LV contains simulated data for a prey--predator (Lotka--Volterra)
bioeconomic model used to illustrate multi-dimensional natural capital
asset pricing (CAPN) methods.
This example corresponds to a two-dimensional deterministic setting.
The dataset includes both approximation data on a Chebyshev grid and
time-series simulation output from the underlying dynamic system.
See vignette("LVDemo") for an example demonstrating the use of this data.
data("LV")A list with two elements:
A data.frame of approximation data evaluated on a \(20 \times 20\) Chebyshev grid:
xs: Prey stock
ys: Predator stock
xdot: Evaluated prey dynamics \(\frac{dx}{dt}\)
ydot: Evaluated predator dynamics \(\frac{dy}{dt}\)
wval: Profit (objective value \(W\) in Fenichel and Abbott (2014))
A data.frame of time-series simulation output from solving the ODE system:
tseq: Time sequence from 0 to 100
xs: Prey stock
ys: Predator stock
The prey--predator system is given by:
Prey (\(X\)): \( \dot{X} = r X \left( 1 - \frac{X}{K} \right) - a X Y - \theta X \)
Predator (\(Y\)): \( \dot{Y} = b X Y - m Y - \gamma Y \)
The biological parameters are:
\(r = 0.025\): Intrinsic growth rate of prey
\(K = 1\): Carrying capacity of prey
\(a = 0.08\): Predation effect on prey
\(b = 0.05\): Prey-to-predator conversion parameter
\(m = 0.01\): Natural mortality rate of predator
\(\gamma = 0.005\): Predator harvest control slope
\(\theta = 0.005\): Prey harvest control slope
The economic objective is defined as: \( W = \text{harv.prey} \, (p_{\text{prey}} - c_{\text{prey}} / X) \, \theta X + \text{harv.pred} \, (p_{\text{pred}} - c_{\text{pred}} / Y) \, \gamma Y \)
The economic parameters are:
\(p_{\text{pred}} = 0\): Price per unit harvest of predator
\(p_{\text{prey}} = 25\): Price per unit harvest of prey
\(c_{\text{prey}} = 0.1 \, p_{\text{prey}}\): Cost per unit of prey effort (Schaefer model, with \(q = 1\))
\(c_{\text{pred}} = c_{\text{prey}}\): Cost per unit of predator effort (Schaefer model, with \(q = 1\))