Assuming \( P \geq 0 \), suppose that a population develops according to the logistic equation \[ \frac{d P}{d t}=0.01 P-5 \times 10^{-5} P^{2} \] where \( t \) is measured in weeks. Answer the following questions. 1. What is the carrying capacity? Carrying Capacity: 2. What is the value of \( k \) ? Answer: \( k= \) Ans 3. For what values of \( P \) is the population increasing? Answer (in interval notation): 4. For what values of \( P \) is the population decreasing? Answer (in interval notation):

To analyze the logistic growth model given by the equation \[ \frac{d P}{d t} = 0.01 P - 5 \times 10^{-5} P^{2}, \] we can find the answers to the questions as follows: 1. **Carrying Capacity:** The carrying capacity, denoted by \( K \), can be determined from the condition \( \frac{d P}{d t} = 0 \). Setting the equation to zero, we have: \[ 0 = 0.01 P - 5 \times 10^{-5} P^{2}. \] Factoring out \( P \) gives us: \[ P(0.01 - 5 \times 10^{-5} P) = 0. \] This leads to \( P = 0 \) or \( 0.01 - 5 \times 10^{-5} P = 0 \). Solving for \( P \) gives: \[ 0.01 = 5 \times 10^{-5} P \implies P = \frac{0.01}{5 \times 10^{-5}} = 200. \] So, the carrying capacity is \( K = 200 \). 2. **Value of \( k \):** In the logistic equation, \( k \) is defined as the growth rate. Based on the equation \[ \frac{dP}{dt} = rP(1 - \frac{P}{K}), \] we identify \( r = 0.01 \) (the intrinsic growth rate) and find that \( k = 0.01 \). 3. **Values of \( P \) where the population is increasing:** The population is increasing when \( \frac{d P}{d t} > 0 \). From our expression, \[ 0.01 P - 5 \times 10^{-5} P^{2} > 0. \] Factoring yields: \[ P(0.01 - 5 \times 10^{-5} P) > 0. \] This inequality is satisfied when \( P > 0 \) and also when \( 0.01 - 5 \times 10^{-5} P > 0 \), or \( P < 200 \). Thus, the population is increasing in the interval \( (0, 200) \). 4. **Values of \( P \) where the population is decreasing:** The population is decreasing when \( \frac{d P}{d t} < 0 \). Thus, we need to solve \[ 0.01 P - 5 \times 10^{-5} P^{2} < 0. \] This is true when \( P < 0 \) (which we discard since \( P \geq 0 \)) or when \( P > 200 \). Therefore, the population is decreasing for \( (200, \infty) \). In summary: 1. Carrying Capacity: \( 200 \) 2. \( k = 0.01 \) 3. Population increasing: \( (0, 200) \) 4. Population decreasing: \( (200, \infty) \)