General Solver for Ordinary Differential Equations

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.

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

Karline Soetaert <karline.soetaert@nioz.nl>

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".

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)
#>                Prey    Predator
#> Min.      1.0000000   1.8632829
#> 1st Qu.   1.9999571   3.9999115
#> Median    2.0000000   4.0000000
#> Mean      2.0317905   3.9606228
#> 3rd Qu.   2.0000751   4.0000418
#> Max.      4.2001812   4.9787222
#> N       201.0000000 201.0000000
#> sd        0.3138139   0.3489079

## 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"))


if (FALSE) { # \dontrun{
## =======================================================================
## 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...
} # }