assignment-07.Rmd

---
title: 'Assignment 7: Modelling (9 marks)'
output:
  pdf_document: default
  html_document: default
---

```{r}

library(deSolve)

```


1. In this exercise, we will be working with the Lotka-Volterra competition
model introduced in class. Use the following parameter set in this question. (3
marks)

- `r` = 0.5
- `K` = 1000
- `alpha12` = 3
- `alpha21` = 1.5


    a. Solve the Lotka-Volterra competition model and plot the trajetory of the
    population for 100 time steps. Briefly explain the population dynamics of these
    species. (1 mark)

    b. Catastrophe hits population 2 at t=10, such that their numbers were
    drastically decreased to a quarter of what it was (hint: at t=10, N2 reset
    to 1/4 its non-catastrophe value)! Use a simulation to show the trajetory of
    these species over the 100 time steps. Explain what you saw. (2 marks)


2. In this exercise, we will be working with the malaria dynamics model we
worked with in class, and we will be thinking of ways in which we can
"implement" various mosquito control methods in this hypothetical population.
For each of the following mosquito control strategies, describe how you would
implement them in terms of math. For example, you may wish to modify some
parameter, add a parameter, or out right change the strucutre of the model.
Explain your choices, and include any new parameters or new equations where
applicable (e.g., include an equation to show where a new parameter would
appear). (3 marks)

    a. Use of insecticide to kill off adult mosquitoes. (1 mark)
    b. Use of bednet to reduce contact between mosquito and host. (1 mark)
    c. Provide hosts with vaccines. (1 mark)


3.  The Allee Effect (3 marks)

    Generally, as population size increases, a population will
    experience a decreased growth rate due to greater competition for
    resources. This is a negative density-dependent growth rate, and one
    example of this is the logistic model.

    The Allee effect introduces positive density dependence, where
    increases in population size result in increased growth rates over a
    certain range of population sizes. One way to incorporate
    the Allee effect into the logistic growth equation is as follows:

    $$\frac{dN}{dt} = rN\left(1-\frac{N}{K}\right)\left(\frac{N-A}{K}\right)$$

    Here $r$ represents the growth rate of the population, $K$ is the
    carrying capacity, and $A$ is the critical population size above which 
    the total growth rate is positive.

    a. Take $r=1$, $A=10$, and $K=50$. Plot $\frac{dN}{dt}$ vs. $N$ for
    $0 \le N \le 55$. For which values of $N$ is the growth rate
    ($\frac{dN}{dt}$) positive or negative? (0.5 marks)

    b. Plot the **per capita** growth rate ($\frac{1}{N}\frac{dN}{dt}$)
    vs. $N$ for this model of the Allee effect and for the logistic growth
    model: $\frac{dN}{dt} = rN(1-\frac{N}{K})$. (1 marks)

    c. What do you notice about the density ($N$) dependence of the per capita
    growth rate in each case? Hint: in the logistic model, the growth rate **per
    capita** (per organism) decreases in a straight line as $N$ increases. (0.5
    marks)
    
     d. What happens to the Allee effect as $A$ decreases? Plot curves for $A=0$
     and a few values of $A>0$. (0.5 marks)

    e. Describe two biological situations in which you might expect to see an Allee
    effect (either weak or strong). (0.5 marks)