4 In this question you are required to consider the family of differential equations $$\frac { d P } { d t } = r P \left( 1 - \frac { P } { K } \right) , \quad t \geqslant 0 , \quad P ( t ) \geqslant 0 \left( ^ { * } \right)$$ where \(r\) and \(K\) are positive constants. This differential equation can be used as a model for the size of a population \(P\) as a function of time \(t\).

1. 1. Determine the values of \(P\) for which
      • \(\frac { \mathrm { dP } } { \mathrm { dt } } = 0\)
2. \(\frac { \mathrm { dP } } { \mathrm { dt } } > 0\)
3. \(\frac { \mathrm { dP } } { \mathrm { dt } } < 0\)
   (ii) Solve the equation (*) subject to the initial condition that \(P = P _ { 0 }\) when \(t = 0\).
   (iii) Find a property common to your solution in (ii) in the cases \(\mathrm { P } _ { 0 } > \mathrm { K }\) and \(\mathrm { P } _ { 0 } < \mathrm { K }\).
   (iv) State a feature of your solution in (iii) for the case \(\mathrm { P } _ { 0 } > \mathrm { K }\) which is different to the case \(P _ { 0 } < K\).
   (v) Interpret the value \(K\) when \(P ( t )\) is the size of a population at time \(t\).
4. In this question you will explore the limitations of using the Euler method to approximate solutions to the differential equation
$$\frac { d P } { d t } = 2 P ^ { 1.25 } \left( 1 - \frac { P } { 1000 } \right) ^ { 1.5 } , t \geqslant 0 , P ( t ) \geqslant 0 ( * * )$$ The diagram shows the tangent field to (**), and a solution in which \(P = 1\) when \(t = 0\), produced using a much more accurate numerical method.
\includegraphics[max width=\textwidth, alt={}, center]{4715d0f0-a860-4189-802f-1d2d019e1115-4_899_1552_1763_319}
(i) The Euler method for the solution of the differential equation \(f ( t , P ) = \frac { d P } { d t }\) is as follows $$P _ { n + 1 } = P _ { n } + h f \left( t _ { n } , P _ { n } \right)$$ It is given that \(t _ { 0 } = 0\) and \(P _ { 0 } = 1\).
• Construct a spreadsheet to solve (**) using the Euler method so that the value of \(h\) can be varied.
• State the formulae you have used in your spreadsheet.
  (ii) Use your spreadsheet with \(h = 0.1\) to approximate
• \(P ( 1 )\)
• \(P ( 2 )\)
• \(P ( 3 )\)
  (iii) Use your spreadsheet with \(h = 0.05\) to approximate
• \(P ( 1 )\)
• \(P ( 2 )\)
• \(P ( 3 )\)
  (iv) State, with reasons, whether the estimates to \(P ( t )\) given in your spreadsheet are likely to be overestimates or underestimates to the exact values.
  (v) With reference to the diagram, explain any noticeable feature identified when comparing the approximations given to \(P ( 2 )\) in (ii) and (iii).