Exercises CH924: Non-linear systems
Exercise 1
For the systems below do the following: (1) Find the equilibrium points and analyse
their stability by linear stability analysis; (2) Scetch the vector field and a plausible
phase portrait.
(a) ẋ = −xy, ẏ = (1 + x)(1 − y)
(b) ẋ = y 2 − x2 , ẏ = x − 1
Exercise 2: Human growth factor
(Exercise 8.12 from the book)
The rate at which a human body grows is governed by a certain growth hormone.
Protein reserves, expressed here as muscularity M , regulate the blood concentration G of the growth factor. The following equations describe this interaction
dM
= φ − ρM − µG(1 + γM )
dt
dG
= ν(M − M0 )G
dt
(1)
(2)
where φ, ρ, µ, γ, ν, M0 are positive parameters. At time t = 0 whe have M (0) > 0
and G(0) > 0.
(a) Draw a phase plane portait of the system for the case where φ > ρM0 . Indicate
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and label clearly the null clines and sketch the direction field qualitatively, employing a few well-chosen arrows. The systems has two equilibrium points. Indicate
these points in the phase plane portrait. One of these two points can readily be
seen to be unstable. Label this point ’unstable’ in your diagram, and label the
remaining equilibrium point ’stable’. Give formulae for the values of M and G at
the equilibrium point which you labelled ’stable’ in the previous part. Use linear
stability analysis to confirm that the point labelled ’stable’ is in fact stable.
(b) Draw a second phase plane portrait of the system, this time for the case
φ < ρM0 . Indicate and label clearly the null clines and sketch the direction
field qualitatively, employing a few well-chosen arrows. This second phase plane
portrait should also exhibit two equilibrium points. Label one of them ’irrelevant’
and the other ’relevant’ (a calculation is not required). Use linear stability analysis
to establish that the equilibrium point labelled ’relevant’ is stable.
(c) Let Ḡ denote the value of G at the stable equilibrium point. Sketch the graph
of Ḡ as a function of the parameter φ, where it is assumed that all other parameters
are held constant. (Hint: pay close attention to the point where phi crosses the
value ρM0 .
Exercise 3: Population dynamics
(From Edelstein-Keshet: Mathematical models in biology)
Species may derive mutual benefit from their association; this type of interaction
is known as mutualism. The following set of equations has been suggested for
describing a pair of mutualists
N1
dN1
= rN1 1 −
dt
κ1 + αN2
dN2
N2
= rN2 1 −
dt
κ2 + βN1
where Ni is the population for the ith species, and αβ < 1.
(a) Explain why the equations describe a mutualistic interaction.
(b) Determine the qualitative behaviour of this system by phase-plane and linearisation methods.
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(c) Why is it necessary to assume αβ < 1?
Exercise 4: Positive feedback to one gene
(Adapted from Edelstein-Keshet: Mathematical models in biology)
Consider a gene that is directly induced by m copies of the protein E for which it
codes. Let M be the concentration of messenger RNA (mRNA). Then the following
equations have been suggested as a (dimensionless) model for this positive feedback
loop
Em
− αM
1 + Em
Ė = M − βE
Ṁ =
(a) Sketch the function
f (E) =
(3)
(4)
Em
1 + Em
for a few values of m.
(b) Show that one equilibrium point is given by E = M = 0, and that others
satisfy
E m−1 = αβ(1 + E m )
For m = 1 show that this steady state exists only if αβ ≤ 1.
(c) Let m = 1 and αβ > 1. Show that the only steady state E = M = 0 is stable.
Draw a phase-plane diagram of the system.
(d) Let m = 2. At the equilibrium point the value of E is given by
p
1 ± 1 − 4α2 β 2
E=
2αβ
Conclude that there are two solutions if 2αβ < 1, one if 2αβ = 1 and none if
2αβ > 1. Sketch these three scenarios in the M E-plane (Hint: recall that the
equilibrium points are the intersections of the M and the E null clines).
(e) Let m = 2 and 2αβ < 1. Draw a phase-plane diagram of the system. Indicate
whether the equilibrium points are stable or unstable.
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