The Lotka-Volterra predator-prey equations:

dV/dt = rV aPV

dP/dt = caPV dP

represent the (simplest possible) dynamics of a predator species P eating prey (victim) species V. For positive parameters they are well known to have a trivial solution (V = P = 0) in addition to a neutrally stable equilibrium with a surrounding limit cycle. The parameters should be reasonably self-explanatory (c is a unitless efficiency parameter that determines how much of the energy from consuming prey can be used by the predators to increase reproduction/decrease mortality).

1. Consider the L-V equations with prey self-regulation, where the preys exponential growth rate is replaced by a logistic term rV(1 V/K), where K is a carrying capacity.

a. Find all of the equilibria of the system, with their associated stability; show the Jacobian computations; you do not need to distinguish saddles from sources/sinks from stable/unstable spirals. (Hint: when evaluating the Jacobian for the non-trivial case, evaluate gP/P|P=P first!)

b. Draw the phase space/nullclines of the system for parameter values in the range that include a non-trivial equilibrium (i.e., all state variables>0). Label all features of interest (equilibria and intersections of nullclines with axes) with their symbolically computed values.

Please code this in python :)