Download Math Biology: Problems III [3.1] Termite nest building. Termites build

Math Biology: Problems III
[3.1] Termite nest building. Termites build nests by regurgitating material from their guts and
applying it either to the surface where the nest will be or on previously applied nest material.
As they do so they emit a pheromone, a volatile chemical that attracts other termites. The
nest starts off as a number of pillars with a characteristic spacing. Eventually arcs are formed
between the pillars, a cover is put over the whole structure and the complex architecture of
the nest is built up. We will consider the initial stage of nest building. Let the density of
termites be n(x, t) and the concentration of pheromone be p(x, t). Pheromone is assumed
to be emitted at a rate proportional to the density n, decays at a constant rate and diffuses
with coefficient Dp . The termites are assumed to attain a typical density K in the absence
of pheromone and diffuse with coefficient Dn . The resulting chemotactic equations take the
form
∂n
∂2n
∂
n
= rn 1 −
+ Dn 2 −
∂t
K
∂x
∂x
∂p
χn
∂x
,
∂2p
∂p
= αn − βp + Dp 2
∂t
∂x
(a) Linearize about the steady state (n∗ , p∗ ) = (K, αK/β), and show that the Jacobian matrix
governing stability with respect to perturbations of wavenumber k is
χKk 2
−r − k 2 Dn
J(k) =
α
−β − k 2 Dp
(b) Show that instability can only occur if a2 (q) < 0 for some q, where q = k 2 and
a2 (q) = Dn Dp q 2 + (βDn + rDp − αχK)q + rβ
(c) Treating the chemotaxis sensitivity χ as a bifurcation parameter, a typical sketch of a2 (λ) =
0 in the (q, χ)–plane is shown in figure 3. Show that there is a Turing bifurcation point at
(qc , χc ) such that if χ > χc there is a range of q for which a2 (q) < 0. Since a2 (q) and
da2 (q)/dq = 0 at the bifurcation point, deduce that
λc = rβ/(Dn Dp )
One thus finds that as long as the sensitivity χ is sufficiently high, the system leads to a
pattern of aggregation of termites and nest material, with the spacing between nest pillars
approximated by lc = 2π/kc where kc2 = qc .
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marginal stability curve
χc
qc
Figure 3: A sketch of a2 (q) = 0 in the (q, χ)–plane.
[3.2] Model of phagocytes. Polymorphonuclear (PMN) phagocytes (white blood cells) are
generally the first defense mechanisms employed in the body in response to bacterial invasion.
PMN phagocytes are rapidly mobilized cells that migrate across the walls of small connective
veins to ingest and eliminate microbes and other foreign bodies in the tissue. Lauffenberger
and Kennedy (1983) introduced a chemotactic model to describe this process, which can be
written in the dimensionless form
∂u
∂
∂2u
= α(1 + σc − u) + 2 − δ
∂t
∂x
∂x
∂c
u
∂x
,
γc
uc
∂2c
∂c
=
−
+ρ 2
∂t
1+c κ+c
∂x
where c is the density of bacteria and u is the density of phagocyte.
(a) Interpret the various terms of the model based on the following assumptions: phagocytes
exhibit chemotaxis towards relatively high bacterial densities, phagocytes enter the domain at
a rate f1 (u, b) and die at a constant rate g1 , bacteria grow at a rate f2 (c) and are eliminated
at a rate g2 (u, c).
(b) Show that there are two types of uniform steady state solutions: (i) b = 0, u = 1 and (ii)
b > 0, u = 1 + σb. Identify the meaning of these steady states.
(c) Determine the Jacobian matrix governing stability of a steady state (u∗ , c∗ ) with respect
to a nonuniform perturbation of wavenumber k.
(d) For the steady state (u∗ , c∗ ) = (1, 0), show that the eigenvalues of the matrix in (c) are
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given by
1
λ1 = −ρk 2 + γ − ,
κ
λ2 = −k 2 − α
Which mode (value of k) is most likely to cause instability and discuss the implication of this
for pattern formation.
(e) For the second class of steady state, u∗ = 1 + σc∗ and c∗ > 0, show that the Jacobian
matrix is
J(k) =
−ρk 2 + F (c∗ ) −H(c∗ )
δu∗ k 2 + ασ
−k 2 − α
where
F (c) =
c(1 + σc)(1 − κ)
,
(1 + c)(κ + c)2
H(c) =
c
κ+c
and that the eigenvalues have negative real parts provided the following inequalities are satisfied:
tr J(0) < (1 + ρ)k 2
and
det J(0) > F (v ∗ ) − ρk 2 − ρα − δu∗ H(v ∗ ) k 2
[3.3] A simplified version of the Gierer–Meinhardt reaction diffusion system for animal coat
patterns is given by
∂u
u2
∂2u
=
− bu + 2 ,
∂t
v
∂x
∂v
∂ 2v
= u2 − v + d 2
∂t
∂x
where b and d are constants.
(a) Find the Jacobian matrix at the positive spatially uniform steady state.
(b) Derive the conditions for stability of the steady state with respect to spatially uniform
perturbations
(c) Determine the condition for marginal stability of the steady state in the presence of diffusion.
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(d) Hence show that the parameter domain for a Turing instability of the steady state is given
by
0 < b < 1,
√
bd > 3 + 2 2
and sketch the region of parameter space in which such an instability occurs.
(e) Show that the critical wavenumber kc for marginal stability is given by kc2 = (1 +
√
2)/d.
[3.4] Consider the Segel–Jackson model for a spatially distributed predator–prey system.
∂V
∂2V
= V R(V ) − AV E + µ1 2 ,
∂t
∂x
∂2E
∂E
= BV E − M E − CE 2 + µ2 2
∂t
∂x
where V (x, t) = the prey (‘victims’), E(x, t) = the predators (‘exploiters’), and
R(V ) = K0 + K1 V
(a) Interpret the various terms in the above differential equations and explain why −CE 2 can
be described as a ‘combat’ term.
(b) Show that the model will not yield a diffusive instability when C = 0 and interpret this
result.
(c) Set M = 0 and rewrite the equations in the following dimensionless form
∂v
∂2v
= (1 + kv)v − aev + δ 2 2 ,
∂t
∂x
∂e
∂2e
= ev − e2 + 2
∂t
∂x
where e = EC/K0 , v = V B/K0 , distances are scaled in units of K0 /µ2 and time is scaled
in units of K0 . Show that q = K1 /B, a = A/C and δ 2 = µ1 /µ2 .
(c) Show that the nontrivial steady state of the system is
e=v=
1
a−q
(d) Show that the Turing instability condition is
√
q − δ2 > 2 a − q
and that the wavenumber of the excitable modes is given by
1
k2 = √
δ a−q
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[3.5] Two–dimensional patterns. Consider the two–dimensional activator-inhibitor system
∂u
= f (u, v) + D1 ∇2 u,
∂t
∂v
= g(u, v) + D2 ∇2 v
∂t
with 0 < x < Lx , 0 < y < Ly and no–flux boundary conditions.
(a) Show that the steady state of a two–dimensional activator-inhibitor system is marginally
stable with respect to excitation of the (m, n) mode cos(mπx/Lx ) cos(nπy/Ly ) if
n2
L2
gv∗
2
∗
m + 2 = 2
fu +
γ
2π D
β
where Lx = L, Ly = γL, D1 = D and D2 = βD.
(b) Suppose that all parameters are fixed except the length ratio γ. Furthermore, assume that
the only excitable modes are those that satisfy the above equation. If the excitable mode is
m = n = 4 when γ = 1, determine the succession of excitable modes (mj , nj ) corresponding
to a sequence of increasing values of γ.
(c) Sketch a few of these patterns in the (x, y)–plane by shading regions for which the corresponding mode cos(mj πx/Lx ) cos(nj πy/Ly ) > 0.
(d) Now suppose that the boundaries are not impermeable but are kept at the steady state
concentrations
u(0, y) = u(L, y) = u(x, 0) = u(x, γL) = u0 ,
v(0, y) = v(L, y) = v(x, 0) = v(x, γL) = v0
How would this change the assumed spatial dependence of the perturbations? How would it
effect the form of the allowed wavenumbers?
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