5.2: A mathematical view

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Differential equations can be amenable to mathematical analysis. To repeat, here is the differential population model.

\(\frac{1}{N}\,\frac{dN}{dt}\,=\,r\,+\,sN\)

It turns out there is something simple about infinity, and when time steps are infinitely small the methods of calculus developed over the centuries can solve this differential equation exactly, mathematically. If you apply a symbolic mathematics computer package, or the methods for integration of functions developed in calculus, you can find the population value N for any future time t. This is called the “solution” to the differential equation.
\(N(t)\,=\frac{1}{(\frac{s}{r}\,+\frac{1}{N_0})\,e^{-rt}\,-\frac{s}{r}}\)

Most differential equations cannot be solved this way but, fortunately, the basic equations of ecology can. This solution becomes useful in projecting forward or otherwise understanding the behavior of a population. If you know the starting N, s, and r, you can plug them into the formula to find the population size at every time in the future, without stepping through the differential equation.