Plot xvs x for h a124a2 h lt a124a2 h gt a124a2 what is

Question

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 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 diagrams?

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^-, where t^-, x^-, h^- are constants.

then use h^-= a₁²/2a₂; t^- =2/a1; x^-= a₁²/2a_{2 to simplify the equation}

6. If h2/4a₂ 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?