---
title: "Optimizing Costs of Fish Stocking"
author: "Paul Frater"
date: "7/13/2020"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library(optistock)
```

## Introduction

Initial propagation efforts by the Wisconsin DNR focused on stocking many small
fish into waterbodies throughout the state. As time has gone by trends have 
shifted from stocking many, small fish to a strategy that inputs fewer, but 
larger fish into systems presumably with the intent that larger fish are less 
susceptible to natural mortality resulting in more fish recruiting to a size
of the minimum legal limit. This is a good strategy, but, as is shown in this
model, may not be the most cost effective.

This model uses knowledge about a fish species or population's growth and 
mortality to determine how many fish should be stocked to achieve a certain 
number of fish that are recruited into the fishery. From this the costs of 
stocking (and changes thereof across time) can be used to determine the most 
cost effective time to stock a particular species of fish into a waterbody.

## Determining Time to Recruitment into the Fishery

With knowledge about a fishes growth, one can determine the time that it takes 
for fish to grow to the minimum legal length (MLL). A basic inversion of the 
von Bertalanffy growth function (VBGF) will predict the time that it takes 
for a fish to reach MLL. Since daily values for hatchery costs will be used in 
this model the time to MLL will be given in days. A straightforward and 
relatively close approximation can be achieved by leaving all VBGF parameters
the same and dividing the *k* parameter by 365. 

## Using natural mortality to gauge stocking number to achieve desired recruits

The number of fish at a future time interval can be determined by the number of
fish at the present time multiplied by the exponent of a natural mortality 
parameter (i.e. N[t]




