Predator-prey dynamics with Holling type II functional resonse

Modelling Exploitation

!mathematical-methods-lecturenotes, p.55

Equilibria:

Trivial equilibrium $\hat{N}=0$
Two quadratic solutions: $\hat{N}=-1/\beta T$ and $\hat{N}=K$
Then, there are more equilibria along the line $f(x)=\beta P$, since this is where the quadratic must be equal to this to solve to zero

• Stable equilibrium at the intersection of this line and the parabola when

Special equilibrium

If there was a very precise number of predators such that $\beta P$ is equal to the exact maximum of our parabola, this equilibrium would be both stable and unstable, and also the only equilibrium other than the trivial one. We call this point $P_F$ the fold bifurcation point

Bifurcation Diagram

This can be used to show the stability of the equilibria:.

We plot N as a function of P:
!mathematical-methods-lecturenotes, p.57
The transcritical bifurcation point $P_T=0$. See Harvested Logistic Model