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MATH0030
Answer all questions.
1. Two interacting species with densities X,Y are modelled by the equations
dX
dt = X(a bX cY )(1)
dY
dt = Y ( d + eX), (2)
where a, b, c, d, and e are positive real constants.
(a) Briefly discuss the model, identifying carefully the types of species-species
interactions involved. What is the carrying capacity of the species with den_xfffe_sity X [Hint: For the carrying capacity, consider the differential equation
for X when Y = 0.]
(b) Re-scale the equations, using the new variables and parameters:
τ = at, x =
bX
a
, y =
cY
a
, d0 =
d
a
, φ =
db
ae
.
(c) Assuming that φ < 1, find all steady states of the re-scaled system of equations that you found in part (b). Determine whether each steady state is locally stable or unstable. (d) Carefully sketch the phase plane for the system that you found in part (b), again assuming that φ < 1. (e) Assuming that a = d, give conditions on the parameters of the original sys tem (equations (1) and (2)) for the interior steady state (i) to be biologically meaningful and (ii) to also be a stable node. (f) For the case a = b = d = 1 and e = 2, briefly sketch the phase plane of the re-scaled system from part (b), indicating the behaviour in the vicinity of the interior steady state. (20 marks) MATH0030 Page 1 of 3 2. In a single-species population, and to first order in δt, d(t)δt is the probability that an individual chosen at random from the population at time t will die in the time interval [t, t + δt). There is no birth. Let pk(t) denote the probability that, at time t, the population size X is k. (a) Show that for k = 1, 2, ..., dpk dt = d(t)((k + 1)pk+1 kpk). (b) If N(t) denotes the expected population size E(X, t) at time t, show that N(t) = exp _x0012_ Z 0 t d(s)ds N0, where N0 = N(0). (c) Let V (t) be the variance of X at time t. Derive a differential equation for V (t). (d) Suppose that V (0) = V0 and that the death rate satisfies: d(t) = 0 0 ≤ t ≤ t1 d1 t > t1,
where d1 is a positive constant. Find N(t) for t > t1. Using this expression
and your solution to part (c), find V (t) for t > t1.
(20 marks)
3. Hackney’s population of eels is described by the following model. The density
of eels in year t is given by Xt, and Xt+1 satisfies the following equation:
Xt+1 =

2Xt 0 ≤ Xt ≤
K
2
2(K Xt)
K
2 < Xt ≤ K 0 Xt > K,
where K is a positive constant.
(a) Rescale the model by setting xt = Xt/K.
Hence find a function f : [0, 1] → [0, 1] such that xt+1 = f(xt), provided
xt ∈ [0, 1].
(b) Plot the function f(x).
(c) Find the fixed points of the map f and determine whether they are stable
or unstable. [You may assume f
0 (0) = 2.]
(d) Find and plot f
2
(x) ≡ f(f(x)). Hence, or otherwise, determine how many
2-cycles there are of the map f.
(e) Plot f
4
(x). You do not have to justify your answer.
(f) Using your answers to parts (c), (d) and (e), determine how many 4-cycles
there are of the f. Are any stable
(20 marks)
MATH0030 Page 2 of 3
4. In an age structured population of Moray eels, each eel lives at most n years.
We denote by Nk(t) the population density of age k at time t, so the population can
be written as N(t) = (N1(t), …, Nn(t)). The expected number of offspring of each
eel of age k is bk, which is known to decrease with the age of the eel as bk = α
β
k ,
with β and α > 1. The probability p that an individual aged k ≥ 0 (with k = 0
corresponding to newborns) survives to age k + 1 is a constant 0 < p < 1. (a) Show that the evolution of the population follows N(t+ 1) = LN(t), where L is a matrix that you should find. (b) Show that the eigenvalues λ of matrix L, and its eigenvectors v, satisfy the equations nX k=1 _x0010_ p αλ _x0011_ k = 1 β v = 1, λ p , λ p 2 , ..., λ p n 1 . (c) In the scenario that the eels do not have a maximum age limit (n → ∞), find the condition in terms of α, β and p for which the population does not collapse. (d) In an alternative scenario in which n = 3, find the conditions in terms of α, β and p for which the population does not collapse. [You may state without proof theorems from the lecture notes.] (20 marks) 5. The generalised Lotka-Volterra system describing N interacting species of den sities X = (X1, X2, ..., XN ) T is given by the equations: ˙Xi = Xi ri + N X j=1 aijXj , i = 1, ..., N. Here aij are the elements of the interaction matrix A and r = (r1, r2, ..., rN ) T is the intrinsic growth vector. Each component of r can be positive, negative or zero. (a) State Lypanunov’s theorem for the asymptotic stability of a steady state. (b) Prove that if the system has a steady state X inside the positive orthant (R +N ), and the matrix A is negative definite, then the steady state is globally attracting inside the positive orthant. Hint: Consider the function V (X) = N X i=1 φi(Xi), where φi(ξ) = ξ Xi Xi log X ξ i . (c) Given the following interaction matrix for a system composed of 3 species A = 0 3 0 2 3 1 1 1 1 , discuss carefully the interactions between the species and show that this interaction matrix does not support a unique interior steady state. (d) For the interaction matrix given in part (c), by considering d dt log(x1x2x 3 3 ) or otherwise, show that interior orbits are unbounded if r1 + r2 + 3r3 > 0.
(20 marks)
MATH0030 Page 3 of 3