of Contents 
Ecology Home 

and Physical Sciences 
and Mathematics 
Ecology 
B. How do populations change; "the motion picture" (population dynamics)
3. Regulation
of
populations
From a previous lecture
N_{t+T}
= (e^{rT})
* N_{t}
dN/dt= rN 
What does this model of
population
growth assume?
Hint:
Bacteria innoculated in a broth (N_{0}=100)
Bacteria double every 20 min (r=1.035 min^{1})
population size after 36 hours = e^{1.035(36*60}^{)}* 100 = 2 X 10^{34}
or enough bacteria to cover the entire earth 1 ft deep!
What really happens to natality and
mortality
as a population grows beyond a certain size? Why?
Resource = "environmental
component
used by an organism" e.g. food, shelter, light, space.

The population size at which population
growth
rate is zero (no slope, dN/dt=0) is the 'carry
capacity' or 'equilibrium population density' (K).
It is the maximum number the environment can support
indefinitely.
At some point, something has to limit population growth.
There are numerous ways that limitations
to population growth can be modelled and the most appropriate model for
a given population would be a function of the type of factors that most
affect that population. Two general types of models:
Deterministic models  The population density at a future time can be predicted based on present density. These models should work best for populations in which intrinsic or densitydependent factors dominate. That is, the density of the population directly determines the rate of births and/or deaths. For example:
 competition for a limiting resource affects the level of that resource, resulting in higher rates of starvation when population density is high.
 higher population density attracts more predators.
 weather events that occassionally kill off some of the population so that population size never reaches the point that resources become limiting.
 the level of the resource determining carrying capacity is not constant but determined by factors other than the population being modelled.
Here are two examples of these models:
A simple deterministic model of population growthTo model this, previous equations must be altered:
dN/dt= rN * (1  ^{N}/_{K}) *
where K is the 'carrying capacity' or 'equilibrium population density'In plain english: population growth is not just a function of the intrinsic rate of growth and population size; it is limited to K number of individuals (i.e. is a function of intrinsic or densitydependent factors).
The term (1  ^{N}/_{K}) esssentially converts r from r_{max} to r_{actual} which is expected to be a function of current population size relative to carrying capacity.
Look at what this equation does mathmatically:
What happens graphically if carry capacity is increased?
 When N is very small relative K, growth is essentially exponential as modelled by dN/dt= rN
 As N approaches K, ^{N}/_{K }appoaches 1 so 1 minus this is near 0 resulting in essentially no population growth (no slope).
 What happens to population growth rate if N is greater than K?
What happens graphically if the per capita intrinsic rate of increase is increased?
Click here to run a simulation using the above equation
This model is 'artifical'(as are all models).
 The term r in these models is constant and is actually r_{max} , the maximum per capita rate of increase under optimal environmental conditions. In reality, the per capita rate of growth is a function of birth and death rates that change of populations size (N).
 K is a theoretical term that forces predicted population growth to not exceed that particular population size, and does not directly model resource levels that affect birth and death rates.
If artifical, what good is the
model?
The model predicts the general
shape
population
growth for some species.

The above model can be used to project change in population size over time by iterating the following equation:
N_{t+1}
= (e^{r}^{(1
 N/K)}) * N_{t}
At higher values of r,
there are some surprizing changes in population size over the longterm
(click here to run a simulation).
A more
recent idea,
Complexity/Chaos Theory:
Simple relationships among a few
variables can also generate complex patterns.
For example, fractals.
http://www2.capcollege.bc.ca/~mfreeman/mand.html
Consequence of Complexity
Theory: Larger
complex processes are highly contingent on initial,
smallscale conditions (the "butterfly
effect").
So what? While other
factors
may contribute to erratic fluctuations, even simple intrinsic factors
influencing
population change can produce complex patterns.
Are there other factors that
might
cause
population size to fluctuate erratically over time?
where €_{t}
is a 'random' variable that represent changes in the intrinsic rate of
increase due to unpredictable extrinsic factors.
In plain english: population growth
is a
function of both intrinsic and extrinsic
factors
Click
here
to run a simulation using the above equation
What factors might be
extrinsic
and tend to be erratic?
More complex
models
might include components that model resource levels as a function of
intrinsic
and extrinsic factors.
1957
Symposium at
Cold Spring Harbor: A. J. Nicholson advocated population regulation by densitydependent influences. In support Elton wrote, "it is becoming increasingly understood by population ecologists that the control of populations, i.e., ultimate upper and lower limits set to increase, is brought about by densitydependent factors" Andrewartha and Birch disagreed: densitydependent factors "are not a general theory because . . . they do not describe any substantial body of empirical facts." 
So how can these processes be
distinguished?
Erratic fluctuations in population size are likely due to DENSITYINDEPENDENT processes where the affects of a particular influence do not change proportionately with population size.
Constant or cyclic fluctuations are likely due to DENSITYDEPENDENT proceses, whose influence is a proportional to it population size (But, theoretically, can densitydependent factors result in erratic fluctuations?)
A hypothetical numerical example:
Which mortality
patterns are densitydependent
and densityindependent?
population size
(N)

?

?

?

100

50 deaths
(50%)

50 deaths
(50%)

20 deaths
(20%)

1000

50 deaths (5%)

500 deaths
(50%)

500 deaths
(50%)

10,000

50 deaths (0.5%)

5000 deaths
(50%)

8000 deaths
(80%)

Do the data in the table below support densitydependent or densityindependent influences?
Mortality in the commercial catch of fish after severe cold weather on the Texas Gulf Coast in Winter of 1940: Locality N before cold weather N after cold weather %Decline Matagorda Bay 16,919 1,089 93.6 Aransas Bay 55,224 2,552 95.4 Laguna Madre 2,016 149 92.6
Causes of densitydependent effects
The study of such effects is sometimes referred to as "Population Regulation"Do individuals purposely decrease births (or commit suicide) as numbers increase? Are there adaptations that arise for "the good of the species"?
For example WynneEdwards proposed that some bird populations regulate their clutch size in hard times to benefit the population as a whole" Can this reduction in clutch size be explained more simply?
A. Ives writes that population regulation wrongly "implies that something is doing the control, and that the actions of the controller have some purpose. In fact, population regulation is used by ecologists in a much more neutral way simply to describe patterns of population dynamics...there is no reason to suppose that a purpose exists. For example, a popular concept of "the balance of nature" implies that natural processes act in a purposeful way to create greater stability and harmony in the natural world. In fact, no scientific evidence supports this."
i.e. What we see in ecological systems can most easily be explained by simple Darwinian selection on individuals.
From your 'quick review of evolution', a community is a collection of individuals evolving to take advantage of available resources resulting in a dynamic 'balance of power'.
Given this, which of the statements below is more accurate way of describing a population competing for a limiting resource?"As population density increases toward a point at which resources are insufficient, individuals in competition reduce their intake of food.""As population density increases toward a point at which resources are insufficient, the intake per individual is reduced due to shortages"
The negative effect of density on population growth can result from decreases in births or increases in deaths or dispersal. This may appear to "be for the good of the population" but can be explained much more easily either through direct causes or Darwinian selection of individuals:
1. Increased mortality
 directly due limited energy/nutrients necessary for survival as a result of intraspecific competition
 due to less direct physiological effects such as weaken immunity, increased transmission of disease, increased predation...
2. Decreased reproducton
 directly due limited energy/nutrients necessary for reproduction.
 due to a physiological mechanism that decrease reproductive rates as density increases. Can this be explained by natural selection or is this an example of an adaptation for the good of the species?
3. Increased dispersal
 due to need to increase search area for resource.
 due to change in territorrial behavior in which some individuals "move out". Can this be explained by natural selection or is this an example of an adaptation for the good of the species?
One more time at applying ecological
thinking
to human populations: How will resources affect the future of human
population
growth?
Will human populations be limited by insufficient resources? Are there sufficient resources (agricultural land, water, energy) left to accommodate the world's population growth ?
Do we need to think about resources needed by humans differently than we think about resources needed by nonhuman populations?
 dependence on nonrenewable resources
 degradation of the environment that produces renewable resources
 differences in resource consumption per individual
Global distribution of wealth:Is consumption of world resources unequal? 20% affluent
 60% selfsufficient
 20% destitute
How affluent are you relative to the rest of the world?
The affluent fifth of the world consumes 2/3 of the metals and 3/4 of the world's energy, and produces 3/4 of the world's pollution.
On a per capita basis, individuals in developing nations use 1.0 kw/person whereas developed nation use 7.5 kw.And the rate of consumption is increasing. In the U.S. since 1950's, the average American consumes twice as much steel, meat, wood, and energy and 21 times as much plastic.
What factors have led to the disparity in consumption rates?Should goals of consumption be questioned?
Relationships between resource consumption and happiness are not obvious.
New York Times October 4, 2005 From Science, 1995, Vol 267 From the film "The Gods Must be Crazy
What drives consumption?Is consumption ingrained in our system of values?Does extreme consumption affect others?
of Contents 
Ecology Home 

and Physical Sciences 
and Mathematics 