univariate, linear affine models 1. if dx/dt = a-bx and x(0) c (a > 0, b > 0, c > 0), find the time-dependent solution by separating variables and integrating both sides of the equation. confirm that your solution simplifies to c when t 0 and converges to a/b as t oo. theta-logistic model The theta-logistic model is a generalization of the logistic equation: r,K,0 >0 2. IN has units of population density (individuals/area), identify the units of r, K, and . 3. Write down a non-dimensionalized version of the theta-logistic by finding the parameters with units of (1) time (or time) and (2) population density (or density ) and equating them to 1 4. Evaluate the equilibria and conditions for stability of this model. Compare the equilibria and stability conditions to those for the standard logistic (i.e. when -1). How does affect the population growth rate when rare (rate of departing from the zero equilibrium ? How does it affect the stability of the non-zero equilibrium e for what values of s it more, less or equally stable? here more stable would mean a negative Jacobian of larger magnitude; less unstable" would mean a positive Jacobian of smaller magnitude; etc.] 5. Use Python to draw a graph of dN/dt vs. N (you can use the non-dimensionalized version for simplicity) showing lines for = 1.0 = 2, and -1/2 on the same plot. Explain how the characteristics of this plot confirm your analytical conclusions. 6. Define a gradient function for the (dimensional) theta-logistic model, taking arguments t, x, and params (a tuple of parameters in the order r, K, theta). Use scipy.odeint.integrate to generate and plot the time dynamics (a) for r-0.5, K 2, and theta equal to 0.5, 1, and 2, with N(0)-0.1. On this graph, also include the closed-form solution of the logistic equation, using the logist_soln function given below, as a cross-check on your solution for theta-1 (b) For r equal to 0.5, 1, and 2 with K-2, theta-2, and N ( 0 )-4