SECTION A

1. a) Red Foxes and European rabbits can be found in ecosystems all over the UK,
often forming a binary predator-prey relationship within the ecosystems. Based on the
assumptions below, write down a pair of differential equations (do not include code) that
describe the changes over time of fox population F, and rabbit population R. Clearly
define the state variables and parameters, and specify their units.
● Rabbits have access to unlimited food
● The only predator of rabbits are foxes
● Foxes fully depend on rabbits as their food supply, and die otherwise
[5%]

b) Discuss how you can find the equilibrium points of the system? Discuss how you
could check the stability properties of the equilibrium points, i.e. whether they are
asymptotically stable, Lyapunov stable or unstable, via simulation (only discuss, don’t
simulate).
[5%]

c) The system in part (a) was simulated using the Euler method and its trajectory
was plotted on the phase plane in Fig 1(i). This was repeated with a different time step Δt
in Fig 1(ii). Which of the two plots used a larger Δt? Justify your answer.

Fig 1(i) Fig 1(ii)
[5%]

d) Write 3 circumstances for which it would be applicable to use an agent-based
model, rather than an equation-based approach in order to simulate this system?
[5%]

2. a) Write the differential equation (do not include code) and the spiking procedure for
a Leaky Integrate and Fire (LIF) neuron with constant input. Define the state variables
and parameters and provide their units.
[5%]

b) Assume that the constant input remains greater than the threshold for the
following two questions.
i) If threshold increases, will the mean time between two consecutive spikes
increase or decrease and explain why?
ii) If τm (the time constant of leaky integration) increases, will the mean time
between two consecutive spikes increase or decrease and explain why?
[5%]

SECTION B

3. The equations of a predator-prey system are simulated using a 2nd order Runge Kutta
method with time step Δt=0.001 and simulated from time t=0 to t=100 (arbitrary time
units). At t=100 units, the simulated predator and prey values are each off from their ideal
ones, in absolute value by approximately 1e-5. With Δt=0.01, calculate how far off the
predator and prey values are expected to be from their ideal ones, in absolute value, at
t=100 units. Explain your working / calculation.
[15%]

4. Suppose that the volume of a plant bud is proportional to the amount of water it contains.
Take the bud to be spherical. Assume that the rate at which it absorbs water from its
parent stem is proportional to the radius of the bud. Further, assume that the bud loses
water at a rate proportional to its surface area.
Write a differential equation (do not include code) that describes the growth of the volume
of this plant bud over time, as a function of its radius. Explain how you constructed this
equation. What is the differential equation for the rate of change of the radius of the bud?
(Note: This is not necessarily a realistic model of bud growth.)
[15%]

5. Complete the code snippet below, to simulate the time evolution of the logistic growth
system: dN/dt = aN(1-N/Nc) using ode45, starting from t=0 to t=100 (arbitrary time units),
with an absolute tolerance of 1e-5, and a relative tolerance of 1e-6. Use a=0.2, Nc=5000,
N(t=0)=1. You only need to type/write the code on paper, not simulate in Matlab (though
you can, to verify it). Insert your own code at the <your code> locations.
Explain briefly, in your own words, how the ode45 function solves the initial value
problem to maintain the error within tolerance.