. Consider a model for the dynamics of two interacting species given by the following

system of ordinary differential equations

du

dt

= a − (b + 1)u + u

γ

v,

dv

dt

= bu − u

γ

v,

u(0) = u

0 > 0, v(0) = v

0 > 0,

where a, b, γ > 0 are strictly positive constants.

(a) Determine the type of interaction between the species, i.e., is the system competitive, mutualistic or predator-prey (note this may vary depending on the population

size). You may assume without proof that the size of the populations remain strictly

positive for all times. [4 marks]

(b) Determine the unique strictly positive steady state of the system. [1 marks]

(c) Show that the steady state above is linearly asymptotically stable if and only if [5 marks]

a

γ >

γ − 1

_x0001_

b − 1.

2. Consider the system of the previous question modified to include spatial effects as follows

∂tu − du∆u = a − (b + 1)u + u

γ

v, in Ω, t ∈ (0, T]

∂tv − dv∆v = bu − u

γ

v, in Ω, t ∈ (0, T]

∇u · ν = 0, ∇v · ν = 0 on ∂Ω, t ∈ (0, T]

u(x, 0) = u

0

(x) > 0, v(x, 0) = v

0 > 0, in Ω,

with Ω ⊂ R

n

a bounded domain with smooth boundary ∂Ω and ν the outward unit normal

to Ω, and where a, b, γ, du, dv > 0 are strictly positive constants.

(a) For γ = 1 prove that solutions to the system remain positive (you may assume the

existence of bounded solutions). [6 marks]

(b) Determine the spatially uniform steady states of the system. [1 marks]

(c) Show that for a Turing instability to occur we require γ > 1 and that du < dv

. [3 marks

MAGIC 091 Mathematical Biology

3. A competitor species C is introduced to control the numbers of a pest P. Suppose the

dynamics of the populations are governed by the system

Pn+1 =

γ1Pn

1 + k11Pn + k12Cn

,

Cn+1 =

γ2Cn

1 + k21Pn + k22Cn

,

(1)

where γ1, γ2, k11, k12, k21, k22 > 0 are all constants.

(a) Show the system (1) may be scaled, with a scaling you should specify, to

xn+1 =

γ1 xn

1 + xn + k1yn

yn+1 =

γ2yn

1 + k2 xn + yn

,

(2)

where xn, yn denote the scaled pest and competitor populations respectively. [1 marks]

(b) Show that the system has a trivial steady state and in a suitable parameter range,

which you should specify, two biologically relevant semitrivial (i.e., when only one

of the two populations is nonzero) steady states. [2 marks]

(c) If the goal is to drive the pests to extinction, deduce the parameter range in which

the introduction of the competitors is not required. [2 marks]

(d) Compute a value γ

c

1

in terms of the parameters such that for γ1 ≥ γ

c

1

the pests can

not be driven to extinction. You must justify your answer. [5 marks]

4. (a) The population dynamics of a species is governed by the following ordinary differential equation:

dN

dt

= f(N) =

N

2

1 + N2

− aN,

where a > 0 is a constant.

i. Determine the parameter range of a for which there exists a trivial steady state

and two biologically relevant non-trivial steady states. [2 marks]

ii. In the parameter range of a which you found in the previous part, investigate

the stability of the steady states. Hence, state the maximum initial size for the

population, N(0), such that any initial population size below this level will be

driven to extinction and justify your answer. [3 marks]

(b) Let a = 1/2 and consider the modification of the model to include spatial effects

corresponding to

∂tn − D∆n =

n

2

1 + n

2

−

n

2

, x ∈ R

with boundary conditions n(−∞) = 1 and n(+∞) = 0 and n

′

(±∞) = 0.

Derive the travelling wave equation associated with this system and deduce the direction of motion of the travelling wave