upca: The Uniform Period Chaotic Amplitude Model

In simecol: Simulation of Ecological (and Other) Dynamic Systems

The Uniform Period Chaotic Amplitude Model

Description

simecol example: resource-predator-prey model, which is able to exhibit chaotic behaviour.

Usage

  data(upca)

Format

S4 object according to the odeModel specification. The object contains the following slots:

main

The differential equations for predator prey and resource with:

u

resource (e.g. grassland or phosphorus),

v

producer (prey),

w

consumer (predator).

equations

Two alternative (and switchable) equations for the functional response.

parms

Vector with the named parameters of the model, see references for details.

times

Simulation time and integration interval.

init

Vector with start values for u, v and w.

solver

Character string with the integration method.

Details

To see all details, please have a look into the implementation below and the original publications.

References

Blasius, B., Huppert, A., and Stone, L. (1999) Complex dynamics and phase synchronization in spatially extended ecological systems. Nature, 399 354–359.

Blasius, B. and Stone, L. (2000) Chaos and phase synchronization in ecological systems. International Journal of Bifurcation and Chaos, 10 2361–2380.

See Also

sim, parms, init, times.

Examples

##============================================
## Basic Usage:
##   explore the example
##============================================
data(upca)
plot(sim(upca))

# omit stabilizing parameter wstar
parms(upca)["wstar"] <- 0
plot(sim(upca))

# change functional response from
# Holling II (default) to Lotka-Volterra
equations(upca)$f <- function(x, y, k) x * y
plot(sim(upca))

##============================================
## Implementation:
##   The code of the UPCA model
##============================================
upca <- new("odeModel",
  main = function(time, init, parms) {
    u      <- init[1]
    v      <- init[2]
    w      <- init[3]
    with(as.list(parms), {
      du <-  a * u           - alpha1 * f(u, v, k1)
      dv <- -b * v           + alpha1 * f(u, v, k1) +
                             - alpha2 * f(v, w, k2)
      dw <- -c * (w - wstar) + alpha2 * f(v, w, k2)
      list(c(du, dv, dw))
    })
  },
  equations  = list(
    f1 = function(x, y, k){x*y},           # Lotka-Volterra
    f2 = function(x, y, k){x*y / (1+k*x)}  # Holling II
  ),
  times  = c(from=0, to=100, by=0.1),
  parms  = c(a=1, b=1, c=10, alpha1=0.2, alpha2=1,
    k1=0.05, k2=0, wstar=0.006),
  init = c(u=10, v=5, w=0.1),
  solver = "lsoda"
)

equations(upca)$f <- equations(upca)$f2

simecol documentation built on June 8, 2025, 10:45 a.m.