Mathematical Biology

MAGIC MAGIC 091 MAGIC EXAM: 2024 Mathematical Biology 1. Consider a model for the dynamics of two interacting species given by the following system of ordinary differential equations du = a − (b + 1)u + uγ v, dt dv = bu − uγ v, dt u(0) = u0 > 0, v(0) = v0 > 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] aγ > γ − 1 b − 1. 2. Consider the system of the previous question modified to include spatial effects as follows ∂t u − du ∆u = a − (b + 1)u + uγ v, in Ω, t ∈ (0, T ] ∂t v − dv ∆v = bu − uγ v, in Ω, t ∈ (0, T ] ∇u · ν = 0, ∇v · ν = 0 on ∂Ω, t ∈ (0, T ] u(x, 0) = u0 (x) > 0, v(x, 0) = v0 > 0, in Ω, with Ω ⊂ Rn 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 γ1 P n , 1 + k11 Pn + k12Cn γ2 C n , Cn+1 = 1 + k21 Pn + k22Cn Pn+1 = (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 γ1 xn 1 + xn + k1 yn γ2 yn yn+1 = , 1 + k2 xn + yn xn+1 = (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 γ1c in terms of the parameters such that for γ1 ≥ γ1c 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: N2 dN = f (N) = − aN, dt 1 + N2 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 n n2 − , x∈R ∂t n − D∆n = 2 1+n 2 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. [5 marks] 2 End of paper