Suppose that a population develops according to the logistic

Suppose that a population develops according to the logistic equation dP/dt = 0.1P - 0.002P^2 where t is measured in weeks. (a) What is the carriying capacity? (b) Is the solution increasing or decreasing when P is between 0 and the carriying capacity? (c) Is the solution increasing or decreasing when P is greater than the carriying capacity?

Solution

Given logistic equation

dP/dt= 0.1P -0.002P^2 Let eqution A

Since time t ( in week) is independent varialble

therefore we can write above logistic equation the form of

dP/(0.1p-0.002P^2) = dt

Taking integration on both side and simplification

-500 (dP/ (P-50)P) = dt

now we would use partial fraction on Left hand side

-500 (dP/50(P-50)-dP/50P)= dt

On simplification

-10 (ln (P-50) -lnP )= t +C1 ( C1 is an arbitrary constant )

Now as we know that P is dependent on time (weeks ) t

thus we can write above equaiton like ( Solve for P(t)

P(t) = 50 e^0.1t /(2.72^50C1 + e^0.1t )

Part (a)

Now for maximum or minimum

we have to caluclate where P is changing

dP/dt= 0

0.1P-0.002P^2 = 0

P= 0

P= 50

We will consider point P= 50 here because domain is natural log there for always greater than 0

therefore capacity capacity is 50 units

Part (b) the solution undefined over the range between 0 and the carrying capacity of 50 units

Part (c) the solution is decreasing when P is greater than the 50-unit carrying capacity