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")