What are the dimensions of a1-a2 and h

Question

Graphing and Harvesting Functions

Solve the following problems:

Harvesting:

x^· = a₁x - a₂x² - h       a₁ > 0, a₂ > 0, h>0

1. |x| = tons and |x^·| = tons/years

What are the dimensions of a₁, a₂, and h?

2. Use the quadratic formula to show that the phase diagrams x^· vs. x)have no intercepts:

    If a₁²/4a₂ 

    If h=a₁²/4a₂ ,what is the intercept?

3. If h < a₁²/4a₂ , x^· vs. x will have 2 intercepts, what are the signs ( + or - )?

4. Plot x^· vs. x for h = a₁²/(4a₂) , h < a₁²/(4a₂) , h > a₁²/(4a₂)

    What is the significance of h= a₁²/4a₂ with respect to the phase diagram?

5. Normalize and non-dimensionalize the harvesting equation using the variable transformation:

        h→h*

        t→t*

        x→x*

for h=h*h¯;t=t*t¯;x=x*x¯, wherex¯,t¯ and h¯ are constants.

   THEN use h¯=a₁²/2a₂;t¯=2/a₁;x0=a₁²/2a₂ to simplify the equation.

6. If h < a₁^2/4a₂, what is the magnitude for h*? What is the equilibrium value for x*?

7. How do the values in number 6 relate to the h = a₁²/4a₂ phase diagram?