * p return(list(dp)) }) } library(deSolve) p <- c(ce=.1, e=.01) time <- 1:10 initialN <- .3 out <- ode(y=initialN, times=time, func=gotelli, parms=p) plot(time, out[,-1], type='l')
A function for the core-satellite metaapopulation dynamics, for use with
ode in the deSolve package.
hanski(t, y, parms)hanski(t, y, parms)
t |
Argument for time |
y |
A scalar for the population variable |
parms |
Vector or list of parameters |
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variable (a scalar for the proportion of sites occupied). |
Hank Stevens <[email protected]>
Hanski, I. (1982) Dynamics of regional distribution: the core and satellite species hypothesis. Oikos, 38, 210–221.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
gotelli, hanski,lande,
MetaSim, clogistic
prms <- c(ci<- 0.15, e=0.05) out <- ode(y=.2, times=1:100, func=hanski, parms=prms ) matplot(out[,1], out[,2], type='l', ylab="p", xlab="time")prms <- c(ci<- 0.15, e=0.05) out <- ode(y=.2, times=1:100, func=hanski, parms=prms ) matplot(out[,1], out[,2], type='l', ylab="p", xlab="time")
A Lotka-Volterra model of intraguild predation, after Holt and Polis (1997).
For use with ode in the deSolve package.
igp(t, y, params)igp(t, y, params)
t |
the time point for a realization of the integration. |
y |
the vector of populations, at each time t. |
params |
a vector or list containing the necessary parameters. |
Returns a list of length one which is the vector of the rates of
increase (required by ode).
Hank Stevens <[email protected]>
Holt, R.D. and Polis, G.A. (1997) A theoretical framework for intraguild predation. The American Naturalist, 149, 745–764.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
library(deSolve) params <- c(bpb= 0.032, abp=10^-8, bpn=10^-5, anp=10^-4, mp=1, bnb=0.04, abn=10^-8, mn=1, r=1, abb=10^-9.5) t=seq(0, 60, by=.1) N.init <- c(B = 10^9, N = 10^4, P = 10^3) igp.out <- ode(N.init, t, igp, params) matplot(t, log10(igp.out[,-1]+1), type="l", ylab="log(Abundance)") legend('right', c("B", "N", "P"), lty=1:3, col=1:3, lwd=2, bty="n")library(deSolve) params <- c(bpb= 0.032, abp=10^-8, bpn=10^-5, anp=10^-4, mp=1, bnb=0.04, abn=10^-8, mn=1, r=1, abb=10^-9.5) t=seq(0, 60, by=.1) N.init <- c(B = 10^9, N = 10^4, P = 10^3) igp.out <- ode(N.init, t, igp, params) matplot(t, log10(igp.out[,-1]+1), type="l", ylab="log(Abundance)") legend('right', c("B", "N", "P"), lty=1:3, col=1:3, lwd=2, bty="n")
A function for the metaapopulation dynamics, for use with ode in the
deSolve package.
lande(t, y, parms)lande(t, y, parms)
t |
Argument for time |
y |
A scalar for the population variable |
parms |
Vector or list of parameters |
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variable (a scalar for the proportion of sites occupied). |
Hank Stevens <[email protected]>
P. Kareiva and Wennergren, U. (1995) Connecting landscape patterns to ecosystem and population processes. Nature, 373, 299–302.
Lande, R. (1987) Extinction thresholds in demographic models of territorial populations. The American Naturalist, 130, 624–635.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
gotelli, hanski,levins,
MetaSim, clogistic
## The function is currently defined as function (t, y, parms) { p <- y[1] with(as.list(parms), { dp <- ci * p * (1 - D - p) - e * p return(list(dp)) }) } library(deSolve) p <- c(ci=.1, e=.01, D=.5) time <- 1:10 initialN <- .3 out <- ode(y=initialN, times=time, func=lande, parms=p) plot(time, out[,-1], type='l')## The function is currently defined as function (t, y, parms) { p <- y[1] with(as.list(parms), { dp <- ci * p * (1 - D - p) - e * p return(list(dp)) }) } library(deSolve) p <- c(ci=.1, e=.01, D=.5) time <- 1:10 initialN <- .3 out <- ode(y=initialN, times=time, func=lande, parms=p) plot(time, out[,-1], type='l')
A function for the classic metaapopulation dynamics, for use with ode
in the deSolve package.
levins(t, y, parms)levins(t, y, parms)
t |
Argument for time |
y |
A scalar for the population variable |
parms |
Vector or list of parameters |
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variable (a scalar for the proportion of sites occupied). |
Hank Stevens <[email protected]>
Levins, R. (1969) Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America, 15, 237–240.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
gotelli, hanski,lande,
MetaSim, clogistic
## The function is currently defined as function (t, y, parms) { p <- y[1] with(as.list(parms), { dp <- ci * p * (1 - p) - e * p return(list(dp)) }) } library(deSolve) p <- c(ci=.1, e=.01) time <- 1:10 initialN <- .3 out <- ode(y=initialN, times=time, func=levins, parms=p) plot(time, out[,-1], type='l')## The function is currently defined as function (t, y, parms) { p <- y[1] with(as.list(parms), { dp <- ci * p * (1 - p) - e * p return(list(dp)) }) } library(deSolve) p <- c(ci=.1, e=.01) time <- 1:10 initialN <- .3 out <- ode(y=initialN, times=time, func=levins, parms=p) plot(time, out[,-1], type='l')
System of ordinary differential equations for two species Lotka-Volterra
competition. For use with ode in the deSolve package.
lvcomp2(t, n, parms)lvcomp2(t, n, parms)
t |
Times points that will return N. |
n |
a vector of length two for the population sizes at time = t. |
parms |
vector or list of model parameters (see details below). |
The parameters include r1, r2, a11, a12, a21, a22 with the usual
meanings. Here the a's are the per capita effects which determine K
(a11 = 1/K).
Returns a list of length one which is the rate of increase (required
by ode).
Hank Stevens <[email protected]>
Lotka, A.J. (1956) Elements of Mathematical Biology. Dover Publications, Inc.
Stevens. M.H.H. (2009) A Primer of Theoretical Population Ecology with R. Use R! Series. Springer.
## The function is currently defined as function (t, n, parms) { with(as.list(parms), { dn1dt <- r1 * n[1] * (1 - a11 * n[1] - a12 * n[2]) dn2dt <- r2 * n[2] * (1 - a22 * n[2] - a21 * n[1]) list(c(dn1dt, dn2dt)) }) } library(deSolve) parms <- c(r1 = 1, r2 = 0.1, a11 = 0.2, a21 = 0.1, a22 = 0.02, a12 = 0.01) initialN <- c(1, 1) out <- ode(y = initialN, times = 1:100, func = lvcomp2, parms = parms) matplot(out[, 1], out[, -1], type = "l")## The function is currently defined as function (t, n, parms) { with(as.list(parms), { dn1dt <- r1 * n[1] * (1 - a11 * n[1] - a12 * n[2]) dn2dt <- r2 * n[2] * (1 - a22 * n[2] - a21 * n[1]) list(c(dn1dt, dn2dt)) }) } library(deSolve) parms <- c(r1 = 1, r2 = 0.1, a11 = 0.2, a21 = 0.1, a22 = 0.02, a12 = 0.01) initialN <- c(1, 1) out <- ode(y = initialN, times = 1:100, func = lvcomp2, parms = parms) matplot(out[, 1], out[, -1], type = "l")
System of ordinary differential equations for three species Lotka-Volterra
competition. For use with ode in the deSolve package.
lvcomp3(t, n, parms)lvcomp3(t, n, parms)
t |
the time for each integration. |
n |
a vector of length three for the population sizes at time = t. |
parms |
vector or list of model parameters (see details below). |
The parameters include r, a with the usual meanings. Here the
a's are the per capita effects which determine K (a11 = 1/K1).
Returns a list of length one which is the rate of increase (required
by ode).
Hank Stevens <[email protected]>
Lotka, A.J. (1956) Elements of Mathematical Biology. Dover Publications, Inc.
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
## The function is currently defined as function (t, n, parms) { with(as.list(parms), { dn1dt <- r1 * n[1] * (1 - a11 * n[1] - a12 * n[2] - a13 * n[3]) dn2dt <- r2 * n[2] * (1 - a22 * n[2] - a21 * n[1] - a23 * n[3]) dn3dt <- r3 * n[3] * (1 - a33 * n[3] - a31 * n[1] - a32 * n[2]) list(c(dn1dt, dn2dt, dn3dt)) }) } library(deSolve) parms <- c(r1 = 0.1, r2 = 0.2, r3 = 0.3, a11 = 0.1, a12 = 0.01, a13 = 0.01, a21 = 0.01, a22 = 0.15, a23 = 0.01, a31 = 0.01, a32 = 0.01, a33 = 0.2) initialN <- c(1, 1, 1) out <- ode(y = initialN, times = 1:100, func = lvcomp3, parms = parms) matplot(out[, 1], out[, -1], type = "l")## The function is currently defined as function (t, n, parms) { with(as.list(parms), { dn1dt <- r1 * n[1] * (1 - a11 * n[1] - a12 * n[2] - a13 * n[3]) dn2dt <- r2 * n[2] * (1 - a22 * n[2] - a21 * n[1] - a23 * n[3]) dn3dt <- r3 * n[3] * (1 - a33 * n[3] - a31 * n[1] - a32 * n[2]) list(c(dn1dt, dn2dt, dn3dt)) }) } library(deSolve) parms <- c(r1 = 0.1, r2 = 0.2, r3 = 0.3, a11 = 0.1, a12 = 0.01, a13 = 0.01, a21 = 0.01, a22 = 0.15, a23 = 0.01, a31 = 0.01, a32 = 0.01, a33 = 0.2) initialN <- c(1, 1, 1) out <- ode(y = initialN, times = 1:100, func = lvcomp3, parms = parms) matplot(out[, 1], out[, -1], type = "l")
A general Lotka-Volterra competition model, for any number of species. For
use with ode in the deSolve package. This function uses a
vector and matrix within the list of parameters.
lvcompg(t, n, parms)lvcompg(t, n, parms)
t |
the time point for a realization of the integration. |
n |
the vector of populations, at each time t. |
parms |
a LIST containing a vector of growth rates (r), and a matrix of interaction coefficients (a). |
Returns a list of length one which is the vector of the rates of
increase (required by ode).
Hank Stevens <[email protected]>
Lotka, A.J. (1956) Elements of Mathematical Biology. Dover Publications, Inc.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
lvcomp3, clogistic, igp,
scheffer
## Specify the number of species S <- 10 alpha <- .01 r <- runif(S)*2 a <- matrix(rnorm(S^2, m=alpha, sd=alpha/10), nrow=S, ncol=S) parms <- list(r,a) t=seq(0,40, by=.1) N0 <- runif(S)/(S*alpha) library(deSolve) lvout <- ode(N0, t, lvcompg, parms) matplot(t, lvout[,-1], type="l", ylab="N", log='y')## Specify the number of species S <- 10 alpha <- .01 r <- runif(S)*2 a <- matrix(rnorm(S^2, m=alpha, sd=alpha/10), nrow=S, ncol=S) parms <- list(r,a) t=seq(0,40, by=.1) N0 <- runif(S)/(S*alpha) library(deSolve) lvout <- ode(N0, t, lvcompg, parms) matplot(t, lvout[,-1], type="l", ylab="N", log='y')
A pedagogical tool for learning about isoclines and stable equilibria.
LVCompGames(Alpha = NULL, r1 = 0.1, r2 = 0.1, num = 20, time = 10, step = 1)LVCompGames(Alpha = NULL, r1 = 0.1, r2 = 0.1, num = 20, time = 10, step = 1)
Alpha |
a 2 x 2 matrix of coefficients; if NULL, then a sensible random matrix will be generated - fun for games. |
r1 |
the intrinsic rate of increase for species 1 (isocline – a solid line). |
r2 |
the intrinsic rate of increase for species 2 (isocline – a dashed line). |
num |
the desired number of random starting points for trajectories. |
time |
the number of time steps for each trajectory. |
step |
the number of time steps for which the integration is estimated (has no effect on accuracy, merely the smoothness of the line). |
First generates a plot of isoclines and initial points; the user is then prompted to "Hit <return>" in the console to see the trajectories. These will indicate the rate and path the trajectories, demonstrating, among other things, whether the equilibrium is stable.
Hank Stevens <[email protected]>
Lotka, A.J. (1956) Elements of Mathematical Biology. Dover Publications, Inc.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
## LVCompGames() # Hit return in the console to see the trajectories.## LVCompGames() # Hit return in the console to see the trajectories.
Originally focused on creating a community of core-satellite species, this function allows simulation of several metapopulation models, where colonization and extinction rates are stochastic draws from uniform distributions, with specified means and ranges.
MetaSim( Time = 50, NSims = 1, method = "hanski", ci = 0.25, e = 0.25, phi = 0.75, p0 = 0.5, D = 0.5 )MetaSim( Time = 50, NSims = 1, method = "hanski", ci = 0.25, e = 0.25, phi = 0.75, p0 = 0.5, D = 0.5 )
Time |
A scalar for the number of time steps over which to simulate each population. |
NSims |
A scalar for the number of simulations, which is analogous to the number of species in the community. |
method |
A character string, in quotes, specifying which metapopulation model to use: "hanksi", "gotelli", "lande", "levins". See "See Also" below. |
ci |
Scalar for mean colonization rate. |
e |
Scalar for mean extinction rate. |
phi |
A scalar for the relative variability in rates. See Details. |
p0 |
Initial proportion of sites occupied for each species. |
D |
Parameter for habitat destruction; applies to only the "lande" model. |
phi is one half of the relative range of each rate (colonization and
extinction). For each time step, each rate is drawn from a uniform
distribution, Unif(rate-phi, rate+phi). Thus, the range is 2*phi, and
center on the specified mean (ci or e).
Function returns a list with these components.
method |
The method used (default is "hanski"). |
time |
The integer sequence of
times, from 0 to the value of the argument |
Ns |
|
Parameters |
A named vector of the parameters used for the simulations. |
Hank Stevens <[email protected]>
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
out <- MetaSim(NSims=2) pops <- out$Ns matplot(out$t, pops, type='l') title(sub=paste(out$method, "model"))out <- MetaSim(NSims=2) pops <- out$Ns matplot(out$t, pops, type='l') title(sub=paste(out$method, "model"))
Data set of moth species richness from Ohio and Indiana, USA.
A data frame with 21 observations on the following 6 variables.
a factor with levels NCT WAP
for the two collection regions.
a factor with levels associated with county names.
a numeric vector for the area of each forest fragment
a numeric vector, the number of moth species.
a numeric vector; latitude
a numeric vector; longitude
Data proided graciously by Keith Summerville and Thomas Crist; see "A Primer of Ecology with R" for specific references.
K. S. Summerville and T. O. Crist. 2003. Determinants of lepidopteran community composition and species diversity in eastern deciduous forests: roles of season, ecoregion and patch size. Oikos, 100:134-148.
K. S. Summerville and T. O. Crist. 2004. Contrasting effects of habitat quantity and quality on moth communities in fragmented landscapes. Ecography, 27:3-12.
data(moths) plot(spp ~ area, data=moths)data(moths) plot(spp ~ area, data=moths)
Generates 1/f noise with a specified power or amplitude.
one_over_f(gamma = 1, N = 200)one_over_f(gamma = 1, N = 200)
gamma |
spectral power, numeric, where 0 generates a white noise time series, 2 generates reddened noise. Defaults to 1 (pink). |
N |
length of the time series. |
Hank Stevens
J. M. Halley. Ecology, evolution and 1/f-noise. Trends in Ecology & Evolution, 11:33-37, 1996. O. L. Petchey, A. Gonzalez, and H. B. Wilson. Effects on population persistence: the interaction between environmental noise colour, intra-specific competition and space. Proceedings of the Royal Society of London Series B, 264:1841-1847, 1997. J. E. Cohen, C. M. Newman, A. E. Cohen, O. L. Petchey, and A. Gonzalez. Spectral mimicry: a method of synthesizing matching time series with different Fourier spectra. Circuits, Systems and Signal Processing, 18:431-442, 1999.
spec_mimic to rearrange one vector, X, to mimic the spectrum of another vector, Y; plot_f to plot the time series and the spectrogram of the series, and estimate the slope.
set.seed(1) time.series <- one_over_f(gamma=2, N=50) plot(1:50, time.series, type='l', main="Reddened noise") time.series <- one_over_f(gamma=0, N=50) plot(1:50, time.series, type='l', main="White noise") one_over_f()set.seed(1) time.series <- one_over_f(gamma=2, N=50) plot(1:50, time.series, type='l', main="Reddened noise") time.series <- one_over_f(gamma=0, N=50) plot(1:50, time.series, type='l', main="White noise") one_over_f()
Used primarily to repeat simulations and analyses of Pimm and Lawton (1977), given a Jacobian matrix. Analyses include eigenanalysis, but also measuring average interaction strength (May 1972), average intraspecific negative density dependence, and the strength of the omnivory interaction, if present.
pimmlawton(mat, N = 1, omni.i = NA, omni.j = NA, omega = NULL)pimmlawton(mat, N = 1, omni.i = NA, omni.j = NA, omega = NULL)
mat |
a numerical matrix; the "maximum" Jacobian matrix. See details below. |
N |
a scalar for the number of randomizations |
omni.i |
if omnivory is present, the row/col index for the prey. |
omni.j |
if omnivory is present, the row/col index for the predator. |
omega |
if not NULL, a scalar $0<x<1$ indicating the relative weight of omniovory (after McCann et al. (1998)) |
This function simulates a constrained randomization of a Jacobian food web
matrix. The matrix it uses mat is of a special form, which assumes
that all non-zero values are drawn from a uniform distribution between zero
and a value of some specified magnitude, either positive or negative.
Returns a data frame, where each row corresponds to a single random Jacobian matrix, with the following columns.
DomEig |
the real part of the dominant eigenvalue |
Im |
the imaginary part of the dominant eigenvalue |
IntraDD |
average magnitude, over all species, of the intraspecific negative density dependence; the square root of the sum of the squared diagonal elements of the random Jacobian matrix |
I |
average interaction strength (after May 1972); the square root of, sum of the squared off diagonal elements divided by the number of off diagonal elements. |
I.omni |
average interaction strength for the omnivory interaction; the square root of, sum of the squared omnivory elements effect of predator on prey and prey on predator divided by two. |
Hank Stevens <[email protected]>
May, R.M. (1973) Stability and Complexity in Model Ecosystems, volume 6 of Monographs in Population Biology. Princeton University Press.
McCann, K., Hastings, A. and Huxel, G.R. (1998) Weak trophic interactions and the balance of nature. Nature, 395, 794–798.
Pimm, S.L and Lawton, J.H. (1997) Number of trophic levels in ecological communities. Nature, 268, 329–331.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer. 2009.
### A relevant style of matrix for the first food chain in Pimm and Lawton (1977). ### Note each non-zero element is the appropriate sign, and the maximum magnitude ### specified by Pimm and Lawton (1977). Aq = matrix(c( -1, -10, 0, 0, 0.1, 0, -10, 0, 0, 0.1, 0, -10, 0, 0, 0.1, 0), nrow=4, byrow=TRUE) pimmlawton(Aq, N=1) out <- pimmlawton(Aq, N=2000) out <- subset(out, -1/DomEig <150) hist(-1/out$DomEig, main="Frequencies of Return Time")### A relevant style of matrix for the first food chain in Pimm and Lawton (1977). ### Note each non-zero element is the appropriate sign, and the maximum magnitude ### specified by Pimm and Lawton (1977). Aq = matrix(c( -1, -10, 0, 0, 0.1, 0, -10, 0, 0, 0.1, 0, -10, 0, 0, 0.1, 0), nrow=4, byrow=TRUE) pimmlawton(Aq, N=1) out <- pimmlawton(Aq, N=2000) out <- subset(out, -1/DomEig <150) hist(-1/out$DomEig, main="Frequencies of Return Time")
Function to plot a time series and its periodogram, and calculate the confidence interval of 2 times the spectral power.
plot_f(z)plot_f(z)
z |
numeric vector. |
Hank Stevens
J. M. Halley. Ecology, evolution and 1/f-noise. Trends in Ecology & Evolution, 11:33-37, 1996.
O. L. Petchey, A. Gonzalez, and H. B.Wilson. Effects on population persistence: the interaction between environmental noise colour, intra-specific competition and space. Proceedings of the Royal Society of London Series B, 264:1841-1847, 1997. J.
E. Cohen, C. M. Newman, A. E. Cohen, O. L. Petchey, and A. Gonzalez. Spectral mimicry: a method of synthesizing matching time series with different Fourier spectra. Circuits, Systems and Signal Processing, 18:431-442, 1999.
one_over_f to generate 1/f noise; spec_mimic to rearrange
one vector, X, to mimic the spectrum of another vector; spectrum for the
hard work.
## white noise plot_f(z=runif(50)) plot_f## white noise plot_f(z=runif(50)) plot_f
Resampling stochastic simulator for a single density-independent population.
PopSim(Rs, N0, years = 50, sims = 10)PopSim(Rs, N0, years = 50, sims = 10)
Rs |
vector of observed annual growth rates (N[t+1]/N[t]). |
N0 |
initial population size. |
years |
number of years to simulate. |
sims |
number of simulated populations. |
Designed to simulate trajectories based on resampled observed N[t+1]/N[t].
Returns a matrix of population sizes for time = t (rows) for each replicated population (columns).
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
lambdas <- runif(10, .5, 1.5) out <- PopSim(Rs=lambdas, years=50, N0=50) matplot(0:50, out, type='l') summary(out[51,])lambdas <- runif(10, .5, 1.5) out <- PopSim(Rs=lambdas, years=50, N0=50) matplot(0:50, out, type='l') summary(out[51,])
The Lotka-Volterra predator-prey model, for use with ode in the
deSolve package.
predpreyLV(t, y, params)predpreyLV(t, y, params)
t |
Argument for time |
y |
A vector of length 2, for population 1 and 2 |
params |
Vector or list of parameters |
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variables, y. |
Hank Stevens <[email protected]>
Lotka, A.J. (1956) Elements of Mathematical Biology. Dover Publications, Inc.
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
params1 <- c(b=.5, a=.01, s=.2, e=.1) Time <- seq(0,100, by=.1) # Set time here LV.out <- ode(c(H0=25,P0=5), Time, predpreyLV, params1) matplot(Time, (LV.out[,2:3]), type="l", ylab="Population Size")params1 <- c(b=.5, a=.01, s=.2, e=.1) Time <- seq(0,100, by=.1) # Set time here LV.out <- ode(c(H0=25,P0=5), Time, predpreyLV, params1) matplot(Time, (LV.out[,2:3]), type="l", ylab="Population Size")
An implementation of a predator-prey model, after Rosenzweig and MacArthur
(1963). Includes prey density-dependence, and a type-II predator functional
response. For use with ode in the deSolve package.
predpreyRM(t, y, p)predpreyRM(t, y, p)
t |
Argument for time |
y |
A vector of length 2, for population 1 and 2 |
p |
Vector or list of parameters |
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variables, y. |
Hank Stevens <[email protected]>
Rosenzweig, M.L. and MacArthur, R.H. (1963) Graphical representation and stability conditions of predator-prey interactions. The American Naturalist, 97, 209–223.
Stevens. M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
library(deSolve) pars <- c(b = .8, e = 0.07, s = .2, w = 5, D = 400, alpha = 0.001) Time <- 50 RM1 <- ode(c(900,120), 1:Time, predpreyRM, pars) matplot(1:Time, RM1[,-1], type='l')library(deSolve) pars <- c(b = .8, e = 0.07, s = .2, w = 5, D = 400, alpha = 0.001) Time <- 50 RM1 <- ode(c(900,120), 1:Time, predpreyRM, pars) matplot(1:Time, RM1[,-1], type='l')
Kermack and McCormick (1927) provided data on the number of plague deaths per week in Bombay in 1905–06 (Bombay is the former name for the Indian coastal city Mumbai. It is the capital of Maharashtra and is one of the largest cities in the world).
A data frame with 32 observations on the following 2 variables.
a numeric vector
a numeric vector
Data provided kindly by S.P. Ellner (Cornell University)
Kermack, W.O. and McCormick, W.G. (1927) A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society, Series A, 115, 700–721.
data(ross) str(ross) ; plot(CumulativeDeaths ~ Week, data=ross, type='b')data(ross) str(ross) ; plot(CumulativeDeaths ~ Week, data=ross, type='b')
A model of floating vs. submerged plant dominance in shallow aquatic
systems, after Scheffeer et al. (2003). For use with ode in
the deSolve package. Floating plants are better competitors for
light, as long as submerged plants cannot drive down nitrogen levels.
scheffer(t, y, p)scheffer(t, y, p)
t |
the time point for a realization of the integration. |
y |
the vector of populations, at each time t. |
p |
a vector or list containing the necessary parameters. |
Returns a list of length one which is the vector of the rates of
increase (required by ode).
Hank Stevens <[email protected]>
Scheffer, M., Szabo, S., Gragnani, A., van Nes, E.H., Rinaldi, S., Kautsky, N., Norberg, J., Roijackers, R.M.M. and Franken, R.J.M. (2003) Floating plant dominance as a stable state. Proceeding of the National Academy of Sciences, U.S.A., 100, 4040–4045.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
p <- c(N=2.5, as=0.01, af=0.01, b=0.02, qs=0.075, qf=0.005, hs=0, hf=0.2, ls=0.05, lf=0.05, rs=0.5, rf=0.5, W=0) t <- 1:200 Initial <- c(F=10, S=10) S.out1 <- ode(Initial, t, scheffer, p) matplot(t, S.out1[,-1], type='l') legend('right', c("F", "S"), lty=1:2, col=1:2, bty='n')p <- c(N=2.5, as=0.01, af=0.01, b=0.02, qs=0.075, qf=0.005, hs=0, hf=0.2, ls=0.05, lf=0.05, rs=0.5, rf=0.5, W=0) t <- 1:200 Initial <- c(F=10, S=10) S.out1 <- ode(Initial, t, scheffer, p) matplot(t, S.out1[,-1], type='l') legend('right', c("F", "S"), lty=1:2, col=1:2, bty='n')
The S-I-R epidemilogical disease model with births and deaths (population
dynamics), for use with ode in the deSolve package. This
model uses mass action transmission.
SIRbd(t, y, p)SIRbd(t, y, p)
t |
times points for which values will be returned |
y |
the vector of disease states of hosts (S, I, R) |
p |
a vector of parameters |
The user does not put these directly into this function, but rather uses
ode in the deSolve package.
Returns of list of one component (required by ode).
Hank Stevens <[email protected]>
Ellner, S.P. and Guckenheimer, J. (2006) Dynamic Models in Biology, Princeton University Press.
Kermack, W.O. and McCormick, W.G. (1927) A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society, Series A, 115, 700–721.
Stevens, M.H.H. (2009) A Primer of Ecology with R, Use R! Series. Springer.
library(deSolve) N <- 10^6; R <- 0; I <- 1; S <- N - I - R g <- 1/(13/365); b <- 1/50; age <- 5; R0 <- 1 + 1/(b*age) B <- R0 * (g + b) / N parms <- c(B = B, g = g, b = b, m=b) years <- seq(0,30, by=.1) SIRbd.out <- data.frame(ode(c(S=S,I=I,R=R), years, SIRbd, parms, hmax=.01)) matplot(SIRbd.out[,1], sqrt(SIRbd.out[,-1]), type='l', lty=1:3, ylab="sqrt(No. of Individuals)", xlab='Years') legend('right', c('S','I','R'), lty=1:3, col=1:3, bty='n')library(deSolve) N <- 10^6; R <- 0; I <- 1; S <- N - I - R g <- 1/(13/365); b <- 1/50; age <- 5; R0 <- 1 + 1/(b*age) B <- R0 * (g + b) / N parms <- c(B = B, g = g, b = b, m=b) years <- seq(0,30, by=.1) SIRbd.out <- data.frame(ode(c(S=S,I=I,R=R), years, SIRbd, parms, hmax=.01)) matplot(SIRbd.out[,1], sqrt(SIRbd.out[,-1]), type='l', lty=1:3, ylab="sqrt(No. of Individuals)", xlab='Years') legend('right', c('S','I','R'), lty=1:3, col=1:3, bty='n')
The S-I-R epidemilogical disease model with density-dependent transmission, for use with ode in the
deSolve package.
SIRd(t, y, p)SIRd(t, y, p)
t |
times points for which values will be returned |
y |
the vector of disease states of hosts (S, I, R) |
p |
a vector of parameters |
The user does not put these directly into this function, but rather uses
ode in the deSolve package.
Returns of list of one component (required by ode).
Hank Stevens <[email protected]>
Ellner, S.P. and Guckenheimer, J. (2006) Dynamic Models in Biology, Princeton University Press.
Kermack, W.O. and McCormick, W.G. (1927) A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society, Series A, 115, 700–721.
Stevens, M.H.H. (2009) A Primer of Ecology with R, Use R! Series. Springer.
N <- 10^3; I <- R <- 1; S <- N - I - R parms <- c(B=.01, g=4) months <- seq(0, 3, by=0.01) require(deSolve) SIR.out <- data.frame( ode(c(S,I,R), months, SIRd, parms) ) matplot(months, SIR.out[,-1], type='l') legend('right', c('R', 'I', 'S'), lty=3:1, col=3:1, bty='n')N <- 10^3; I <- R <- 1; S <- N - I - R parms <- c(B=.01, g=4) months <- seq(0, 3, by=0.01) require(deSolve) SIR.out <- data.frame( ode(c(S,I,R), months, SIRd, parms) ) matplot(months, SIR.out[,-1], type='l') legend('right', c('R', 'I', 'S'), lty=3:1, col=3:1, bty='n')
The S-I-R epidemilogical disease model with frequency dependent
transmission, for use with ode in the deSolve package.
SIRf(t, y, p)SIRf(t, y, p)
t |
times points for which values will be returned |
y |
the vector of disease states of hosts (S, I, R) |
p |
a vector of parameters |
The user does not put these directly into this function, but rather uses
ode in the deSolve package.
Returns of list of one component (required by ode).
Hank Stevens <[email protected]>
Ellner, S.P. and Guckenheimer, J. (2006) Dynamic Models in Biology, Princeton University Press.
Kermack, W.O. and McCormick, W.G. (1927) A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society, Series A, 115, 700–721.
Stevens, M.H.H. (2009) A Primer of Ecology with R, Use R! Series. Springer.
R <- 0; S <- 1000; I <- 1000; N <- S+I+R parmsf <- c(B=1, g=1) Months <- seq(0, 8, by=0.1) outf <- ode(c(S,I,R), Months, SIRf, parmsf) matplot(Months, outf[,-1], type='l', ylab="Prevalence (I/N)") legend('right', legend=c('S','I','R'), lty=1:3, col=1:3, bty='n')R <- 0; S <- 1000; I <- 1000; N <- S+I+R parmsf <- c(B=1, g=1) Months <- seq(0, 8, by=0.1) outf <- ode(c(S,I,R), Months, SIRf, parmsf) matplot(Months, outf[,-1], type='l', ylab="Prevalence (I/N)") legend('right', legend=c('S','I','R'), lty=1:3, col=1:3, bty='n')
The S-I-R epidemiological disease model with births and deaths (population
dynamics), for use with ode in the deSolve package. This
model uses scaled transmission, where z controls the degree of density- and frequency-dependence.
SIRmod(t, y, p)SIRmod(t, y, p)
t |
times points for which values will be returned |
y |
the vector of disease states of hosts (S, I, R) |
p |
a vector of parameters |
The user does not put these directly into this function, but rather uses
ode in the deSolve package.
Returns of list of one component (required by ode).
Hank Stevens <[email protected]>
Ellner, S.P. and Guckenheimer, J. (2006) Dynamic Models in Biology, Princeton University Press.
Kermack, W.O. and McCormick, W.G. (1927) A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society, Series A, 115, 700–721.
Stevens, M.H.H. (2009) A Primer of Ecology with R, Use R! Series. Springer.
library(deSolve) N <- 10^6; R <- 0; I <- 1; S <- N - I - R g <- 1/(13/365); b <- 1/50; z <- 0; age <- 5; R0 <- 1 + 1/(b*age) B <- R0 * (g + b) / N parms <- c(B = B, g = g, b = b, mu=b) years <- seq(0,30, by=.1) SIR.out <- data.frame(ode(c(S=S,I=I,R=R), years, SIRmod, parms, hmax=.01)) matplot(SIR.out[,1], sqrt(SIR.out[,-1]), type='l', lty=1:3, ylab="sqrt(No. of Individuals)", xlab='Years') legend('right', c('S','I','R'), lty=1:3, col=1:3, bty='n')library(deSolve) N <- 10^6; R <- 0; I <- 1; S <- N - I - R g <- 1/(13/365); b <- 1/50; z <- 0; age <- 5; R0 <- 1 + 1/(b*age) B <- R0 * (g + b) / N parms <- c(B = B, g = g, b = b, mu=b) years <- seq(0,30, by=.1) SIR.out <- data.frame(ode(c(S=S,I=I,R=R), years, SIRmod, parms, hmax=.01)) matplot(SIR.out[,1], sqrt(SIR.out[,-1]), type='l', lty=1:3, ylab="sqrt(No. of Individuals)", xlab='Years') legend('right', c('S','I','R'), lty=1:3, col=1:3, bty='n')
Song Sparrow (Melospiza melodia) counts in Darrtown, OH, USA. From Sauer, J. R., J.E. Hines, and J. Fallon. 2005. The North American Breeding Bird Survey, Results and Analysis 1966–2004. Version 2005.2. USGS Patuxent Wildlife Research Center, Laurel, MD.
A data frame with 36 observations on the following 3 variables.
a numeric vector
a numeric vector
a numeric vector
http://www.pwrc.usgs.gov/BBS/
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
data(sparrows) ## maybe str(sparrows) plot(Count ~ Year, sparrows)data(sparrows) ## maybe str(sparrows) plot(Count ~ Year, sparrows)
This function rearranges one vector, X, to mimic the spectrum of another vector, Y.
spec_mimic(X, Y = NULL, gamma = 1)spec_mimic(X, Y = NULL, gamma = 1)
X |
numeric vector. |
Y |
if not NULL, a numeric vector. |
gamma |
power of the 1/f noise; used only if Y is NULL. |
If Y is NULL, this function will use [one_over_f()] to generate a random series with power = gamma.
Hank Stevens
J. M. Halley. Ecology, evolution and 1/f-noise. Trends in Ecology & Evolution, 11:33-37, 1996.
O. L. Petchey, A. Gonzalez, and H. B. Wilson. Effects on population persistence: the interaction between environmental noise colour, intra-specific competition and space. Proceedings of the Royal Society of London Series B, 264:1841-1847, 1997.
J. E. Cohen, C. M. Newman, A. E. Cohen, O. L. Petchey, and A. Gonzalez. Spectral mimicry: a method of synthesizing matching time series with different Fourier spectra. Circuits, Systems and Signal Processing, 18:431-442, 1999.
one_over_f to generate 1/f noise; plot_f to plot the time
series and the spectrogram of the series, and estimate the slope.
N = 50 set.seed(1) X1 <- runif(N) Y <- one_over_f(gamma=2, N=N) X2 <- spec_mimic(X1, Y) series <- cbind(X1, Y, X2) matplot(1:50, series, type='l') legend("bottomright", legend=colnames(series), col=1:3, lty=1:3)N = 50 set.seed(1) X1 <- runif(N) Y <- one_over_f(gamma=2, N=N) X2 <- spec_mimic(X1, Y) series <- cbind(X1, Y, X2) matplot(1:50, series, type='l') legend("bottomright", legend=colnames(series), col=1:3, lty=1:3)
This is the five-state, two-species model of the succession-niche model,
after Pacala and Rees (1998). For use with ode in the deSolve
package.
succniche(t, y, params)succniche(t, y, params)
t |
Argument for the time point at integration |
y |
A vector of length four, for states, E, M, S, and R. |
params |
Vector or list of parameters |
Returns a list of length one, for use with ode in the
deSolve package.
Component 1 |
vector of the state variable (a scalar for the proportion of sites occupied). |
Hank Stevens <[email protected]>
Pacala, S.W. and Rees, M. (1998) Models suggesting field experiments to test two hypotheses explaining successional diversity. The American Naturalist, 152, 729–737.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
params.suc <- c(a=7, c=0.2, g=.1, m=0.04, D=0) t=seq(0,50,.1) init.suc <- c(S=0, E=0.5, M=0.5, R=0.00) # Free space is implicit, F=1-(S+E+M+R). ccg.out <- data.frame(ode(init.suc, t, succniche, params.suc)) matplot(t, ccg.out[,-1], type="l", ylab="Relative Frequency", xlab="Time", ylim=c(0,1) ) legend("right", colnames(ccg.out)[5:2], lty=4:1, bty="n")params.suc <- c(a=7, c=0.2, g=.1, m=0.04, D=0) t=seq(0,50,.1) init.suc <- c(S=0, E=0.5, M=0.5, R=0.00) # Free space is implicit, F=1-(S+E+M+R). ccg.out <- data.frame(ode(init.suc, t, succniche, params.suc)) matplot(t, ccg.out[,-1], type="l", ylab="Relative Frequency", xlab="Time", ylim=c(0,1) ) legend("right", colnames(ccg.out)[5:2], lty=4:1, bty="n")
A function for continuous theta-logistic growth, for use with ode in
the deSolve package.
thetalogistic(times, y, parms)thetalogistic(times, y, parms)
times |
Times points that will return N |
y |
N |
parms |
a vector of logistic growth parameters |
The user does not put these directly into this function, but rather uses
ode in the deSolve package. See ode in the
deSolve package.
Returns of list of one component (required by ode).
Hank Stevens ([email protected])
Sibly, R.M., Barker, D., Denham, M.C., Hone, J., and Pagel, M. (2005) On the regulation of populations of mammals, birds, fish, and insects. Science, 309, 607–610.
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
library(deSolve) p <- c(r=1,alpha=.01, theta=.5) time <- seq(1,10, by=.1) initialN <- 10 out <- ode(y=initialN, times=time, func=thetalogistic, parms=p) plot(time, out[,-1], type='l')library(deSolve) p <- c(r=1,alpha=.01, theta=.5) time <- seq(1,10, by=.1) initialN <- 10 out <- ode(y=initialN, times=time, func=thetalogistic, parms=p) plot(time, out[,-1], type='l')
Percent cover of six of the most common herbaceous perennial species from the Buell-Small long term succession study (http://www.ecostudies.org/bss/). Data are plot-level visual estimates of percent cover.
A data frame with 15140 observations on the following 8 variables.
a numeric vector
a factor
with levels Aster, Euthamia, Solidago
a factor with levels canadensis,
gigantea, graminifolia, novae-angliae, pilosus,
rugosa
a factor with levels C3,
C4, C5, C6, C7, D1, D2, D3,
E1, E2
a numeric vector indicating the number of years of succession, since abandonment from agriculture.
a numeric vector
a numeric vector; percent cover, estimated visually.
a factor
with levels A.novae-angliae, A.pilosus, E.graminifolia,
S.canadensis, S.gigantea, S.rugosa
Data graciously provided by Scott Meiners (Eastern Illinois University, http://www.ecostudies.org/bss/).
Stevens, M.H.H. (2009) A Primer of Ecology with R. Use R! Series. Springer.
data(weeds) str(weeds); # lattice::xyplot(Cover ~ Age, data=weeds, groups=Species, # type=c("a"), auto.key=list(lines=TRUE, points=FALSE, columns=3), # ylim=c(-1,20))data(weeds) str(weeds); # lattice::xyplot(Cover ~ Age, data=weeds, groups=Species, # type=c("a"), auto.key=list(lines=TRUE, points=FALSE, columns=3), # ylim=c(-1,20))