# BIO120H1 Lecture Notes - Lecture 5: Exponential Growth, Population Ecology, Inflection Point

17 views3 pages

Lecture 5: Population ecology: models without age structure

Continuous VS. Discrete generations

Continuous-time approach

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:

or generally:

Continuous time can be expressed as exponential growth.

r=exponential growth rate

Geometric growth

Exponential growth

the geometric and exponential growth are related by

&

Geometric growth (λ)

Exponential growth(r)

Decreasing population

Constant population size

Increasing population

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

event)

Logistic equation

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

capacity.

The curve is symmetrical about the inflection

point (K/2) accelerating and decelerating

phases of population growth have the same

shape

Solving Nt vs. t gives famous “sigmoid

growth curve”S-shaped

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).