ode.2D {deSolve} | R Documentation |
Solves a system of ordinary differential equations resulting from 2-Dimensional transport-recation models that include transport only between adjacent layers.
ode.2D(y, times, func, parms, nspec=NULL, dimens, ...)
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:
yprime = func(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 .
|
parms |
parameters passed to func |
nspec |
the number of *species* (components) in the model. |
dimens |
2-valued vector with the number of *boxes* in two dimensions in the model. |
... |
additional arguments passed to lsodes |
This is the method of choice for 2-dimensional models, that are only subjected to transport between adjacent layers.
Based on the dimension of the problem, the method first calculates the sparsity pattern of the Jacobian, under the assumption
that transport is onely occurring between adjacent layers. Then lsodes
is called to solve the problem.
As lsodes
is used to integrate, it will probably be necessary to specify the length of the real work array, lrw
.
Although a reasonable guess of lrw
is made, it is likely that this will be too low.
In this case, ode.2D
will return with an error message telling
the size of the work array actually needed. In the second try then, set lrw
equal to this number.
See lsodes
for the additional options
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 useful elements.
The first element of istate returns the conditions under which the last call to the integrator returned. Normal is istate = 2.
If verbose
= TRUE, the settings of istate and rstate will be written to the screen. See the help for the selected integrator for details.
It is advisable though not mandatory to specify BOTH nspec
and dimens
. In this case, the solver can check whether the input makes sense
(as nspec*dimens[1]*dimens[2] = length(y))
do NOT use this method for problems that are not 2D
Karline Soetaert <k.soetaert@nioo.knaw.nl>
ode
ode.band
for integrating models with a banded Jacobian
ode.1D
for integrating 1-D models
lsodes
for the integration options.
#=============================================================================== # A Lotka-Volterra predator-prey model with predator and prey # dispersing in 2 dimensions #=============================================================================== ###################### # Model definitions # ###################### lvmod2D <- function (time, state, pars, N, Da, dx) { NN <- N*N Prey <- matrix(nr=N,nc=N,state[1:NN]) Pred <- matrix(nr=N,nc=N,state[(NN+1):(2*NN)]) with (as.list(pars), { # Biology dPrey <- rGrow* Prey *(1- Prey/K) - rIng* Prey *Pred dPred <- rIng* Prey *Pred*assEff -rMort* Pred zero <- rep(0,N) # 1. Fluxes in x-direction; zero fluxes near boundaries FluxPrey <- -Da * rbind(zero,(Prey[2:N,]-Prey[1:(N-1),]),zero)/dx FluxPred <- -Da * rbind(zero,(Pred[2:N,]-Pred[1:(N-1),]),zero)/dx # Add flux gradient to rate of change dPrey <- dPrey - (FluxPrey[2:(N+1),]-FluxPrey[1:N,])/dx dPred <- dPred - (FluxPred[2:(N+1),]-FluxPred[1:N,])/dx # 2. Fluxes in y-direction; zero fluxes near boundaries FluxPrey <- -Da * cbind(zero,(Prey[,2:N]-Prey[,1:(N-1)]),zero)/dx FluxPred <- -Da * cbind(zero,(Pred[,2:N]-Pred[,1:(N-1)]),zero)/dx # Add flux gradient to rate of change dPrey <- dPrey - (FluxPrey[,2:(N+1)]-FluxPrey[,1:N])/dx dPred <- dPred - (FluxPred[,2:(N+1)]-FluxPred[,1:N])/dx return (list(c(as.vector(dPrey),as.vector(dPred)))) }) } ###################### # Model applications # ###################### 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 =5 ) # mmol/m3, carrying capacity R <- 20 # total length of surface, m N <- 50 # number of boxes in one direction dx <- R/N # thickness of each layer Da <- 0.05 # m2/d, dispersion coefficient NN <- N*N # total number of boxes # initial conditions yini <- rep(0,2*N*N) cc <- c((NN/2):(NN/2+1)+N/2,(NN/2):(NN/2+1)-N/2) yini[cc] <- yini[NN+cc] <- 1 # solve model (5000 state variables... times <- seq(0,50,by=1) out<- ode.2D(y=yini,times=times,func=lvmod2D,parms=pars, dimens=c(N,N),N=N,dx=dx,Da=Da,lrw=5000000) # plot results Col<- colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan", "#7FFF7F", "yellow", "#FF7F00", "red", "#7F0000")) for (i in seq(1,length(times),by=1)) image(matrix(nr=N,nc=N,out[i,2:(NN+1)]), col=Col(100),xlab="x",ylab="y",zlim=range(out[,2:(NN+1)]))