optistock (version 0.0.2)

daily_cost_fun: Compute the instantaneous cost of raising hatchery fish

Description

This is a multivariable function of both time and number of recruits raised. Cost-per-time and cost-per-recruit can be calculated as a quadratic where the slope and exponent can be specified.

Usage

daily_cost_fun(
  time,
  recruits,
  daily_cost,
  time_slope = 0,
  time_exp = 1,
  rec_slope = 1,
  rec_exp = 1,
  type = "multiplicative"
)

Value

A numeric value representing the cost of rearing the number of recruits at a given time and given the number of recruits raised

Arguments

time

The time at which fish are raised in hatchery

recruits

The number of recruits raised

daily_cost

Baseline daily cost to raise a single fish

time_slope

The slope term on the amount of time (see details)

time_exp

The exponent on the amount of time

rec_slope

The slope term on the number of recruits

rec_exp

The exponent on the number of recruits

type

Either multiply the number of recruits times the cost-at-time or add to it (see Details).

Details

The cost-per-fish based on time and number of recruits uses the function:

C = s_Tt^α * s₂R^β + b

if type = "multiplicative". Otherwise it uses:

C = s_Tt^α + s₂R^β + b

if type = "additive"

where C = the cost to rear R number of recruits at time T, the s values are the slopes, α and β, are the exponents on time (T) and recruits (R), respectively, and b is the intercept. The instantaneous cost is really what is of interest, and the number of recruits essentially adjusts the intercept on that dimension of the equation.

Increasing the exponent will dramatically increase the cost of raising hatchery fish as time goes on. Increasing the exponent dramatically increases the cost of raising a greater number of fish. Integrating this equation across time will compute the total cost to raise the number of recruits to time T. Use the total_daily_cost function to do this automatically.

Examples

# compute the instantaneous cost of raising 1000 fish on day 100
daily_cost_fun(time = 100, recruits = 1000, daily_cost = 0.05,
         time_slope = 0, time_exp = 1,
         rec_slope = 0.01, rec_exp = 1)
# plot a curve of instantaneous cost against time
curve(daily_cost_fun(x, 1000, 0.05, 0.01, 1.2, 0.05, 1), 0, 1000,
      xlab = "Time", ylab = "$")
if (FALSE) {
# 3d plot of costs by time and recruit
emdbook::curve3d(daily_cost_fun(x, y, 0.05, 0.01, 1.2, 0.05, 1),
                 from = c(0, 0),
                 to = c(1000, 1000),
                 xlab = "Time", ylab = "Recruits",
                 zlab = "$", sys3d = "wireframe")
}