Predator-Prey Model with logistic prey and Satiation: Bifurcation study

Equations for x-predator, y -prey

x' = -a*x  + (b*y/(1 + k*y))*x
y' = d*(1 - y/n)*y - (f*y/(1 + k*y))*x

Description of the equations:

This model for predator-prey populations incorporates predator satiation.
The predation rate levels-off as they satiate, no matter the prey population size. 
As the prey population increases in size, the predators have a limited capacity 
to reduce the prey numbers. This can lead to oscillations in predator-prey 
population sizes that grow in time, to stable limit cycles, and other effects.  

Parameters:
b - max growth rate of predator, 
f - max predation rate, 
k - 1/"satiation threshold"
(they determine the interaction between the two species).

d - max growth rate of prey
n - carrying capacity of prey


Set parameters:

a = .5; d = 1; n = 1; b = 3; f = 3,

and experiment with 0 < k < 6.
Use common IC (Iinitial condition) x = 0.25, y = 0.5, for all solutions. 

You will see the solutions change their behavior as k passes through 
certain critical bifurcation values. 

The goals of the project are 

1) determine (approximately) the bifurcation values k; 
2) compare your conclusions with the analysis of linearized equilibria (Jacobians)
3) describe the qualitative changes in the behavior of solutions and discuss their effect 
on coexistence of predator - prey.

Hint: there are 5 ranges of parameter k, 2 of them - "unstable". 

4) Discuss the type of bifurcations when 'spiral sink' -> 'spiral source' 
Can linearized system predict long-range behavior of non-linear system in that case ?