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					                POPULATION GROWTH, CARRYING CAPACITY, AND COMPETITION


Exponential Growth.
Population growth, with an infinite resource supply, is described by this equation:

                                         dN
                                             rN         (1)
                                         dt

where N = the population size, r = the rate of increase per individual (i.e., the number of births
minus deaths divided by the population size), and t = time (dN/dt means “change in N over time
t).

As a simple example, imagine a population of plants or animals
that reproduce seasonally. Assume N = 100 individuals, and t =             Population Growth
                                                                                r = 0.1
one year. During that year, 20 individuals die, but 50 are born,
resulting in the net addition of 30 individuals.
                                                                          800,000
                                                                          700,000
That’s a growth rate of 30 additions per 100 individuals, or 0.3.         600,000
The change in population over the year would, therefore, be               500,000
100 x 0.3 = 30. The total population would be 130.                        400,000




                                                                      N
                                                                          300,000
In year two, we’d start with N = 130, so the change in                    200,000
                                                                          100,000
population would be 130 x 0.3 = 39 more individuals (total                      0
population = 169), and so on. After 35 years, the population                          0    10 20 30
would be 748,297 individuals.                                                                   Year

Carrying Capacity.
In reality, of course, resources are finite: a given environment can only support so many
individuals, and this limit is called carrying capacity.

As a population reaches carrying capacity, each individual has
a harder time getting the resources it needs to survive and                 Population Growth
reproduce. Fertility decreases (fewer births) and/or mortality            r = 0.1 and K = 5,000
increases (more deaths), and the overall growth rate slows.
                                                                          6,000
We can modify equation (1) to represent this decrease:                    5,000
                                                                          4,000
                   dN      K N
                       rN 
                                                                      N




                                
                                                                          3,000
                                           (2)
                   dt       K                                           2,000
                                                                          1,000
where K is the carrying capacity. For the example above, if the               0
carrying capacity is 5000 individuals, population growth begins                   0       10    20     30
to slow after N = 2500.                                                                        Year
                POPULATION GROWTH, CARRYING CAPACITY, AND COMPETITION


Competition.
Competition between individual organisms of the same species follows from the two general
principles described above: population growth is exponential and resources are finite. As
resources become scarce, individuals must compete for them.

An organism’s ability to survive and reproduce depends on its ability to get the resources it needs
and to tolerate the range of conditions that exist in its environment. Because no two individual
organisms are exactly alike, those individuals best able to get resources and tolerate
environmental conditions will leave the most offspring.

Similarly, no two species are exactly alike in their resource needs and/or environmental
tolerances. Species living in the same environment must either utilize different resources or
compete for the same (finite) resource. Because each species has different tolerances and
abilities, one species will always be better able to capture resources in a given environment.

We can see how this works for two species whose population dynamics are described by
equation (2) above by modifying their population equations to model the effect that each species
has on the other. We’ll call one species A and the other B and assume that both have the same
carrying capacity and growth rates. For species A:

                                dN A         K  N A  CB 
                                      rN A                    (3)
                                 dt               K       

where CB is the competitive effect that B has on A. (This impact is calculated as the population of
species B times a coefficient, alpha, representing the negative effect of one individual of species
B on an individual of species A.) To model the populations as they interact, we would write an
equation for species B as well, and calculate the changing populations of each.

This graph shows the results when               Population Dynamics with Competition
A is a much stronger competitor.                r = 0.1 and K = 5,000 for both species
Note that species A starts with only
30 individuals, and B starts with            6,000
500. Both populations grow at first,                                                Species A
but as the population of A                   5,000
                                                                                    Species B
increases to about 800 individuals,          4,000
growth rates for species B become
                                         N




                                             3,000
negative (deaths exceed births) and                                             Sp. A alpha = 3
the population of B begins to                2,000                              Sp. B alpha = 0
decline. By the end of the                   1,000
simulation, species A has
                                                 0
completely displaced species B
                                                     0    10     20      30
from the environment. This is an
example of competitive exclusion.                              Year

				
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posted:11/10/2011
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