8.01h Modelling with recurrence: birth/death rates, INT function

6 When \(10 ^ { 6 }\) of a certain type of bacteria are detected in a blood sample of an infected animal, a course of treatment is started. The long-term aim of the treatment is to reduce the number of bacteria in such a sample to under 10000 . At this level the animal's immune system can fight the infection for itself. Once treatment has started, if the number of bacteria in a sample is 10000 or more, then treatment either continues or restarts. The model suggested to predict the progress of the course of treatment is based on the recurrence system \(P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }\) for \(n \geqslant 0\), with \(P _ { 0 } = 1000\), where \(P _ { n }\) denotes the number of bacteria (in thousands) present in the animal's body \(n\) days after the treatment was started. The table below shows the values of \(P _ { n }\), for certain chosen values of \(n\). Each value has been given correct to 2 decimal places (where appropriate).

1. Find the value of \(P _ { 6 }\) correct to 2 decimal places.
2. Using the given values for \(P _ { 0 }\) to \(P _ { 9 }\), and assuming that the model is valid,
   1. describe the effects of this course of treatment during the first 9 days,
   2. state the number of days after treatment is started when the animal's own immune system is expected to be able to fight the infection for itself.
3. 1. Using information from the above table, suggest a function f such that, for \(n > 10 , \mathrm { f } ( n )\) is a suitable approximation for \(P _ { n }\).
   2. Use your suggested function to estimate the number of days after treatment is started when the animal may once again require medical intervention in order to help fight off this bacterial infection.
   3. Using information from the above table and the recurrence relation, verify or correct the estimate which you found in part (c)(ii).
4. One criticism of the system \(P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }\), with \(P _ { 0 } = 1000\), is that it gives non-integer
   values of \(P\). values of \(P _ { n }\). Suggest a modification that would correct this issue.

7 In a conservation project, a batch of 100000 tadpoles which have just hatched from eggs is introduced into an environment which has no frog population. Previous research suggests that for every 1 million tadpoles hatched only 3550 will live to maturity at 12 weeks, when they become adult frogs. It is assumed that the steady decline in the population of tadpoles, from all causes, can be explained by a weekly death-rate factor, \(r\), which is constant across each week of this twelve-week period. Let \(\mathrm { T } _ { \mathrm { k } }\) denote the total number of tadpoles alive at the end of \(k\) weeks after the start of this project.

1. 1. Explain why a recurrence system for \(\mathrm { T } _ { \mathrm { k } }\) is given by \(T _ { 0 } = 100000\) and \(\mathrm { T } _ { \mathrm { k } + 1 } = ( 1 - \mathrm { r } ) \mathrm { T } _ { \mathrm { k } }\) for \(0 \leqslant k \leqslant 12\).
   2. Show that \(r = 0.375\), correct to 3 significant figures. The proportion of females within each batch of tadpoles is \(p\), where \(0 < p < 1\). In a simple model of the frog population the following assumptions are made.
      • The death rate factor for adult frogs is also \(r\) and is the same for males and females.
      • The frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project.
      • 1. Find the smallest value of \(p\) for which the frog population will survive according to the model.
        2. Write down one assumption that has been made in order to obtain this result.
      Each surviving female will then lay a batch of eggs from which 2500 tadpoles are hatched.
2. By considering the total number of tadpoles hatched, give one criticism of the assumption that the frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project. \section*{END OF QUESTION PAPER}

5 In a conservation project in a nature reserve, scientists are modelling the population of one species of animal. The initial population of the species, \(P _ { 0 }\), is 10000 . After \(n\) years, the population is \(P _ { n }\). The scientists believe that the year-on-year change in the population can be modelled by a recurrence relation of the form \(\mathrm { P } _ { \mathrm { n } + 1 } = 2 \mathrm { P } _ { \mathrm { n } } \left( 1 - \mathrm { k } \mathrm { P } _ { \mathrm { n } } \right)\) for \(n \geqslant 0\), where \(k\) is a constant.

1. The initial aim of the project is to ensure that the population remains constant. Show that this happens, according to this model, when \(k = 0.00005\).
2. After a few years, with the population still at 10000 , the scientists suggest increasing the population. One way of achieving this is by adding 50 more of these animals into the nature reserve at the end of each year. In this scenario, the recurrence system modelling the population (using \(k = 0.00005\) ) is given by \(P _ { 0 } = 10000\) and \(\mathrm { P } _ { \mathrm { n } + 1 } = 2 \mathrm { P } _ { \mathrm { n } } \left( 1 - 0.00005 \mathrm { P } _ { \mathrm { n } } \right) + 50\) for \(n \geqslant 0\).
   Use your calculator to find the long-term behaviour of \(P _ { n }\) predicted by this recurrence system.
3. However, the scientists decide not to add any animals at the end of each year. Also, further research predicts that certain factors will remove 2400 animals from the population each year.
   1. Write down a modified form of the recurrence relation given in part (b), that will model the population of these animals in the nature reserve when 2400 animals are removed each year and no additional animals are added.
   2. Use your calculator to find the behaviour of \(P _ { n }\) predicted by this modified form of the recurrence relation over the course of the next ten years.
   3. Show algebraically that this modified form of the recurrence relation also gives a constant value of \(P _ { n }\) in the long term, which should be stated.
   4. Determine what constant value should replace 0.00005 in this modified form of the recurrence relation to ensure that the value of \(P _ { n }\) remains constant at 10000 .

7 In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, \(t\) years after their introduction, is an integer denoted by \(N _ { t }\). The initial number of breeding pairs is given by \(N _ { 0 }\). An initial discrete population model is proposed for \(N _ { t }\). $$\text { Model I: } N _ { t + 1 } = \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right)$$

1. (a) For Model I, show that the steady state values of the number of breeding pairs are 0 and 150 .
   (b) Show that \(N _ { t + 1 } - N _ { t } < 150 - N _ { t }\) when \(N _ { t }\) lies between 0 and 150 .
   (c) Hence find the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when \(N _ { 0 } \in ( 0,150 )\). An alternative discrete population model is proposed for \(N _ { t }\). $$\text { Model II: } N _ { t + 1 } = \operatorname { INT } \left( \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right) \right)$$
2. (a) Given that \(N _ { 0 } = 8\), find the value of \(N _ { 4 }\) for each of the two models.
   (b) Which of the two models gives values for \(N _ { t }\) with the more appropriate level of precision?

5

1. (a) Solve the recurrence relation $$X _ { n + 2 } = 1.3 X _ { n + 1 } + 0.3 X _ { n } \text { for } n \geqslant 0$$ given that \(X _ { 0 } = 12\) and \(X _ { 1 } = 1\).
   (b) Show that the sequence \(\left\{ X _ { n } \right\}\) approaches a geometric sequence as \(n\) increases. The recurrence relation in part (i) models the projected annual profit for an investment company, so that \(X _ { n }\) represents the profit (in \(\pounds\) ) at the end of year \(n\).
2. (a) Determine the number of years taken for the projected profit to exceed one million pounds.
   (b) Compare your answer to part (ii)(a) with the corresponding figure given by the geometric sequence of part (i)(b).
3. (a) In a modified model, any non-integer values obtained are rounded down to the nearest integer at each step of the process. Write down the recurrence relation for this model.
   (b) Write down the recurrence relation for the model in which any non-integer values obtained are rounded up to the nearest integer at each step of the process.
   (c) Describe a situation that might arise in the implementation of part (iii)(b) that would result in an incorrect value for the next \(X _ { n }\) in the process.

1

1. (a) Show that the number of arrangements of 25 distinct objects is an integer multiple of \(5 ^ { 6 }\).
   (b) Explain how this shows that the number of arrangements of 25 distinct objects is a whole number of millions.
2. (a) Calculate the values of

In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, \(t\) years after their introduction, is denoted by \(N_t\). The initial number of breeding pairs is given by \(N_0\). An initial discrete population model is proposed for \(N_t\). Model I: \(N_{t+1} = \frac{6}{5}N_t\left(1 - \frac{1}{900}N_t\right)\)

1. 1. For Model I, show that the steady state values of the number of breeding pairs are 0 and 150. [3]
   2. Show that \(N_{t+1} - N_t < 150 - N_t\) when \(N_t\) lies between 0 and 150. [3]
   3. Hence determine the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when \(N_0 \in (0, 150)\). [2]
   An alternative discrete population model is proposed for \(N_t\). Model II: \(N_{t+1} = \text{INT}\left(\frac{6}{5}N_t\left(1 - \frac{1}{900}N_t\right)\right)\)
2. Given that \(N_0 = 8\), find the value of \(N_4\) for each of the two models and give a reason why Model II may be considered more suitable. [3]