Download ECOLOGY EVOLUTION

The stabilizing and destabilizing effects of eco-­evolutionary feedbacks
ECOLOGY
EVOLUTION
1
2
1
Swati Patel , Michael Cortez , Sebastian Schreiber ESA 2016 1: University of California Davis 2: Utah State University
Eco-­‐evolutionary feedbacks affect communities
Eco-­‐evolutionary feedbacks affect communities
Example: Rotifer-­‐algae experiment alters predator-­‐prey cycles
Becks et al. 2010 Ecology Letters
Eco-­‐evolutionary feedbacks affect communities
Example: Rotifer-­‐algae experiment alters predator-­‐prey cycles
prey evolve to form clumps for defense
Becks et al. 2010 Ecology Letters
Eco-­‐evolutionary feedbacks affect communities
Example: Rotifer-­‐algae experiment alters predator-­‐prey cycles
genetic variation
low
high
measure of relative rate of evolution
Becks et al. 2010 Ecology Letters
(c)
(d)
(c)
Eco-­‐evolutionary feedbacks affect communities
Example: Rotifer-­‐algae experiment alters predator-­‐prey cycles
genetic variation
low
(e)
high
(f)
Figure 5 Dynamics
prey density
predator density
clump size
STABLE
Figure 4 Dynamics
of population abundances and mean algal
clump size in six replicate predator–prey chemostat experiments
with ungrazed Chlamydomonas as the prey (panels a–f). Predator
Becks et al. 2010 Ecology )1 Letters
clump size in four
with grazed Chlamy
a function of p
availability no lo
when two prey ty
present, resulting
cyclical relationsh
by undefended c
negative to positiv
(c)
(d)
(c)
(c)
(d)
Eco-­‐evolutionary feedbacks affect communities
Example: Rotifer-­‐algae experiment alters predator-­‐prey cycles
genetic variation
al.
low
(e)
high
(b)
(e)
(a)
(f)
(f)
(
Figure 5 Dynamics
prey density
predator density
clump size
clump size in four
with grazed Chlamy
a function of p
availability no lo
when two prey ty
(d)
(c)
present, resulting(
STABLE
Figure 4 Dynamics
of population
and
meanrelationsh
algal
Figure 4 Dynamics
of population
abundances
and meanabundances
algal UNSTABLE
cyclical
clump
size in six chemostat
replicate predator–prey
experiments
clump size in six replicate
predator–prey
experimentschemostat
by undefended c
with ungrazed
Chlamydomonas
as the
prey (panels a–f). Predator
with ungrazed Chlamydomonas
as the prey
(panels
a–f).
Predator
negative
to positiv
Becks et al. 2010 Ecology )1 Letters
density (rotifers mL)1) – red curve and solid circles;
prey density
(c)
(d)
(c)
(c)
(d)
Eco-­‐evolutionary feedbacks affect communities
Example: Rotifer-­‐algae experiment alters predator-­‐prey cycles
genetic variation
al.
low
(e)
high
(b)
(e)
(a)
(f)
(f)
(
Figure 5 Dynamics
prey density
predator density
clump size
clump size in four
with grazed Chlamy
a function of p
availability no lo
when two prey ty
(d)
(c)
present, resulting(
STABLE
Figure 4 Dynamics
of population
and
meanrelationsh
algal
Figure 4 Dynamics
of population
abundances
and meanabundances
algal UNSTABLE
cyclical
clump
size in six chemostat
replicate predator–prey
experiments
clump size in six replicate
predator–prey
experimentschemostat
by undefended c
ungrazed
Chlamydomonas
as the
Predator
suggests that rwith
elative o(panels
f evolution cprey
an a(panels
lter sa–f).
tability
with ungrazed
Chlamydomonas
as therate prey
a–f).
Predator
negative
to positiv
)1
density (rotifers mL)1) – red curve and solid circles;
prey density
Eco-­‐evolutionary feedbacks affect communities
predator-­‐prey
competition
P
N2
N1
N
Abrams and Matsuda 1997
Cortez 2016
apparent competition
intraguild predation
P
P
N1
Vasseur et al. 2011
N
N2
R
Schreiber et al. 2011
Patel and Schreiber 2015
2
Density or
Density or
Density or
3
2
Relative rate of evolution can stabilize and destabilize
0
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0
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Genetic Variation
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D
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Figure 4: With increased additive genetic variation, stabilizing selection can stabilize ecological oscillations (A–C) or destabilize a stable ecological system (D–F). Destabilization occurs when individual fitness decreases with higher mean defense (F a 2 Ga 1 0), and stabilization
occurs when individual fitness decreases with higher mean defense (F a 2 Ga ! 0). (De)stabilization occurs for all sufficiently large genetic variation when F a 2 Ga is large in magnitude (A, D) and for intermediate genetic variation values when F a 2 Ga is small in magnitude (B, E).
A, B, D, E, Maximum and minimum long-term predator density (solid black), prey density (dashed black), and mean prey trait (dashed-dotted
gray). For each genetic variation value, a single curve for each variable denotes the stable equilibrium value, and two curves denote the maximum and minimum values during the cycle. C, Numerical example of the cycle at V p 0:05 in A; the predator-prey phase lag is less than a
quarter period. F, Numerical example of the cycle at V p 0:25 in D; the predator-prey phase lag is between one-quarter and one-half of a
period. See appendix E for models and parameter values.
cost of reduced sharing of resources among conspecifics are
destabilizing.
Cortez 2016
(De)stabilization Occurs When the Amount of Genetic
stability of the ecological subsystem (measured by the leading eigenvalue of the Jacobian of the ecological subsystem;
for stable subsystems this is the subsystem resilience sensu
Pimm and Lawton [1977]). Rewriting this equation as
V y p 2S reveals that at the critical value of V, the
2
Density or
Density or
Density or
3
2
Relative rate of evolution can stabilize and destabilize
0
0.8
1.4
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2
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Genetic Variation
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50
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1
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1
Genetic Variation
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Genetic Variation
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000100
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Genetic Variation
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Cortez 2016
3
0
Genetic Variation
cost of reduced sharing of resources among conspecifics are
destabilizing.
1.5
2
2
D
it
(De)stabilization Occurs When the Amount of Genetic
50
Time
100
stability of the ecological subsystem (measured by the leading eigenvalue of the Jacobian of the ecological subsystem;
2
for stable E
subsystems this is the subsystem resilience
F sensu
Pimm and Lawton [1977]). Rewriting this equation as
V y p 2S reveals that at the critical value of V, the
it
0.8
Density or Trait
Density or Trait
Density or Trait
6 With increased additive genetic variation, stabilizing selection can stabilize ecological oscillations (A–C) or destabilize a stable ecoFigure 4:
logical system (D–F). Destabilization occurs when individual fitness decreases with higher mean defense (F a 2 Ga 1 0), and stabilization
occurs when individual fitness decreases with higher mean defense (F a 2 Ga ! 0). (De)stabilization occurs for all sufficiently large genetic variation when F a 2 Ga is large in magnitude (A, D) and
2 for intermediate genetic variation values2when F a 2 Ga is small in magnitude (B, E).
A, B, D, 3E, Maximum and minimum long-term predator density (solid black), prey density (dashed black), and mean prey trait (dashed-dotted
gray). For each genetic variation value, a single curve for each variable denotes the stable equilibrium value, and two curves denote the maximum and minimum values during the cycle. C, Numerical example of the cycle at V p 0:05 in A; the predator-prey phase lag is less than a
quarter period. F, Numerical example of the cycle at V p 0:25 in D; the predator-prey phase lag is between one-quarter and one-half of a
period. See
0 appendix E for models and parameter values.
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0
it
N
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2
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When there is oscillatory phenotype coexisten
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Density or
Density or
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and culminatesN
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(!x ≈ vR ), the IGP module resembles exploitative competition,00.8
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2
Relative rate of evolution can stabilize and destabilize
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6
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A variation, stabilizing selection
B
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6 With increased additive genetic
Figure 4:
can stabilize ecological
oscillations
(A–C)
or
destabilize
predator
0.8
1.4
2
0
Genetic Variation
1
2
Density or Trait
6
2
4
Density or Trait
4
2
0
Density or Trait
logical system (D–F). Destabilization occurs when individual fitness decreases with higher mean defense (F a 2 Ga 1 0), and stabilization
occurs when individual fitness decreases with higher mean defense (F a 2 Ga ! 0). (De)stabilization occurs for all sufficiently large genetic variation when F a 2 Ga is large in magnitude (A, D) and
−1.0 values2when F a 2 Ga is small in magnitude (B, E). −1.0
2 for intermediate genetic variation
preyprey density (dashed
A, B, D, 3E, Maximum and minimum long-term predator density (solid black),
prey black), and mean prey trait (dashed-dotted
gray). For each genetic variation value, a single curve for each variable
denotes the stable equilibrium value, and two curves denote the maxresource
imum and minimum values during the cycle. C, Numerical example of the cycle at V p 0:05
in A; the predator-prey
is less than
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1.0a
resource
quarter period. F, Numerical example of the cycle at V p 0:25 in D; the predator-prey phase lag is between one-quarter and one-half of a
heritability
period. See
appendix
E
for
models
and
parameter
values.
Genetic
Variation
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1.0
0
0
0
3
0
Genetic Variation
50
Time
100
it
it
x
the ecological
(measured
the leadcost of reduced sharing of resources among conspecifics
are 6: stability
Figure
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ingphase
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the Jacobian
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Patel nd Schreiber 2015phenotype
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as
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it
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0
Main Objective:
General theory on how eco-­‐
evolutionary feedbacks affect community stability
ECOLOGY
EVOLUTION
Main Objective:
General theory on how eco-­‐
evolutionary feedbacks affect community stability
ECOLOGY
population densities
EVOLUTION
population traits
Main Objective:
General theory on how eco-­‐
evolutionary feedbacks affect community stability
ECOLOGY
population densities
EVOLUTION
population traits
Stability: response to perturbations
Main Objective:
General theory on how eco-­‐
evolutionary feedbacks affect community stability
Outline
1. general model 2. simple mathematical stability conditions for slow and fast evolution and intuition 3. implications of conditions through example
Model
•
•
many species interacting in a community: -­‐
where N
i is population density of species i
-­‐
N
= (N1 , . . . , Nk )
some or all evolving in one or more traits: -­‐
where x j is the trait value of trait j
-­‐
x = (x1 , . . . , x )
ECOLOGY
dNi
= Ni fi (N, x)
dt
EVOLUTION
dxj
= εgj (N, x)
dt
Example: quantitative genetics framework; Lande’s approach
per-­‐capita Vitness
time scale separation
Model
•
•
many species interacting in a community: -­‐
where N
i is population density of species i
-­‐
N
= (N1 , . . . , Nk )
some or all evolving in one or more traits: -­‐
where x j is the trait value of trait j
-­‐
x = (x1 , . . . , x )
ECOLOGY
dNi
= Ni fi (N, x)
dt
EVOLUTION
dxj
= εgj (N, x)
dt
Example: quantitative genetics framework; Lande’s approach
per-­‐capita Vitness
time scale separation
Model
•
•
many species interacting in a community: -­‐
where N
i is population density of species i
-­‐
N
= (N1 , . . . , Nk )
some or all evolving in one or more traits: -­‐
where x j is the trait value of trait j
-­‐
x = (x1 , . . . , x )
ECOLOGY
dNi
= Ni fi (N, x)
dt
EVOLUTION
dxj
= εgj (N, x)
dt
Example: quantitative genetics framework; Lande’s approach
per-­‐capita Vitness
time scale separation
Model
•
•
many species interacting in a community: -­‐
where N
i is population density of species i
-­‐
N
= (N1 , . . . , Nk )
some or all evolving in one or more traits: -­‐
where x j is the trait value of trait j
-­‐
x = (x1 , . . . , x )
ECOLOGY
dNi
= Ni fi (N, x)
dt
EVOLUTION
dxj
= εgj (N, x)
dt
Example: quantitative genetics framework; Lande’s approach
per-­‐capita Vitness
time scale separation
Model
•
•
many species interacting in a community: -­‐
where N
i is population density of species i
-­‐
N
= (N1 , . . . , Nk )
some or all evolving in one or more traits: -­‐
where x j is the trait value of trait j
-­‐
x = (x1 , . . . , x )
ECOLOGY
dNi
= Ni fi (N, x)
dt
EVOLUTION
dxj
= εgj (N, x)
dt
Example: quantitative genetics framework; Lande’s approach
per-­‐capita Vitness
time scale separation
Model
•
•
many species interacting in a community: -­‐
where N
i is population density of species i
-­‐
N
= (N1 , . . . , Nk )
some or all evolving in one or more traits: ECOLOGY
-­‐
where x j is the trait value of trait j
-­‐
x = (x1 , . . . , x )
dNi
= Ni fi (N, x)
dt
per-­‐capita Vitness
Example: classic Lotka Volterra equations modiVied with trait-­‐
dependent parameters
Model
•
•
many species interacting in a community: -­‐
where N
i is population density of species i
-­‐
N
= (N1 , . . . , Nk )
some or all evolving in one or more traits: -­‐
where x j is the trait value of trait j
-­‐
x = (x1 , . . . , x )
ECOLOGY
dNi
= Ni fi (N, x)
dt
EVOLUTION
dxj
= εgj (N, x)
dt
Model
•
•
many species interacting in a community: -­‐
where N
i is population density of species i
-­‐
N
= (N1 , . . . , Nk )
some or all evolving in one or more traits: -­‐
where x j is the trait value of trait j
-­‐
x = (x1 , . . . , x )
ECOLOGY
dNi
= Ni fi (N, x)
dt
EVOLUTION
dxj
= εgj (N, x)
dt
Example: quantitative genetics framework; Lande’s approach
selection equation
Model
•
•
many species interacting in a community: -­‐
where N
i is population density of species i
-­‐
N
= (N1 , . . . , Nk )
some or all evolving in one or more traits: -­‐
where x j is the trait value of trait j
-­‐
x = (x1 , . . . , x )
ECOLOGY
dNi
= Ni fi (N, x)
dt
EVOLUTION
dxj
= εgj (N, x)
dt
relative rate of evolution
Lande’s Approach
single trait in species i (freq. independent selection)
dxj
fi
= Vi
dt
xj
Vitness
•
trait
Lande’s Approach
single trait in species i (freq. independent selection)
dxj
fi
= Vi
dt
xj
Vitness
•
trait
Traits change in direction of Vitness gradient
Lande’s Approach
single trait in species i (freq. independent selection)
dxj
fi
= Vi
dt
xj
Vitness
•
Evolutionary equilibria occur at Vitness peaks or valleys
Vitness
trait
trait
Lande’s Approach
single trait in species i (freq. independent selection)
dxj
fi
= Vi
dt
xj
Vitness
•
Genetic variance inVluences relative rate of evolution to ecology
Vitness
trait
trait
Model
ECOLOGY
EVOLUTION
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
equilibrium
(N , x )
Model
ECOLOGY
EVOLUTION
dNi
= Ni fi (N, x)
dt
equilibrium
dxj
= εgj (N, x)
dt
(N , x )
STABILITY-­‐ three different perspectives:
1. Mathematically 2. Eco, evo, and eco-­‐evo feedbacks 3. Trajectory after perturbation
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability: Mathematically
Reminder: determined from the eigenvalue of the Jacobian with the largest real part
Ṅi
Nj
J=
...
..
.
x˙k
Nj
Ṅj
x
..
.
...
x˙k
x
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability: Mathematically
Reminder: determined from the eigenvalue of the Jacobian with the largest real part
Ṅi
Nj
J=
...
..
.
x˙k
Nj
Ṅj
x
..
.
...
x˙k
x
s(J) = stability modulus
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability: Mathematically
Reminder: determined from the eigenvalue of the Jacobian with the largest real part
Ṅi
Nj
J=
...
..
.
x˙k
Nj
Ṅj
x
..
.
...
x˙k
x
s(J) = stability modulus
s(J) < 0
STABLE
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability: Mathematically
Reminder: determined from the eigenvalue of the Jacobian with the largest real part
Ṅi
Nj
J=
...
..
.
x˙k
Nj
Ṅj
x
..
.
...
x˙k
x
s(J) = stability modulus
s(J) < 0
s(J) > 0
STABLE
UNSTABLE
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability
MATHEMATICALLY
A
J=
εC
B
εD
FEEDBACKS
trait
pop. dens.
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability
MATHEMATICALLY
A
J=
εC
B
εD
FEEDBACKS
trait
pop. dens.
A
DIRECT ECOLOGICAL INTERACTIONS: how population densities affect population Vitness
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability
MATHEMATICALLY
A
J=
εC
B
εD
FEEDBACKS
D
trait
pop. dens.
A
DIRECT EVOLUTIONARY INTERACTIONS: how traits affect selection dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability
MATHEMATICALLY
A
J=
εC
FEEDBACKS
B
εD
D
trait
B
pop. dens.
A
ECO-­‐EVOLUTIONARY INTERACTIONS: how traits affect population Vitness dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability
MATHEMATICALLY
A
J=
εC
B
εD
FEEDBACKS
D
trait
C pop. dens.
A
ECO-­‐EVOLUTIONARY INTERACTIONS: how population densities affect selection
B
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability: Eco and Evo Uncoupled
MATHEMATICALLY
A
J=
0
0
εD
1.) ecologically stable:
s(A) < 0
2.) evolutionarily stable:
s(D) < 0
FEEDBACKS
D
trait
pop. dens.
A
dNi
= Ni fi (N, x)
dt
dxj
= εgj (N, x)
dt
Stability: Eco and Evo Coupled
FEEDBACKS
two feedback pathways:
D
trait
B
1.) evo-eco-evo pathway
2.) eco-evo-eco pathway
C pop. dens.
A
EVO-ECO-EVO PATHWAY
EVO-ECO-EVO PATHWAY
traits change
EVO-ECO-EVO PATHWAY
traits change
alters Vitness
EVO-ECO-EVO PATHWAY
traits change
population densities change
alters Vitness
EVO-ECO-EVO PATHWAY
traits change
population densities change
alters Vitness
alters selection
EVO-ECO-EVO PATHWAY
traits change
population densities change
alters Vitness
traits change
alters selection
EVO-ECO-EVO PATHWAY
traits change
population densities change
alters Vitness
traits change
alters selection
STABILIZING
EVO-ECO-EVO PATHWAY
traits change
population densities change
alters Vitness
traits change
alters selection
DESTABILIZING
EVO-ECO-EVO PATHWAY
traits change
population densities D
changetrait
alters Vitness
B
traits change
C pop. dens. alters selection
A
feedback effects captured by matrix product:
CA
1
( B)
DESTABILIZING
ECO-EVO-ECO PATHWAY
ECO-EVO-ECO PATHWAY
population densities change
traits change
alters selection
population densities change
STABILIZING
alters Vitness
DESTABILIZING
ECO-EVO-ECO PATHWAY
population densities change
traits changeD
alters selection
trait
B
C pop. dens. A
alters population densities change
Vitness
feedback effects captured by matrix product:
BD
1
( C)
STABILIZING
DESTABILIZING
Stability: Eco and Evo Coupled
QUESTION: How do these eco-­‐evolutionary feedbacks affect stability? Stability: Eco and Evo Coupled
QUESTION: How do these eco-­‐evolutionary feedbacks affect stability? relative rate of evolution
slow
(small ε )
fast
(large ε )
Stability: Eco and Evo Coupled
QUESTION: How do these eco-­‐evolutionary feedbacks affect stability? relative rate of evolution
slow
(small ε )
evo-eco-evo
feedback
fast
(large ε )
eco-evo-eco
feedback
Slow Evolution: Evo-­‐Eco-­‐Evo feedback important for stability
MATHEMATICALLY
1. ecologically stable: s(A) < 0
2. sum of evolution with evo-­‐eco-­‐evo feedback stable:
s(D + CA
1
( B)) < 0
FEEDBACKS D
trait
pop. dens.
A
Slow Evolution: Evo-­‐Eco-­‐Evo feedback important for stability
MATHEMATICALLY
1. ecologically stable: s(A) < 0
2. sum of evolution with evo-­‐eco-­‐evo feedback stable:
s(D + CA
1
( B)) < 0
FEEDBACKS D
trait
pop. dens.
A
CA
1
( B)
Slow Evolution: Evo-­‐Eco-­‐Evo feedback important for stability
MATHEMATICALLY
1. ecologically stable: s(A) < 0
2. sum of evolution with evo-­‐eco-­‐evo feedback stable:
s(D + CA
1
( B)) < 0
FEEDBACKS D
trait
pop. dens.
A
CA
pop. dens.
TWO FOLD RESPONSE TO PERTURBATION:
eco
eq
uili
bri
a
a
i
r
b
i
l
i
u
q
e
o
ev
trait
1
( B)
Slow Evolution: Evo-­‐Eco-­‐Evo feedback important for stability
MATHEMATICALLY
1. ecologically stable: s(A) < 0
2. sum of evolution with evo-­‐eco-­‐evo feedback stable:
s(D + CA
1
( B)) < 0
FEEDBACKS D
trait
pop. dens.
A
CA
pop. dens.
TWO FOLD RESPONSE TO PERTURBATION:
eco
eq
uili
bri
a
a
i
r
b
i
l
i
u
q
e
o
ev
trait
1
( B)
Slow Evolution: Evo-­‐Eco-­‐Evo feedback important for stability
MATHEMATICALLY
1. ecologically stable: s(A) < 0
2. sum of evolution with evo-­‐eco-­‐evo feedback stable:
s(D + CA
1
( B)) < 0
FEEDBACKS D
trait
pop. dens.
A
CA
1
( B)
pop. dens.
TWO FOLD RESPONSE TO PERTURBATION:
eco
eq
uili
bri
a
a
i
r
b
i
l
i
u
q
e
o
ev
trait
1. FAST ECOLOGICAL RESPONSE
Slow Evolution: Evo-­‐Eco-­‐Evo feedback important for stability
MATHEMATICALLY
1. ecologically stable: s(A) < 0
2. sum of evolution with evo-­‐eco-­‐evo feedback stable:
pop. dens.
s(D + CA
1
( B)) < 0
FEEDBACKS D
trait
pop. dens.
A
1
CA ( B)
TWO FOLD RESPONSE TO PERTURBATION:
eco
eq
uili
bri
a
a
i
r
b
i
l
i
u
q
e
o
ev
trait
1. FAST ECOLOGICAL RESPONSE
2. SLOW ECO-EVOLUTIONARY
RESPONSE (with feedbacks)
Fast Evolution: Eco-­‐Evo-­‐Eco feedback important for stability
Fast Evolution: Eco-­‐Evo-­‐Eco feedback important for stability
MATHEMATICALLY
1. evolution stable: s(D) < 0
2. sum of ecology with eco-­‐evo-­‐eco feedback stable:
s(A + BD
1
( C)) < 0
FEEDBACKS
D
trait
pop. dens.
Fast Evolution: Eco-­‐Evo-­‐Eco feedback important for stability
1
BD
(
C)
MATHEMATICALLY
FEEDBACKS
1. evolution stable: s(D) < 0
2. sum of ecology with eco-­‐evo-­‐eco feedback stable:
s(A + BD
1
( C)) < 0
D
trait
pop. dens.
A
Fast Evolution: Eco-­‐Evo-­‐Eco feedback important for stability
1
BD
(
C)
MATHEMATICALLY
FEEDBACKS
1. evolution stable: s(D) < 0
trait
2. sum of ecology with eco-­‐evo-­‐eco feedback stable:
pop. dens.
s(A + BD
1
D
( C)) < 0
pop. dens.
A
TWO FOLD RESPONSE TO PERTURBATION:
eco
eq
uili
bri
a
a
i
r
b
i
l
i
u
q
e
o
ev
trait
1. FAST EVOLUTION RESPONSE
Fast Evolution: Eco-­‐Evo-­‐Eco feedback important for stability
1
BD
(
C)
MATHEMATICALLY
FEEDBACKS
1. evolution stable: s(D) < 0
trait
2. sum of ecology with eco-­‐evo-­‐eco feedback stable:
pop. dens.
s(A + BD
1
D
( C)) < 0
pop. dens.
A
TWO FOLD RESPONSE TO PERTURBATION:
eco
eq
uili
bri
a
a
i
r
b
i
l
i
u
q
e
o
ev
trait
1. FAST EVOLUTION RESPONSE
2. SLOW ECO-EVOLUTIONARY
RESPONSE (with feedbacks)
Summary: Stability
Uncoupled
Coupled with slow evolution
1. ecologically 1. ecologically stable stable 2. sum of evolution with 2. evolutionarily evo-­‐eco-­‐evo feedback stable
stable
1. evolutionarily stable 2. sum of ecology with eco-­‐evo-­‐eco feedback stable
s(D) < 0
1
s(A + BD ( C)) < 0
pop. dens.
s(D) < 0
s(A) < 0
1
s(D + CA ( B)) < 0
pop. dens.
s(A) < 0
Coupled with fast evolution
trait
trait
Applications to a competition model
N1
N2
•
two competing species •
one species evolving in a trait that affects intra-­‐ and inter-­‐ speciVic competition •
trade off between optimal trait Applications to a competition model
N1
N2
•
two competing species •
one species evolving in a trait that affects intra-­‐ and inter-­‐ speciVic competition •
trade off between optimal trait Pruitt et al. 2008 Animal Behavior
Applications to a competition model
N2
N1
•
two competing species •
one species evolving in a trait that affects intra-­‐ and inter-­‐ speciVic competition •
trade off between optimal trait USE THIS INFORMATION TO DETERMINE SIGNS OF MATRICES A, B, C
D
trait
B
C pop. dens.
A
Pruitt et al. 2008 Animal Behavior
Applications to a competition model
BIOLOGY
•
D
trait
B
C pop. dens.
A
two competing species with inter-­‐ and intra speciVic competition
N1
N2
SIGNS OF MATRICES
A=
Applications to a competition model
BIOLOGY
•
two competing species with inter-­‐ and intra speciVic competition
•
at equilibrium, evolving species is at Vitness peak or valley aggressiveness has negative impact on species 2
D
trait
B
C pop. dens.
A
•
N2
N1
SIGNS OF MATRICES
A=
B=
0
Applications to a competition model
BIOLOGY
•
two competing species with inter-­‐ and intra speciVic competition
•
at equilibrium, evolving species is at Vitness peak or valley aggressiveness has negative impact on species 2
D
trait
B
C pop. dens.
A
•
•
•
N2
N1
SIGNS OF MATRICES
high density of species 1 selects for less aggressiveness high density of species 2 selects for more aggressiveness
A=
B=
C=
0
+
Use signs of matrices to determine effects of eco-­‐evolutionary feedbacks
relative evolutionary time scale
slow
fast
Use signs of matrices to determine effects of eco-­‐evolutionary feedbacks
relative evolutionary time scale
slow
fast
Assuming ecological stability, evo-­‐eco-­‐evo feedback is stabilizing:
CA
1
( B) =
Use signs of matrices to determine effects of eco-­‐evolutionary feedbacks
relative evolutionary time scale
slow
fast
Assuming ecological stability, evo-­‐eco-­‐evo feedback is stabilizing:
CA
1
( B) =
s(D + CA
1
( B)) < 0
Use signs of matrices to determine effects of eco-­‐evolutionary feedbacks
relative evolutionary time scale
slow
fast
Assuming ecological stability, evo-­‐eco-­‐evo feedback is stabilizing:
CA
1
( B) =
s(D + CA
1
( B)) < 0
Even if evolution is unstable, adding the evo-­‐eco-­‐evo feedback can lead equilibria to be stable
Use signs of matrices to determine effects of eco-­‐evolutionary feedbacks
relative evolutionary time scale
slow
Assuming ecological stability, evo-­‐eco-­‐evo feedback is stabilizing:
CA
1
( B) =
s(D + CA
fast
Assuming evolutionary stability, eco-­‐evo-­‐eco feedback is stabilizing:
BD
1
( B)) < 0
Even if evolution is unstable, adding the evo-­‐eco-­‐evo feedback can lead equilibria to be stable
1
0
( C) =
+
0
Use signs of matrices to determine effects of eco-­‐evolutionary feedbacks
relative evolutionary time scale
slow
Assuming ecological stability, evo-­‐eco-­‐evo feedback is stabilizing:
CA
1
( B) =
s(D + CA
fast
Assuming evolutionary stability, eco-­‐evo-­‐eco feedback is stabilizing:
BD
1
( B)) < 0
Even if evolution is unstable, adding the evo-­‐eco-­‐evo feedback can lead equilibria to be stable
1
s(A + BD
0
( C) =
+
1
0
( C)) < 0
Even if ecology is unstable, adding the eco-­‐evo-­‐eco feedback can lead equilibria to be stable
Applications to a competition model
simulated model from Vasseur et al. 2011
1.2
0.8
1.0
trade off
1.4
1.6
•
0.2
0.4
0.6
0.8
strength of competitor 2 intraspecific competition
Applications to a competition model
•
simulated model from Vasseur et al. 2011
1.2
0.8
1.0
trade off
1.4
1.6
unstable
stable
eco
0.2
0.4
0.6
0.8
strength of competitor 2 intraspecific competition
Applications to a competition model
•
simulated model from Vasseur et al. 2011
1.6
unstable
stable
eco
1.2
0.8
1.0
trade off
1.4
unstable
stable
evo
0.2
0.4
0.6
0.8
strength of competitor 2 intraspecific competition
Applications to a competition model
•
simulated model from Vasseur et al. 2011
1.6
unstable
stable
eco
1.4
1.2
stable for slow and fast evolution
stable for fast evolution
0.8
1.0
trade off
stable for slow evolution
unstable
stable
evo
unstable for slow and fast evolution
0.2
0.4
0.6
0.8
strength of competitor 2 intraspecific competition
Applications to a competition model
•
simulated model from Vasseur et al. 2011
1.6
unstable
stable
eco
1.4
1.2
stable for slow and fast evolution
stable for fast evolution
0.8
1.0
trade off
stable for slow evolution
unstable
stable
evo
unstable for slow and fast evolution
For slow evolution, equilibrium stable despite being evolutionarily unstable 0.2
0.4
0.6
0.8
strength of competitor 2 intraspecific competition
Applications to a competition model
•
simulated model from Vasseur et al. 2011
1.6
unstable
stable
eco
1.4
1.2
stable for slow and fast evolution
stable for fast evolution
0.8
1.0
trade off
stable for slow evolution
unstable
stable
evo
unstable for slow and fast evolution
For slow evolution, equilibrium stable despite being evolutionarily unstable 0.2
0.4
0.6
0.8
strength of competitor 2 intraspecific competition
For fast evolution, equilibrium stable despite being ecologically unstable
Applications to a competition model
•
simulated model from Vasseur et al. 2011
1.6
unstable
stable
eco
1.4
1.2
stable for slow and fast evolution
stable for fast evolution
0.8
1.0
trade off
stable for slow evolution
0.2
0.4
0.6
0.8
unstable
stable
evo
unstable for slow and fast evolution
For slow evolution, equilibrium stable despite being evolutionarily unstable For fast evolution, equilibrium stable despite being ecologically unstable
strengthstabilizing of competitore2ffects intraspecific
competitionwere really important
of feedbacks Applications to a competition model
•
simulated model from Vasseur et al. 2011
1.6
unstable
stable
eco
1.2
stable for slow evolution
stable for slow and fast evolution
stable for fast evolution
0.8
1.0
trade off
1.4
unstable for slow and fast evolution
0.2
0.4
0.6
0.8
strength of competitor 2 intraspecific competition
Applications to a competition model
•
simulated model from Vasseur et al. 2011
1.5
1.0
0.5
stable for slow evolution
trait
species 2
0.0
1.4
unstable for slow and fast evolution
density/density
trait value
1.6
unstable
stable
eco
0.8
2.0
0.4
0.6
0.5
1.0
1.5
N2 density
trait
0.4
0.6
genetic variance
0.4
0.6
0.8
strength of competitor 2 intraspecific competition
1.0
heritability
0.2
0.2
0.8
0.0
density
density/
trait value
1.2
stable for slow and fast evolution
stable for fast evolution
1.0
trade off
0.2
0.8
1.0
Applications to a competition model
•
simulated model from Vasseur et al. 2011
1.5
1.0
0.5
stable for slow evolution
trait
species 2
0.0
1.4
unstable for slow and fast evolution
density/density
trait value
1.6
unstable
stable
eco
0.8
2.0
0.4
0.6
0.8
heritability
0.5
1.0
1.5
N2 density
trait
0.2
0.4
0.6
0.8
genetic variance
If either ecology or evolution is unstable, then this can lead to difference in stability for different relative rates of evolution 0.2
0.4
0.6
0.8
1.0
0.0
density
density/
trait value
1.2
stable for slow and fast evolution
stable for fast evolution
1.0
trade off
0.2
1.0
Main Conclusions
•
there are three important components to stability: ecology, evolution and eco-­‐evolutionary feedbacks •
two types of feedback pathways •
when evolution is slow, evo-­‐eco-­‐evo feedback is important •
•
can lead to stability even if equilibria are evolutionarily unstable when evolution is fast, eco-­‐evo-­‐eco feedback is important •
can lead to stability even if equilibria are ecologically unstable
Main Conclusions
•
there are three important components to stability: ecology, evolution and eco-­‐evolutionary feedbacks •
two types of feedback pathways •
when evolution is slow, evo-­‐eco-­‐evo feedback is important •
•
1
can are ( C)
D lead to stability even if equilibria BD
trait
evolutionarily unstable D
when evolution is fast, eco-­‐evo-­‐eco feedback is important trait
•
can lead to stability even if equilibria are pop. dens.
ecologically unstable
A
CA
evo-eco-evo
feedback
1
( B)
pop. dens.
A
eco-evo-eco
feedback
Main Conclusions
•
there are three important components to stability: ecology, evolution and eco-­‐evolutionary feedbacks •
two types of feedback pathways •
when evolution is slow, evo-­‐eco-­‐evo feedback is important •
•
can lead to stability even if equilibria are evolutionarily unstable when evolution is fast, eco-­‐evo-­‐eco feedback is important •
can lead to stability even if equilibria are ecologically unstable
Main Conclusions
•
there are three important components to stability: ecology, evolution and eco-­‐evolutionary feedbacks •
two types of feedback pathways •
when evolution is slow, evo-­‐eco-­‐evo feedback is important •
•
can lead to stability even if equilibria are evolutionarily unstable when evolution is fast, eco-­‐evo-­‐eco feedback is important •
can lead to stability even if equilibria are ecologically unstable
Main Conclusions
•
there are three important components to stability: ecology, evolution and eco-­‐evolutionary feedbacks •
two types of feedback pathways •
when evolution is slow, evo-­‐eco-­‐evo feedback is important •
•
when evolution is fast, eco-­‐evo-­‐eco feedback is important •
•
can lead to stability even if equilibria are evolutionarily unstable can lead to stability even if equilibria are ecologically unstable
differences in stability for different rates of evolution may be due to whether equilibria are ecologically or evolutionarily unstable
Thank you
Organizers:
Casey terHorst Peter Zee Collaborators:
Sebastian Schreiber Michael Cortez Feedback:
Jacob Moore Masato Yamamichi Axel Saenz Thomas Schoener Funding: