Note: a HW assignment for Wed below (*)Note
Models:
How populations grow.....
When we consider the two
scenarios above, the concept of exponential population growth becomes
a reality. Where before we had a limited sustaninable populations
size ( similar to early human populations), now we have the masses.
How can this happen? we definetly know of the consequences, and we
also know that this type of growth pattern is not the "norm" , at
least for most of the species out there... so under what conditions,
can this occur?
The exponential model
is associated with the name of Thomas Robert Malthus (1766-1834) who
first realized that any species can potentially increase in numbers
according to a geometric series.
For example, if a species
has non-overlapping populations (e.g., annual plants or insects
), and each organism produces R offspring, then, population numbers
N in generations t=0,1,2,... is equal to:
When t ( number of generations
or time ) is large, then this equation can be approximated by an exponential
function:
There are 3 possible model
outcomes:
if r = 0 then
no new growth
if r > 0 population size increases
if r < 0 population size decreases
The parameter r
can be called: Malthusian parameter = intrinsic rate of increase= instantaneous
rate of natural increase = population growth rate
Assumptions of the Exponential
Model:
- Continuous reproduction
(e.g., no seasonality)
- All organisms are identical
(e.g., no age structure)
- Environment is constant
in space and time (e.g., resources are unlimited)
In general the exponential
model is robust -- it gives good approximations even if these conditions
are not met. Organisms may differ in their age, survival, and mortality.
But if the population consists of a large number of organisms, and
thus their birth and death rates are averaged given the differences
are not too great among them then the assumptions are approximated.
Parameter r in the exponential
model can be interpreted as a difference between the birth (reproduction)
rate and the death rate
- r = b -d +(+ immigration
- emmigration)
where b is the birth
rate and m is the death rate.
- Birth rate is the
number of offspring organisms produced per one existing organism
in the population per unit time.
- Death rate is the
probability of dying per one organism per unit time.
Solve a problem:
if you start a population with 10 individuals
*** and each
pair has 8 offspring/year
*** and each individual lives on the average 2 years
then how many rabbits will there be in 3 years?
1. Calculate what b and
d are:
b 4 per individual reproductive
effort
d every year half die off use 1/longevity 1/2
then r = 4 - .5 = 3.5
and No = 10
N1 = No (original) + plus
the new additions
dN/dT = 10 + (3.5)(10)
= 45 and so on.
How fast can populations
grow if they grow exponentially?
Rabbit population
in Australia:
1859 12 pairs
1865 they had increased
by a factor of 10,000 in 6 years.
This with an r of 1.5 and a doubling time of 5.5 months.
However this is very unusual......why?
| Generally populations
have regulated growth, whether by predators*, parasites*, weather*,
lack of food*/interspecific competition*, lack of breeding sites*,
social interactions*, etc..... |
How can these
parameters be included in the model of population growth?
We must decrease the intrinsic rate of growth itself whether directly
or by a modifier: This is done in the logistic
model which
models sigmoidal growth
Logistic model was developed
by Belgian mathematician Pierre Verhulst (1838) who suggested that
the rate of population increase may be limited, i.e., it may depend
on population density:
 where
r is equal to
Assume a carrying capacity
K ( the maximum sustainable level of N)
as long as
N/K <1 population
grows at an increasingly lower rate over time
N/K > 1 population size will decrease
N/K = 1 stable size
See for yourself, as N
--> K the ro coefficient will get smaller. Solved the
equations is equal to:
At low densities (N <
< 0), the population growth rate is maximal and equals to ro. Parameter
ro can be interpreted as population growth rate in the absence of
intra-specific competition.
Population growth
rate declines with population numbers, N, and reaches 0 when N = K.
Parameter K is the upper limit of population growth and it is called
carrying capacity. It is usually interpreted as the amount of resources
expressed in the number of organisms that can be supported by these
resources. If population numbers exceed K, then population growth rate
becomes negative and population numbers decline.
The logistic equation
would be okay as a general growth descriptor if:
- 1. Individuals could
react immediately to increases in Nt---- but can they, especially
if they breed only once a year?
- 2. Parameter K or the
carrying capacity has biological meaning for populations with a
strong interaction among individuals that controls their reproduction.
- 3. There were no environmental
fluctuations which altered K - but unfortunately K does vary. In
a good year many more individuals may be suppported than in a bad
year .
- 4. There were no interactions
between them and other species which alter their b, d values....
like predators, parasites...ut that is almost impossible.. if prey
levels are high, obviously predators will attempt to kill more unless
they are very wise.
Thus many populations oscillate in size with time.... WHETHER in a
regular or chaotic pattern.
How much they oscillate or whether they can stabilize these ocscillations
depends on part on the populations 'r' value....
If r<1 then
the increase in population size between t and t+1 will be less then
the difference between N and K and the population will adjust monotonically
( in a smooth directed pattern; see above
If 1< r < 2 the population will have a dampened oscillation:
see above
When r > 2 especially if r > 2.52 oscillations will actually
increase and the population growth will become chaotic
IF r >> 2 the population will likely crash genrally in short
time
IF r > 2 but less than 2.5 population may display a stable ( regular
with same amplitude ) limit cycle
To portray a more realistic
growth model, we use the concepts of time delays or lags: featuring
via a new parameter ...
You can modify the logistic equation to compensate for lag time (
time required to recognize that the population has overshot the K)
with the following modification
dN/dT = rNt [ 1- (N (t- lag) / K)]
- dampened oscillations-
r*lag pi/2 ~ 1.6
- monotonic r*lag or
= .37
- limit cycle r*lag >
pi/2
This all approximates real growth if in the populations:
- All individuals are
the same size- this model can't take into account that 1 large individual
may be equivalent to 25 small so it isn't effective for long-lived
animals or plants where size differences make a difference.
- It totally ignores
genetic differences between individuals- or any reproductive/mortality
difference etc.
- Variablity is sacrificied
for the simplicity of the model.
If you want to model population growth to account for some variability
( gentetic, size, sex) you need to either use the Leslie Matrix model
which allow differences in birth rates and mortality between the age
or size classes.
Still simple but more powerful.or develop your own simulation model:
see below:
**Set up on paper a STELLA black box model
for the explosion of the mitten crab using the following information:
Please bring in this assignment on Wed!
(from: http://www.delta.dfg.ca.gov/mittencrab/life_hist.html)
Chinese Mitten Crabs:
Life and History
Life History and Background Information on the Chinese Mitten Crab
August 5, 1998
The Chinese mitten crab (Eriocheir sinensis), so named for the dense
patches of hairs on the claws of larger juveniles and adults, is native
to the coastal rivers and estuaries of the Yellow Sea. It was accidentally
introduced to Germany in the early 1900s and spread to many northern
European rivers and estuaries. In San Francisco Estuary, the mitten
crab was first collected in 1992 by commercial shrimp trawlers in
South San Francisco Bay and has spread rapidly throughout
the estuary. Mitten crabs were first collected in San Pablo Bay in
fall 1994, Suisun Marsh in February 1996, and the Delta in September
1996. As of August 1998, the known distribution of the Chinese mitten
crab extends north of Colusa to Hunter's Creek (near Delevan National
Wildlife Refuge) in the Sacramento River drainage, east to Roseville
(Cirby Creek) and eastern San Joaquin County near Calaveras County
(Mormon Slough and Littlejohns Creek) and south in the San Joaquin
River to Hiway 165, near San Luis National Wildlife Refuge. The most
probable mechanism of introduction to the estuary was either deliberate
release to establish a fishery or accidental release via ballast water.
In Asia, the mitten crab is a delicacy and crabs have been imported
live to markets in Los Angeles and San Francisco.
The mitten crab is catadromous - adults reproduce in salt water and
the offspring migrate to fresh water to rear. In the San Francisco
Estuary, the mitten crab probably matures in 2 to 3 years, although
it reportedly matures from 1 to 5 years elsewhere, depending on water
temperature. Males and females grow to a maximum carapace width of
approximately 80 mm (3 inches) in the estuary. Mating and fertilization
occurs in late fall and winter, generally at salinities >20‰.
The females carry their eggs until hatching and both sexes die soon
after reproduction. A single female can carry 250,000 to 1 million
eggs. After hatching, larvae are planktonic for approximately 1 to
2 months. The small juvenile crabs settle in salt or brackish water
in late spring and migrate to freshwater to rear.
Young juvenile mitten crabs are found in tidal freshwater areas, and
usually burrow in banks and levees between the high and low tide marks.
Mitten crabs apparently do not burrow as extensively in non-tidal
areas, probably because they are not subject to desiccation during
low tides. Older juveniles are found further upstream than younger
juveniles, and in China and Europe they have been reported several
hundred miles from the sea. We do not understand what cues this upstream
migration, although high densities were reportedly a factor in Germany
and the upstream migration may be tied to the monsoon season in southern
China. Maturing crabs move from shallow areas to the channels in late
summer and early fall and migrate to salt water in late fall and early
winter to complete the life cycle.
Mitten crabs are adept walkers on land, and, in their upstream migration,
they readily move across banks or levees to bypass obstructions, such
as dams or weirs. In Germany, large numbers of mitten crabs were reported
to leave the water at night when they encountered an obstruction and
occasionally wandered the streets and entered houses. In Stockton,
2 adult mitten crabs climbed over a levee and into a swimming pool
when they encountered a small dam blocking their downstream migration.
Mitten crabs are omnivores, with juveniles eating mostly vegetation,
but preying upon animals, especially small invertebrates, as they
grow. In the Delta, adult crabs have been incidentally caught by anglers
using a variety of baits, ranging from ghost shrimp to shad. Relatively
little is known about the predators of the mitten crab, although white
sturgeon, striped bass, bullfrogs, loons, and egrets have been reported
to prey upon them in the estuary. We assume that other predatory fishes,
including largemouth bass and larger sunfishes, river otters, racoons,
and other wading birds will consume mitten crabs.
Based on the impacts of mitten crabs in their native range and Europe,
they pose several possible threats. The mitten crab is the secondary
intermediate host for the Oriental lung fluke, with mammals, including
humans, as the final host. Humans become infested by eating raw or
poorly cooked mitten crabs. However, neither the lung fluke nor any
of the freshwater snails that serve as the primary intermediate host
for the fluke in Asia have been found in the Estuary. It has been
noted that several species of freshwater snails which could possibly
serve as an intermediate host are present in the watershed.
The burrowing activity of mitten crabs may accelerate the erosion
of banks and levees. In Germany, burrows were reported to be up to
50 cm (20 inches) deep and some damage to levees and structures has
occurred. Mitten crab burrow densities as high as 30/m2 (2.7/ft2)
have been reported from South Bay creeks, with most burrows no more
than 20-30 cm (8-12 inches) deep. The highest density of juvenile
crabs was approximately 6/m2 (0.8/ft2) in Suisun Marsh and 1/m2 (0.1/ft2)
in the Delta in summer 1997. In the Delta large numbers of juvenile
mitten crabs were also reported in water hyacinth, which is not found
in Suisun Marsh, San Francisco Bay, or it's tributaries.
In China and Korea, juvenile mitten crabs have been reported to damage
rice crops by consuming the young rice shoots and burrowing in the
rice field levees. Rice fields in tidally influenced areas apparently
are most subject to damage.
The most widely reported economic impact of mitten crabs in Europe
has been damage to commercial fishing nets and the catch when the
crabs are caught in high numbers. The mitten crab has become a nuisance
for commercial Bay shrimp trawlers in South Bay, as it is time consuming
to remove the crabs from the nets (one trawler has reported catching
over 200 crabs in a single tow several times). Shrimp trawlers have
also reported that a large catch of mitten crabs damages and even
kills the shrimp, making them unsuitable for the bait market. Shrimp
trawlers have been able to move to areas with fewer crabs, but, as
the mitten crab population grows, this option diminishes.
The mitten crab overlaps in dietary and habitat preferences with the
introduced red swamp crayfish (Procambarus clarkii) in South San Francisco
Bay creeks and negative interactions between the two species have
been observed in the field. In the Delta, the mitten crab may reduce
abundance and growth rates of the introduced signal crayfish (Pacifastacus
leniusculus), which supports a commercial fishery.
The ecological impact of a large mitten crab population is the least
understood of all the potential impacts. Although juveniles primarily
consume vegetation, they do prey upon animals, especially invertebrates,
as they grow. A large population of mitten crabs could reduce populations
of native invertebrates through predation and change the structure
of the Estuary's fresh and brackish water benthic invertebrate communities.
In Germany, extensive efforts were undertaken by the government in
the 1920s and 1930s to control mitten crab populations in some rivers.
Control measures often took advantage of the mitten crab's migratory
behavior; traps were placed on the upstream side of dams to capture
juvenile crabs as they migrated upstream. At one site, as many as
113,960 crabs were trapped in a single day. It was hypothesized that
this population explosion may have coincided with a reduction of predators,
especially fishes, in the rivers. In recent years, European mitten
crab populations have apparently been stable, although there are occasional
reports of "invasions". In 1981, the mitten crab population
in the Netherlands increased substantially, resulting in serious damage
to fishing nets.
Information on the impacts of the mitten crab in China and Korea has
been more difficult to obtain. Although the mitten crab damages rice
crops, no control measure have been reported. In some rice fields,
they are cultured with fish. Apparently, mitten crabs are stocked
at a rate that does not damage the rice crop.
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