# 5.5: Orthologistic solution

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Finally, let *s* be positive. This creates a vertical asymptote and orthologistic growth. The position in time of the vertical asymptote is the “singularity” mentioned earlier. The interesting question is, when *s* is positive, what is the time of the singularity? That is, when will the population grow beyond all bounds in this model?

What must happen to the denominator for the population to grow to unbounded values? It has to get closer and closer to zero, for then the *N *(*t *) will grow closer and closer to infinity. So to find the singularity, you only have to set the denominator to zero, and then solve for the time *t*. You can go through the intermediate steps in the algebra below, or use a mathematical equation solver on your computer to do it for you.

Setting the denominator in the equation from 5.2 to zero will lead along this algebraic path:

\(\frac{s}{r}\,=\,(\frac{s}{r}\,+\frac{1}{N_0})\,e^{-rt}\)

Multiply through by (r/s)e^{rt}, to obtain

\(e^{rt}\,=\,(\,1\,+\frac{r}{s}\frac{1}{N_0})\)

Next take logarithms of both sides

\(rt\,=\,ln\,(1\,+\frac{r}{s}\frac{1}{N_0})\)

Finally, divide through by r to find the time of the singularity.

\(t\,=\frac{1}{r}\,ln\,(1\,+\frac{r}{s}\frac{1}{N_0})\)

In the 1960s, Heinz von Foerster wrote about this in the journal *Science*. Though the consequences he suggested were deadly serious, his work was not taken very seriously at the time, perhaps in part because the time was so far away (about a human lifetime), but perhaps also because he put the date of the singularity on Friday the 13^{th}, 2026, his 115^{th} birthday. In the title of his paper he called this “doomsday”, when the human population would have demolished itself.

Von Foerster used a more complicated model than the *r*+*sN* model we are using, but it led to the same result. Some of the ideas were picked up by Paul Ehrlich and others, and became the late-1960s concept of the “population bomb”— which was taken seriously by many.