辅导案例-MA4MB-Assignment 20

MA3MB/MA4MB - Mathematical Biology Assignment 2020
Your attempts to the following questions should be handed into the JJT Support Centre by
12.00hrs on Friday 13th March 2020 (Friday of 9th week of Spring Term). The assignment
constitutes 20% of the final mark for MA3MB/MA4MB and will be marked out of 50
(Question 1 - 20 marks, Question 2 - 30 marks) with further details on the marking
breakdown provided beside each question.
The usual University rules for late work apply - 10% of the total marks available (50 here)
for every day for the first 5 days, thereafter 0 marks unless extenuating circumstances are
granted. Marked assignments will be available for collection from the JJT Support Centre
on Friday 3rd April 2020.
Question 1
Consider the single species population model defined by
dR
dt
=
gR
R +Rm
− dR, ∀t > 0,
where g, Rm and d are all positive parameters and R(0) = R0. Here as in lectures, we
consider R ≥ 0.
(a) Describe the biological meaning of each term on the right-hand side of the equation.
[4 marks]
(b) Determine the steady-states of the system. Discuss any constraints on the model
parameters in order for the model to admit biologically meaningful solutions. [6 marks]
(c) Determine the steady-state stability and discuss any variation in this with respect to
the model parameter values. [8 marks]
(d) Discuss the biological significance of the steady-state stability and how this affects
the overall population levels. [2 marks]
1
Question 2
Consider the following non-dimensional model
du

= u(1− u− αabv), (1)
dv

= ρv(1− v + αbau), ∀t > 0, (2)
which describes the interaction between two species denoted u = u(τ) and v = v(τ), with
initial conditions u(0) = u0 and v(0) = v0. Here all parameters and variables are taken
to be positive.
(a) Show that two of the steady-states of equations (1) and (2) are given by
(u∗1, v

1) = (0, 0) (u

2, v

2) = (0, 1)
and determine any remaining steady-states which the system may exhibit. Don’t forget to
discuss any bounds on the system parameters to ensure the steady-states are biologically
meaningful. [9 marks]
(b) Discuss the steady-state stability of (u∗1, v

1) and (u

2, v

2). [10 marks]
(c) Discuss the stability of any remaining steady-states determined in 2(a). [6 marks]
(d) Outline any model scenarios based upon your analysis (your responses to 2(a)-2(c))
which may occur based upon how parameter values in the model may or may not affect
the occurrence of steady-states and their stability. [5 marks]
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