MATH0300 Mathematical Ecology Problem Sheet 1
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MATH0300 Mathematical Ecology Problem Sheet 1
1. Suppose that the per-capita net reproductive rate of a species with population density N is r(1 - N2 /K), where r, K > 0 are constants
a) Find the population density N(t)
b) Sketch N as a function of t for (i) N(0) > ^K and (ii) 0 < N(0) < ^K
c) Discuss the biological implications of the solutions
r
2. If the per-capita net reproductive rate of a species with density x is 1 + x/K , where r, K > 0 are constants, is the population bounded if x(0) > 0? (You may use that x(t) 2 0 for all t 2 0.)
3. In lectures we derived the Master Equation for the probability pn (t) of having n indi- viduals at a time t given a constant reproduction rate b and null mortality rate,
dpn (t) dt
= b (-npn (t) + (n - 1)pn_1 (t)) .
Given that at a time t = 0 there are m individuals, we can define the initial population as pn (0) = δn,m where δn,m is Kronecker delta:
a) What is the probability pn (t) for n < m?
b) Find the probability pn (t) for n = m.
c) Find the probability pn (t) for n = m + 1.
d) Using this result find the general probability pn (t) for n > m.
4. Let the death rate be d(t), so that the net intrinsic growth rate (per individual) is r(t) = b(t) - d(t). Using pk (t) to denote the probability that the population is size k at time t:
a) Derive a new set of differential equation for pk (t) for each k = 0, 1, . . .
b) Show that the expected value for the number of individuals (n(t)) = 
kpk (t) is given by (n(t)) = (n(0)) exp ╱ 0(t) r(s) ds、.
c) When r(t) = e_λt , so that the net intrinsic growth rate goes to zero as t → o, what happens to the mean population size in the long run?
2023-04-29