Assignment:
Part I
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 II
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)
Provide complete and step by step solution for the question and show calculations and use formulas.