# Exercises

1. Consider the modified Lotka-Volterra predator-prey system in which the predator disperses via diffusion much faster than the prey; the dimensionless equations are du dv d2v

where a > 0, 0 < b < l and u and v represent the predator and prey respectively. Investigate the existence of realistic travelling wavefront solutions of speed c in terms of the travelling wave variable x + ct, in which the wavefront joins the steady states u = v = 0 and u = b, v = l — b. Show that if c satisfies 0 < c < [4a (l — b)]l/2 such wave solutions cannot exist whereas they can if c > [4a(1 — b)]l/2. Further show that there is a value a* such that for a > a* (u, v) tend to (b, 1 — b) exponentially in a damped oscillatory way for large x + ct.

2. Consider the modified Lotka-Volterra predator-prey system dU = AU (l — UQ — BUV + DiUxx, d V

— = CUV — DV + D* Vxx, d t where U and V are respectively the predator and prey densities, A, B, C, D and K, the prey carrying capacity, are positive constants and Dl and D2 are the diffusion coefficients. If the dispersal of the predator is slow compared with that of the prey, show that an appropriate nondimensionalisation to a first approximation for D2/Dl ~ 0 results in the system du d2u dv

Investigate the possible existence of travelling wavefront solutions.

3. Quadratic and cubic autocatalytic reaction steps in a chemical reaction can be represented by

A + B ^ 2 B, reaction rate = kqab, A + 2 B ^ 3 B, reaction rate = kcab2, where a and b are the concentrations of A and B and the k's are the rate constants. With equal diffusion coefficients D for A and B these give rise to reaction diffusion systems of the form d a 2 2

First nondimensionalise the system and then show that it can be reduced to the study of a single scalar equation with polynomial reaction terms.

Consider in turn the situations when there is only a cubic autocatalysis, that is, kq = 0, kc = 0, and then when there is only a quadratic autocatalysis term, that is, when kq = 0, kc = 0. In the one-dimensional situation investigate possible travelling waves in terms of the travelling wave coordinate z = x — ct where c is the wavespeed.

4. A primitive predator-prey system is governed by the model equations d u „d2 u