Differential Equations : Predator / Prey Models

Part a)

Given the following predator prey model where x(t) is the predator population and y(t) is the prey population:

      dx/dt = - ax + bxy + (z1)*x

       dy/dt = cy - gxy +(z2)*y

Here both z1 and z2 can be positive or negative; parameters a, b, c, g are all defined to be positive.

Parameters z1 and z2 can represent an external introduction of species at a constant rate (positive values) or an external harvesting of the species at a constant rate (a negative values).

Under what conditions of  z1 and z2 can the trivial fixed point (both species evolve to extinction) become stable (in both directions not a saddle node)

Interpret these results.


Part b)  

Given the following predator prey model where x(t) is the predator population and y(t) is the prey population:

      dx/dt = - a*x + b*xy -----Q

       dy/dt = c*y - g*xy ------P


Solve the second equation, equation P, for the predator population (x)  and simplify. Then show that the average predator population over one cycle or period T is equal to c/g.

Note that the populations are periodic so y(T) = y(0)   and x(T) = x(0)