Question

The population model given in (1) in Section 1.3 dP/dt \propto P or dP dt = kP (1)

fails to take death into consideration; the growth rate equals the birth rate. In another model of a changing population of a community it is assumed that the rate at which the population changes is a net rate that is, the difference between the rate of births and the rate of deaths in the community. Determine a model for the population P(t) if both the birth rate and the death rate are proportional to the population present at time t > 0. (Assume the constants of proportionality for the birth and death rates are k1 and k2 respectively. Use P for P(t).)

dP dt = __.

Answer 1

`(dP)/(dt)=k_1 P -k_2 P= P(k_1 -k_2)`

Explanation:

For this case we know that the birth rate is given by
`b` and the death rate is given by
`d`.

We also know that these rates are proportional to the population size, so then we have this:

`b \propto P(t)`

`d \propto P(t)`

And in order to have expression with the sign= we have the proportional constants given
`k_1` for b and
`k_2` for d, so then we convert the system of equations on this:

`b = k_1 P(t)`

`d = k_2 P(t)`

And then the change in the population respect to the time would be calculated on this way:

`(dP)/(dt) = b-d`

And if we replace what we found we got:

`(dP)/(dt)=k_1 P -k_2 P= P(k_1 -k_2)`

And we can solve the differential equation reordering the terms like this:

`(dP)/(P)= (k_1 -k_2) dt`

And if we integrate both sides we got:

`ln |P| = (k_1 -k_2) t +C`

Using exponentials we got:

`P(t) = e^((k_1 -k_2)t) *e^c`

And we can rewrite this expression like this:

`P(t) = P_o e^((k_1 -k_2)t)` where
`e^c = P_o`