The LV provides the data and functions to simulate prey-predator (Lotka-Volterra) model. The original code was written by Joshua Abbott in MATLAB and Seong Do Yun adapted it to a package example. The prey-predator model is:
Prey (\(X\)): \( \dot{X} = rX \left( 1 - \frac{X}{K} \right) - aXY - \theta X \), and
Predator (\(Y\)): \( \dot{Y} = bXY - mY - \gamma Y \).
The parameters are given as: \(r = 0.025\): intrinsic growth rate for prey, \(K = 1\): carrying capacity for prey, \(a = 0.08\): predator-related mortality parameter for prey, \(b = 0.05\): predator/prey uptake parameter for predator, \(m = 0.01\): natural mortality for predator, \(\gamma = 0.005\): slope for linear predator harvest control rule, and \(\theta = 0.005\): slope for linear prey harvest control rule
The predator with no economic value (unharvested) is designed for the economic program as:
\( W = harv.prey(p.prey-c.prey/X)\theta X + harv.pred*(p.pred-c.pred/Y)\gamma Y\).
The paramters are: \(p.pred = 0\): price per unit harvest of predator, \(p.prey = 25\): price per unit harvest of prey, \(c.prey = 0.1 p_prey\): cost /per unit of prey effort in Schaefer model (really c/q with q=1), and \(c.pred = c_prey\): cost per unit of predator effort in Schaefer model (really c/q with q=1).
## Load dataset
data("lvdata")
## Demonstration of example
# demo(LV, package="capn")
## R-script location
# system.file("demo", "LV.R", package = "capn")lvaproxdata: a data.frame for approximation (evaluated on (20 x 20) Chebyshev nodes)
xs prey stock
ys predator stock
xdot evaluated xdot \(\frac{dx}{dt}\)
ydot evaluated ydot \(\frac{dy}{dt}\)
wval profit (W in Fenichel and Abtott (2014))
lvsimdata.time: a data for time simulation (101 ODE solution)
tseq time sequence from 0 to 100
xs prey stock
ys predator stock