Solves a system of ordinary differential equations; a wrapper around the implemented ODE solvers

```
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 deSolve
print(x, …)
# S3 method for deSolve
summary(object, select = NULL, which = select,
subset = NULL, …)

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.

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.

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"`

.

`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,`diagnostics`

to print diagnostic messages.

# NOT RUN { ## ======================================================================= ## 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... # }