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
ANSWERED
asked 2021-06-12
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?

Expert Answers (1)

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:
\(k=0.05\quad M=100\)
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