Question

# Suppose that a population develops according to the logistic equation \frac{dP}{dt}=0.05P-0.0005P^2 where t is measured in weeks. What is the carrying capacity? What is the value of k?

Differential equations
Suppose that a population develops according to the logistic equation
$$\frac{dP}{dt}=0.05P-0.0005P^2$$
where t is measured in weeks. What is the carrying capacity? What is the value of k?

2021-06-13
Logistic differential equation:
$$\frac{dP}{dt}=kP(1-\frac{P}{M})$$
where M is carrying capacity
Hence, to obtain M and k we need to rewrite the given equation:
$$\frac{dP}{dt}=0.05P-0.0005P^2$$
$$=0.05P(1-0.01P)$$
$$=0.05P(1-\frac{P}{100})$$
Therefore we have:
$$\frac{dP}{dt}=0.05P(1-\frac{P}{100})$$
And from this equation we get:
$$k=0.05\quad M=100$$