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Ecology 8310
Population (and Community)
Ecology
Seguing into from populations to communities
• Species interactions
• Lotka-Volterra equations
• Competition
• Adding in resources
Species
interactions:
Competition (- , -)
Predation
(+ , -)
(Herbivory, Parasitism,
Disease)
Mutualism
(+ , +)
None
(0 , 0)
Species
interactions:
aij>0
5
a16
a15
4
a14=0
1
a13
3
aij<0
6
a11
a12
aij gives the per
capita effect of
species j on species i’s
per capita growth
rate, dNi/Nidt
2
aij =
(
¶ dNi / Ni dt
¶N j
)
Generalized LotkaVolterra system:
dN1 / N1dt  r1  a11 N1  a12 N 2  a13 N 3  ...
dN 2 / N 2 dt  r2  a21 N1  a22 N 2  a23 N 3  ...
dN 3 / N 3 dt  r3  a31 N1  a32 N 2  a33 N 3  ...

Special cases:
1. Exponential model: all a’s=0
2. Logistic model: aii<0; others =0
dN1/N1dt
N2
N1
dN1/N1dt
N2
N1
Species
interactions:
aij>0
1
a31
a13
3
a43
a34
a11
aij<0
a21
a12
2
4
a44
What can you say about the interactions between
dN1 / N1dt  r1  a11N1  a12 N 2  a13 N 3  a14 N 4
these species?
dN
/
N
dt

r

a
N

a
N

a
N

a
N
2
2
2
21
1
22
2
23
3
24
4
Which are interspecific competitors?
dN 3 / Nare
 r3  a31and
N1 prey?
 a32 N 2
3 dtpredator
Which
 a33 N 3  a34 N 4
dN 4 / Nare
 r4  a41NWhich
N 2  self
a43 Nlimitation?
Which
4 dtmutualists?
1  a42show
3  a44 N 4
Competition:
a21
1
a12
Arises when two organisms use
the same limited resource, and
deplete its availability
(intra. vs. interspecific)
2
a11
Alternate terminology:
a22
1
2
R
 ij = aij/aii , the effect of interspecific
competition relative to the intraspecific effect
(e.g., how many of species i does it take to have
the same effect as 1 individual of species j?)
Competition:
a21
1
a12
a11
dN1 / N1dt  r1  a11 N1  a12 N 2 
r1 ( K1  N1  12 N 2 ) / K1
dN 2 / N 2 dt  r2  a21 N1  a22 N 2 
r2 ( K 2  N 2   21 N1 ) / K 2
2
a22
Competition:
a21
1
a12
a11
2
a22
Can we use this model to understand patterns of
competition among two species (e.g., coexistence
and competitive exclusion)?
E.g., Paramecium experiments by Gause…
Classic studies of resource
competition by Gause (1934, 1935)
Paramecium
caudatum
Paramecium aurelia
Paramecium bursaria
Competitive
exclusion:
P. aurelia excludes P. caudatum
In contrast…
Paramecium caudatum
Paramecium bursaria
Why this disparity, and can
we gain insights via our
model?
Competition:
a21
1
a12
a11
dN1 / N1dt  r1  a11 N1  a12 N 2 
r1 ( K1  N1  12 N 2 ) / K1
dN 2 / N 2 dt  r2  a21 N1  a22 N 2 
r2 ( K 2  N 2   21 N1 ) / K 2
2
a22
Competition:
dN1 / N1dt  r1 ( K1  N1  12 N 2 ) / K1
dN 2 / N 2 dt  r2 ( K 2  N 2   21N1 ) / K 2
At equilibrium, dN/Ndt=0:
N  K1  12 N 2
*
1
N  K 2   21N1
*
2
Phase planes:
Graph showing regions where
dN/Ndt=0 (and +, -); used to
infer dynamics
Species 1’s zero growth
isocline…
K1/ 12
N  K1  12 N 2
N2
*
1
N1
dN1/N1dt=0
K1
Phase planes:
What if the system is not on
the isocline. Will what N1 do?
N2
K1/ 12
N1
dN1/N1dt=0
K1
Phase planes:
N  K 2   21N1
*
2
N2
K2
dN2/N2dt=0
N1
K2/ 21
Phase planes:
Putting it together…
K2
dN2/N2dt=0
Species 2 “wins”:
N2* =K2, N1* =0
K1/ 12
N2
(reverse to get
Species 1 winning)
N1
dN1/N1dt=0
K1
K2/ 21
Phase planes:
Your turn…. For A and B:
1) Draw the trajectory on
the phase-plane
K2
2) Draw the dynamics (N
vs. t) for each system.
dN2/N2dt=0
A
N2
K1/ 12
B
N1
dN1/N1dt=0
K1
K2/ 21
Phase planes:
K2
Another possibility…
dN2/N2dt=0
“It depends”: either
species can win,
depending on starting
conditions
N2
K1/ 12
dN1/N1dt=0
N1
K2/ 21
K1
Phase planes:
Your turn….
Draw the dynamics (N vs. t)
for the system that starts
at:
• Point A
K2
A
N2
K1/ 12
• Point B
B
dN1/N1dt=0
N1
K2/ 21
K1
Phase planes:
Now do it for many
starting points:
Separatrix or
manifold
K2
N2
K1/ 12
dN1/N1dt=0
N1
K2/ 21
K1
Phase planes:
A final possibility…
N2
K1/ 12
dN1/N1dt=0
Coexistence!
K2
N1
dN2/N2dt=0
K1
K2/ 21
Phase planes:
“Invasibility”…
N2
K1/ 12
Mutual invasibility 
coexistence!
dN1/N1dt=0
Why: because each
species is self-limited
below the level at
which it prevents
growth of the other
K2
N1
dN2/N2dt=0
K1
K2/ 21
Invasibility:
Contrast that with…
K2
dN2/N2dt=0
Neither species can
invade the other’s
equilibrium (hence no
coexistence).
N2
K1/ 12
dN1/N1dt=0
N1
K2/ 21
K1
Coexistence:
N2
K1/12
K 1 / a 12 > K 2
and
K 2 / a 21 > K 1
dN1/N1dt=0
substitute...
rearrange...
a 12a 21 < 1
K2
N1
dN2/N2dt=0
K1
K2/21
Coexistence:
“intra > inter”
Coexistence requires that the strength
of intraspecific competition be stronger
than the strength of interspecific
competition.
 Resource partitioning
 Two species cannot coexist on a single
limiting resource
Can we now explain Gause’s results?
Bacteria in
water column
Yeast on
bottom
Paramecium
caudatum
Paramecium aurelia
Paramecium bursaria
Resources:
But what about resources?
(they are “abstracted” in LV model)
Research by David Tilman
Resources:
Data = points.
Lines = predicted from model
Followed population growth
and resource (silicate) when alone:
Resources:
What will happen when growth
together: why?
Resources:
R*: resource concentration after
consumer population equilibrates
(i.e., R at which Consumer shows
no net growth)
Species with lowest R* wins
(under idealized scenario: e.g.,
one limiting resource).
If two limiting resources, then
coexistence if each species
limited by one of the resources
(intra>inter): trade-off in R*s.
Next time: Tilman's R* framework