The Exponential Model

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N(t) - Number of population at time t, which is measured in appropriate units.

b(t) - Birth Rate, c(t) - Death rate

a(t) = b(t) - c(t) is the Growth Rate.

This gives dN/dt = aN. This is separable and solves to give solution logN = at + C (C is an arbitrary constant).

If we are given an inital condition, such as N(0)=n, we can find a particular solution to the model, N=nexp(at). These solutions depend on two parameters, the growth rate a and the initial condition n.

Qualitative observations;

a>0 (b>c), N tends to infinity as time tends to infinity

aa=0 (b=c), N=n as time tends to infinity