ode.band: Solver for Ordinary Differential Equations; Assumes a Banded Jacobian

Description

Solves a system of ordinary differential equations. Assumes a banded jacobian matrix, but does not rearrange the state variables (in contrast to ode.1D). Suitable for 1-D models that include transport only between adjacent layers and that model only one species.

Usage

ode.band(y, times, func, parms, nspec = NULL, 
  bandup = nspec, banddown = nspec, method = "lsode", ...)

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.

parms

parameters passed to func.

nspec

the number of *species* (components) in the model.

bandup

the number of nonzero bands above the jacobian diagonal.

banddown

the number of nonzero bands below the jacobian diagonal.

method

the integrator to use, one of "vode", "lsode", "lsoda", "lsodar".

...

additional arguments passed to the integrator.

Value

A matrix 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. The output will have the attributes istate and rstate, two vectors with several elements. See the help for the selected integrator for details. the first element of istate returns the conditions under which the last call to vode returned. Normal is istate = 2. If verbose = TRUE, the settings of istate and rstate will be written to the screen.

Details

This is the method of choice for single-species 1-D reactive transport models. For multi-species 1-D models, this method can only be used if the state variables are arranged per box, per species (e.g. A[1], B[1], A[2], B[2], A[3], B[3], .... for species A, B). By default, the model function will have the species arranged as A[1], A[2], A[3], .... B[1], B[2], B[3], .... in this case, use ode.1D. See the selected integrator for the additional options.

Examples

Run this code

## =================================================
## The Aphid model from Soetaert and Herman, 2008.
## A practical guide to ecological modelling.
## Using R as a simulation platform. Springer.
## =================================================

## 1-D diffusion model

## ================
## Model equations
## ================
Aphid <- function(t, APHIDS, parameters)
{
  deltax  <- c (0.5, rep(1, numboxes-1), 0.5)
  Flux    <- -D*diff(c(0, APHIDS, 0))/deltax
  dAPHIDS <- -diff(Flux)/delx + APHIDS*r

  list(dAPHIDS)   # the output
}
  
## ==================
## Model application
## ==================

## the model parameters:
D         <- 0.3    # m2/day  diffusion rate
r         <- 0.01   # /day    net growth rate
delx      <- 1      # m       thickness of boxes
numboxes  <- 60 

## distance of boxes on plant, m, 1 m intervals
Distance  <- seq(from = 0.5, by = delx, length.out = numboxes)

## Initial conditions, ind/m2
## aphids present only on two central boxes
APHIDS        <- rep(0, times = numboxes)
APHIDS[30:31] <- 1
state         <- c(APHIDS = APHIDS)      # initialise state variables 
                  
## RUNNING the model:
times <- seq(0, 200, by = 1)   # output wanted at these time intervals
out   <- ode.band(state, times, Aphid, parms = 0, nspec = 1)

## ================
## Plotting output
## ================

## the data in 'out' consist of: 1st col times, 2-41: the density
## select the density data
DENSITY   <- out[,2:(numboxes + 1)]

filled.contour(x = times, y = Distance, DENSITY, color = topo.colors,
               xlab = "time, days", ylab = "Distance on plant, m",
               main = "Aphid density on a row of plants")

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