predator-prey model
- Consider the predator-prey model represented by the system
… (1)
… (2)
where represent the predator and prey populations respectively at time . In this, and are all positive constants and
(a) If there were no predators, state without solving the ODE in , what happens to the prey population in the long-term.
(5 Marks)
(b) Show that the system has critical points at and . Given that there is a third critical point at , find and in terms of and .
(15 Marks)
(c) By deriving the Jacobian, determine the nature of the points at and , and then show that critical point at is stable, but that it could be either a node or a spiral point.
(20 Marks)
(d) Find an inequality for in terms of and , for to be a stable spiral point.
(10 Marks)
(e) From your findings in part (d), explain why will be a stable spiral when is large (there is no need to specify a lower limit for this value of ).
(5 Marks)
(Total 55 marks)
- In lecture 6.3 we considered two functions and whose Laplace transforms exist and are respectively and . We then defined their convolution to be
(a) Using a simple change of variables, prove that the convolution is commutative – that is
(8 Marks)
(b) In that lecture we also stated the powerful Laplace transform convolution theorem, which states that
Use the convolution theorem with to show that
(10 Marks)
(c) An important form of calculus-based equation you have not yet encountered in the course is an integral equation, such as the following
Taking the Laplace transform of this equation, find the relevant equation for in the -space, and hence find the solution to the integral equation
(17 Marks)
(d) Assuming the following (it is tedious to find by hand) indefinite integral result
show, by substituting your answer to part (c) back into the integral equation from there, that your answer is a valid solution
(10 Marks)
(Total 45 marks)
END OF COURSEWORK
