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Bio 0200
Ecology I - Population Growth
Principal Concepts
Roots of the Science of Ecology
Mathematics of Population
Growth Limits: Carrying Capacity
Survivorship & Life History
Human Population Growth
Predation & Life History
• Labs begin Next Week.
• Homework Assignment #2 (due Friday!) on web site.
• Two “virtual” handouts today (Pop. Growth & Guppies)
• Send e-mail today to request Lab Section.
ecol-o-gy \ i-'klj\ n. pl. -gies [G kologie, fr. kec- + -logie -logy] (1858) 1. A branch of
science concerned with the interrelationship of
organisms and their environments. 2. The
totality or pattern of relations between
organisms and their environment.
Species - collection of individuals interbreeding
under natural conditions to produce fertile
offspring, reproductive isolated from other such
groups. (“biological species” definition)
Population - individuals of the same species
occupying a given area.
Habitat - place where a population lives.
Community - the population of all species
interacting in a habitat. (including primary
producers, and several levels of consumers)
Ecosystem - a community and its physical and
chemical environment
Biosphere - the realm occupied by all living
organisms on Earth
Your textbook uses
two slightly different
notations for
“discrete growth” &
“continuous growth.”
For accuracy and
simplicity, we will use
a single system, as
described in the
handout for today’s
class.
N = number of individuals in a population
I = increase in population over a period of time
I = births - deaths
N = I = B - D
t
Determining birth and death rates:
b=
births
N
d=
deaths
N
Because b and d are rates, they automatically
include the unit of time over which the data
were collected (per year, per month, etc…)
Population Growth - an Example:
In a town of 1,000 people, over the course of a year 46
babies are born, and 36 people die.
Increase: I = 10
Birth Rate: b = 46 / 1000 = 0.046
Death Rate: d = 36 / 1000 = 0.036
And, we can determine a new value that takes births
and deaths into consideration: the net Rate of
Population Growth, r:
r=b-d
r = 0.046 - 0.036 = 0.010
Suppose that we know the rate of population growth (r) and
we'd like to figure out how much that population will
increase over a period of time. To determine the increase to
be expected we simply multiply r by the size of the
population (N):
I = rN
I = 0.010 x 1000 = 10
We could also ask the question in a slightly different way.
How large is the population at the end of a year when the
growth rate (r) is 0.010?
N = No + I
(or)
N = No + rNo
N = 1000 + (0.010 x 1000) = 1000 + 10 = 1010
If we wanted to do this over several years, we could
run this equation for each and every year:
N = No + rNo
Or, we could simplify the equation:
N = No + rNo
N = No (1 + r)
And then, we could simply place an exponent over
the bracketed term... allowing the equation to
apply for any number of years (or intervals):
N = No (1 + r)
n
Example: a population of 250 animals is
increasing in size by 6% a year (rate = 0.06).
At that growth rate, how many of them will
be around in 10 years?
N = No (1 + r)
n
10
N = 250 (1.06)
N = 448 (actually, 447.71)
Key Point: Although birth rates are (of
course) affected by death rates, so long as r is
positive, the population will always grow in
an exponential fashion:
- growth of bacteria in a flask.
- doubling occurs every 30 minutes
- place 1 bacterium in flask
- after 10 hours: 1,048,576
bacteria
- at the end of 20 hours:
1.0995116 x 1012
- After 2.5 days? Mass of
bacteria exceeds mass of
Earth!
Key Point: Although birth rates are (of course)
affected by death rates, so long as r is positive, the
population will always grow in an exponential
fashion:
bacteria
Question:
What happens
if there’s a 10
or 20% death
rate between
generations?
No
deaths
10% die
between
doublings
25% die
between
doublings
Freeman 3/e
Fig. 52.5
High r
Moderate r
Low r
Key Point: Exponential growth occurs at any value of
r ... so unchecked growth will eventually produce high
population density.
Even though
exponential growth
can be observed
under certain
conditions . . .
. . . eventually such
growth is stopped
under real conditions.
What stops it?
Survival declines at high population density
Fecundity declines at high population density
Density-Dependent checks on
Population Growth
• Birth and death rates usually fluctuate in response
to population density; that is, they are densitydependent.
• As a population increases in size, it may deplete its
food supply, reducing the amount of food each
individual gets. Poor nutrition may increase death
rates and decrease birth rates.
• If predators are able to capture a larger proportion
of the prey when prey density increases, the per
capita death rate of the prey rises.
• Diseases, which may increase death rates, spread
more easily in dense populations than in sparse
populations.
Density-Independent checks on
Population Growth
• Factors that affect birth and death
rates in a population independent of
its density are said to be densityindependent.
• Severe weather (like a cold winter)
may kill large numbers of a
population regardless of its density.
• Other examples: some diseases,
earthquakes, volcanic eruptions,
floods, storms.
Exponential growth only
occurs under idealized
(unlimited) conditions
Real conditions
always seem to
approach a limit:
We call this apparent limit a “Carrying Capacity.”
(environmental resistance)
Carrying Capacity
(biotic
potential)
Equilibrium
Exponential
Growth
Growth starts out
exponential:
and then seems to
approach a limit:
This means that the observed growth rate (r)
varies throughout this curve. Early on, it seems
purely exponential. Later, it seems limited. Can
we use math to model these characteristics?
I = ro N
ro = Intrinsic Growth Rate (maximum growth rate).
“The rate observed in the presence of unlimited
resources.” Also: rmax (in your text)
Important! The intrinsic rate (ro) is generally close
to (or identical to) the observed rate (r) during the
early (non-limited) phase of population growth.
We can model
what happens in
the later phase by
assuming that the
system has a
carrying capacity
(K) ...
What
it
represents
does this
the
proportion of
unexploited
actually
resources in K.
represent?
K
I = ro N
[
K-N
K
]
N
K-N
= rmax N
t
K
[
]
I = ro N
[
K-N
K
]
What happens when N is much smaller than K?
What happens when N is 1/2 the value of K?
What happens when N is nearly as big as K?
What happens when N exceeds the carrying
capacity (K)?
Part of your homework assignment is to
work with a simple logistic growth model...
... and then to record and explain your
results. (The model is accessed from the Week 2 web page)
The distinct effects of “r” (growth rate) and “K”
(carrying capacity) on populations give rise to
two general “strategies” of long-term survival:
R-strategists: many offspring, little care, fast
maturity (reliance on fast growth: “r”)
K-strategists: few offspring, plenty of care, slow
maturity.
K-strategists: few offspring, plenty of care
Type I
Type II
Type III
R-strategists: many offspring, little care
In the absence of strong predation
pressure, r-strategists show frequent
population fluctations.
So do K-strategists (over
a longer term)
Pribilof Is.
THE INTRODUCTION, INCREASE, & CRASH OF REINDEER ON ST. MATTHEW
ISLAND David R. Klein Alaska Cooperative Wildlife Research Unit, University of Alaska
What provides for long-term
stability in a population of
consumers?
What provides for long-term
stability in a population of
consumers?
A key study of natural populations:
• guppies in two small streams in Trinidad, as
studied by David Reznick and his associates (UC
Riverside):
killifish
~ 3 cm
Pike cichlid
~ 30 cm
The pike cichlid doesn’t chase small,
immature guppies (which it seldom catches
anyway). The pike concentrates on the
largest guppies in the stream
The killifish preys upon the smaller
guppies. Mature fish are too big for it
to take on.
Guppies in the pikecichlid stream are
smaller (male & female)
at sexual maturity.
Guppies in the pikecichlid stream reproduce
more quickly, and
produce smaller
embryos.
Small predator (killifish).
Attacks only small, immature fish.
Effects on life history:
Guppies mature later, females are large,
produce larger eggs and embryos.
Rationale: Once out of danger, best
reproductive strategy is to produce larger eggs
that will outgrow the predator as quickly as
possible.
Large predator (Pike-cichlid) attacks large,
mature fish.
Effects on life history:
Guppies mature earlier, females are small,
produce smaller eggs and embryos, reproduce
more frequently.
Rationale: Since maximum danger comes at
maturity, mature more quickly and reproduce as
often as possible (with smaller eggs).
Transplantation experiments confirm that a change in
predator eventually results in a change in life cycles.
• Life History Patterns are
heritable.
• They show variation within
a population.
• The variants best able to
evade the predator will be
most successful in
reproduction.
• Over time, the effects of the
predator will change the life
history pattern of the fish
population.