# ode: General Solver for Ordinary Differential Equations In deSolve: Solvers for Initial Value Problems of Differential Equations ('ODE', 'DAE', 'DDE')

## Description

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

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```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>

• `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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171``` ```## ======================================================================= ## 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 H <- y 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) ```

deSolve documentation built on Oct. 7, 2021, 9:24 a.m.