R: General Solver for Ordinary Differential Equations
ode
R Documentation
General Solver for Ordinary Differential Equations
Description
Solves a system of ordinary differential equations; a wrapper
around the implemented ODE solvers
Usage
ode(y, times, func, parms,
method = c("lsoda", "lsode", "lsodes", "lsodar", "vode", "daspk",
"euler", "rk4", "ode23", "ode45", "radau",
"bdf", "bdf_d", "adams", "impAdams", "impAdams_d", "iteration"), ...)
## S3 method for class 'deSolve'
print(x, ...)
## S3 method for class 'deSolve'
summary(object, select = NULL, which = select,
subset = NULL, ...)
Arguments
y
the initial (state) values for the ODE system, a vector. If
y has a name attribute, the names will be used to label the
output matrix.
times
time sequence for which output is wanted; the first
value of times must be the initial time.
func
either an R-function that computes the values of the
derivatives in the ODE system (the model definition) at time t, or a
character string giving the name of a compiled function in a
dynamically loaded shared library.
If func is an R-function, it must be defined as:
func <- function(t, y, parms,...). t is the current time
point in the integration, y is the current estimate of the
variables in the ODE system. If the initial values y has a
names attribute, the names will be available inside func.
parms is a vector or list of parameters; ... (optional) are
any other arguments passed to the function.
The return value of func should be a list, whose first
element is a vector containing the derivatives of y with
respect to time, and whose next elements are global values
that are required at each point in times. The derivatives
must be specified in the same order as the state variables y.
If func is a string, then dllname must give the name
of the shared library (without extension) which must be loaded
before ode is called. See package vignette
"compiledCode" for more details.
parms
parameters passed to func.
method
the integrator to use, either a function that performs
integration, or a list of class rkMethod, or a string
("lsoda",
"lsode", "lsodes","lsodar","vode",
"daspk", "euler", "rk4", "ode23",
"ode45", "radau", "bdf", "bdf_d", "adams",
"impAdams" or "impAdams_d" ,"iteration").
Options "bdf", "bdf_d", "adams", "impAdams" or "impAdams_d" are the backward
differentiation formula, the BDF with diagonal representation of the Jacobian,
the (explicit) Adams and the implicit Adams method, and the implicit Adams
method with diagonal representation of the Jacobian respectively (see details).
The default integrator used is lsoda.
Method "iteration" is special in that here the function func should
return the new value of the state variables rather than the rate of change.
This can be used for individual based models, for difference equations,
or in those cases where the integration is performed within func).
See last example.
x
an object of class deSolve, as returned by the
integrators, and to be printed or to be subsetted.
object
an object of class deSolve, as returned by the
integrators, and whose summary is to be calculated. In contrast to R's default,
this returns a data.frame. It returns one summary column for a multi-dimensional variable.
which
the name(s) or the index to the variables whose summary should be
estimated. Default = all variables.
select
which variable/columns to be selected.
subset
logical expression indicating elements or rows to keep when
calculating a summary: missing values are taken as FALSE
...
additional arguments passed to the integrator or to the
methods.
Details
This is simply a wrapper around the various ode solvers.
See package vignette for information about specifying the model in
compiled code.
See the selected integrator for the additional options.
The default integrator used is lsoda.
The option method = "bdf" provdes a handle to the backward
differentiation formula (it is equal to using method = "lsode").
It is best suited to solve stiff (systems of) equations.
The option method = "bdf_d" selects the backward
differentiation formula that uses Jacobi-Newton iteration (neglecting the
off-diagonal elements of the Jacobian (it is equal to using
method = "lsode", mf = 23).
It is best suited to solve stiff (systems of) equations.
method = "adams" triggers the Adams method that uses functional
iteration (no Jacobian used);
(equal to method = "lsode", mf = 10. It is often the best
choice for solving non-stiff (systems of) equations. Note: when functional
iteration is used, the method is often said to be explicit, although it is
in fact implicit.
method = "impAdams" selects the implicit Adams method that uses Newton-
Raphson iteration (equal to method = "lsode", mf = 12.
method = "impAdams_d" selects the implicit Adams method that uses Jacobi-
Newton iteration, i.e. neglecting all off-diagonal elements (equal to
method = "lsode", mf = 13.
For very stiff systems, method = "daspk" may outperform
method = "bdf".
Value
A matrix of class deSolve with up to as many rows as elements in
times and as many
columns as elements in y plus the number of "global" values
returned in the second element of the return from func, plus an
additional column (the first) for the time value. There will be one
row for each element in times unless the integrator returns
with an unrecoverable error. If y has a names attribute, it
will be used to label the columns of the output value.
Author(s)
Karline Soetaert <karline.soetaert@nioz.nl>
See Also
plot.deSolve for plotting the outputs,
dede general solver for delay differential equations
ode.band for solving models with a banded
Jacobian,
ode.1D for integrating 1-D models,
ode.2D for integrating 2-D models,
ode.3D for integrating 3-D models,
aquaphy, ccl4model, where
ode is used,
lsoda, lsode,
lsodes, lsodar, vode,
daspk, radau,
rk, rkMethod for additional
Runge-Kutta methods,
forcings and events,
diagnostics to print diagnostic messages.
Examples
## =======================================================================
## Example1: Predator-Prey Lotka-Volterra model (with logistic prey)
## =======================================================================
LVmod <- function(Time, State, Pars) {
with(as.list(c(State, Pars)), {
Ingestion <- rIng * Prey * Predator
GrowthPrey <- rGrow * Prey * (1 - Prey/K)
MortPredator <- rMort * Predator
dPrey <- GrowthPrey - Ingestion
dPredator <- Ingestion * assEff - MortPredator
return(list(c(dPrey, dPredator)))
})
}
pars <- c(rIng = 0.2, # /day, rate of ingestion
rGrow = 1.0, # /day, growth rate of prey
rMort = 0.2 , # /day, mortality rate of predator
assEff = 0.5, # -, assimilation efficiency
K = 10) # mmol/m3, carrying capacity
yini <- c(Prey = 1, Predator = 2)
times <- seq(0, 200, by = 1)
out <- ode(yini, times, LVmod, pars)
summary(out)
## Default plot method
plot(out)
## User specified plotting
matplot(out[ , 1], out[ , 2:3], type = "l", xlab = "time", ylab = "Conc",
main = "Lotka-Volterra", lwd = 2)
legend("topright", c("prey", "predator"), col = 1:2, lty = 1:2)
## =======================================================================
## Example2: Substrate-Producer-Consumer Lotka-Volterra model
## =======================================================================
## Note:
## Function sigimp passed as an argument (input) to model
## (see also lsoda and rk examples)
SPCmod <- function(t, x, parms, input) {
with(as.list(c(parms, x)), {
import <- input(t)
dS <- import - b*S*P + g*C # substrate
dP <- c*S*P - d*C*P # producer
dC <- e*P*C - f*C # consumer
res <- c(dS, dP, dC)
list(res)
})
}
## The parameters
parms <- c(b = 0.001, c = 0.1, d = 0.1, e = 0.1, f = 0.1, g = 0.0)
## vector of timesteps
times <- seq(0, 200, length = 101)
## external signal with rectangle impulse
signal <- data.frame(times = times,
import = rep(0, length(times)))
signal$import[signal$times >= 10 & signal$times <= 11] <- 0.2
sigimp <- approxfun(signal$times, signal$import, rule = 2)
## Start values for steady state
xstart <- c(S = 1, P = 1, C = 1)
## Solve model
out <- ode(y = xstart, times = times,
func = SPCmod, parms = parms, input = sigimp)
## Default plot method
plot(out)
## User specified plotting
mf <- par(mfrow = c(1, 2))
matplot(out[,1], out[,2:4], type = "l", xlab = "time", ylab = "state")
legend("topright", col = 1:3, lty = 1:3, legend = c("S", "P", "C"))
plot(out[,"P"], out[,"C"], type = "l", lwd = 2, xlab = "producer",
ylab = "consumer")
par(mfrow = mf)
## =======================================================================
## Example3: Discrete time model - using method = "iteration"
## The host-parasitoid model from Soetaert and Herman, 2009,
## Springer - p. 284.
## =======================================================================
Parasite <- function(t, y, ks) {
P <- y[1]
H <- y[2]
f <- A * P / (ks + H)
Pnew <- H * (1 - exp(-f))
Hnew <- H * exp(rH * (1 - H) - f)
list (c(Pnew, Hnew))
}
rH <- 2.82 # rate of increase
A <- 100 # attack rate
ks <- 15 # half-saturation density
out <- ode(func = Parasite, y = c(P = 0.5, H = 0.5), times = 0:50, parms = ks,
method = "iteration")
out2<- ode(func = Parasite, y = c(P = 0.5, H = 0.5), times = 0:50, parms = 25,
method = "iteration")
out3<- ode(func = Parasite, y = c(P = 0.5, H = 0.5), times = 0:50, parms = 35,
method = "iteration")
## Plot all 3 scenarios in one figure
plot(out, out2, out3, lty = 1, lwd = 2)
## Same like "out", but *output* every two steps
## hini = 1 ensures that the same *internal* timestep of 1 is used
outb <- ode(func = Parasite, y = c(P = 0.5, H = 0.5),
times = seq(0, 50, 2), hini = 1, parms = ks,
method = "iteration")
plot(out, outb, type = c("l", "p"))
## Not run:
## =======================================================================
## Example4: Playing with the Jacobian options - see e.g. lsoda help page
##
## IMPORTANT: The following example is temporarily broken because of
## incompatibility with R 3.0 on some systems.
## A fix is on the way.
## =======================================================================
## a stiff equation, exponential decay, run 500 times
stiff <- function(t, y, p) { # y and r are a 500-valued vector
list(- r * y)
}
N <- 500
r <- runif(N, 15, 20)
yini <- runif(N, 1, 40)
times <- 0:10
## Using the default
print(system.time(
out <- ode(y = yini, parms = NULL, times = times, func = stiff)
))
# diagnostics(out) shows that the method used = bdf (2), so it it stiff
## Specify that the Jacobian is banded, with nonzero values on the
## diagonal, i.e. the bandwidth up and down = 0
print(system.time(
out2 <- ode(y = yini, parms = NULL, times = times, func = stiff,
jactype = "bandint", bandup = 0, banddown = 0)
))
## Now we also specify the Jacobian function
jacob <- function(t, y, p) -r
print(system.time(
out3 <- ode(y = yini, parms = NULL, times = times, func = stiff,
jacfunc = jacob, jactype = "bandusr",
bandup = 0, banddown = 0)
))
## The larger the value of N, the larger the time gain...
## End(Not run)