General Ecology: Lecture 5
October 7, 2005
I. Properties of populations (cont.)
A.
Mortality and natality
1.
Various ways to calculate—we’ll get into the specifics of these rates as needed.
2.
Life tables (briefly)
a.
What can they tell you?
b.
NOTE: We won’t go through the particulars of the calculations here. Will
touch upon as needed in next lecture.
3.
Survivorship curves [Fig. 10.13]
a.
General patterns

4.
5.
II.
III.
Type 1: Description and examples

Look carefully—usually an initial drop due to infant mortality

Type 2: Description and examples.

Type 3: Description and examples
b.
Variability of survivorship pattern within a population:

Summer annual garden rocket exhibits different curves for different
cohorts [Fig. 10.19]

Reflects rainfall conditions in different years
Mortality curves [Fig. 10.20 compared to 10.15]

Based on mortality of each successive age class.

NOTE: Survivorship curve is based on initial number in the cohort,
while mortality curve is not. What does this mean in terms of the
curve’s accuracy?
Fecundity curve for red deer [Fig. 10.21]
a.
Shows relative reproductive potential of the cohort at each age

Takes both fertility and numbers into account
Geometric (discrete-time) model of population growth
A.
Assumptions
1.
Models for growth in discrete time are used for populations that have young during
a particular time of year, rather than continuously
a.
Examples
2.
If the age distribution of a population is stable (recall what that means!), the
population is small, and the resources are relatively unlimiting, the population will
grow (theoretically) at a constant rate.
a.
Realistic? Why develop these models if not?
B.
The geometric finite rate of increase, λ, can be calculated simply as the total number of
individuals in year x+1 (all cohorts) divided by the total number of individuals in year x
(all cohorts). Mathematically:
1.
λ = Nt+1/Nt ; or…. Nt+1 = λ Nt
a.
The λ value can be used to predict the population sizes of successive
generations.
b.
Example:
Generation
Nt
λ
0
50
3
1
150
3
2
3
4
IV.
V.
450
1350
4050
c.
Can write equation in a more general way: Nt = λtN0
C.
What does this really mean in terms of what is happening to individuals in the
population?
1.
Nt+1 = Nt +birth -death + immigration -emigration
a.
Usually simplify and remove immigration and emigration from these
equations
b.
Also, use “per capita” birth and death rates

bt = total births/person

dt = total deaths/person
2.
So, re-write mathematically:
3.
Nt+1 = Nt + bt Nt - dt Nt = (1+ bt - dt) Nt
4.
Thus, λ = 1+ bt - dt.
a.
In the spreadsheet exercises, the term R is introduced.

R=b–d

λ = 1+ R
Exponential (continuous time) models of population growth.
A.
Nt = N0ert
1.
Assumptions
a.
Generations are added continuously rather than discretely

Examples: humans and bacteria
2.
Note similarity to the previous equation:
a.
Nt = N0ert compared to Nt = λt N0

So if a geometric and a discrete population were growing at the
same rate, λ = er [Fig. 11.1]
b.
r = b-d
B.
One form of the equation is: dN/dt = rN
1.
This is the derivative of Nt = N0ert; if you have not had calculus you won’t
understand the derivation, but you should know what it means…
a.
The “d” is not a variable, but rather much like the symbol Δ. So you could
think ΔN/ Δt, that is the “change in N” divided by the “change in t”. The
“d” denotes an instantaneous rate, and corresponds to the slope at a
particular point in time.
b.
Note that different values of r provide a family of exponential curves [Fig.
11.2]

For negative “r”, the populations are shrinking
Logistic growth
A.
This model takes into account some realistic constraints on population growth. In
particular, as a population grows, effects of increased density become apparent:
1.
Think about disease, food availability and predators
B.
There is a carrying capacity (K): The population size (for a particular area) above which
an equilibrium population cannot be supported.
C.
The exponential growth equation can be modified to take the carrying capacity into
account:
1.
dN/dt = rN((K-N)/K)
a.
D.
Think about this mathematically.

When N is negligible, the equation basically collapses to the
exponential growth equation because K/K = 1.

When N = K, K-N = 0, and there is no population growth.

When N > k, K-N is negative, as is the change in the size of the
population.
b.
Think about this visually, particularly what is meant by dN/dt

dN/dt is highest at the midrange. Why? [Fig. 11.5, top]
Time lags [Fig. 11.8]
1.
The response of a population to its own change in numbers is not
instantaneous.
a.
The logistic growth equation can be altered to reflect that time lag.


E.
dN/dt = rNt-g((K-Nt-w)/K)
w = reaction time lag

It associated with the “N” term of (K-N) because it reflects
the fact that it takes awhile for the population to detect and
respond to population pressure (or lack thereof).

g = reproductive time lag

It is associated with its particular “N” term because it reflects
the fact that the offspring just produced take time to produce
their own offspring.

It associated with the “N” term because it reflects the fact that it
takes awhile for the environment to “see” the increased (or
decreased) population.
b.
Time lags result in fluctuations around K.
Is K really a constant?
1.
What situations can alter K?
Study questions
1.
What is plotted on the x-axis of a survivorship curve? On the y-axis (be clear as to the scale used
here)?
2.
Sketch/describe the three different types of survivorship curves, and provide an example for each.
3.
Does a given species always show the same characteristic pattern of survivorship? Use an example
to explain your answer.
4.
Compare a mortality curve to a survivorship curve.
5.
What is a fecundity curve, and what does it take into account?
6.
Spreadsheet exercises 7-8 (=Problem Set #2). Include associated questions. You should have a
good, intuitive understanding of how particular changes to parameters of the equation will affect
the population through time.
7.
Know the geometric, exponential and logistical growth equations. You should also know the
meaning of the various variables and constants, and state in words what the equations say.
8.
Be able to recognize and sketch the basic curves for geometric, exponential and logistic growth.
9.
Know the assumptions of the geometric and logistic growth equations.
10. What does the logistic growth equation take into account that the exponential growth does not
11. How does a time lag term make the exponential growth equation more realistic? Sketch the size of
a population through time that is experiencing moderate time lag.
12. Is K really a constant? Explain your answer.