Transcript Slide 1
Competition
Photo of hyenas and lioness at a carcass from https://www.flickr.com/photos/davidbygott/4046054583
Pairwise Species Interactions Influence of species A
A - (negative) B 0 + A Competition 0 B Amensalism A + Antagonism B (Predation/Parasitism) 0 (neutral/null) 0 A B Amensalism A 0 0 Neutralism B (No interaction) 0 A B + Commensalism + (positive) + A B Antagonism (Predation/Parasitism) + A B 0 Commensalism A + + Mutualism B Redrawn from Abrahamson (1989); Morin (1999, pg. 21)
Intra-specific vs. Inter-specific Competition
Interaction between individuals in which each is harmed by their shared use of a
limiting resource
(which can be consumed or depleted) for growth, survival, or reproduction Photo of hyenas and lioness at a carcass from https://www.flickr.com/photos/davidbygott/4046054583
Intra-specific vs. Inter-specific Competition “Complete competitors cannot coexist.”
(Hardin 1960)
Paramecium aurelia Paramecium caudatum
Cain, Bowman & Hacker (2014), Fig. 12.11, after Gause (1934); photomicrographs from Wikimedia Commons
Intra-specific vs. Inter-specific Competition Resource partitioning
– differences in use of
limiting resources
– can allow species to
coexist
P. aurelia P. bursaria
&
P. caudatum
ate mostly floating bacteria; ate mostly yeast cells on the bottoms of the tubes Cain, Bowman & Hacker (2014), Fig. 12.11, after Gause (1934)
Lotka – Volterra Phenomenological Competition Models Alfred Lotka & Vito Volterra (1880-1949) (1860-1940)
Photo of Lotka from http://blog.globe-expert.info; photo of Volterra from Wikimedia Commons
Lotka – Volterra Phenomenological Competition Models Lotka-Volterra Competition Equations: Logistic population growth model
– growth rate is reduced by
intraspecific competition
: Species 1:
dN 1
/
dt
=
r 1 N 1
[(
K 1
-
N 1
)/
K 1
] Species 2:
dN 2
/
dt
=
r 2 N 2
[(
K 2
-
N 2
)/
K 2
] Functions added to further reduce growth rate owing to
interspecific competition
: Species 1:
dN 1
/
dt
=
r 1 N 1
[(
K 1
-
N 1
-
f
(
N 2
) )/
K 1
] Species 2:
dN 2
/
dt
=
r 2 N 2
[(
K 2
-
N 2
-
f
(
N 1
) )/
K 2
]
Lotka – Volterra Phenomenological Competition Models Lotka-Volterra Competition Equations:
The function (
f
) could take on many forms,
e.g.
: Species 1:
dN 1
/
dt
=
r 1 N 1
[(
K 1
-
N 1
-
αN 2
)/
K 1
] Species 2:
dN 2
/
dt
=
r 2 N 2
[(
K 2
-
N 2
-
βN 1
)/
K 2
] The
competition coefficients
α
&
β
measure the
per capita
effect of one species on the population growth of the other, measured relative to the effect of intraspecific competition If
α
= 1, then
per capita
intraspecific effects = interspecific effects If
α
< 1, then intraspecific effects are more deleterious to Species 1 than interspecific effects If
α
> 1, then interspecific effects are more deleterious
Lotka – Volterra Phenomenological Competition Models
Find equilibrium solutions to the equations,
i.e
., set
dN
/
dt
= 0: Species 1:
N 1
=
K 1
-
αN 2
Species 2:
N 2
=
K 2
-
βN 1
This makes intuitive sense: The equilibrium for
N 1
is the carrying capacity for Species 1 (
K 1
) reduced by some amount owing to the presence of Species 2 (
αN 2
) However, each species’ equilibrium depends on the equilibrium of the other species! So, by substitution… ^ Species 1:
N 1
=
K 1
-
α
(
K 2
-
βN 1
) ^ Species 2:
N 2
=
K 2
-
β
(
K 1
-
αN 2
)
Lotka – Volterra Phenomenological Competition Models
The equations for equilibrium solutions become: Species 1:
N 1
= [
K 1
-
αK 2
] / [1 -
αβ
] Species 2:
N 2
= [
K 2
-
βK 1
] / [1 -
αβ
] These provide some insights into the conditions required for coexistence under the assumptions of the model
E.g
., the product
αβ
must be < 1 for N to be > 0 for both species (a necessary condition for coexistence) But they do not provide much insight into the dynamics of competitive interactions,
e.g.
, are the
equilibrium points stable
?
Lotka – Volterra Phenomenological Competition Models
4 time steps State-space graphs help to track population trajectories (and assess stability) predicted by models From Gotelli (2001)
Lotka – Volterra Phenomenological Competition Models
4 time steps State-space graphs help to track population trajectories (and assess stability) predicted by models 4 time steps Mapping state-space trajectories onto single population trajectories From Gotelli (2001)
Remember that equilibrium solutions require
dN
/
dt
= 0 Species 1:
N 1
=
K 1
-
αN 2
Therefore: When
N 2
= 0,
N 1
=
K 1
When
N 1
= 0,
N 2
=
K 1 / α K 1 / α
Lotka-Volterra Model
Isocline for Species 1
dN 1
/
dt
= 0
N 1 K 1
Remember that equilibrium solutions require
dN
/
dt
= 0 Species 2:
N 2
=
K 2
-
βN 1
Therefore: When
N 1
= 0,
N 2
=
K 2
When
N 2
= 0,
N 1
=
K 2 / β K 2
Lotka-Volterra Model
Isocline for Species 2
dN 2
/
dt
= 0
N 1 K 2 / β
Plot the isoclines for 2 species together to examine population trajectories
K 1 / α
>
K 2 K 1
>
K 2 / β
For species 1:
K 1
>
K 2 α
(intrasp. > intersp.) For species 2:
K 1 β
>
K 2
(intersp. > intrasp.) = stable equilibrium
K 1 / α K 2
Lotka-Volterra Model Competitive exclusion
of Species 2 by Species 1
K 2 / β N 1 K 1
Plot the isoclines for 2 species together to examine population trajectories
K 2
>
K 1 / α K 2 / β
>
K 1
For species 1:
K 2 α > K 1
(intersp. > intrasp.) For species 2:
K 2
>
K 1 β
(intrasp. > intersp.) = stable equilibrium
K 2 K 1 / α
Lotka-Volterra Model Competitive exclusion
of Species 1 by Species 2
N 1 K 1 K 2 / β
Plot the isoclines for 2 species together to examine population trajectories
K 2 K 1
>
K 1 / α
>
K 2 / β
For species 1:
K 2 α > K 1
(intersp. > intrasp.) For species 2:
K 1 β
>
K 2
(intersp. > intrasp.) = stable equilibrium = unstable equilibrium
K 2 K 1 / α
Lotka-Volterra Model Competitive exclusion
with an
unstable equilibrium
K 2 / β N 1 K 1
Plot the isoclines for 2 species together to examine population trajectories
K 1 / α K 2 / β
>
K 2
>
K 1
For species 1:
K 1
>
K 2 α
(intrasp. > intersp.) For species 2:
K 2
>
K 1 β
(intrasp. > intersp.) = stable equilibrium
K 1 / α K 2
Lotka-Volterra Model Coexistence
at a
stable equilibrium
K 1 N 1 K 2 / β
Mechanisms of Competition Exploitation competition
Dissecting exploitation competition reveals its indirect nature H + H + P
Interference competition
(direct
aggression
,
allelopathy
,
etc
.) H H P P Solid arrows =
direct effects
; dotted arrows =
indirect effects
Redrawn from Menge (1995)
Mechanisms of Competition David Tilman
Synedra Asterionella
Cain, Bowman & Hacker (2014), Fig. 12.4, after Tilman
et al
. (1981); photos of diatoms from Wikimedia Commons; photo of Tilman from http://www.princeton.edu/morefoodlesscarbon/speakers/david-tilman/
Mechanisms of Competition David Tilman
Cain, Bowman & Hacker (2014), Fig. 12.4, after Tilman
et al
. (1981); photo of Tilman from http://www.princeton.edu/morefoodlesscarbon/speakers/david-tilman/
Asymmetric vs. Symmetric Competition
Cain, Bowman & Hacker (2014), Fig. 12.7
Classic Pattern Interpreted as Evidence for Competitively-Structured Assemblages Robert MacArthur (1930-1972)
Painting of “MacArthur’s warblers” by D. Kaspari for M. Kaspari (2008); anniversary reflection on MacArthur (1958)
Character Displacement
The “
Ghost of Competition Past
” (
sensu
Connell 1980) is hypothesized to be the cause of the beak size difference on Pinta Marchena Cain, Bowman & Hacker (2014), Fig. 12.19, after Lack (1947)