Vary parameter through time in ODE

Question

I am using the deSolve package to solve a differential equation describing predator-prey dynamics. As an example, below is a simple L-V predator-prey model. I would like some of the parameters in the model to vary through time. I can vary state variable (e.g. prey density) no problem using the event argument in the ode function.

But I cannot use the event argument to alter parameters.

Here is the simple L-V model with no events added (works fine)

# Lotka-Volterra Predator-Prey model from ?deSolve::ode

# define model
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)))
  })
}

# parameters
pars  <- c(rIng   = 0.2,    # rate of ingestion
           rGrow  = 1.0,    # growth rate of prey
           rMort  = 0.2 ,   # mortality rate of predator
           assEff = 0.5,    # assimilation efficiency
           K      = 10)     # carrying capacity


# initial densities (state variables)
yini  <- c(Prey = 1, Predator = 2)
# time steps
times <- seq(0, 200, by = 1)

# run model
out   <- ode(yini, times, LVmod, pars)

## plot
plot(out)

Here is the L-V model with state variable Prey multiplied by some rnorm()every 5 timesteps (works fine).

# add prey every 5 timesteps using events
add_prey <- function(t, var, parms){
  with(as.list(var),{
    Prey <- Prey * rnorm(1, 1, 0.05)
    return(c(Prey, Predator))
  })
}

# run ode - works fine
out <- ode(y = yini,
           times = times,
           func = LVmod,
           parms = pars,
           method = "ode45",
           events = list(func = add_prey, time = seq(0, 200, by = 5)))

plot(out)

Here is my attempt to increase K every 5 timesteps (does not work)

# vary K through time
add_k <- function(t, var, parms){
  with(as.list(var),{
    K <- K + 2
    return(c(Prey, Predator))
  })
}

# run ode
out <- ode(y = yini,
           times = times,
           func = LVmod,
           parms = pars,
           method = "ode45",
           events = list(func = add_k, time = seq(0, 200, by = 5)))

Which produces this error:

Error in eval(substitute(expr), data, enclos = parent.frame()) : 
object 'K' not found

Based on the error K is not being passed to add_k, in add_k the line with(as.list(var) is obviously only accessing the variables Prey and Predator. In the ode and event helpfiles I can only find information regarding altering state variables (Prey and Predator in this case), and no information about altering other parameters. I am new to ODEs, so maybe I am missing something obvious. Any advice would be much appreciated.

Answer 1

Events are used to modify state variables. This means that at each event time, the ode solver is stopped, then states are modified and then the solver is restarted. In case of time-dependent *parameters, this can be done much easier without events. We call this a "forcing function" (or forcing data).

A modified version of the original code is shown below. Here a few explanations:

• approxfun is a function that returns a function. We name it K_t that interpolates between the data timesteps t and the parameter values K. The argument rule = 2 is important. It describes how interpolation is to take place outside the range of t, where 2means that the closest value is returned, because some solvers overshoot the end of the simulation and then interpolate back. The method can be constant or linear, whatever is more appropriate.
• The interpolation function of the variable model parameter K_t can be added to the function as an optional argument at the end. It is also possible to define it globally or to include it in parmsif it is defined as a list, not a vector.
• The LVmod function checks if this additional parameter exists and if yes overwrites the default of K.
• At the end of the LVmodfunction we return the actual value of K as an optional (auxiliary) variable, so that it is included in the model output.

library(deSolve)

LVmod <- function(Time, State, Pars, K_t) {
  with(as.list(c(State, Pars)), {
    if (!is.null(K_t)) K <- K_t(Time)
    Ingestion    <- rIng  * Prey * Predator
    GrowthPrey   <- rGrow * Prey * (1 - Prey/K)
    MortPredator <- rMort * Predator
    
    dPrey        <- GrowthPrey - Ingestion
    dPredator    <- Ingestion * assEff - MortPredator
    
    list(c(dPrey, dPredator), K = K)
  })
}

pars  <- c(rIng   = 0.2,    # rate of ingestion
           rGrow  = 1.0,    # growth rate of prey
           rMort  = 0.2 ,   # mortality rate of predator
           assEff = 0.5,    # assimilation efficiency
           K      = 10)     # carrying capacity
yini  <- c(Prey = 1, Predator = 2)
times <- seq(0, 200, by = 1)

# run model with constant parameter K
out <- ode(yini, times, LVmod, pars, K_t=NULL)

## make K a function of t with given time steps
t   <- seq(0, 200, by = 5)
K   <- cumsum(c(10, rep(2, length(t) - 1)))
K_t <- approxfun(t, K, method = "constant", rule = 2)

out2 <- ode(yini, times, LVmod, pars, K_t=K_t)

plot(out, out2, mfrow=c(1, 3))