LV {capn}

Prey-Predator (Lotka-Volterra) example: two stocks

Description

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).

Usage

## Load dataset
data("lvdata")
## Demonstration of example
# demo(LV, package="capn")
## R-script location
# system.file("demo", "LV.R", package = "capn")

Format

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