Week 10 - Nov 12th and 14th

Topics Covered (Lynch, Chapter 5)

11th

Download Tutorial Quiz

12th and 14th

  1. Download Limit Sets, Attractors and Separatrix Cycles 
  2. Download Limit Cycles and Poincaré-Bendixon Theory

Recommended Exercises 

1. Read Lynch, Chapter 5, and do problems: Section 5.6 -- 1, 2, 3, 4,  6, 7 (g),(a) and 10.
2. Consider the following modification of the predator–prey equations called the Holling-Tanner model (See Example 3 in Section 4.2 of Lynch)

\dot{x}=x(1-x)-\frac{axy}{x+c},\,\,\,\, \dot{y}=by\left(1-\frac{y}{x}\right)

     where a, b, and c are positive constants.

    (a) Determine the region in the parameter space of a,b,c for which this system has a stable equilibrium with both x and y non-zero. 

    (b) Using ideas we discussed in the lecture about limit cycles, show that, if the equilibrium point is unstable, this system has a stable limit cycle.

    (c) Intepret each of the two cases above in terms of the predator-prey interactions and population dynamics point of view. Explain why this system is a more realistic model of the predator-prey behaviour.