Philosophy of Biology

Formalisations of Evolutionary Biology 517

Population Growth

It is an elementary task to describe mathematicallyunconstrained population
growth. This, of course, was the starting point for the Malthusian claim that,
unconstrained, a population will grow exponentially and for Darwin’s application
of it to evolutionary dynamics.
Letrbe the net population growth rate (i.e., Births minus deaths in a time
step) and letntbe thetth time step (generation, although that term lacks the
required precision), then:
nt+1=rnt

If one assumes, simplistically, that the growth rateris constant, then:

dn/dt=rn

The solution to this differential equation is:

nt+1=n 0 ert

Ifris greater than 0, the population grows exponentially. If one assumesris less
than 0, the population decreases exponentially to 0. If one assumesr=0,the
population is static at itsn 0 level.
This, of course is too unconstrained. A key element of Malthus’ and Darwin’s
models is population density. For example, at some point the population outstrips
its resource supply and physical space to inhabit. To take this into accountr
cannot be constant. A simple, but as it turns out empirically rich way to take
density effects into account is to makera decreasing function of population density.
Hence:
rt+1=rt(1−n/S)

Sis the optimal, or stable, population size. Some refer to this as the carry capacity
of the environment for the population. When the population density exceedsS,
the growth rate,rt+1, will be negative. When the population density is belowS,
the growth rate will be positive. When the population size isS, the population
will be in a stable equilibrium andrt+1will equal 0.
The differential equation for growth then becomes:

dn/dt=r(1−n/S)n

This is the powerfulPearl-Verhulstor logistic equation whose solution is:

nt+1=S/(1 + (S/n 0 −1)e−rt)

A population obeying the logistic equation will oscillate fromn 0 aroundS, even-
tually settling at the stable equilibriumS.
To this basic framework of growth one can add other features that increase
the complexity of the model and its faithfulness to empirical phenomena. In