R: General Solver for Ordinary Differential Equations

ode {deSolve} | 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)

[Package deSolve version 1.32 Index]