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MTH307 2022/23

Population Dynamics

Assessed Coursework

There are three questions worth 50 marks in total. Please submit your hand-written solutions as a single pdf file before the deadline of 16:00 on Tuesday 14th November.

For full marks you must explain your answers carefully and make clear graphs and diagrams.

For the numerical investigations in parts Q2(c) and Q3(c), please insert your computer output into your solution. Upload separately your computational file e.g. Excel .xls file, Matlab .m file

Question 1 (10 Marks)

Consider single species continuous-time population models of the general form

where the per capita reproduction rate r(N) depends on the population density N ≥ 0. A model is said to represent the Allee e↵ect if r0 (0) > 0, and is said to represent competition if there exists a constant Q ≥ 0 such that for all N>Q we have r0 (N) < 0.

Consider all possible models with both Allee e↵ect and competition for which r(N) is a quadratic function of N. Describe the long-term behaviours for this class of models: that is, find all the possible phase portraits. In which cases will the population survive and in which cases does extinction occur?

Question 2 (20 Marks)

Consider the discrete-time model T : [0, 1] ! [0, 1] which is given by

(a) Draw the graphs of T, T(2) = T ◦ T, and T(3) = T ◦ T ◦ T for x 2 [0, 1]. Find the equilibrium points of T. Find the number of periodic points of T of (minimal) period p for p = 2, 3, 4, 5, 6. What can you say about the local stability of periodic points in this model?

(b) A point x is called pre-periodic if there exists k > 1 such that T(k) (x) is a periodic point. Prove that if x is a pre-periodic point for T, then x must be a rational number. [Hint: You could use induction to show that all iterates T(n) (x) have the form Ax + B for some integers A and B.]

(c) Numerical investigation: Let D 2 {0,..., 99} be the integer formed by the last two digits of your student ID number. Let x 2 [0, 1] be given by x = ⇡D/320.

Using computer software (e.g. Excel or Matlab), calculate the decimal expansions of the first 60 terms of the sequence (x, T(x), T(2)(x), ...). Be sure to perform the computations numerically (NOT symbolically or algebraically).

Describe and explain the observed behaviour of the sequence, comparing with your answer to part (b).Question 3 (20 Marks)

The spotted knapweed Centaurea biebersteinii, whose natural habitat stretches from Central Eu-rope to Western Siberia, is now a serious problem weed in North America where it was accidentally introduced in 1880s. The population of plants consists of five age groups: seeds (age group 0), seedlings (age group 1), young adults (age group 2), medium adults (age group 3) and old adults (age group 4). Each year:

• a fraction B0 2 (0, 1) of seeds remain in the soil, and a fraction P0 2 (0, 1) germinates into seedlings;

• a fraction P 2 (0, 1) of seedlings become young adults;

• the same fraction P of young adults become medium adults;

• the same fraction P of medium adults become old adults,

• young, medium and old adults each produce B > 0 seeds on average; seedlings do not produce any seeds.

The dynamics of this population can be described by a Leslie model

Nt+1 = LNt,                                                (1)

where Nt is the column vector in R5 describing the population in year t, with entries N(0, t), N(1, t), ..., N(4, t), describing the sizes of the age groups in that year.

(a) Write down the system of discrete-time evolution equations for the plant age groups. Thus construct the Leslie transition matrix L. Find the characteristic polynomial χ(µ) for the matrix L.

(b) Observations have shown that an unchecked population of spotted knapweed increases by 25% each year. Assuming that P0 = B0 = 1/4 and P = 1/2 estimate, to 1 decimal place, the average value B of seeds produced by an adult plant.

(c) Numerical investigation: Suppose the initial population vector N0 2 R5 has entries given by the last 5 digits of your student ID number.

Assuming the parameter values specified in part (b) and using computer software (e.g. Excel or Matlab), make a chart to show how the sizes of each of the five ages classes varies over the next 10 years. Comment on your results, comparing with the observations in part (b).

(d) Two species of seed head flies, Urophora affinis and U. quadrifasciata, are well established on spotted knapweed. The larvae of these species reduce seed production by 50% by feeding on spotted knapweed seed heads. Determine if using these flies would alone be an e↵ective method of biocontrol of the knapweed population, for the values of parameters specified in part (b).