Other patterns in communities
• Macroecology: relationships of
– geographic distribution and body size
– species number and body size
• Latitudinal gradients: changes in S with
latitude
• Species-Area relations: Island
biogeography and related questions
Species-area relationships
• Islands, either oceanic or habitat
• Selected areas within continents
• How is number of species related to area?
S
A
Mathematics
S=c
z
A
– S is number of species
– A is area sampled
– c is a constant depending on the taxa
& units of area
– z is a dimensionless constant
• often 0.05 to 0.37
Often linearized
• ln (S ) and ln (A )
• ln (S ) = ln (c ) + z ln (A )
– z is now the slope
– ln (c ) is now the intercept
ln (S )
ln (A )
Theory & Hypotheses
• Area per se hypothesis
– why S goes up with A
– why S = c A z
– why z takes on certain values
• Habitat heterogeneity hypothesis
– why S goes up with A
• Passive sampling hypothesis
– why S goes up with A
Area per se
Sn
• large heterogeneous assemblage  log
normal distribution of species abundances
• assume log normal ("canonical log normal")
– Abundance class for most abundant species =
abundance class with most individuals
– constrains variance (s2) of the distribution
• assume that N increases linearly with A
• Yield: unique relationship: S = c Az
• for "canonical" with S > 20: S = c A0.25
ni
Area per se
• z varies systematically
– larger for real islands vs. pieces of contiguous
area
• z does not take on any conceivable value
– if log normal had s2 = 0.25 (very low)
– then z  0.9 … which is virtually unknown in
nature
– implies constraints on log normal distributions
Dynamics of the area per se hypothesis
• open island of a given area
• rate of immigration
(sp. / time) = I initially high
• once a species is added, I
declines
• nonlinear:
I
– 1st immigrants best dispersers
– last are poorest dispersers
S
ST
Dynamics of the area per se hypothesis
• rate of extinction
(sp. / time) = E initially 0
• as species are added, E
increases
• nonlinear:
E
– lower n as S increases
– more competition as S
increases
S
ST
Dynamic equilibrium
RATE
• equilibrium
when E = I
• determines S*
• how are rates
related to
area?
E
I
S
S*
Effect of area on S*
RATE
• 2 islands equally
far from mainland
• large & small
• extinction rate
greater on small
Esmall
Elarge
I
– smaller n’s
– greater competition
• under this
hypothesis I is not
related to area
S
S*small
S*large
Area per se
• Neutral hypotheses vs. Niche hypotheses
• Neutral hypotheses – presume that biological
and ecological differences between species,
though present, are not critical determinants
of diversity
• Area per se is a neutral hypothesis
– S depends only on the equilibrium between
species arrival and extinction
– Large A  large populations  low prob.
extinction
Niche-based hypotheses
• Niche hypotheses - presume that that
biological and ecological differences
between species are the primary
determinants of diversity
• Niche differences enable species to
coexist stably
• Does not require equilibrium between
extinction and arrival
Habitat heterogeneity
• Niche-based hypothesis
• Larger islands  more habitats
– Why?
• More habitats  more species
– does not require competition
– does not require equilibrium
– does not exclude competition or equilibrium
• Larger islands 
bigger “target”
• Neutral hypothesis
• More immigration 
more species
– competition &
equilibrium not
necessary (but
possible)
– under this hypothesis
E is not related to
area
Passive sampling
RATE
E
Ilarge
Ismall
S
S*small
S*large
Processes
• Interspecific competition
Competition
• Competition occurs when:
–a number of organisms use and
deplete shared resources that are
in short supply
–when organisms harm each other
directly, regardless of resources
–interspecific, intraspecific
Resource
competition
competitor #1
Interference
competition
competitor #1
+
competitor #2
-
+
-
competitor #2
-
-
resource
Competition
• Interference
– Direct attack
– Murder
– Toxic
chemicals
– Excretion
• Resource
– Food, Nutrients
– Light
– Space
– Water
• Depletable,
beneficial, &
necessary
Competition & population
• Exponential
growth
• dN / dt = r N
N
– r = exponential
growth rate
– unlimited growth
• Nt = N0 ert
t
Competition & population
• Logistic growth:
[K-N]
dN / dt = r N 
K
• r = intrinsic rate of
increase
• K = carrying capacity
K
N
t
Carrying capacity
• Intraspecific competition
– among members of the same species
• As density goes up, realized growth rate
(dN / dt) goes down
• What about interspecific competition?
– between two different species
Lotka-Volterra Competition
N1 N2
r1 r2
K1 K2
[ K1 - N1 - a2 N2 ]
dN1 / dt = r1 N1 
K1
[ K2 - N2 - a1 N1 ]
dN2 / dt = r2 N2 
K2
Lotka-Volterra Competition
• a1 = competition coefficient
– Relative effect of species 1 on species 2
• a2 = competition coefficient
– Relative effect of species 2 on species 1
• equivalence of N1 and N2
Effects of Ni & Ni’ on growth
[ K1 - N1 - a2 N2 ]
dN1 / dt = r1 N1 
K1
In the numerator, a single individual of N2
has a equivalent effect on dN1 / dt to a2
individuals of N1
Competition coefficients: a’s
• Proportional constants relating the effect
of one species on the growth of a 2nd
species to the effect of the 2nd species on
its own growth
– a2 > 1  impact of sp. 2 on sp. 1 greater
than the impact of sp. 1 on itself
– a2 < 1  impact of sp. 2 on sp. 1 less than
the impact of sp. 1 on itself
– a2 = 1  impact of sp. 2 on sp. 1 equals the
impact of sp. 1 on itself
Notation
dNi / dt
vs.
• total population
growth
• dNi / dt
= riNi [Ki-Ni-ai’Ni’]/Ki
dNi / Nidt
• per capita
population growth
• dNi / Nidt
= ri [Ki-Ni-ai’Ni’]/Ki
Lotka-Volterra equilibrium
• at equilibrium
– dN1 / N1dt = 0 & dN2 / N2dt = 0
– also implies dN1 / dt = dN2 / dt = 0, so...
• 0 = r1N1 [ (K1-N1-a2N2)/ K1]
• 0 = r2N2 [ (K2-N2-a1N1)/ K2]
• true if N1 = 0 or N2 = 0 or r1= 0 or r2 = 0
Lotka-Volterra equilibrium
• for 0 = r1N1 [ (K1-N1-a2N2)/ K1]
• true if 0 = (K1-N1-a2N2)/ K1
• if N2 = 0, implies N1 = K1 (logistic
equilibrium)
• as N1  0, implies a2N2=K1 or N2 = K1 / a2
• plot as graph of N2 vs. N1
Equilibrium
• dNi / dt = 0 for both species
• K1 - N1 -a2N2 = 0 and K2 - N2 -a1N1 = 0
N2
Zero Growth Isocline
(ZGI) for species 1
K1/a2
dN1/dt<0
dN1/dt>0
N1
K1
Zero growth isocline
for sp. 2
N2
dN2 /N2 dt < 0
K2
Zero Growth
Isocline (ZGI)
dN2 /N2 dt = 0
dN2 /N2 dt > 0
0
K2/a1
N1
Zero growth isocline
for sp. 1
N2
dN1 / N1 dt < 0
K1 / a2
Zero Growth
Isocline (ZGI)
dN1/N1dt = 0
dN1 /N1 dt > 0
0
K1
N1
dN1 / N1dt
r1
0
K1 / a2
N2
Isocline in
3 dimensions
K1
N1
Zero
Growth
Isocline ...
dN1/N1dt = 0
Isocline in
3 dimensions
0
K1 / a2
N2
K1
N1
Zero
Growth
Isocline ...
dN1/N1dt = 0
N2
K1 / a2
Isocline
Zero
Growth
Isocline ...
dN1/N1dt = 0
0
K1
N1
Two Isoclines on same graph
• May or may not cross
• Indicates whether two competitors can coexist
• For equilibrium coexistence, both must have
– Ni > 0
– dNi / Ni dt = 0
Lotka-Volterra Zero Growth Isoclines
N2
K1/a2

K2
0


K2/a1
Species 1 “wins”
K1
N1
• K1 / a2 > K 2
• K1 > K2 / a1
• Region 
dN1/N1dt>0 &
dN2/N2dt>0
• Region 
dN1/N1dt>0 &
dN2/N2dt<0
• Region 
dN1/N1dt<0 &
dN2/N2dt<0
Lotka-Volterra Zero Growth Isoclines
N2
K2

K1/a2

0

K1
Species 2 “wins”
K2/a1
N1
• K2 > K1 / a2
• K2 / a1 > K1
• Region 
dN1/N1dt>0 &
dN2/N2dt>0
• Region 
dN1/N1dt<0 &
dN2/N2dt>0
• Region 
dN1/N1dt<0 &
dN2/N2dt<0
Competitive Asymmetry
• Competitive Exclusion
• Suppose K1  K2. What values of a1 and a2
lead to competitive exclusion of sp. 2?
• a2 < 1.0 (small) and a1 > 1.0 (large)
• effect of sp. 2 on dN1 / N1dt less than effect of
sp. 1 on dN1 / N1dt
• effect of sp. 1 on dN2 / N2dt greater than
effect of sp. 2 on dN2 / N2dt
Lotka-Volterra Zero Growth Isoclines
N2
K1/a2
• K1 / a2 > K2
• K2 / a1 > K1
• Region  both
species increase
• Regions  & 
one species
decreases & one
species increases
• Region  both
species decrease
Stable coexistence


K2
0


K1
K2/a1 N1
Stable Competitive Equilibrium
• Competitive Coexistence
• Suppose K1  K2. What values of a1 and a2 lead to
coexistence?
• a1 < 1.0 (small) and a2 < 1.0 (small)
• effect of each species on dN/Ndt of the other is less
than effect of each species on its own dN/Ndt
• Intraspecific competition more intense than
interspecific competition
Lotka-Volterra
Zero
Growth
Isoclines
N
2
K2
• K2 > K1 / a2
Unstable two
species equilibrium
• K1 > K2 / a1
• Region  both
species increase
• Regions  & 
one species
decreases & one
species increases
• Region  both
species decrease


K1/a2
0


K2/a1
K1
N1
Unstable Competitive Equilibrium
• Exactly at equilibrium point, both species survive
• Anywhere else, either one or the other “wins”
• Stable equilibria at:
– (N1 = K1 & N2 = 0)
– (N2 = K2 & N1 = 0)
• Which equilibrium depends on initial numbers
– Relatively more N1 and species 1 “wins”
– Relatively more N2 and species 2 “wins”
Unstable Competitive Equilibrium
• Suppose K1  K2. What values of a1 and lead to
coexistence?
• a1 > 1.0 (large) and a2 >1.0 (large)
• effect of each species on dN/Ndt of the other is
greater than effect of each species on its own
dN/Ndt
• Interspecific competition more intense than
intraspecific competition
Lotka-Volterra competition
• Four circumstances
–
–
–
–
Species 1 wins
Species 2 wins
Stable equilibrium coexistence
Unstable equilibrium; winner depends on initial N’s
• Coexistence only when interspecific competition
is weak
• Morin, pp. 34-40
Competitive Exclusion Principle
• Two competing species cannot coexist
unless interspecific competition is weak
relative to intraspecific competition
• What makes interspecific competition
weak?
– Use different resources
– Use different physical spaces
– Use exactly the same resources, in the same
place, at the same time  Competitve
exclusion
Model assumptions
• All models incorporate assumptions
• Validity of assumptions determines validity
of the model
• Different kinds of assumptions
• Consequences of violating different kinds
of assumptions are not all the same
Simplifying environmental assumption
• The environment is, with respect to all properties
relevant to the organisms:
– uniform or random in space
– constant in time
• realistic?
• if violated  need a better experimental system
Simplifying biological assumption
• All the organisms are, with respect to their
impacts on their environment and on each other:
– identical throughout the population
• clearly must be literally false
• if seriously violated  need to build a different
model with more realistic assumptions
Explanatory assumptions
• What we propose as an explanation of nature
(our hypothesis)
– r1, r2, K1, K2, a1, a2 are constants
– competition is expressed as a linear decline in per
capita growth (dN / N dt ) with increasing N1 or N2
– Some proportional relationship exists between the
effects of N1 and N2 on per capita growth
• If violated  model (our hypothesis) is wrong
Interspecific competition: Paramecium
• George Gause
• P. caudatum goes
extinct
• Strong
competitors, use
the same
resource (yeast)
• Competitve
asymmetry
• Competitive
exclusion
Interspecific competition: Paramecium
• P. caudatum &
P. burseria
coexist
• P. burseria is
photosynthetic
• Competitive
coexistence
• Apparently
stable
Experiments in the laboratory
•
•
•
•
•
Gause’s work on protozoa
Flour beetles (Tribolium)
Duck weed (Lemna, Wolffia)
Mostly consistent with Lotka Volterra
No clear statement of what causes
interspecific competition to be weak
Alternative Lotka-Volterra competition
• Absolute competition coefficients
dNi / Nidt = ri [1 – bii Ni - bij Nj]
equivalent to:
dNi / Nidt = ri [Ki - Ni - aj Nj] / Ki
= ri [Ki/Ki - Ni/Ki - ajNj/Ki]
= ri [1- (1/Ki)Ni – (aj/Ki)Nj]
Absolute Lotka-Volterra
N2
1/b12
Stable coexistence


1/b22
0


1/b11
1/b21 N1
Competitive effect vs. response
• Effect: impact of density of a species
– Self density (e.g., b11)
– Other species density (e.g., b21)
• Response: how density affects a species
– Self density (e.g., b11)
– Other species’ density (e.g., b12)
• Theory: effects differ (b11 > b21)
• Experiments: responses (b11, b12)
Absolute Lotka-Volterra
N2
1/b12
Stable coexistence


1/b22
0


1/b11
1/b21
N1
Not ecological models
• No mechanisms of competition in the model
– Phenomenological
• Environment not explicitly included
• Mechanistic models of Resource competition