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MTH307 2022/23

Population Dynamics

Coursework Questions

This coursework counts for 15% of the module credit. Write you solutions by hand (on paper or tablet) and save as a single pdf file. Please submit your solutions before the deadline of 11:00am on Wednesday 23rd November through Learning Mall.

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

Question 1 (10 Marks) Consider continuous-time population models of the general form

dN

where the reproduction rate r(N) depends on the population density N  ≥  0. Recall that a this equation models the Allee e↵ect if r(N) is strictly increasing when N is small and models competition if r(N) is strictly decreasing when N is large.

Consider models with both these features: namely, assume that there exists N*   > 0 such that r(N) is a smooth function that is strictly increasing for N 2 [0,N* ) and strictly decreasing for N 2 [N* , 1). Find all the di↵erent possible types of long-term behaviour for such models, that is, all possible phase portraits.

Question 2 (20 Marks) Consider the discrete-time population model Nt+1   = F(Nt ), where

F(N) = RN2 (1 − N)    for R > 0.

(a) What is the maximum value of R for which all solutions remain biologically feasible? (b) Find the equilibria and the values of R for which they exist.

0 < N0 < 1 and

(c) In the case R = 9/2, determine the basin of attraction of each equilibrium.

(d) Describe the bifurcation that occurs for R = 16/3. Compare the long-term behaviour of typical solutions for the two cases R slightly less than 16/3 and R slightly greater than 16/3.

Question 3 (20 Marks) A population of giant tortoises is modelled by the delay di↵erential equation

dN

dt

where m, b and c are positive parameters and T is the time delay between egg laying and hatching.

(a) Give a biological interpretation for each of the three terms in the right hand side of the equation. Explain how we can see that the tortoises have a naturally long lifespan when the population is small.

(b) Show that there is a unique positive equilibrium N* , and find a linear di↵erential equation

(t) = bN(t −     −T) c[N(t − T)]2 − m[N(t)]2 ,

equation.

(c) Using parameter continuation, show that as T is increased from zero, instability of the equilibrium N* will first be observed (if at all) when the following system has a real solution

for ! and T:

(c − m)sin(!T) = (c + m)!/b.

(d) What is the meaning of the parameter !? Find an expression for ! in terms of b, c and m only. Show that N*  is in fact locally stable for all T ≥ 0 if c/m is not too large.