ode: 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"), ...)
## S3 method for class 'deSolve':
print(x, \dots)

Arguments

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

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","vo

an object of class deSolve, as returned by the integrators, and to be printed.

...

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.

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".

Examples

Run this code

## =======================================================================
## Example1: Predator-Prey Lotka-Volterra model
## =======================================================================

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     <- as.data.frame(ode(func = LVmod, y = yini,
                         parms = pars, times = times))

matplot(out$time, 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 <- as.data.frame(list(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, input = sigimp)

## Default plot method
plot(out, type = "l")

## 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)

Run the code above in your browser using DataLab