BIO120H1 Lecture Notes - Lecture 5: Exponential Growth, Population Ecology, Inflection Point
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Lecture 5: Population ecology: models without age structure
Continuous VS. Discrete generations
An approach to population modeling assuming that time flows continuously and
that change can occur at every instant.
Differential equations are used, time steps are infinitely small: use concept of
limits &calculus; growth is smooth.
Discrete time approach
An approach to population modeling that uses discrete time intervals, generally
corresponding to intervals between reproductive periods
Difference equations are used, time steps are discrete units (days, years, etc.):
use iterated recursion equations; growth is stepwise and bumpy.
Calculating population growth rates
Per capita: the rate of growth on a per-individual basis.
General bookkeeping model:
Geometric growth: Increase/decrease in a population as measured over discrete
intervals in which the increment is proportional to the number of individuals at
the beginning of the interval.
When we combine death, birth, emigrant and immigrant, we will have a ratio λ
(factor by which population changes over one time unitnet reproductive rate).
Discrete time can be expressed as geometric growth:
Continuous time can be expressed as exponential growth.
r=exponential growth rate
the geometric and exponential growth are related by
Geometric growth (λ)
Constant population size
Factors regulating population size
Density-dependent factors: Having an influence on a population that varies with
the size of the population.(population↑ food supplies ↓ places to live ↓)
Density-independent factors: Having an influence on a population that does not
very with the size of the population.(temperature, precipitation, catastrophic
The differential equation describing restricted population growth:
Simplest form of density
dependence. K=carrying capacity
The population grows slowly at first,
then more rapidly as the number of
individuals increases, and finally more
slowly as it approaches to carrying
The curve is symmetrical about the inflection
point (K/2) accelerating and decelerating
phases of population growth have the same
Solving Nt vs. t gives famous “sigmoid
Trajectories are sigmoid only when
starting from low numbers.
The overall rate of population
growth (dN/dt) is the product of the
per capita exponential rate of
increase (r=1/N dN/dt) and
population size (N). The value of r
declines as a linear function of
population size (N), from r0 at N=0
to 0 when N=K. the overall growth
rate of a population reaches a
maximum at the inflection point, at
which the population size is one-half the carrying capacity (K/2).