MATH3063: Nonlinear ODEs & Applications
Nonlinear ODEs & Applications
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MATH3063: Nonlinear ODEs & Applications
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Ecosystems such as lakes, forests, coral reefs, and deserts, are continuously exposed to gradual changes in
climate, available nutrients, habitat destruction or exploitation. Some of these effects are caused by humans,
others are a natural phenomenon. It is usually assumed that the ecosystem will repond to these gradual changes
in a smooth way - that is, small changes in the environment lead to small changes in the ecosystem. However,
in a 2001 review by Scheffer et. al [1], it was revealed that ecosystems do not always change smoothly. Strong
evidence was put forward to suggest that the ecosystems can experience drastic switches to a different state.
Often this drastic situation is catastophic in that the process can not be easily reversed.
Arid and semi-arid ecosystems may display the castrophic behaviour suggested by Scheffer et. al. Desertifi-
cation is the loss of vegetation in arid and semi-arid ecosystems. Desertification is often listed as the primary
ecological threat facing the modern world. In this assignment, we will investigate a relatively simple model
of desertification and how small changes in the ecosystem might lead to a catastrophic outcome for the arid
environment, one with no hope of return.
Let A be the amount of barren soil in an arid ecosystem. We will assume a simple model of the change in
barren soil over time as
dA
dt
= c− rA+ δ A
4
K4 +A4
,
where c, r, δ,K are positive parameters. The ecological meaning of each parameter is as follows.
c: the rate of constant destruction of vegetation.
r: the relative rate of recolonisation of soil by vegetation.
δ: the relative rate of erosion by wind and runoff.
We will perform bifurcation analysis to investigate whether the rate of constant destruction can lead to a
catastrophic increase in barren soil, and hence, an irreversible desertification process.
1. (a) For each of the parameters c, r, δ, give one human process and one natural process that would change
the value of the parameter. (You might find the section in [1] on desertification helpful. )
(b) Show that the model can be non-dimensionalised to the form
dx
dτ
= a− bx+ x
4
1 + x4
Give an expression for x, τ, a, b in terms of the original variables A, t and constants c, r, δ,K.
We want to analyse the effect of the value of c and r on the barren soil. Hence, we will examine the parameters
a, b. Only consider ecologically relevant values of x (so non-negative values) and parameter values a > 0 and
b > 0.
2. (a) Suppose that a = 0, that is, there is no constant rate of increase in barren soil. In this case, show
that x = 0 is a fixed point and use linear stability analysis to determine the stability for different
values of b > 0.
(b) When a = 0, draw a bifurcation diagram for the parameter b. Give the value bc such that, if b > bc
there is a single fixed point at x = 0, and if b < bc there are two more positive fixed points. List any
bifurcations that occur and the corresponding bifurcation value.
3. (a) The right hand side of the equation for
dx
dτ
can be rewritten as f(x) − g(x) where g(x) = −a + bx
and f(x) = x
4
1+x4 . Use this decomposition to plot the phase line for the system in the following three
cases.
1. b = 1, a = 0.1
2. b = 0.5, a = 0.1
3. b = 0.2, a = 0.1
(b) Find parametric equations for the bifurcation curves in (a, b) parameter space.
(c) Use the parametric equations to plot a stability diagram in (a, b) space. (You will need to use a
computer to paremetrically plot the bifurcation curve. I suggest Mathematica or Desmos).
4. (a) Suppose a particular arid ecosystem begins with x(0) close to 0 and with a = 0 and b = 0.2.
Qualitatively describe what happens to the particular solution x as a is increased. What happens if
a then goes back to zero?
(b) Explain what the results of 4.a imply for desertification. Do the results agree with the comment
by Scheffer et. al. [1] that ecosystems may experience drastic, irreversible changes due to gradual
changes in the environment? Why or why not?