RK4adapt {spuRs}

A function which uses the Fourth order Runge-Kutta method with adaptive step size to solve a system of ODE's.

Description

This function simulates a discrete time Markov chain with transition matrix P, state space 0,1,..,n and and initial state i for nsteps transitions.

Usage

RK4adapt(dydt, t0, y0, t1, h0 = 1, tol = 1e-10, ...)

Arguments

dydt

a function giving the gradient of y(t).

t0

initial value of t.

y0

initial value of y(t).

t1

system solved up to time t1.

h0

initial step size

tol

tolerance for adapting step size.

...

pass arguments to function dydt.

Details

We assume that P is well defined transition matrix with rows summing to 1.

Value

Returns a list with elements t, a vector giving times, and y, a matrix whose rows give the solution at successive times.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

Examples

LV <- function(t=NULL, y, a, b, g, e, K=Inf)
  c(a*y[1]*(1 - y[1]/K) - b*y[1]*y[2], g*b*y[1]*y[2] - e*y[2])

xy <- RK4adapt(LV, 0, c(100, 50), 200, 1, tol=1e-3, 
               a=0.05, K=Inf, b=0.0002, g=0.8, e=0.03)

par(mfrow = c(2,1))
plot(xy$y[,1], xy$y[,2], type='p', 
     xlab='prey', ylab='pred', main='RK4, adaptive h')
plot(xy$t, xy$y[,1], type='p', xlab='time', 
     ylab='prey circles pred triangles', main='RK4, adaptive h')
points(xy$t, xy$y[,2], pch=2)
par(mfrow=c(1,1))