Question

Consider the differential equation dx/dt =x(1−x)−h, which describes logistic population growth with harvesting. The existence of equilibrium solutions depends on the harvesting parameter h.

a) Determine the value of h which corresponds to a bifurcation point.
b) For values of h below and above the bifurcation point plot the directional field of the equation together with several integral curves.
c) Draw the bifurcation diagram of this DE, i.e. plot the location of the critical points versus the parameter h.

Answer 1

The value of h corresponding to a bifurcation point in the differential equation dx/dt =x(1−x)−h can be determined by examining the critical points.

Step-by-step explanation:

The value of h corresponding to a bifurcation point in the differential equation dx/dt =x(1−x)−h can be determined by examining the critical points. To find the critical points, set the derivative equal to zero:

(1−2x)−h = 0

Solving for x, we get:

x = (1−h)/2

A bifurcation occurs when the derivative changes sign, so we can substitute the critical points back into the derivative. If the derivative is positive for one value of x and negative for another value of x, a bifurcation point exists. Therefore, substituting the critical points into the derivative:

(1−(1−h)/2)(1−(1−h)/2)−h = ((1−h)/2)((1+h)/2)−h = 0

Simplifying and solving for h, we find the value of h corresponding to a bifurcation point.