import numpy as np import matplotlib. pyplot as plt from scipy. integrate import odeint # Define the system of differential equations def system(y, t): [z,p] = y dzdt= 2 * z - z* *2- z*p dpdt = p**2 p * (z**2) return [dzdt, dpdt] # Function to calculate the Jacobian matrix at an equilibrium point def jacobian_matrix (point): z, p = point return np.array ( [ [2 - 2+ z-p, -z], [-2 pz, 2 p - z**2]]) # Equilibrium points equilibrium_points = [(0, 0), (1, 1), (2, 0), (-2, 4)] # Set up the plot fig, axs plt.subplots (2, 2, figsize=(10, 8)) fig.suptitle('Phase Portraits for Linearized Solutions') # Plot linearized solutions at each equilibrium point for i, point in enumerate(equilibrium_points): # Calculate the Jacobian matrix at the equilibrium point A = jacobian_matrix (point) # Eigenvalues and eigenvectors eigenvalues, eigenvectors = np. linalg.eig (A) # Initial conditions for the linearized system ye = np.concatenate ([point, eigenvectors[:, 0]]) # Time points t = np. linspace(0, 5, 100) # Solve the linearized system linearized_solution = odeint (system, ye, t) #Plot the phase portrait axs[i // 2, 1 % 2].plot(linearized_solution[:, 0], linearized_solution[:, 1]) axs [i // 2, 1 % 2].set_title(f'Equilibrium Point: {point}') #Set common labels for ax in axs.flat: ax.set (xlabel='z', ylabel='p') # Adjust layout plt.tight_layout(rect= [0, 0.03, 1, 0.95]) # Show the plot plt.show()

Ок Denote the owl and wood rat populations at time k by x = Rk where k is in months, Ok is the number of owls, and R is the number of rats (in thousands). Suppose OK and RK satisfy the equations below. Determine the evolution of the dynamical system. (Give a formula for XK.) As time passes, what happens to the sizes of the owl and wood rat populations? The system tends toward what is sometimes called an unstable equilibrium. What might happen to the system if some aspect of the model (such as birth rates or the predation rate) were to change slightly? Ok+1 = (0.2)0k + (0.5)RK Rk+1=(-0.16)0k + (1.1)Rk Give a formula for xk- xk = 4 ( D +C₂1