Question

3. A population is modeled by the differential equation . (Obviously)  

    For what values of P is the population increasing? Why?

Answer 1

That is absolutely true, that population is modeled by the differential equation.

Let we understand modelling population first before taking an example.

Let P = population, t = time that has passed in days or years.

The reasonable modelling is, the rate of change of population with respect to time dP/dt is proportional to population.

Implies, dP/dt = KP (K = constant of proportionality)

Because larger the population means larger the rate with respect to time.We will solve this for a particular solution.

This is a separable differential equation.So we want one variable and all the differentials involving that variable in one side , and the other variable and all the differentials involving the other variable on the other side then we can integrate the both sides.

dP/dt = KP-----------------eq 1

divide both sides of eq 1 by P,

implies, dP/Pdt = K---------------eq 2

Now multiply both sides of eq 2 by dt,

implies, dP/P = Kdt---------------------eq 3

by integrating both sides of eq 3 we get,

dP/P = Kdt ------------------------eq 4

implies, lnIPI = Kt + c   (c = constant of integration) -------------------------eq 5

implies, absolute value of P = IPI = exp(Kt + c ) = e^Kte^c = Ce^Kt---------------------------eq 6

P = Ce^Kt (solution)

so if we have the initial conditions then we can easily calculate the population from above solution.

Next i have taken an example to solve this problem.

If we need to calculate the value of P or population for which it should be increasing or decreasing or let it be in equilibrium, then firstly we need a differential equation. Let me assume a differential equation for population which is as follow:

dP/dt = P(1-P/5400)----------------------------equation 7 (I have taken 5400 ,just a number to explain this concept but the differential equation of population is same as shown.This number may only vary)

since, dP/dt 0 when P0 or P5400

But population cannot be negative

so, dP/dt 0 when or P5400

Now the differential equation that i have taken for consideration is, dP/dt = P(1-P/5400) (from eq 7)

as we know that by maxima-minima theorem if we need to find maximum value of P then   dP/dt=0

implies, P(1-P/5400) = 0

implies, P = 0 and P = 5400,so the population is increasing for the value P=0 and  P=5400