Lotka-Volterra 2-D Demonstration

This vignette illustrates the use of V-approximation to approximate the value function of a 2-D deterministic Lotka-Volterra (prey-predator) system.

This example, adapted from Joshua Abbot, is described in the LV dataset.

## 2-D Deterministic: Prey--Predator example
data("LV")

lvspace <- aproxdef(
  deg = c(20, 20),
  lb = c(0.1, 0.1),
  ub = c(1.5, 1.5),
  delta  = 0.03
)

vCLV <- vaprox(
  lvspace,
  LV$lvaproxdata[, c("xs", "ys")],
  LV$lvaproxdata[, c("xdot", "ydot")],
  LV$lvaproxdata[, "wval"]
)

lvsim <- vsim(vCLV,LV$lvsimdata[,c('xs','ys')])

# plot Biomass
plot(LV$lvsimdata[,"tseq"], LV$lvsimdata[,"xs"], type='l', lwd=2, col="blue",
     xlab="Time",
     ylab="Biomass")
lines(LV$lvsimdata[,"tseq"], LV$lvsimdata[,"ys"], lwd=2, col="red")
legend("topright", c("Prey", "Predator"), col=c("blue", "red"),
       lty=c(1,1), lwd=c(2,2), bty="n")


# plot shadow (accounting) prices
plot(LV$lvsimdata[,"tseq"],lvsim[["shadowp"]][,1],type='l', lwd=2, col="blue",
     ylim = c(-8,7),
     xlab="Time",
     ylab="Shadow price")
lines(LV$lvsimdata[,"tseq"],lvsim[["shadowp"]][,2], lwd=2, col="red")
legend("topright", c("Prey", "Predator"), col=c("blue", "red"),
       lty=c(1,1), lwd=c(2,2), bty="n")


# plot inclusive weath and value function
plot(LV$lvsimdata[,"tseq"],lvsim[["iw"]],type='l', lwd=2, col="blue",
     ylim = c(-0.5,2.5),
     xlab="Time",
     ylab="Inclusive Wealth / Value Function ($)")
lines(LV$lvsimdata[,"tseq"],lvsim[["vfun"]], lwd=2, col="red")
legend("topright", c("Inclusive Wealth", "Value Function"),
       col=c("blue", "red"), lty=c(1,1), lwd=c(2,2), bty="n")

LVDomo

Seong D. Yun

Lotka-Volterra 2-D Demonstration