joelmiller
commented
9 months ago Let us assume that $dP/dt = f(P)$ where $P$ is the size of a population.
Making reasonable assumptions about birth/death, when $P=0$, $dP/dt=0$, so $f(0)=0$.
Then if we expand $f$ as a Taylor Series, at linear order we have $P’=aP$. This yields exponential growth. As things grow exponentially, at some point the next term in the Taylor series shows up.
how do we interpret this biologically? At small population size individuals are basically on their own and reproducing or dying without being affected by crowding. However, as the density of individuals increases, they start to interact with one another. In many biological situations, this would suggest that they are competing for resources. So we would expect the interaction to have a negative impact. So we expect something like $P’=aP - b P^2$ where $a$ and $b$ are positive
DP/dt = f(P)
f(0)=0. Then first term is positive (if growth from small values exists) and assuming competition is next effect, the quadratic term must be negative.