For each differential equation find all equilibria and deter

For each differential equation, find all equilibria and determine whether they are asymptotically stable or unstable (assume all parameters r, K, a, and d > 0). (10 points)

Solution

Equilibrium solutions [ or critical points ] occur whenever x¹ = f(x) =0.

That is they are the roots of f(x). Equilibrium solutions are constant functions that satisfy the equation ,that is they are the constant solutions of the differential equation.

x¹ = rxln(k/x)

To find equilibrium points, we solve x¹=rxln(k/x) = 0

Then either x =0 or x = ln(k/x) =0.,and so our equilibrium points are x^* =0 ,k

To determine the stability of the critical points , we look at f¹(x).

f¹(x) = rln (k/x) - r

since f¹(0) > 0 ,then x^* = 0 is an unstable critial point .

when x^* = k, f¹(k) < 0 and so the critical point x^* = k is asymptotically stable.

similerly we can solve the other three cases.