1. (a) The size of the population of a certain type of cell in an individual at age t satisfies dN dt Ae-at kN, = where A= 107, a = 0.1year-1 and k lyear-1. Assuming N(0) = 5 x 106, find N(1) and N(10). Sketch the population size as a function of age. (b) In a mathematical description of phytoplankton and zooplankton, the two popula- tions obey the ODES d P2 P= BP(1 – P) – BZ dt V2 + P2 d P2 Z. dt V2 + P2 2 (1) Z=/ 2. Assume that B > 0, 7 > 0 and v = determine whether they are stable. 20 Find the steady states of (1), and 2. (a) The size of a population, H(t), is governed by the ordinary differential equation: d H = -aH+ BH?. dt The parameters a and B are positive, and the initial condition is H(0) = 1. Describe how the fate of the population depends on the values of the parameters a and B. Sketch the population size as a function of time for each case you identify. (b) In a spatial model of a rabies epidemic in a fox population, the susceptible and infected populations, S(x, t) and 1(x, t), obey the partial differential equations as = -r SI at al =rSI – al + D at Әr2* 221 (2) A travelling wave solution describes the spread of the populations. Use the substi- tutions S(x, t) = f(x – ct) and 1(x,t) = g(x – ct) to show that cf' g (3) rf Find the corresponding differential equation for g", substitute (3) into it and inte- grate to obtain a relation between g' and g, on one hand, and f and log f, on the other. Sketch the form of the solution that you would expect for the function f. 3. (a) A hen lays a number of eggs that has the Poisson distribution with mean 4. Each egg hatches (a chick is born) with probability p, independently of the other eggs. Let S be the total number of chicks born. Find the probability generating function of S. What type of distribution does S have? (b) Suppose each individual plant in a plant species lives for a fixed time and leaves behind k offspring with probability pk where 1 1 Po = Pi = 1 P2 = 4 and P3 = 1 4' 4 and pk = 0) if k > 4. Let the random variable Zn be the number of plants in the nth generation. Suppose that Zo = 1. i. Find E(Zn). ii. Write down the probability generating function of Z2. Hence find P[Z2 = 1). iii. Let un = P[Zn = 0). Find U2 and 43. iv. As n +00, Un +v. Find a function f(u) such that 1 V= + f(u). 3 Hence, without using a calculator, show that v> 31 81 4. (a) Suppose that insects fall randomly on a pond's surface, independently with rate .. Let Xt be the total number of insects that fall on the pond's surface up to time t, and let fn(t) = P(X+ = n). i. Write down fo(t), the probability that no insect falls on the pond's surface up to time t. ii. Explain why, for n > 1, d Sn(t) = -1(fr(t) – fn-1(t)). dt a Find fi(t). iii. Let G(z,t) = E(2X1). Show that ƏG(z,t) = g(z)G(z,t) )G and find g(2). Hence find G(z,t). (b) A cell population is governed by a continuous-time birth-and-death process. The number of cells at time t is Ct and the probability generating function, defined as G(z,t) =P[Ct = k]zk, is given by k=0 2 G(2,8) = (08 u(1 – 2) – (– dz)e-(1–4)t) 1(1 – 2) – (u – Iz)e-(1-4) i. Find the probability that the population ever goes extinct. ii. Find the mean number of cells at time t.
Purchase answer to see full attachment