Question

Consider the system of differential equations

dt/dp =p(1-p-q), dt/dq=q(2-3p-q),p,q≥0
Which of the following are equilibria? Select all that apply.
Select all that apply:
(p^,q^) = (0,0)
(p^,q^) = (0,2)
(p^,q^) = (1,0)
(p^,q^) = (1/2,1/2)
(p^,q^) = (1,2)
(p^,q^) = (1/2,0)
(p^,q^) = (1/2,2)

Answer 1

The equilibria of the given system of differential equations are the points where both derivatives equal zero. After solving the system, the equilibrium points are found to be (0,0), (1,0), (1/2,1/2), and (1/2,0).

To determine the equilibria of the given system of differential equations, we must find the points where the derivatives dt/dp and dt/dq are both zero. Setting dt/dp equal to zero gives the equation p(1-p-q) = 0, and setting dt/dq equal to zero gives q(2-3p-q) = 0. We need to solve these equations simultaneously to find the values of p and q that satisfy both conditions, representing the equilibria points.

The possible equilibrium solutions are (p,q) pairs that make both equations true. Looking at the options provided:

• (0,0) - Satisfies both equations.
• (0,2) - Does not satisfy the first equation.
• (1,0) - Satisfies both equations.
• (1/2,1/2) - Satisfies both equations.
• (1,2) - Does not satisfy the first equation.
• (1/2,0) - Satisfies both equations.
• (1/2,2) - Does not satisfy the first equation.

Therefore, the equilibria of the system are: (0,0), (1,0), (1/2,1/2), and (1/2,0).