Influx at specific time with changing time step in DeSolve

Question

I am making a population distribution model with ODEs and DDEs in DeSolve with the method lsoda. In this model I want to have an influx of the population at a specific time (a specific day). A very simple example:

dn1dt=influx - mortality

The influx (x) needs to occur at time (t) = y (in days). If it is not day y, I don't want an influx. At the moment I have written the influx as influx=function(t,y,x){ifelse((t==y), x, 0)), but I am running into the problem of a changing time step due to the method I am using (lsoda). Due to the changing time step, I will not hit the specific time (y) that will trigger the influx. I am just starting out to work with a changing time step, so I am not sure how to deal with this problem. Please let me know if anything is unclear.

Answer 1

The influx can be implemented as time dependent input (also called "forcing", that lasts a certain time, e.g. a day) or as event, where a state variable is changed immediately. Here an example using a rectangular signal as forcing. Interpolation of time is implemented with approxfun. The flux function can simply be added as additional argument to ode and to the model:

library("deSolve")

model <- function(t, y, p, flux) {
  influx <- flux(t)
  dN <- influx - p["d"] * y["N"]
  list(dN, influx = influx)  
}

y0 <- c(N = 1)
p  <- c(d = 0.2)
times <- seq(0, 10, length.out = 100)
flux <- approxfun(x = c(0, 5, 6, 10), 
                  y = c(0, 1, 0,  0), rule = 2, method = "constant")

out <- ode(y = y0, times = times, model, parms = p, 
           flux = flux, method = "lsoda")

plot(out, las=1)

More can be found on the ?forcings help page or at this page.