We now provide you with code for simulating the model above

$($

see below

$)$

$.$

$$This consists of two functions, which you can inspect below. All can be seen in the live docs. Read them to help answer the remainder of question

$5$

$,$

$$which is situated below the code.

parameterise

$\_$

basic

$\_$

model

solve

We also provide a variable data, that provides noisy data on the predator and prey populations taken by a biologist...

data

$[$

:

$,$

$1$

$]$

$$contains the timepoints

$($

measured in years

$)$

$$over which the data was taken

data

$[$

:

$,$

$2$

$]$

$$represents the estimated prey population

$($

measured in hundreds, i

$.$

e

$.$

$$data

$[$

$2$

$,$

$2$

$]$

$=$

$3$

$$means

$3$

$0$

$0$

$$estimated animals at timepoint

$2$

$)$

$$over these timepoints

data

$[$

:

$,$

$3$

$]$

$$represents the predator population over these timepoints, measured in tens.

$1$

$0$

$0$

$1$

$\backslash$

times

$3$

$$Matrix

$\{$

Float

$6$

$4$

$\}$

:

$$

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$.$

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$$

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$$solve

$($

f::Function, tspan, x

$0$

::Vector

$)$

Numerically solves the ODE x

$$

$($

t

$)$

$=$

$$f

$($

x

$($

t

$)$

$,$

$$t

$)$

on the timespan whose start

$/$

end points are enclosed in tspan. Recall that x

$$

$$denotes the time derivative of x

$($

t

$)$

$,$

$$i

$.$

e

$.$

$$dx

$/$

dt

$.$

Example

solve

$($

f

$,$

$($

$0$

$,$

$1$

$0$

$)$

$,$

$[$

$1$

$,$

$0$

$]$

$)$

solves between t

$=$

$0$

$$and t

$=$

$1$

$0$

$,$

$$using initial conditions x

$0$

$=$

$[$

$1$

$,$

$0$

$]$

$.$

$$ e

$)$

$$How would the basic model change if we instead measured predators and prey in units of a single animal?

Hint: take

$$

$=$

$$

$1$

$0$

$0$

$$and

$$

$=$

$$

$1$

$0$

$.$

$$

Can you rewrite the differential equation in terms of these two variables instead, using the chain rule?

f

$)$

$$Build a function simulation

$($

p

$)$

$.$

$$It must take in a vector of

$4$

$$parameters

$($

e

$.$

g

$.$

$$simulation

$($

$[$

$1$

$,$

$2$

$,$

$3$

$,$

$4$

$]$

$)$

$.$

$$It must output a

$1$

$0$

$0$

$1$

$\backslash$

times

$2$

$$matrix holding the solution of the differential equation: i

$.$

e

$.$

$$the populations of prey

$($

$1$

st column

$)$

$$and predators

$($

$2$

nd column

$)$

$$over the timepoints range

$($

$0$

$,$

$1$

$0$

$,$

step

$=$

$0$

$.$

$0$

$1$

$)$

$.$

$$The initial conditions can be

$[$

$1$

$,$

$1$

$]$

$.$

$$

g