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Computer Lab PFS.Rmd
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---
title: 'Computer Lab PFS: Impacts of fishing in an ecological context'
author: "Max Lindmark & Anna Gårdmark"
date: "7 december 2018"
output: pdf_document
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
# Overview
In this practical you will use dynamic population models of a predator and its prey, based on Yodzis (1994), in which he discusses the model by Leslie (1948) and later May et al (1979). These models were applied to explore how predator-prey interactions affects potential harvesting of a population. You will be using R to explore the dynamics of the model. R is a statistical software that is very popular in especially biological sciences and can be downloaded for free. If you have not installed R and R-studio already, please see the document **"First time R user"**. R-studio is just an interface for R with many helpful utilities. It is an advantage if you know some Basic R, but you should be able to follow along this lab without any prior knowledge if you consult **"First time R user"**. In this document R-code is text with grey background. You can copy these chunks of code to your R-script in R-studio (instead of pasting it directly into the console because it quickly becomes hard to overview your work!). For some it might also be a repetition of calculus and algebra, and you will also practice some basic population dynamics and equilibrium theory.
There are hints for most questions **at the end of this document**. Do have a look at them if you get stuck! You can also look at the solutions or ask an instructor if you get really stuck!
You might also find it useful to check out the solutions to a similar lab in a previous course (*Ecology for fish management and conservation*) to get a feeling for how these questions can be addressed.
# The Model
Here we will implement and analyze the model by Yodzis (1994), which is a simplistic predator-prey model with harvesting (fishing) on the prey species. The model received attention when it was used by May et al. (1979) to study Baleen whale predators (parvorder *Mysticeti*) and their prey, krill (order *Euphausiacea*). This is a continuous-time model, based on the logistic growth equation (the Verhulst model). The following differential equation describes the rate of change in number of individuals (density) in the prey (krill) population, $N_k$ (subscript $k$ for krill):
\begin{equation}
\label{eq:1}
dN_k/dt = r_kN_k(1-N_k/K) - aN_kN_w - EN_k
\end{equation}
where $r_k$ is the intrinsic growth rate and $K$ is the carrying capacity (when $N_k=K$ growth stops due to resource limitation) and $E$ is the constant fishing effort.
The dynamics of the predator population is described by a logistic growth equation in which their carrying capacity is proportional to the amount of prey available (note, with a different proportionality constant ($\alpha$), not $a$ as in the predation term!) (subscript $w$ is for whale):
\begin{equation}
\label{eq:2}
dN_w/dt = r_wN_w(1-N_w/\alpha N_k)
\end{equation}
* **Question 1: In ecology, the functional response describes the intake rate of food as a function of food density. Identify the term for the predator's functional response, and explain which type it is (you can search the webs if you need to!). **
* **Question 2: The functional response is a key function when studiyng predator-prey interactions. Draw Holling's functional responses (Type 1, 2 and 3) (on a piece of paper or in the computer). Describe the assumptions behind each function and discuss for which species they might be relevant**
We have an equilibrium when the rate of change of both equations equal 0. One find to visualize equilibria is to find the so called zero-growth isoclines (the lines where each single differential equation's rate of change is 0) and plot them in the phase plane (i.e. with the predator and prey densities on the y- and x-axis).
* **Question 3: Find the expressions (there are two!) for $N_k$ that satisfy $dN_k/dt = r_kN_k(1-N_k/K) - aN_kN_w - EN_k = 0$ **.
* **Question 4: Repeat question #3 for the whale species**
* **Question 5: a) Plot the whale-isocline as a function of prey density. b) Using the lines()-function, add in the prey isoclines as well. Use the following parameters: $r_k=0.1$, $K=10$, $a=0.005$ and $\alpha=1$. Use three different values of effort: 0, 0.005, 0.008. Set the plotting range like this: plot(Y ~ X, ylim = c(0, 15), xlim = c(0, 15)). Consider predator and prey densities between 0 and 30. c) Locate the points of stable equilibrium. d) Explain how the stable equilibrium varies with effort. e) How are the whales affected by harvesting of krill?**
* **Question 6: How does whale abundance affect yield of krill? Yield can be defined as: $Y = EN_k$. Explore fishing efforts between 0 and 0.1 and whale densities of 1, 3 and 5**
* **Question 7: Discuss how the results in 5 and 6 depend on the assumptions of the functional response. Would they differ with another type, and what would that imply for fisheries managers and conservatioists? **
## References
May et al, P. 1979. Management of Multispecies Fisheries. *Science*, 205, 4403.
Yodzis, P. 1994. Predator-Prey Theory and Management of Multispecies Fisheries. *Ecological Applications*, 4, 1, 51-58.
## Hints
Question 1: No hint here!
Question 2: No hint here!
Question 3: No hint here!
Question 4: No hint here!
Question 5: Since we want to plot a curve, we need several values of predator and prey densities. We can simply replace $N_k$ and $N_w$ with vectors of the whole range of $N$ values. You can create a sequential vector using the seq()-function or with c() and give the values manually.
Question 6: Substitute N in the equation for yield with the equation for the krill-isocline.
Question 7: It is likely that if the predator would have had e.g. a type three functional response that it would not be able to survive on extremely small krill densities, as in the current model. Note also that the whale-carrying capacity declines with resource densities, which is why it can survive on very small krill densities. If they whales require a specific threshold density to persist, this might introduce conflicts between conservation of whales and maximizing exploitation of krill.