Stanford University Mod 4 Differential Equations Questions You can check attachment for a pdf and that looks better. 1. Consider the following populati

Stanford University Mod 4 Differential Equations Questions You can check attachment for a pdf and that looks better.

1. Consider the following population model: dx/dt=9−x2 , where x(t) is the population density at

time t.

(a) Sketch the phase line for this model (you only need to show the phase line for x > 0). Use the phase

line to identify the positive fixed points, and indicate the type of stability.

(b) Suppose we incorporate a constant harvest h to this population, where h is any positive number.

(i) Write a new differential equation that incorporates the harvest into the original model.

(ii) Sketch the phase line for this model. You only need to show the phase line for x > 0. Include

different cases to show all types of possibilities for different numbers of fixed points and types

of stability.

(iii) In each case you described in (ii), identify the non-negative fixed points, and indicate the type

of stability for each fixed point.

(iv) Find the critical harvest level. That is, find the number such that the population approaches 0

for any initial condition if h is above this number.

(c) Suppose we incorporate a percentage harvest Hx to the original model, where 0