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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
6
aij<0
5
a16
a15
4
a14=0
1
a13
3
aij gives the per capita
effect of species j on
species i’s per capita
growth rate, dNi/Nidt
a12
a11
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
One interpretation:
Sp. 1 and Sp. 2 are
competitors
Species
interactions:
aij>0
1
a31
a13
3
aij<0
a21
a12
a11
2
a43
a34
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 / N
 r3  aand
3 dt
31 Nprey?
1  a32 N 2
Which
are
predator
 a33 N 3  a34 N 4
dN 4 / N
 r4  a41Which
N1  ashow
a43limitation?
N 3  a44 N 4
Which
are
mutualists?
self
4 dt
42 N 2 
Arises when two organisms use
the same limited resource, and
deplete its availability
(intra. vs. interspecific)
Competition:
a21
1
a12
a11
Alternate terminology:
2
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
Why this disparity, and can we
gain insights via our model?
Paramecium bursaria
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
dN1/N1dt=0
N1
K1
Phase planes:
What if the system is not on the
isocline. Will what N1 do?
N2
K1/ 12
dN1/N1dt=0
N1
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)
dN1/N1dt=0
N1
K1
K2/ 21
Your turn….
Phase planes:
Draw the trajectors on the
phase plane; then draw the
dynamics (N vs. t)…for the
system that starts at:
K2
A
• Point B
N2
K1/ 12
• Point A
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
K1
Phase planes:
A final possibility…map out the
trajectories:
K1/ 12
dN1/N1dt=0
N2
Coexistence!
K2
dN2/N2dt=0
N1
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
dN2/N2dt=0
N1
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:
K 1 / a 12 > K 2
and
K 2 / a 21 > K 1
N2
K1/ 12
substitute...
rearrange...
1 > K 2a 12 / K 1 > a 12a 21
dN1/N1dt=0
K2
thus :
a11a22 > a12a21
dN2/N2dt=0
N1
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
("competitive exclusion principle")
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:
Followed population growth
and resource (silicate) when alone:
Data = points.
Lines = predicted from model
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 for
two consumers & two resources