MAC297 MATHEMATICAL BIOLOGY

1.

Consider the dimensionless model involving two species,where u(t) and v(t) are population densities and α and β are positive constants.

(a) (i) What type of interaction does this model describe for α ∈ (0, 1)? Briefly justify your answer. [2]

(ii) Briefly explain role of species u in the death rate of v. [2]

(iii) Briefly describe the interaction described by the model for α > 1. [1]

In the remainder of the question assume α ∈ (0, 1).

(b) Find all of the biologically relevant steady-states (u ∗ , v∗ ) of equations (∗∗). Why is (0, 0) not a valid steady-state? State the biological interpretation of those that are valid. [4]

[Hint: the steady-states are very simple in form.]

(c) Find the stability matrix for equations (∗∗) and determine the stability of each of the valid steady-states found in part (b). [6]

(d) Regarding the stability of the coexistent state in part (c), you should find that βc = (1 − 2α − α 2 )/(1 + α) is a significant value of β, provided βc > 0. Find the largest value of αc such that a Hopf bifurcation is possible for any α ∈ (0, αc). [3]

(e) Write down the null-clines and sketch them in the region u ≥ 0 and v ≥ 0 in a phaseplane. By deducing the signs of f(u, v) and g(u, v) in each of the regions bounded by the null-clines, indicate the general direction of the trajectories between and on the null-clines and use this information to sketch representative trajectories. [8]

(f) Assuming that α ∈ (0, αc) and β is such that the coexistent state is unstable,

(i) draw in your diagram a plausible confined set based on the general direction of the trajectories shown in your phase-plane. [2]

[Hint: look for a simple shape to use as the boundary; the plausibility of the confined set should be clear from the arrows in your diagram without the need of further analysis.]

(ii) having established a confined set, how do the solutions u(t) and v(t) behave in time if the coexistent state is unstable? State the name of the theorem that confirms this. [2]

2.

The killer shrimp (Dikerogammarus villosus) is a 3 cm long freshwater shrimp with a voracious appetite for small invertebrates (including other shrimps), fish eggs and fry. They have been accidentally introduced into river systems around the world, sometimes driving native species to extinction. In a particular river system, the spread of the killer shrimp can be described using the following equation, where P(X, T) is the shrimp density, T is time, X is the distance along the river (river flow being in the direction of increasing X) and a, b, c and D are positive constants.

(a) Given X is measured in km, T in months and P in population/km, write down the units of a, b, c and D. Explain briefly what it means biologically to increase D in the model. [3]

(b) Show by suitable non-dimensionalisation that this PDE can be rewritten to the form[4]

(c) This part of the question corresponds to the spatially homogeneous version of (‡‡), where u(x, t) = u(t). Write down the steady-states of this ODE. Find the minimum value µmin such that there exists a positive steady-state for µ > µmin. Find the stability of each of the steady-states for the case. [4]

The remainder of this question concerns the equation (‡‡), in particular the travelling wave form of the equation to describe the long term spread of the killer shrimp. Assume µ > µmin in what follows.

(d) Derive the travelling wave form of the dimensionless PDE using the change of variables z = x − c t (where constant c is the velocity of the wave) and u(x, t) = U(z). [2]

(e) Write down a set of suitable boundary conditions as z → −∞ and z → +∞ for the travelling wave problem. Show that the sign of c is the same as the sign of R U(−∞) U(∞) f(U) dU and hence deduce the direction of the wave. Will the shrimp successfully spread along the river? Briefly justify your answer. [6]

(f) By writing the dimensionless equation as a system of two first-order differential equations. Use linear analysis about U = 0 as z → ±∞ (depending on your orientation of the boundary conditions in part (e)) to deduce a lower bound for c 2 . Hence, write down the corresponding inequality for the speed c, using your answer to part (e). [6]

(g) New data suggests that other factors are influencing the spread of the killer shrimp. Modify the original dimensional equation ($$) to account for the following: • the flow of the river means that the shrimp additionally move with an effective constant velocity W > 0. [2] • the shrimp is repulsed by river pollutants, which has a spatially varying concentration given by S(X). [3]

3.

The striped patterning of a type of coral snake is found to be governed by three interacting chemical species that stimulates the pigment release in the snake’s skin. The three chemicals X, Y and Z interact through the following reactions,where the k’s are the rate constants, and the reactant concentrations of species A, B, C, E and F are assumed constant.

(a) State clearly the Law of Mass Action and use this law to derive the differential equation system for the concentrations of X, Y and Z. [7]

(b) Let R(X, Y, Z) be the function you derived for the rate of change of Z, so that

dZ/dt = R(X, Y, Z)

The rest of the question concerns a dimensionless version of the reduced system for species X and Y (represented by dimensionless variables u and v, respectively), that interact in the spatial domain according to[3]

∂u /∂t = ∂2u/∂x2 + γ − (γ + δ) u + u2/v = ∂2u/∂x2 + f(u, v)

∂v/∂t = D ∂2v/∂x2 + u2 − δv = D ∂2v/∂x2 + g(u, v)

(c) Determine the spatially homogeneous steady state (u ∗ , v∗ ) of system (††). Derive the stability matrix Q(k 2 ) that results from the substitution of the perturbation (u, v) = (u ∗ , v∗ ) + (u0, v0)e ikx+λt into (††) for this steady-state. [4]

[Hint: do not to derive the matrix via substituting the indicated perturbation into the PDEs, you may start by using the formula for the Jacobian of f(u, v) and g(u, v) and a matrix multiplied by k 2 .]

(d) Show that the steady-state is stable with respect to the spatially homogeneous perturbations. [2]

(e) Show that diffusion-driven instability is possible if and only if

det(Q(k 2 )) = Dk4 + (D(γ − δ) + δ) k 2 + γ(δ + γ) < 0

for some k 2 > 0. Explain how you use this to deduce necessary conditions for diffusion driven instability (DDI); you may use diagrams if necessary. Hence, find a complete set of necessary conditions on D, γ and δ for DDI, showing that one of these conditions is given by

[Hint: at some stage of your working you should be solving a quadratic equation, you may neglect the negative root in this calculation].

(f) Let u ∗ be the steady-state of u(x, t) found in part (c), then the various pigments on the snake’s skin are produced as follows:

u(x, t) < u∗/2 ⇒ yellow skin.

u ∗/2 < u(x, t) < 3u ∗/2 ⇒ red skin.

3u ∗/2 < u(x, t) ⇒ black skin.

Assuming that the conditions of DDI are satisfied, and suppose in time there are sections of the skin-domain in which u(x, t) < u∗/2 and sections with u(x, t) > 3u ∗/2, what is the expected pattern on the snake’s skin? [2]