Lotka-Volterra model for N species — lv

The default simulates a system of Ndim = 4 coupled competitive Lotka-Volterra equations studied by Vano et al. (2006) using RK4 numerical integration. Vano et al. describe the dynamics of the resulting attractor as chaotic (bounded, quasi-periodic, sensitive dependence on initial conditions), also see this Wiki page.

lv_Ndim(
  Nsim = 1000,
  Ndim = 4,
  Y0 = rep(0.1, Ndim),
  r = c(1, 0.72, 1.53, 1.27),
  A = matrix(c(1, 1.09, 1.52, 0, 0, 1, 0.44, 1.36, 2.33, 0, 1, 0.47, 1.21, 0.51, 0.35,
    1), nrow = 4, byrow = TRUE),
  K = rep(1, Ndim),
  h = 0.5
)

Arguments

Nsim: How many data points to simulate (default = 1000)
Ndim: How many dimensions (coupled L-V equations) to use (default = 4)
Y0: A vector with Ndim elements representing the initial value (default = rep(.1, Ndim))
r: A vector with Ndim elements representing the growth rates for each dimension. The default settings are taken from Vano et al. (2006) (default = c(1, 0.72, 1.53, 1.27))
A: The interaction matrix of Ndim X Ndim representing the nature of the coupling between each dimension. A[1,2] will appear as a parameter in the equation for Y1 setting the magnitude of the interaction with Y2. Conversely A[2,1] will appear in the equation for Y2 setting the magnitude of the interaction with Y1. The diagonal elements have to be 1. The default settings are taken from Vano et al. (2006)
K: A vector with Ndim elements representing the carrying capacity for each dimension. (default = rep(1, Ndim))
h: The Euler parameter for RK4 integration.
returnLongData: Return the data in long format for easy plotting (default = FALSE)

Value

A matrix of Ndim X Nsim with simulated values.

References

Vano, J. A., Wildenberg, J. C., Anderson, M. B., Noel, J. K., & Sprott, J. C. (2006). Chaos in low-dimensional Lotka–Volterra models of competition. Nonlinearity, 19(10), 2391.

Examples


library(plot3D)

Y <- lv_Ndim()
#> Error in laply(1:Ndim, function(i) {    matrix(c(Y0[i], rep(NA, Nsim - 1)), nrow = 1)}): could not find function "laply"
df <- as.data.frame(apply(t(Y), 2, ts_standardise))
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'x' in selecting a method for function 't': object 'Y' not found
lines3D(df$Y1, df$Y2, df$Y3, colvar = df$Y4, clab = "Y4", xlab= "Y1", ylab = "Y2", zlab ="Y3", ticktype = "detailed")
#> Error in df$Y4: object of type 'closure' is not subsettable