Question

A) The updating function f(M) in plain text:

M_(t+1) = M_t * r * (3 - (M_t / 4))
f(M) = r * M * (3 - (M / 4))

b) The positive equilibrium M* in plain text:
M* = ?

c) The derivative of the updating function with respect to M in plain text:
f'(M) = ?

d) The range of values of r for which the positive equilibrium M* is stable in plain text:
r_stable = ?

Answer 1

The updating function f(M) is given by f(M) = r * M * (3 - (M / 4)). The positive equilibrium M* is 12. The derivative of the updating function with respect to M is f'(M) = r * (3 - (M / 4)) - (r * M / 4). The stability of the equilibrium M* depends on the sign of f'(M*) when M = M*.

Step-by-step explanation:

The updating function f(M) is given by the equation f(M) = r * M * (3 - (M / 4)).

To find the positive equilibrium M*, we set f(M) equal to zero and solve for M. So, r * M * (3 - (M / 4)) = 0. This equation is satisfied when M = 0 or (3 - (M / 4)) = 0. Solving the second equation, we find M = 12.

The derivative of the updating function with respect to M is f'(M) = r * (3 - (M / 4)) - (r * M / 4).

To determine the range of values of r for which the positive equilibrium M* is stable, we need to analyze the stability of M* by taking the derivative f'(M) and evaluating its sign when M = M*. If f'(M*) is negative, the equilibrium is stable, while if f'(M*) is positive, the equilibrium is unstable.