Everybody dead by 2050? Not so fast

Calculus Level 3

Your friend comes to you with a simple model for the growth of the human population, \(N(t)\). Quite reasonably, he suggests that the birth of new humans has to come from pairs of existing humans, and since the number of pairings of monogamous humans is at most half the current number of humans, the birth rate is proportional to the human population, \(r=\alpha N\). Likewise, humans die one at a time, and so, the death rate is also proportional to population, \(d = \beta N\). This leads to the differential equation \(\frac{dN}{dt} = r-d=\alpha N - \beta N\) with \(\alpha \neq \beta\).

Now he starts speaking quickly and claims that the model only has one plausible steady state value for \(N\), zero! He proves this by setting the growth rate \(\dot{N}\) equal to zero and solving for \(N\)

\[\left(\alpha-\beta\right)N = 0\]

Now you're also worried and are about to cry when you realize a fatal flaw with your friend's model, what is it?


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