**Lecture 25: Competition -- classic
theory**

I. Background

A. How do we predict the outcome of
competition?

1. One successful
approach has been to represent competition mathematically,
in a
manner that derives directly from models that we have already
learned
in class.

2. Begin
with
exponential growth and logistic growth

3. With
simple
modifications to this model, we can represent mathematically the
effects
of competition between two species.

4. The easiest
way
is to have the competitor influence the population level of a
species
in the same way that the species influences its own density.

--
in the absence of a competitor, the population grows to K

5. Thus, change in population size equals the intrinsic rate of growth, minus density-dependent effects, minus effects due to competition

B. Lotka Volterra competition models
(won't
reproduce in the notes, but can be found on page 374-378 in
your
book)

1. The only terms that
differ from the exponential and logistic equations are:

alpha
= which is the effect of an individual of species 2 on the
exponential
growth rate of the population of species 1.

Likewise, beta is simply the effect of an individual of species 1
on
the exponential growth rate of the population of species 2.

K1,
K2, N1, and N2 are the same as in previous equations, but the
subscript
stands for each different population of a different species

II. Predicting the outcome of competition

A. What is the outcome of
competition?
Four possibilities — sp1 wins, sp2 wins, both coexist, or both go
extinct.

B. To determine which species wins in
competition, we will use a graphical approach

C. First ask: What are equilibrium
conditions?

1. Solve
Lotka-Volterra equations – one species at a time for
solutions at
equilibrium (when population size is not changing) -- Start with
species one

2. Graph the
line
represented by the equilibrium solutions

Note:
an
ISOCLINE
is a line that defines all of the possible
equilibrium population sizes of one species, given the population
size
of a second species

3. Determine
population growth from any initial population size

VI. The outcomes of competition in words:

A. Species 1 wins when it inhibits
species
2 more
than it inhibits itself AND

When
species
2
inhibits itself more than it inhibits species 1

B. Species 2 wins when it inhibits
species
1 more
than it inhibits itself AND

When
species
1
inhibits itself more than it inhibits species 2

C. When each species inhibits
itself
more than it inhibits the other species:

STABLE
COEXISTENCE

D. When each species inhibits the
other species more than it inhibits itself:

UNSTABLE
COEXISTENCE