Is it possible to use rk4 and rootfun in ode (package deSolve) - r

Question

I'm trying to modeling a prey-prey-predator system using differential equations based on the LV model. For the sake of the precision, i need to use the runge-kutta4 method.
But given the equations, some of the populations become quickly negative.
So I tried to use the events/root system of ODE but it seems that rk4 and rootfun are not compatibles...
eventFunc <- function(t, y, p){
if (y["N1"] < 0) { y["N1"] = 0 }
if (y["N2"] < 0) { y["N2"] = 0 }
if (y["P"] < 0) { y["P"] = 0 }
return(y)
}
rootFunction <- function(t, y, p){
if (y["P"] < 0) {y["P"] = 0}
if (y["N1"] < 0) {y["N1"] = 0}
if (y["N2"] < 0) {y["N2"] = 0}
return(y)
}
out <- ode(func=Model_T2.2,
method="rk4",
y=state,
parms=parameters,
times=times,
events = list(func = eventFunc,
root = TRUE),
rootfun = rootFunction
)
This code give me the followin error :
Error in checkevents(events, times, Ynames, dllname) :
either 'events$time' should be given and contain the times of the events, if 'events$func' is specified and no root function or your solver does not support root functions
Is there any solution to use rk4 and forbid the functions to go under 0?
Thanks in advance.
For those who might ask, here is what works :
if(!require(ggplot2)) {
install.packages("ggplot2"); require(ggplot2)}
if(!require(deSolve)) {
install.packages("deSolve"); require(deSolve)}
Model_T2.2 <- function(t, state, par){
with(as.list(c(state, par)), {
response1 <- (a1 * N1)/(1+(a1*h1*N1)+(a2*h2*N2))
response2 <- (a2 * N2)/(1+(a1*h1*N1)+(a2*h2*N2))
dN1 = r1*N1 * (1 - ((N1 + A12 * N2)/K1)) - response1 * P
dN2 = r2*N2 * (1 - ((N1 + A21 * N2)/K2)) - response2 * P
dP = ((E1 * response1) + (E2 * response2)) * P - Mp
return(list(c(dN1, dN2, dP)))
})
}
parameters<-c(
r1=1.42, r2=0.9,
A12=0.6, A21=0.5,
K1=50, K2=50,
a1=0.77, a2=0.77,
b1 = 1, b2=1,
h1=1.04, h2=1.04,
o1=0, o2=0,
Mp=0.22,
E1=0.36, E2=0.36
)
## inital states
state<-c(
P=10,
N1=30,
N2=30
)
times <- seq(0, 30, by=0.5)
out <- ode(func=Model_T2.2,
method="rk4",
y=state,
parms=parameters,
times=times,
events = list(func = eventFunc,
root = TRUE),
rootfun = rootFunction
)
md <- melt(as.data.frame(out), id.vars=1, measure.vars = c("N1", "N2", "P"))
pl <- ggplot(md, aes(x=time, y=value, colour=variable))
pl <- pl + geom_line() + geom_point() + scale_color_discrete(name="Population")
pl
And the result in a graph :
Evolution of prey1, prey2 and predator populations
As you can see, the population of predators become negative which is clearly impossible in the real world.
Edit : missing variables, sorry about that.

Answer 1

This is a problem you will have with all explicit solvers like rk4. Reducing the time step will help, up to a point. Better use a solver with an implicit method, lsoda seems universally available in one form or another.
Another way to explicitly force positive values is to parametrize them as exponentials. Set N1=exp(U1), N2=exp(U2) then the ODE function code translates to (as dN = exp(U)*dU = N*dU)
N1 <- exp(U1)
N2 <- exp(U2)
response1 <- (a1)/(1+(a1*h1*N1)+(a2*h2*N2))
response2 <- (a2)/(1+(a1*h1*N1)+(a2*h2*N2))
dU1 = r1 * (1 - ((N1 + A12 * N2)/K1)) - response1 * P
dU2 = r2 * (1 - ((N1 + A21 * N2)/K2)) - response2 * P
dP = ((E1 * response1*N1) + (E2 * response2*N2)) * P - Mp
For the output you have then of course to reconstruct N1, N2 from the solutions U1, U2.

Answer 2

Thanks to J_F, I am now able to run my L-V model.
The radau (not randau as you mentionned) function indeed accept root function and events ans implicitly implements the runge-kutta method.
Thanks again, hope this will help someone in the future.