Download Modeling coevolution in predator


Modeling coevolution in predator-prey systems
Research conducted in partial fulfillment
of the Honors Program in Biology
By: Cristina Herren
Advisor: Mark McPeek
Submitted for consideration for
The Neukom Prize for Outstanding Undergraduate
Research in Computational Science
May 14, 2012
1
Abstract
Differential equations models used to describe the trajectories of interacting species often
used constants for the vital rates of the species and the attack rate of a predator. However, the
attack rate is the result of a physiological interaction between the predator and the prey, which
depends on the phenotypic traits of the two species.
In predator-prey systems, the traits related to prey capture and predator evasion are under
strong selection, meaning that the phenotypic traits that mediate the predator-prey interaction are
constantly evolving. Recent work on coevolution in predator-prey systems has demonstrated
that phenotypic evolution occurs on time scales that are relevant to population dynamics, and
thus should be included in models. These models assume that because all organisms are
constrained by scarce resources, an increased allocation of resources to traits pertaining to either
prey capture or predator evasion results in a decreased growth rate. In the models presented here,
phenotypic traits change over time due to the local selection gradient on the trait, which is a
function of the abundances and phenotypes of the two species.
In this paper I analyze a differential equations model that allows for changes in the
phenotypes and abundances of both species under two different assumptions about how attack
rate changes as a function of the relative phenotypes of the predator and the prey. In the model
where the predator attains its maximal attack rate by having a large trait value relative to that of
the prey, the predator’s trait value must remain high in order for the predator to persist in the
system. In systems where the predator maximizes its attack rate by having a trait that matches the
trait of the prey, the system is more prone to oscillations because the prey can escape the
predator by evolving its trait to be larger or smaller than that of the predator. Additionally, the
frequency-dependent nature of the selection gradient generates oscillations in instances where a
non-evolutionary model would reach a stable equilibrium. This is due to the optimal strategy of
each species changing as a function of the phenotype of the other species.
2
INTRODUCTION:
Anti‐predator
defenses
take
on
a
variety
of
forms.
Adaptations
in
prey
species
demonstrate
the
diversity
of
evolutionary
strategies
that
lead
to
reduced
predation.
These
strategies
include
synthesis
of
toxic
compounds
(Brodie
and
Brodie
1991,
Hanifin
et
al.
1999),
counterattacks
on
the
predator
(Pratt
1974,
Moitoza
and
Phillips
1979),
evasion
by
speed
(Watanabe
1983,
Watkins
1995,
McPeek
et
al.
1996),
and
camouflage
(Merilaita
and
Lind
2005,
Ioannou
and
Krause
2009).
Though
the
mechanisms
are
different,
the
consequence
of
each
of
these
adaptations
is
to
reduce
mortality
of
the
prey
due
to
consumption
by
the
predator.
From
a
modeling
standpoint,
effective
prey
defenses
decrease
the
attack
rate
of
the
predator.
The
population
dynamics
of
a
predator
and
a
prey
species
are
linked
by
the
attack
rate
of
the
predator
on
the
prey.
In
the
Lotka‐Volterra
framework
of
differential
equation
modeling,
the
attack
rate
is
the
proportion
of
the
prey
population
that
the
predator
can
consume
in
a
given
time
step
(Barryman
1992,
Gotelli
2001).
However,
the
attack
rate
of
the
predator
can
change
as
a
function
of
prey
availability.
The
functional
response
of
the
predator
describes
the
predator’s
rate
of
prey
consumption
as
the
prey
density
varies
(Solomon
1949,
Murdoch
1973).
Holling
(1959)
identified
three
functional
responses
that
express
the
predator’s
total
prey
consumption
as
a
function
of
prey
density.
In
Holling
Type
II
and
Type
III
functional
responses,
the
total
consumption
of
the
predator
saturates
as
prey
density
increases
because
of
the
time
taken
to
“handle”
each
prey.
This
is
one
example
of
attack
rates
changing
as
a
function
of
prey
availability,
because
each
predator
consumes
an
increasingly
smaller
fraction
of
the
total
prey
population
as
prey
density
increases.
3
However,
the
ability
of
a
predator
to
capture
its
prey
is
also
influenced
by
the
phenotypic
traits
of
the
predator
and
the
prey
that
contribute
to
the
predator’s
attack
rate
(Abrams
1995,
Nuismer
et
al.
2005
and
2007).
Thus,
there
are
two
different
kinds
of
interactions
that
determine
the
strength
of
the
predator‐prey
relationship.
The
number
of
prey
consumed
by
the
predators
depends
on
1)
the
number
of
predators
and
their
functional
responses
and
2)
the
effectiveness
of
the
prey’s
evasion
strategy.
However,
the
effectiveness
of
the
prey’s
defenses
depends
on
the
current
phenotypic
traits
of
the
predator
and
the
prey
(Saloniemi
1993).
Additionally,
phenotypic
and
behavioral
traits
often
change
as
a
result
of
repeated
interactions
between
two
species
(Preisser
et
al.
2005).
Trait
changes
can
occur
rapidly
in
response
to
both
environmental
cues
(i.e.
phenotypic
plasticity)
(Reznick
1990,
McCollum
1997,
Van
Buskirk
1999
and
2000,
Van
Buskirk
and
Schmidt
2000,
Agrawal
2001,
Krivan
and
Schmitz
2004),
or
over
many
generations
in
response
to
selective
pressures
(i.e.
phenotypic
evolution)
(Abrams
1986,
Abrams
1990,
Reznick
et
al.
1990,
Arnold
1994,
Abrams
2000).
Predator‐prey
models
with
fixed
parameters
make
the
assumption
that
the
species’
vital
rates
and
interactions
remain
constant
through
time
and
ignore
the
effects
of
trait‐
dependent
interactions
(Bolker
et
al.
2003).
This
statement
implies
that
neither
population
is
under
selection
for
any
trait
that
affects
the
predator‐prey
interaction
(Werner
and
Peacor
2004).
However,
many
empirical
systems
have
demonstrated
that
evolutionary
adaptations
occur
quickly
enough
to
influence
population
dynamics
(Hairston
Jr.
et
al.
1999
and
2005,
Yoshida
et
al.
2003
and
2007,
Mougi
2012).
These
rapid
evolutionary
changes
are
especially
common
in
predator‐prey
and
host‐parasite
systems
(Shertzer
et
al.
2002,
Yoshida
et
al.
2003
and
2007,
Mougi
and
Iwasa
2010,
Morran
et
al.
2011).
The
predator’s
4
fitness,
defined
as
its
per‐capita
growth
rate,
increases
with
increasing
attack
rate,
which
generates
a
selective
pressure
to
maximize
attack
rate.
Similarly,
the
prey’s
per‐capita
growth
rate
decreases
with
increasing
vulnerability
to
the
predator,
which
generates
a
selective
pressure
to
minimize
attack
rate.
Thus,
selection
on
traits
in
the
predator
and
the
prey
that
contribute
to
the
predator’s
attack
rate
should
be
strong
(Abrams
1995).
This
has
been
demonstrated
in
a
number
of
empirical
predator‐prey
systems
(Benkman
1999,
Brodie
and
Brodie
1999,
Abrams
2000,
Benkman
et
al.
2001).
Because
organisms
have
finite
resources
available
to
them,
there
are
trade‐offs
in
their
ability
to
respond
to
biotic
and
abiotic
constraints
(Huston
1979,
Smith
and
Huston
1989,
Tilman
1990).
For
instance,
prey
will
often
change
their
foraging
behavior
in
the
presence
of
predators
in
order
to
balance
the
costs
of
predation
with
the
costs
of
food
scarcity
(Krivan
and
Schmitz
2004).
Thus,
the
effects
of
trait‐mediated
interactions
reduce
population
size,
as
the
prey
is
forced
to
allocate
a
larger
amount
of
its
scarce
resources
to
evasive
behavior
or
defenses
(Ramos‐Jiliberto
2003).
In
some
cases,
the
effects
of
trait‐
mediated
interactions
reduce
the
prey
population
more
than
the
effects
of
direct
consumption
by
the
predator
(Krivan
and
Schmitz
2004,
Preisser
2005).
These
trade‐offs
hold
true
in
the
case
that
the
trait
is
evolutionarily
derived,
as
well
as
in
the
cases
of
behavioral
and
phenotypic
plasticity
(Arnold
1992,
Brodie
and
Brodie
1999,
Yoshida
et
al.
2003).
Predators
also
experience
phenotypic
changes
over
time
in
response
to
the
prey’s
phenotypic
evolution
(Schaffer
and
Rosenzweig
1978,
West
et
al.
1991).
Due
to
pleiotropic
effects,
selection
on
the
genes
controlling
the
predator’s
phenotype
often
reduces
the
population
of
the
predator
through
simultaneous
negative
affects
on
other
aspects
that
5
contribute
to
the
predator’s
fitness
(Abrams
1995,
Abrams
2000,
Werner
and
Peacor
2004).
For
instance,
selection
for
a
phenotypic
trait
that
increases
a
predator’s
attack
rate
on
one
prey
species
may
decrease
its
ability
to
capture
other
prey,
or
may
incur
additional
energetic
costs
to
maintain
that
trait.
These
secondary
effects
of
trait
evolution
demonstrate
why
the
predator’s
growth
rate
decreases
with
the
evolution
of
novel
traits.
Because
the
phenotypes
of
the
predator
and
the
prey
evolve
over
time
and
influence
the
abundances
of
both
species,
coevolutionary
models
allow
for
changes
in
the
phenotype
and
population
of
the
predator
and
the
prey.
In
these
models,
the
phenotypes
of
the
species
respond
to
selective
pressures
by
changing
in
the
direction
of
the
local
selection
gradient.
Lande
(1982)
and
Abrams
et
al.
(1993)
demonstrated
that
the
local
selection
gradient
is
a
product
of
the
additive
genetic
variance
of
the
trait
and
the
partial
derivatives
of
a
species’
abundance
with
respect
to
the
trait.
The
additive
genetic
variance
of
a
trait
describes
the
ability
of
a
phenotype
to
respond
to
selective
pressures,
and
the
partial
derivative
of
abundance
with
respect
to
the
trait
describes
the
quantitative
effect
of
a
phenotypic
trait
on
a
species’
population.
The
direction
of
the
change
in
the
phenotype
will
always
increase
the
species’
fitness,
and
the
magnitude
of
the
change
of
fitness
is
proportional
to
the
strength
of
selection
on
the
trait
and
the
additive
genetic
variance
of
the
phenotype.
Assuming
that
the
phenotypes
are
controlled
by
many
genes
of
small
effect,
the
change
in
the
phenotype
can
be
approximated
by
a
continuous
process
that
can
be
modeled
with
differential
equations
(Lande
1982).
Because
the
phenotypes
of
the
species
affect
the
abundances
of
both
the
species,
a
stable
equilibrium
for
the
system
is
only
achieved
when
both
the
abundances
and
the
traits
have
reached
equilibrium.
Thus,
models
with
coevolutionary
dynamics
exhibit
oscillations
6
in
some
instances
when
the
equivalent
model
without
phenotype
evolution
would
reach
a
stable
equilibrium
(Abrams
2000,
Mougi
and
Iwasa
2011,
Mougi
2012).
This
is
due
to
the
inherent
frequency‐dependent
selection,
as
the
selection
gradient
of
a
species’
phenotype
changes
based
on
the
mean
values
of
the
phenotypes
of
both
species
(Abrams
et
al.
1993,
Saloniemi
1993,
de
Mazancourt
and
Dieckmann
2004).
The
fitness
of
the
predator
and
the
prey
are
determined
by
the
phenotypic
trait
of
the
other
species,
meaning
that
the
optimal
phenotype
of
each
species
changes
as
the
trait
value
of
the
other
species
evolves.
Most
models
of
coevolution
have
utilized
one
of
two
general
approaches
for
generating
the
predator’s
attack
rate
as
a
function
of
the
predator
and
prey
phenotypes.
The
first
is
that
the
attack
rate
grows
larger
as
the
predator
increases
its
phenotype
over
that
of
the
prey
(Salionemi
1993,
Nuismer
et
al.
2005,
Mougi
and
Iwasa
2010).
Instances
where
predator‐prey
systems
follow
the
unidirectional
attack
rate
model
include
the
following
interactions:
venom
versus
antivenom,
speed
versus
speed,
and
armor
versus
strength
(Nuismer
et
al.
2007,
Mougi
and
Iwasa
2010).
This
model
provides
a
unidirectional
axis
by
which
the
prey
can
avoid
the
predator,
because
the
prey
can
only
reduce
the
pressure
of
predation
by
increasing
its
trait
value.
Models
using
a
unidirectional
axis
of
predator
evasion
utilize
a
logistic
curve
to
represent
the
predator’s
attack
rate
as
a
function
of
the
relative
difference
between
the
predator
and
the
prey’s
phenotypes.
Large
relative
differences
in
phenotype,
which
correspond
to
a
larger
quantitative
trait
value
in
the
predator,
produce
the
maximum
possible
attack
rate.
The
second
approach
is
that
the
predator
must
match
the
prey
in
some
capacity
in
order
to
attain
the
maximum
attack
rate,
and
that
attack
rate
declines
if
the
predator’s
phenotype
is
larger
or
smaller
than
that
of
the
prey
(Dieckmann
and
Law
1996,
Gavrilets
7
1997,
Abrams
2000,
Nuismer
et
al.
2007,
Mougi
2012).
Thus,
this
model
provides
a
bidirectional
axis
by
which
the
prey
can
avoid
predation
because
any
mismatch
between
the
predator
and
the
prey’s
traits
lowers
the
attack
rate.
This
model
for
the
predator’s
attack
rate
applies
to
interactions
where
the
predator
selectively
attacks
prey
within
a
specific
size
range
(such
that
the
prey’s
body
size
matches
the
predator’s
jaw
size)
and
to
host‐parasite
systems
where
a
parasite
must
mimic
the
morphology
of
its
host
(Benkman
1999,
Nuismer
et
al.
2007).
Models
assuming
a
bidirectional
axis
of
predator
evasion
utilize
a
Gaussian
curve
to
describe
the
predator’s
attack
rate
as
a
function
of
the
difference
between
the
predator
and
the
prey’s
phenotypes.
When
the
difference
is
zero,
meaning
that
the
predator
perfectly
matches
the
prey,
the
maximum
attack
rate
is
achieved.
Previous
studies
of
coevolutionary
predator‐prey
systems
have
found
that
stabilizing
selection
on
the
trait
is
necessary
for
oscillations
to
occur
(Nuismer
et
al.
2005),
but
that
strong
stabilizing
selection
contributes
to
the
stability
of
equilibria
(Mougi
2012).
Strong
negative
density‐dependence
in
the
prey
also
leads
to
stable
equilibria
(Mougi
and
Iwasa
2011).
In
the
case
of
persisting
oscillations,
similar
and
intermediate
rates
of
evolution
of
the
predator
and
the
prey
generate
oscillations
with
the
largest
amplitude
(Mougi
and
Iwasa
2010).
In
this
analysis,
I
compare
the
results
of
predator‐prey
coveolutionary
systems
where
species
interactions
are
mediated
by
traits
that
generate
a
unidirectional
axis
of
predator
evasion
versus
systems
where
the
axis
is
bidirectional.
Although
it
is
possible
to
analyze
the
existence
and
stability
of
equilibria
in
coevolutionary
systems
analytically,
the
solutions
are
mathematically
intensive
and
complex,
so
I
have
instead
chosen
to
simulate
the
model
over
a
broad
range
of
plausible
parameter
space.
8
MODEL:
Population
Dynamics
In
the
following,
I
simultaneously
model
the
population
dynamics
and
phenotypic
evolution
of
a
predator
(P)
interacting
with
a
single
prey
(N).
The
population
dynamics
of
the
species
are
modeled
in
continuous
time
as
functions
of
their
per
capita
birth
and
death
rates.
These
per
capita
demographic
rates
are
also
influenced
by
the
mean
value
of
one
phenotypic
trait
of
each
species;
these
are
modeled
as
continuous
traits.
The
basic
model
for
each
species
is
then
dN
= N ⎡⎣ BN ( N ,Z N ) − DN ( N , P,Z N ,Z P ) ⎤⎦
dt
dP
= P ⎡⎣ BP ( N , P,Z N ,Z P ) − DP ( P,Z P ) ⎤⎦
dt
(1)
where
the
birth
rate
of
the
prey
(BN)
is
a
function
of
its
abundance
(N)
and
mean
phenotype
(
),
and
the
death
rate
of
the
predator
(DP)
is
a
function
of
its
abundance
(P)
and
mean
phenotype
(
).
Interactions
between
the
species
link
the
birth
rate
of
the
predator
(BP)
with
the
death
rate
of
the
prey
(DN).
The
prey’s
per
capita
birth
rate
is
modeled
as
(
(
BN = c 1− γ Z N* − Z N
) ) − dN
2
(2)
Given
this
function,
the
prey’s
population
abundance
is
limited
in
the
absence
of
the
predator
by
assuming
that
its
per
capita
birth
rate
is
a
linearly
decreasing
function
of
its
own
abundance
(i.e.,
negative
density‐dependence).
The
parameter
d
defines
the
strength
of
density‐dependence.
The
first
term
of
equation
(2)
defines
the
maximum
birth
rate
when
N
≈
0.
defines
the
phenotypic
value
giving
the
highest
maximum
birth
rate,
and
c
9
is
the
maximum
birth
rate
at
as
deviates
farther
from
decline
faster
away
from
.
This
maximum
birth
rate
declines
as
a
quadratic
function
.
Greater
values
of
cause
the
maximum
birth
rate
to
.
As
with
the
prey’s
birth
rate,
I
model
density‐dependence
in
the
predator’s
per
capita
death
rate
using
the
function
(
(
D p = f 1+ θ Z P* − Z P
) ) + gP .
2
(3)
The
first
term
of
equation
(3)
defines
the
minimum
death
rate
at
P
≈
0.
defines
the
phenotypic
value
giving
the
lowest
minimum
death
rate,
and
f
is
the
minimum
death
rate
at
.
Analogous
to
the
prey,
the
predator’s
minimum
death
rate
increases
as
a
quadratic
function
as
deviates
farther
from
.
The
strength
of
the
trade‐off
between
phenotypic
evolution
and
predator
birth
rate
is
represented
by
θ,
where
larger
values
of
θ
cause
the
minimum
death
rate
to
increase
more
rapidly
away
from
.
The
parameter
g
defines
the
strength
of
density‐dependence
in
the
predator’s
death
rate.
The
predator
and
prey
interact
through
the
functional
response
that
defines
the
rate
at
which
predators
kill
prey
(i.e.
)
and
the
numerical
response
of
the
predator
which
defines
how
captured
prey
are
converted
into
new
predators
(i.e.
).
The
attack
rate
function
occurs
in
both
the
predator’s
and
the
prey’s
growth
equation
and
provides
the
link
that
couples
the
abundances
of
the
species.
The
death
rate
of
the
prey
due
to
predation
describes
the
amount
of
the
prey
population
that
is
utilized
as
resources
for
the
predator.
Thus,
the
predator’s
growth
rate
should
be
proportional
to
the
amount
of
prey
consumed.
However,
biomass
is
not
perfectly
assimilated
through
trophic
levels,
and
energy
is
10
required
in
order
for
the
predator
to
reproduce.
The
conversion
constant
b
is
used
in
the
predator’s
birth
rate
function
(
)
to
represent
the
average
reproductive
capacity
gained
by
a
predator
as
a
result
of
consuming
one
prey.
Throughout
this
analysis
I
assume
a
linear
functional
response
in
which
the
attack
coefficient
is
a
function
of
the
predator
and
prey
phenotypes
(i.e.
DN = a ( Z N ,Z P ) P ),
and
the
per
capita
numerical
response
of
the
predator
is
a
constant
fraction
of
prey
killed
(i.e.
).
There
are
two
main
ways
in
which
a
prey’s
phenotype
can
contribute
to
its
evasion
of
a
predator.
The
first
is
that
the
predator
is
most
successful
at
consuming
the
prey
when
its
trait
is
larger
than
the
prey’s
trait,
such
that
the
prey
can
only
escape
predation
by
increasing
the
value
of
its
phenotype.
The
direction
of
selection
on
these
traits
due
to
predation
is
unidirectional
and
always
selects
for
an
increasing
trait
value.
An
example
of
this
kind
of
attribute
would
be
running
speed.
As
the
prey
increases
its
running
speed
relative
to
that
of
the
predator,
the
attack
rate
of
the
predator
decreases.
When
the
predator
has
a
larger
running
speed
than
the
prey,
the
attack
rate
is
high,
but
when
the
prey
is
able
to
outrun
the
predator,
the
attack
rate
is
low.
When
the
predator
and
prey
are
closely
matched,
the
attack
rate
is
intermediate
and
changes
most
rapidly
with
incremental
changes
in
the
relative
phenotypes
of
the
species.
Thus,
the
attack
function
in
the
unidirectional
model
is
given
by
a
logistic
curve
with
the
following
equation:
⎛ a ⎞
a ( Z N ,Z P ) = ⎜
⎝ 1+ e− α Δ ⎟⎠
The
difference
between
the
predator’s
and
prey’s
phenotypes
is
Δ
(
(4)
).
The
predator
approaches
its
maximum
attack
rate
(a)
when
Δ
is
large.
The
steepness
of
the
attack
11
function
near
Δ
=
0
is
controlled
by
the
scaling
factor
α,
where
larger
values
of
α
increase
the
steepness
of
the
function
(Figure
1).
Figure
1.
The
effective
attack
rate
of
the
predator
based
on
the
difference
between
the
predator
and
prey
phenotypes,
with
a
=
1
and
α
=
3.
The
second
way
the
phenotypes
of
the
two
species
could
affect
attack
rate
is
if
the
predator
must
match
the
prey
in
some
aspect
in
order
to
capture
the
prey.
The
direction
of
selection
on
these
traits
due
to
predation
can
be
to
either
increase
or
decrease
the
phenotype
of
the
prey,
and
depends
upon
the
current
value
of
the
predator’s
trait.
An
example
of
such
a
characteristic
would
be
a
predator’s
jaw
size
matching
the
prey’s
body
size.
In
this
case,
the
prey
could
avoid
predation
by
being
either
smaller
or
larger
than
the
body
size
preferred
by
the
predator.
Under
this
assumption,
the
attack
rate
is
maximized
at
Δ
=
0
and
declines
with
increasing
or
decreasing
values
of
Δ.
The
attack
curve
in
the
bidirectional
model
is
given
by
a
Gaussian
function
with
the
equation
a ( Z N ,Z P ) = ae
12
⎛ Δ⎞
−⎜ ⎟
⎝ β⎠
2
(5)
where
β
is
a
scaling
parameter
for
the
steepness
of
the
function
(Figure
2).
Larger
values
of
β
decrease
the
steepness
of
the
function.
Figure
2.
Effective
attack
rate
as
a
function
of
the
difference
between
the
predator
and
prey
phenotypes,
where
a
=
1
and
β
=
1
Phenotypic
Evolution
To
allow
the
phenotypes
to
evolve,
two
additional
equations
are
required
to
describe
the
change
in
the
quantitative
traits
of
the
predator
and
prey.
The
magnitude
and
direction
of
the
change
in
these
quantitative
traits
are
a
result
of
the
local
selection
gradient.
Lande
(1982)
and
Abrams
et
al.
(1993)
demonstrated
that
the
local
selection
gradient
for
either
species
is
the
additive
genetic
variance
multiplied
by
the
gradient
of
the
growth
rate
of
the
species
with
respect
to
its
quantitative
traits.
The
additive
genetic
variance
is
a
constant
describing
the
rate
of
change
of
the
trait
and
is
a
product
of
the
variance
in
the
phenotypic
trait
across
the
population
and
the
heritability
of
the
trait.
The
gradient
of
the
population
growth
rate
is
a
vector
of
partial
derivatives
where
each
entry
is
a
function
for
the
rate
of
change
of
the
population’s
growth
rate
with
respect
to
the
13
quantitative
phenotypic
trait
(Lande
1982).
This
model
considers
one
phenotypic
trait,
so
the
corresponding
selection
gradient
is
a
vector
of
length
1.
In
a
differential
equations
model,
the
gradient
of
the
population
growth
rate
with
respect
to
the
quantitative
trait
is
easy
to
evaluate
because
the
growth
rates
of
the
species
are
explicitly
stated
in
the
model
equations.
Taking
the
partial
derivative
of
the
population
growth
rate
of
the
prey
yields
two
terms,
one
of
which
describes
the
change
in
growth
rate
due
to
predation
and
the
other
of
which
describes
the
change
in
growth
rate
due
to
increasing
mortality
as
a
result
of
phenotype
evolution.
The
selection
gradient
of
the
predator
is
similar,
with
terms
describing
the
change
in
the
predator’s
growth
rate
due
to
attack
rate
and
increased
mortality
as
a
result
of
phenotype
evolution.
The
sign
of
the
sum
of
the
costs
and
benefits
to
phenotype
evolution
determines
whether
the
incremental
change
in
phenotype
in
the
next
generation
will
be
toward
or
away
from
the
phenotype
where
the
maximum
birth
rate
occurs
genetic
variance
of
the
trait
(
.
Multiplying
this
sum
by
the
additive
or
)
gives
the
realized
change
in
phenotype
from
one
generation
to
the
next.
Unidirectional
Prey
⎡
⎤
dZ N
aPe− α Δ
* ⎥
⎢
= VZ
− 2cγ (Z N − Z N )
2
N ⎢
⎥
−α Δ
dt
⎢⎣ 1+ e
⎥⎦
(
)
(6)
Predator
⎡
⎤
dZ P
baN α e− α Δ
* ⎥
⎢
= VZ
− 2 f Θ(Z P − Z P )
2
P ⎢
⎥
−α Δ
dt
1+
e
⎢⎣
⎥⎦
(
)
14
Bidirectional
Prey
Predator
2
⎛ Δ⎞
⎡
⎤
−⎜ ⎟
β
⎝
⎠
⎢
⎥
dZ N
−2aPΔe
*
= VZ ⎢
−
2c
γ
(Z
−
Z
)
N
N ⎥
N
dt
β2
⎢
⎥
⎢⎣
⎥⎦
⎡
⎢ −2baN Δe
dZ P
= VZ ⎢
P
dt
β2
⎢
⎢⎣
⎛ Δ⎞
−⎜ ⎟
⎝β⎠
2
⎤
⎥
− 2 f Θ(Z P − Z P* ) ⎥ ⎥
⎥⎦
(7)
Coupling
equations
(1)
with
the
respective
trait
change
equations
(6)
or
(7)
comprise
the
differential
equations
necessary
for
the
two
models
to
allow
the
population
dynamics
to
change
as
a
function
of
the
phenotypes
of
the
two
species
while
the
phenotypes
themselves
evolve.
RESULTS:
I
evaluated
each
model
under
the
following
conditions:
the
predator’s
quantitative
trait
value
where
its
lowest
death
rate
occurred
was
higher
than
the
prey’s
quantitative
trait
value
for
maximizing
birth
rate
(
),
the
predator’s
minimum
death
rate
occurred
at
a
trait
value
lower
than
that
of
the
prey’s
maximum
birth
rate
(
the
two
growth
rate
maximizing
trait
values
were
equal
(
),
and
).
For
the
cases
where
the
trait
values
of
the
species
were
different,
I
used
differences
of
0.15,
0.5,
and
2.0
quantitative
trait
units
for
the
separation
of
and
(
).
Within
each
of
these
categories
of
,
I
varied
each
parameter
incrementally
to
assess
the
effect
of
the
parameter
on
the
long‐term
behavior
of
the
system.
I
chose
to
analyze
the
system
at
a
combination
of
parameters
that
15
generated
damped
oscillations
so
that
any
change
in
the
model
due
to
parameter
changes
would
be
immediately
apparent.
Effects
of
Parameters
on
System
Behavior:
Dynamic
Behavior
of
Equilibria
In
both
models,
parameters
that
increased
the
growth
rate
of
either
species
were
destabilizing
to
the
system
when
the
values
were
increased.
When
other
parameters
in
the
model
were
fixed,
the
system
rapidly
approached
equilibrium
when
the
prey’s
maximum
birth
rate,
c,
was
small.
As
the
value
of
c
increased,
the
system
began
to
overshoot
the
eventual
equilibrium,
and
both
the
abundances
and
the
trait
values
of
both
species
entered
a
regime
of
damped
oscillations.
As
c
increased
further,
the
fluctuations
before
the
eventual
equilibrium
persisted
for
a
longer
amount
of
time.
When
the
prey’s
maximum
birth
rate
was
large,
the
system
exhibited
persisting
oscillations
(i.e.
a
stable
limit
cycle)
in
the
abundances
and
the
trait
values
of
the
species.
The
maximum
attack
rate
of
the
predator,
a,
and
the
predator’s
conversion
rate
b,
determine
the
birth
rate
of
the
predator
based
on
the
abundance
of
the
prey.
When
increasing
the
value
of
either
of
these
two
parameters,
the
birth
rate
of
the
predator
also
grows
larger.
Both
a
and
b
induce
the
same
progression
of
system
dynamics
as
the
prey’s
birth
rate,
c,
when
the
values
are
increased.
The
abundances
of
the
predator
and
the
prey
initially
approach
a
stable
equilibrium
rapidly,
then
enter
a
region
of
parameter
space
where
the
system
exhibits
damped
oscillations
in
the
approach
to
equilibrium,
and
finally
the
system
oscillates
perpetually.
Any
parameter
that
decreases
the
growth
rate
of
one
species
without
affecting
the
growth
rate
of
the
other
species
is
stabilizing
to
the
system.
When
the
negative
density‐
16
dependence
of
the
prey
is
weak,
meaning
d
is
small,
the
system
oscillates
in
both
the
traits
and
abundances
of
the
two
species.
When
the
negative
density‐dependence
is
moderate
(i.e.,
as
d
is
increased),
the
oscillations
decrease
over
time,
and
the
system
eventually
reaches
a
stable
equilibrium.
If
d
is
large
(i.e.,
negative
density‐dependence
is
strong),
the
trait
values
and
abundances
of
the
two
species
rapidly
approach
a
stable
equilibrium.
The
case
of
the
negative
density‐dependence
of
the
predator,
represented
by
f,
is
the
same.
When
negative
density‐dependence
in
the
predator
is
weak
(i.e.,
f
is
small)
the
abundances
and
traits
of
the
two
species
oscillate
in
a
stable
limit
cycle.
As
f
increases
and
the
negative
density‐dependence
in
the
predator
becomes
stronger,
the
abundances
and
the
trait
values
of
the
two
predator
and
the
prey
show
damped
oscillations
around
a
stable
equilibrium.
When
f
is
large,
the
trait
values
and
abundances
of
both
species
rapidly
approach
equilibrium.
The
predator’s
density‐independent
mortality
rate,
g,
parallels
the
behavior
of
d
and
f
such
that
increasing
values
of
g
stabilize
the
system.
The
parameters
that
affect
the
rate
of
evolution
can
be
stabilizing
or
destabilizing
to
the
system
depending
on
the
rate
of
evolution
of
the
other
species.
The
effects
of
the
evolutionary
parameters
also
differ
slightly
in
the
two
models.
In
both
models,
increasing
the
additive
genetic
variances
of
the
populations,
and
,
shortens
the
period
of
oscillations
when
the
system
is
unstable
and
reduces
the
time
taken
in
reaching
equilibrium
when
the
system
is
stable.
Increasing
the
additive
genetic
variances
of
the
species
in
tandem
effectively
speeds
up
the
dynamics
of
the
system
such
that
the
same
dynamics
occurs
on
a
shorter
time
span.
However,
if
the
additive
genetic
variance
is
large
for
one
species
and
small
for
the
other,
the
system
will
be
stable.
This
is
equivalent
to
giving
one
species
a
much
faster
rate
of
evolution
than
the
other
species,
if
all
other
17
parameters
are
equal.
In
these
scenarios,
either
the
prey’s
rate
of
phenotypic
evolution
is
rapid
enough
to
effectively
evade
the
predator,
or
the
predator
is
able
to
evolve
its
phenotype
to
effectively
capture
and
regulate
the
prey.
Making
the
additive
genetic
variances
of
the
two
species
comparable
in
magnitude
makes
the
system
increasingly
unstable,
as
neither
species
is
able
to
evolve
their
phenotype
quickly
enough
to
escape
the
constraints
imposed
by
the
other
species.
The
parameters
that
describe
the
costs
of
phenotypic
evolution
in
terms
of
the
prey’s
birth
rate,
rate,
,
and
the
costs
of
phenotypic
evolution
in
terms
of
the
predator’s
death
,
have
similar
dynamics.
In
the
unidirectional
model,
if
the
cost
to
one
population
is
much
greater
than
to
the
other
population,
either
the
prey
will
be
able
to
evolve
effective
defenses
or
the
predator
will
be
able
to
regulate
the
prey
population
(Figure
3).
However,
the
bidirectional
model
does
not
have
a
stable
equilibrium
when
the
cost
of
evolution
is
much
higher
in
the
prey
than
in
the
predator.
Even
when
the
cost
of
evolution
in
the
prey
is
high,
the
system
oscillates
so
long
as
the
predator
is
able
to
track
the
prey’s
trait
value
and
impose
strong
costs
of
predation.
If
the
cost
to
the
predator
is
much
larger
than
the
cost
to
the
prey,
the
system
is
stable.
In
both
models,
when
the
rates
of
phenotypic
evolution
in
the
predator
and
the
prey
are
roughly
comparable,
the
system
is
unstable
because
the
predator
is
able
to
effectively
track
the
prey’s
phenotype
and
the
prey
is
able
to
change
its
phenotype
once
the
predator
has
evolved.
18
Figure
3:
Bifurcation
diagrams
showing
regions
of
stable
equilibrium
and
regions
of
persisting
oscillations
in
the
abundances
of
the
predator
and
the
prey.
Data
comes
from
the
unidirectional
model
with
parameters
c
=
3,
d
=
0.05,
a
=
0.2,
b
=
0.1,
f
=
0.1,
g
=
1,
! = 2.2, " = 0.15, Z N* = 1, Z P* = 0, vZ N = .1, vZ P = .1 The
parameters
that
scale
the
steepness
of
the
attack
function
also
have
a
strong
influence
on
the
stability
of
the
predator‐prey
system.
In
the
unidirectional
model,
19
controls
the
slope
of
the
attack
function
when
the
quantitative
trait
values
of
the
predator
and
the
prey
are
equal,
meaning
Z N = Z P .
An
increasing
value
of
means
that
the
attack
function
changes
more
quickly,
such
that
the
predator
gains
a
greater
advantage
over
the
prey
for
the
same
incremental
change
in
the
predator’s
quantitative
trait
value.
When
small
the
system
is
stable
and
rapidly
approaches
equilibrium,
but
as
is
increases
the
system
becomes
unstable
and
oscillates.
In
the
bidirectional
model,
side
of
Z N = Z P .
When
controls
the
steepness
of
the
attack
function
on
either
is
high,
the
function
is
less
steep
and
there
is
little
change
in
the
attack
rate
for
each
incremental
movement
in
trait
space.
At
low
values
of
unstable,
and
as
increases
the
traits
and
abundances
first
enter
a
region
of
damped
oscillations
and
eventually
rapidly
approach
equilibrium
when
the
system
is
The
difference
of
the
predator’s
minus
the
prey’s
is
large.
(Z*P ! Z N* = " * ) also
strongly
affects
the
stability
of
the
models.
In
both
the
unidirectional
and
bidirectional
attack
models,
the
system
was
stable
when
stable
as
was
much
less
than
zero
and
became
less
approached
zero.
This
corresponds
to
scenarios
where
the
prey
has
its
optimal
birth
rate
at
a
trait
value
that
is
much
higher
than
that
of
the
predator.
In
the
unidirectional
system,
this
would
correspond
to
a
prey
that
is
stronger
or
faster
than
the
predator
it
interacts
with.
In
a
bidirectional
model,
this
corresponds
to
a
prey
that
is
much
larger
in
body
size
than
the
predator’s
preferred
prey
size.
In
the
bidirectional
model,
values
of
much
larger
than
zero
also
contribute
to
system
stability.
If
a
predator
attacks
its
prey
based
on
how
close
the
animal
is
to
its
preferred
body
size,
being
smaller
than
that
size
is
equally
as
beneficial
for
a
prey
as
being
larger.
However,
in
the
unidirectional
model,
20
increasing
the
value
of
near
=
0
is
initially
destabilizing
and
causes
oscillations
to
grow
in
amplitude
and
decrease
in
period.
This
area
of
parameter
space
is
very
dynamic
in
terms
of
the
increased
attack
rate
gained
by
the
predator
for
every
small
increase
in
its
relative
phenotype.
In
this
region
of
,
the
predator
is
able
to
run
slightly
faster
than
the
prey,
which
confers
a
large
advantage
for
each
increase
in
the
predator’s
phenotype
over
that
of
the
prey.
Past
small
increases
in
,
larger
values
of
contribute
to
stability
in
the
unidirectional
system
as
well
as
the
predator
is
able
to
maintain
a
high
attack
rate
and
phenotypic
evolution
in
the
prey
has
little
benefit.
Location
of
Equilibria
In
both
models,
increasing
the
maximum
attack
rate
of
the
predator,
a,
decreased
the
equilibrial
abundance
of
the
prey
when
the
system
reached
a
stable
equilibrium.
With
increasing
values
of
maximum
attack
rate,
the
predator’s
equilibrium
abundance
initially
increased
until
intermediate
values
of
a,
but
then
decreased
due
to
a
lower
abundance
of
the
prey.
This
same
pattern
in
abundances
versus
attack
rate
is
seen
in
the
analogous
predator‐prey
population
model
that
does
not
allow
for
evolution.
Identifying
patterns
in
the
equilibrial
trait
values
of
the
two
species
is
more
complex,
since
the
phenotypes
of
minimal
cost
( Z N* and
Z P* )
are
parameters
that
are
subject
to
change.
Thus,
I
describe
the
patterns
in
the
locations
of
the
trait
values
as
converging
toward
or
diverging
away
from
Z N* or
Z P* .
Increasing
the
predator’s
maximum
attack
rate
resulted
in
the
equilibrial
trait
values
of
the
predator
and
the
prey
diverging
from
their
phenotypes
of
minimum
cost.
Increasing
the
conversion
efficiency
of
the
predator,
b,
decreased
the
equilibrium
abundance
of
the
prey
and
increased
the
equilibrial
abundance
of
the
predator.
Similar
to
increasing
the
maximum
attack
rate
of
the
predator,
larger
values
of
b
caused
the
species’
21
trait
values
to
diverge
from
Z N* and
Z P* ,
respectively.
Increasing
the
maximum
birth
rate
of
the
prey,
c,
caused
increases
in
the
equilibrial
abundances
of
both
species,
though
the
slope
of
the
increase
was
larger
in
the
predator
than
in
the
prey.
Larger
values
of
c
caused
the
species’
trait
values
to
converge
toward
Z N* and
Z P* at
equilibrium.
Increasing
the
density‐
independent
death
rate
of
the
predator,
f,
caused
the
equilibrial
abundance
of
the
predator
to
decrease
and
the
equilibrial
abundance
of
the
prey
to
increase.
However,
the
species’
traits
both
converged
toward
Z N* and
Z P* .
Increasing
the
density‐dependent
death
rate
of
the
predator,
g,
showed
the
same
pattern
as
increases
in
f,
though
the
initial
increase
in
prey
abundance
and
decrease
in
predator
abundance
were
more
pronounced.
Increasing
the
cost
of
evolution
to
the
prey,
γ ,
decreased
the
equilibrial
abundance
of
the
prey
and
increased
the
equilibrial
abundance
of
the
predator
in
both
models.
The
trait
values
of
both
species
converged
toward
their
phenotypes
of
lowest
cost.
However,
the
unidirectional
and
bidirectional
models
differed
in
how
the
predator’s
cost
of
evolution,
Θ ,
affected
the
equilibria
of
the
systems.
In
both
models,
as
Θ increased,
the
prey’s
equilibrial
abundance
increased
while
the
predator’s
equilibrial
abundance
decreased.
In
the
bidirectional
model,
the
predator’s
trait
value
at
equilibrium
converged
toward
Z P* whereas
the
prey’s
trait
value
diverged
away
from
Z N* .
In
the
unidirectional
model,
the
prey’s
trait
value
converged
toward
Z N* under
most
circumstances,
but
under
some
conditions
(such
as
Z N* ≈ Z P* ),
the
prey’s
trait
slightly
diverged
from
Z N* .
The
predator’s
trait
value
at
equilibrium
always
tracked
that
of
the
prey,
such
that
the
predator’s
Z P* had
little
effect
on
the
eventual
trait
value
of
the
predator.
22
The
shape
parameters
used
in
the
attack
rate
functions
in
the
two
models
also
affected
the
location
of
the
abundance
and
trait
value
equilibria.
When
the
attack
function
in
the
unidirectional
model
grew
steeper,
meaning
that
α increased,
the
predator’s
equilibrial
abundance
decreased.
The
prey’s
abundance
initially
grew
over
increasing
values
of
α ,
but
then
decreased
at
higher
values
of
α .
As
the
steepness
of
the
attack
function
increased,
the
equilibrial
trait
values
of
the
predator
and
the
prey
both
diverged
from
their
phenotypes
of
lowest
cost.
As
the
steepness
of
the
attack
function
in
the
bidirectional
model
increased,
meaning
that
β grew
larger,
the
predator’s
equilibrial
abundance
increased
while
the
prey’s
equilibrial
abundance
decreased.
However,
the
trait
values
of
the
predator
and
the
prey
converged
toward
Z N* and
Z P* as
β increased.
Unidirectional
Model:
In
the
unidirectional
model
at
equilibrium,
the
predator’s
trait
was
higher
than
the
prey’s
trait
in
all
the
parameter
space
explored.
Under
the
condition
that
the
prey’s
decrease
in
birth
rate
as
a
result
of
phenotype
evolution,
,
is
roughly
equal
to
the
predator’s
increase
in
death
rate,
,
the
cost
to
the
prey
for
phenotypic
evolution
is
much
larger
than
the
cost
to
the
predator.
The
given
by
total
cost
functions
of
phenotypic
evolution
are
for
the
prey
and
.
These
terms
are
analogous,
except
that
the
cost
for
the
prey
is
in
terms
of
its
birth
rate,
c,
whereas
the
cost
to
the
predator
is
in
terms
of
its
density‐dependent
mortality
rate,
f.
However,
the
prey’s
birth
rate
per
unit
time
is
a
much
larger
value
than
the
predator’s
death
rate
per
unit
time
(up
to
one
to
two
orders
of
magnitude),
and
so
the
total
cost
of
phenotypic
evolution,
in
terms
of
growth
rate,
is
larger
for
the
prey
than
for
the
predator
if
is
roughly
equal
to
23
.
The
prey’s
birth
rate
is
necessarily
larger
than
the
predator’s
density‐dependent
death
rate
in
systems
with
both
species
present
because
the
predator’s
conversion
of
prey
into
new
predators
is
not
perfectly
efficient.
For
the
predator
to
have
any
region
of
positive
per
capita
growth,
the
availability
of
prey
must
be
much
greater
than
the
predator’s
death
rate.
Disregarding
the
population
effects
of
evolution,
abN
must
be
greater
than
fP
for
at
least
small
values
of
P,
which
is
only
achieved
under
the
condition
that
c ! f .
In
a
system
without
evolution,
the
equilibrium
predator
abundance
is
abc − fd
,
which
is
only
positive
when
abc > fd .
In
the
dg + a 2b
coevolutionary
model,
a
is
the
maximum
possible
attack
rate
and
f
is
the
minimum
possible
death
rate,
meaning
that
the
condition
of
abc > fd becomes
more
difficult
to
satisfy
as
the
predator’s
phenotype
evolves
away
from
Z P* .
Thus,
c
must
be
much
greater
than
f
in
this
model
in
order
to
ensure
that
the
predator
has
a
positive
equilibrium
value.
The
phenotypes
of
the
predator
and
the
prey
reach
equilibrium
when
the
costs
to
evolution
equal
the
benefits
of
evolution.
Thus,
in
the
unidirectional
model,
the
trait
value
at
which
the
prey’s
benefits
of
phenotypic
evolution
equal
the
costs
of
trait
evolution
occurs
at
a
lower
quantitative
value
than
for
the
predator.
Making
the
cost
to
the
predator
numerically
equal
to
the
cost
to
the
prey
involves
either
decreasing
c
or
,
or
increasing
f
or
.
Changing
any
of
these
parameters
enough
to
equalize
the
costs
of
evolution
of
the
two
species
decreases
the
equilibrium
abundance
of
the
predator
and
eventually
drives
the
predator
out
of
the
system.
Bidirectional
Model:
In
the
bidirectional
model,
when
the
growth
rate
maximizing
quantitative
trait
values
of
the
predator
and
the
prey
are
the
same
(
24
)
the
system
always
reached
an
equilibrium
with
.
When
this
occurs,
the
values
of
,
,
and
are
all
equal
to
zero.
These
terms
are
all
involved
in
the
equations
for
the
change
in
the
phenotypes
of
the
predator
and
the
prey,
and
consequently
the
change
in
the
phenotypes
of
the
species
are
also
equal
to
zero
(
=
=
0).
Without
phenotypic
change,
the
system
follows
the
dynamics
of
a
predator‐prey
model
without
evolution.
In
the
case
that
the
predator’s
phenotypic
optimum
and
the
prey’s
phenotypic
optimum
did
not
occur
at
the
same
quantitative
trait
value
(either
there
was
a
value
of
or
)
that
produced
the
largest
divergence
in
the
prey’s
trait
from
its
birth‐maximizing
phenotype
when
the
system
reached
equilibrium
(Figure
4).
Figure
4:
Deviation
of
and
from
and
increases.
c
=
2,
d
=
.1,
a
=
.25,
b
=
.2,
f
=
.1,
g
=
0,
25
as
the
difference
between
and
*
=
5,
=
.05,
=
.15,
Z N = 0 This
occurs
because
the
total
divergence
from
is
equal
to
the
integral
of
the
trait
change
equation
from
the
starting
point
until
the
system
reaches
equilibrium.
Maximizing
this
integral
means
finding
the
region
of
quantitative
trait
space
where
the
difference
between
the
selective
pressures
and
the
costs
of
evolution
is
largest;
when
this
quantity
is
large,
the
phenotype
of
the
prey
will
change
rapidly.
Using
a
Gaussian
function
for
attack
rate
and
a
quadratic
function
for
costs
in
terms
of
birth
rate,
the
following
figure
shows
that
this
difference
is
largest
between
zero
and
the
maximum
or
minimum
of
the
derivative
of
the
attack
function.
Although
the
fitness
surface
and
the
benefits
to
phenotypic
evolution
change
depending
on
the
predator’s
current
phenotypic
value,
results
of
the
model
showed
that
a
starting
value
of
where
the
benefit
of
evolution
was
much
greater
than
the
cost
to
evolution
led
to
the
greatest
amount
of
prey
evolution
when
the
system
reached
equilibrium.
Notably,
this
point
occurred
at
intermediate
levels
of
predation
where
the
prey’s
abundance
was
not
strongly
affected
by
the
presence
of
the
predator
(Figure
5).
Figure
5:
Selective
pressure
acting
to
move
away
from
preventing
quantitative
trait
divergence
from
26
.
and
costs
of
evolution
Oscillatory
Dynamics
in
the
Two
Models:
The
relationship
between
the
trait
and
abundance
oscillations
in
these
two
models
are
fundamentally
different.
In
the
unidirectional
model,
the
prey
trait
and
population
cycles
are
nearly
in
phase,
meaning
that
peaks
in
the
prey’s
abundance
and
trait
values
occur
at
nearly
the
same
time
(Figure
6).
However,
the
predator
trait
and
population
cycles
have
a
phase
shift
of
approximately
half
a
period,
such
that
abundance
maxima
and
trait
value
minima
occur
at
roughly
the
same
time.
In
any
given
oscillation,
the
prey’s
trait
value
is
the
first
variable
to
reverse
direction.
The
sharp
increase
in
the
prey’s
trait
marks
the
beginning
of
each
oscillation.
Soon
after,
the
prey’s
population
also
increases
as
the
prey
is
able
to
effectively
avoid
capture.
Then,
the
predator
is
under
strong
selective
pressure
to
increase
its
trait
value
in
order
to
capture
the
prey,
and
the
predator’s
quantitative
trait
value
increases
quickly.
However,
this
rapid
evolution
in
the
predator
has
the
cost
of
increased
mortality,
and
the
predator’s
population
declines
as
its
trait
evolves.
When
the
predator’s
trait
value
is
large
enough
such
that
the
prey
suffers
heavy
predation,
the
cost
of
maintaining
a
high
phenotype
in
the
prey
is
not
offset
by
the
benefits
of
predator
evasion,
and
the
prey’s
trait
value
decreases.
The
prey’s
population
decreases
with
increased
predation,
and
the
predator’s
population
increases
due
to
a
heightened
attack
rate.
Additionally,
the
predator’s
trait
value
decreases
because
the
predator
can
attain
a
similarly
high
attack
rate
with
a
lower
trait
value
because
the
attack
rate
is
a
function
of
the
relative
phenotypes
of
the
two
species.
27
Figure
6:
Abundance
and
trait
value
oscillations
in
the
unidirectional
model.
Blue
indicates
the
abundance
and
trait
value
of
the
predator,
whereas
green
indicates
the
abundance
and
trait
value
of
the
prey.
In
contrast,
the
abundances
of
the
predator
and
prey
cycle
twice
as
quickly
as
the
trait
values
of
the
species
the
bidirectional
model
(Figure
7).
The
abundances
of
the
predator
and
the
prey
increase
before
each
minimum
or
maximum
in
the
prey’s
trait
value
and
then
decline
until
immediately
before
the
next
minimum
or
maximum.
In
a
bidirectional
system
where
the
predator’s
ability
to
capture
the
prey
is
based
on
how
closely
the
predator
matches
the
prey’s
quantitative
trait
value,
the
prey’s
trait
is
the
first
variable
to
show
a
dramatic
increase.
The
prey’s
phenotype
changes
in
response
to
the
moderate
level
of
predation
present
due
to
the
high
abundance
of
predators.
At
this
time,
the
benefits
of
the
extreme
trait
value
of
the
prey
do
not
balance
the
costs
of
this
extreme
phenotype.
When
the
prey’s
trait
increases,
the
attack
rate
of
the
predator
initially
increases
as
the
prey’s
phenotype
gets
closer
to
the
preferred
phenotype
of
the
predator.
The
predator’s
abundance
increases
because
its
attack
rate
grows
larger
without
the
cost
of
the
predator’s
trait
evolving.
However,
the
prey
also
benefit
from
their
evolution
back
toward
their
birth
rate
maximizing
trait
value
because
they
are
released
from
the
high
cost
(in
terms
of
birth
rate)
of
maintaining
an
extreme
phenotype.
The
prey’s
abundance
then
28
also
increases,
and
reaches
its
maximum
when
the
prey
has
an
intermediate
trait
value.
At
this
point
in
the
cycle,
the
costs
to
maintaining
this
phenotype
are
relatively
small
and
the
predator
has
not
yet
evolved
to
track
the
prey’s
trait
value,
so
there
is
little
predation.
However,
as
soon
as
the
prey’s
trait
value
surpasses
the
predator’s
current
preferred
trait
value
the
predator’s
phenotype
begins
to
evolve
in
the
opposite
direction
in
order
to
track
the
prey’s
current
trajectory.
When
the
cost
to
the
prey
of
having
a
large
phenotype
outweighs
the
benefits
of
predator
avoidance,
the
prey’s
trait
value
will
reverse
and
the
same
dynamics
will
repeat
when
the
prey’s
trait
decreases.
The
population
dynamics
have
the
same
period
of
oscillation
as
the
trait
values,
but
the
abundances
of
the
predator
and
the
prey
have
two
maxima
and
two
minima
within
each
period.
Each
time
the
direction
of
the
prey’s
trait
evolution
reverses,
the
prey’s
trait
and
the
predator’s
trait
are
closely
matched
for
a
short
period
of
time.
This
increases
the
population
of
the
predator
due
to
an
increased
attack
rate
and
increases
the
population
of
the
prey
due
to
smaller
costs
of
their
phenotype
in
terms
of
their
birth
rate.
Thus,
each
change
in
direction
of
the
phenotype
corresponds
to
an
abundance
peak
in
the
two
species,
which
means
that
abundance
peaks
occur
twice
as
often
as
trait
peaks.
However,
the
two
abundance
maxima
within
each
period
are
not
equal.
The
two
peaks
are
of
different
magnitude
and
change
size
depending
on
the
value
of
magnitude
of
relative
to
and
the
.
The
mismatch
in
the
size
of
the
abundance
peaks
is
determined
by
the
amount
of
asymmetry
in
the
costs
of
evolution
in
various
parts
of
the
trait
value
cycle.
The
trait
oscillations
of
both
the
predator
and
the
prey
are
centered
around
the
prey’s
,
meaning
that
the
prey
always
incur
symmetric
costs
to
evolution
whether
their
trait
is
increasing
or
decreasing
because
it
deviates
the
same
amount
from
its
29
in
either
case.
When
=
0
the
predator’s
is
also
in
the
center
of
the
oscillations,
and
the
predator
incurs
symmetric
costs
as
well.
In
this
case,
the
peaks
in
species’
abundances
are
identical
because
both
species
have
symmetric
costs
of
phenotype
evolution
around
.
However,
if
is
not
equal
to
zero,
the
predator
incurs
asymmetric
costs
because
increasing
its
trait
value
to
track
the
prey
brings
it
closer
to
its
direction
and
further
from
its
in
one
in
the
opposite
direction.
Thus,
the
effect
of
trait
evolution
on
the
abundance
of
the
predator
depends
on
whether
the
prey
trait
is
increasing
or
decreasing.
This
asymmetric
cost
leads
to
the
difference
in
abundance
peaks
in
the
predator,
which
contributes
to
difference
in
the
abundance
peaks
in
the
prey.
Changing
relative
to
changes
the
duration
of
the
abundance
peaks
and
can
generate
erratic
abundance
cycles
when
the
cost
of
evolution
to
one
species
is
much
greater
than
the
cost
to
the
other
species.
Figure
7:
Abundance
and
trait
value
oscillations
in
the
bidirectional
model.
Blue
indicates
the
abundance
and
trait
value
of
the
predator,
whereas
green
indicates
the
abundance
and
trait
value
of
the
prey.
Fitness
Surfaces
During
Oscillations:
Although
the
oscillations
in
the
two
models
are
different,
the
fitness
surfaces
of
the
prey
exhibited
the
same
pattern
when
the
trait
values
oscillated.
The
prey’s
fitness
surface
30
is
bimodal
when
the
system
is
in
a
stable
limit
cycle
(Figure
8).
Each
local
maximum
in
the
fitness
surface
corresponds
to
a
phenotype
where
the
prey
would
have
increased
reproductive
success
when
compared
with
the
success
of
another
prey
with
a
slightly
higher
or
lower
phenotypic
trait
value.
These
two
maxima
correspond
to
the
phenotypes
where
the
prey
maximizes
fitness
by
either
1)
maintaining
a
high
birth
rate
or
2)
effectively
avoiding
predation.
Figure
8:
Fitness
surface
of
the
prey
in
an
oscillating
system
Selection
on
a
phenotypic
trait
will
always
act
to
increase
per
capita
reproduction,
meaning
that
the
change
in
the
trait
value
from
one
generation
to
the
next
will
act
to
increase
the
average
fitness
of
the
population.
Because
the
movement
of
the
phenotype
is
constrained
to
be
continuous
along
the
fitness
surface
and
to
always
increase
fitness,
the
prey
in
this
model
cannot
traverse
local
fitness
minima.
However,
because
the
growth
rate
of
the
prey
depends
on
its
own
phenotype
and
the
phenotype
of
the
predator
(as
well
as
their
abundances),
the
optimal
strategy
of
the
prey
changes
as
the
phenotype
of
the
predator
changes.
Thus,
the
fitness
of
the
prey
is
frequency
dependent
based
on
the
31
predator’s
phenotype,
and
the
shape
of
the
fitness
surface
changes
depending
on
the
current
value
of
.
In
systems
where
oscillations
occur,
the
shape
of
the
fitness
surface
changes
over
time
such
that
the
second
fitness
peak
grows
larger
until
it
extends
out
to
the
point
where
the
prey’s
phenotype
currently
resides.
The
slope
of
the
local
fitness
surface,
and
therefore
the
direction
of
the
local
selection
gradient,
change
sign
when
this
occurs.
The
prey
then
begin
to
reverse
the
direction
of
their
evolution
because
their
fitness
increases
when
evolving
their
phenotype
to
head
toward
the
other
fitness
peak.
When
the
second
fitness
peak
extends
outward
and
encapsulates
the
prey’s
current
trait
value,
the
location
of
the
fitness
trough
crosses
to
the
other
side
of
the
prey’s
trait
value
on
the
phenotype
axis.
The
fitness
surface
is
still
bimodal,
but
the
prey
now
follow
the
path
toward
the
second
fitness
maximum,
as
they
are
no
longer
prevented
by
the
fitness
trough
from
evolving
toward
the
larger
fitness
peak
(Figure
9).
32
Figure
9:
The
fitness
surfaces
of
the
prey
in
an
oscillating
system
when
the
trait
is
increasing
(panel
1)
and
when
the
trait
is
decreasing
(panel
2).
The
current
value
is
shown
by
the
star.
In
the
case
of
damped
oscillations,
the
prey
have
the
same
bimodal
fitness
surface,
but
the
oscillations
grow
smaller
over
time
because
the
prey
switch
strategies
earlier
in
quantitative
trait
space
as
time
goes
on.
This
is
a
result
of
the
fitness
surface
changing
more
rapidly
due
to
the
predator’s
increased
ability
to
track
the
prey
as
compared
to
models
where
the
traits
and
abundances
perpetually
oscillate.
When
the
oscillations
cease,
the
prey’s
traits
reaches
equilibrium
at
a
stable
fitness
minimum
directly
in
the
center
of
the
fitness
trough
separating
the
two
peaks
in
the
fitness
surface
(Figure
10).
Though
movement
in
either
direction
would
increase
the
fitness
of
the
prey,
the
point
is
stable
because
the
fitness
surface
changes
rapidly
enough
around
this
equilibrium
that
the
previous
phenotype
(the
stable
fitness
minimum)
has
greater
fitness
than
the
newly
derived
phenotype
once
the
trait
has
changed
(Abrams
et
al.
1993).
In
this
system,
if
the
prey’s
trait
were
to
evolve
in
either
direction
from
the
stable
fitness
minimum,
the
prey
would
have
increased
fitness
for
a
brief
period
of
time
before
the
predator’s
trait
evolved
in
response.
However,
when
the
predator’s
trait
value
changes,
the
fitness
surface
of
the
prey
33
also
changes.
In
the
case
of
a
stable
fitness
minimum,
the
previous
phenotype
of
the
prey
has
increased
fitness
over
the
current
phenotype
of
the
prey
after
this
fitness
surface
change,
and
the
prey
evolves
toward
its
previous
phenotype.
Though
the
point
is
a
minimum
in
the
prey’s
current
fitness
surface,
it
is
a
maximum
in
the
dynamic
system
because
any
deviation
from
that
point
induces
rapid
changes
in
the
fitness
surface
that
lead
the
prey
to
evolve
back
toward
its
previous
trait
value.
The
predator’s
trait,
however,
equilibrates
at
a
fitness
maximum
in
a
system
where
there
are
damped
oscillations.
Figure
10:
Equilibrium
trait
value
of
the
prey
in
a
system
exhibiting
damped
oscillations.
c
=
3,
d
=
.1,
a
=
.25,
b
=
.2,
f
=
.1,
g
=
0,
α =
1.49,
=
.05,
=
.10
DISCUSSION:
Dynamic
Behaviors
of
Coevolutionary
Models
Coevolutionary
dynamics
can
generate
oscillatory
behavior
in
predator‐prey
systems
where
the
same
system,
lacking
evolution,
would
reach
equilibrium.
This
occurs
34
because
phenotypic
evolution
in
the
two
species
allows
the
predator
to
regulate
the
prey
to
low
levels
through
a
combination
of
density
mediated
and
trait
mediated
interactions.
Previous
investigations
of
predator‐prey
dynamics
have
demonstrated
that
oscillations
in
these
systems
are
most
likely
to
occur
when
a
predator
is
able
to
decrease
the
prey’s
population
size
to
well
below
its
equilibrial
abundance
in
the
absence
of
the
predator
(Maynard
Smith
and
Slatkin
1973;
Ginzburg
1986,
Turchin
2001,
Holt
p.
139).
The
results
of
this
coevolutionary
model
largely
support
this
conclusion.
The
population‐related
parameters
that
augmented
the
predator’s
growth
rate
in
relation
to
that
of
the
prey
were
the
attack
rate,
a,
and
the
conversion
rate
of
prey
into
predators,
b.
Increasing
the
values
of
either
of
these
parameters
destabilized
the
system
and
led
to
oscillations.
The
parameters
that
decreased
the
predator’s
growth
rate
were
the
predator’s
density‐independent
death
rate,
f,
and
the
predator’s
density‐dependent
death
rate,
g.
When
either
of
these
terms
increased,
the
system
became
more
stable.
These
models
also
show
that
increasing
the
prey’s
birth
rate,
c,
or
decreasing
the
strength
of
the
prey’s
negative
density‐dependence,
d,
induces
oscillations
in
the
system.
This
is
in
accordance
with
the
theory
of
the
paradox
of
enrichment,
which
predicts
that
higher
prey
productivity
can
lead
to
instability
in
predator‐prey
systems
(Rosenzweig
1971,
Roy
et
al.
2007).
In
the
case
of
the
paradox
of
enrichment,
the
increased
birth
rate
of
the
prey
causes
the
system
to
shift
from
having
stable
equilibria
to
exhibiting
persisting
oscillations
in
the
abundance
of
the
predator
and
the
prey.
Empirical
studies
of
predator‐
prey
systems
have
demonstrated
a
crossing
of
the
Hopf
bifurcation
in
algae‐rotifer
systems
corresponding
to
an
increase
in
nutrient
input
into
the
system
(Fussman
et
al.
2000).
Reaching
this
bifurcation
point
corresponds
to
the
place
at
which
the
system’s
dominant
35
eigenvalue
transitions
from
having
a
negative
real
part
to
having
only
imaginary
parts,
meaning
that
the
long‐term
trajectories
of
the
system
switch
from
reaching
stable
equilibria
to
oscillating
indefinitely.
These
models
suggest
that
the
same
dynamics
are
present
in
predator‐prey
models
that
allow
for
evolution.
The
evolutionary
parameters,
however,
were
more
complex
in
their
effects
on
system
stability.
When
the
prey
have
low
costs
to
evolution,
meaning
γ is
small
in
comparison
to
Θ ,
the
prey
is
able
to
avoid
predation
pressure
due
to
a
low
attack
rate
of
the
predator,
and
the
system
equilibrates
with
a
large
prey
population
and
a
small
predator
population.
These
results
largely
support
previous
findings
of
predator‐prey
models,
as
the
system
becomes
unstable
as
the
predator’s
rate
of
evolution
increases
relative
to
that
of
the
prey,
meaning
that
the
predator
increases
its
ability
to
regulate
the
population
of
the
prey
via
predation.
When
the
costs
of
evolution
are
similar
in
both
species,
the
prey
evolve
rapidly
when
under
strong
pressure
from
the
predator.
However,
in
order
to
induce
cycles
in
trait
values,
the
predator
must
be
able
to
evolve
rapidly
enough
to
consistently
impose
high
predation
costs
on
the
prey.
Thus,
the
prey
is
able
to
temporarily
evade
the
predator,
but
the
predator’s
cost
to
evolution
is
low
enough
that
the
predator
can
successfully
track
the
prey’s
trait
value.
This
provides
a
plausible
explanation
for
why
intermediate
and
similar
rates
of
evolution
produce
oscillations
of
the
largest
magnitude
in
predator‐prey
systems
(Mougi
and
Iwasa
2010).
In
the
unidirectional
system,
when
the
predator
is
able
to
efficiently
exploit
the
prey
due
to
a
low
cost
of
evolution
of
its
phenotype
(i.e.
Θ is
small),
there
is
a
stable
equilibrium
for
the
traits
and
abundances
of
both
species.
This
result
is
unexpected,
as
the
predator
in
this
case
is
able
to
strongly
limit
the
prey’s
abundance.
In
this
scenario,
the
prey
is
36
regulated
at
such
a
low
level
that
its
per
capita
birth
rate
is
nearly
at
its
maximum,
c.
With
a
high
birth
rate,
a
small
population
of
prey
can
sustain
a
large
population
of
predators.
This
is
often
the
case
for
the
stable
equilibrium
that
occurs
in
the
unidirectional
model
when
Θ is
smaller
than
γ .
Though
this
region
of
stability
was
only
found
in
the
unidirectional
model,
it
is
possible
that
this
pattern
emerges
in
the
bidirectional
model
under
some
conditions.
The
bidirectional
model
under
these
same
conditions
( Θ is
much
smaller
than
γ )
demonstrated
persisting
oscillations,
but
the
number
of
parameters
in
the
model
prevents
a
complete
investigation
of
all
possible
parameter
space.
Though
the
two
models
behave
similarly
in
response
to
many
parameters,
the
locations
of
the
trait
equilibria
are
strongly
dependent
on
the
specific
attack
rate
used
in
the
model.
The
unidirectional
system
is
much
more
likely
to
produce
patterns
similar
to
the
“arms‐race”
description
of
coevolution.
This
framework
of
evolutionary
models
describes
coevolution
as
a
race
between
the
predator
and
the
prey,
with
each
perpetually
evolving
to
increase
its
phenotype
(Rosenzweig
1973,
Van
Valen
1973,
Slobodkin
1974,
Dawkins
&
Krebs
1979).
The
unidirectional
model
provides
a
single
option
for
the
predator
to
increase
its
attack
rate
and
for
the
prey
to
decrease
pressure
from
predation;
in
both
species
this
strategy
is
to
increase
the
trait
value.
This
unidirectional
pressure
resulting
from
the
shape
of
the
attack
rate
curve
selects
for
higher
trait
values
in
both
species,
which
is
in
accordance
with
the
arms
race
analogy.
However,
in
the
bidirectional
model,
the
prey
have
two
options
for
avoiding
predation:
they
can
increase
or
decrease
their
trait
value,
so
long
as
it
evolves
away
from
the
mean
trait
value
of
the
predator.
Similarly,
the
predator
can
attain
an
identical
attack
rate
if
its
phenotypic
value
is
the
same
amount
above
or
below
the
prey’s
phenotypic
value.
37
The
bidirectional
attack
rate
curve
therefore
contributes
to
a
coevolutionary
model
where
arms
races
are
less
likely
to
occur.
Consequently,
the
bidirectional
model
is
more
prone
to
oscillatory
behavior
than
is
the
unidirectional
model
because
both
the
predator
and
the
prey
have
two
evolutionary
strategies
that
result
in
an
identical
attack
rate.
Trait
oscillations
are
the
result
of
a
change
in
the
evolutionary
strategy
of
the
prey.
The
prey
can
maximize
their
growth
rate
in
one
of
two
ways:
they
can
effectively
avoid
predation
by
phenotypic
evolution
or
they
can
optimize
their
birth
rate
by
incurring
low
costs
of
evolution.
The
prey
will
follow
the
trajectory
of
their
current
strategy
until
the
alternate
strategy
allows
for
higher
instantaneous
fitness.
In
terms
of
the
fitness
surface
of
the
prey,
the
prey’s
trait
value
will
move
toward
the
closer
fitness
peak
until
the
slope
of
the
local
selection
gradient
changes
sign.
For
example,
the
prey
will
evolve
an
increasing
trait
value
until
the
point
where
their
birth
rate
is
so
negatively
affected
by
phenotypic
evolution
that
the
tradeoff
between
evasion
and
birth
rate
selects
for
a
higher
birth
rate.
In
the
bidirectional
model,
the
prey
can
evade
predation
by
having
a
larger
or
smaller
phenotype
than
the
predator.
Thus,
switching
between
strategies
will
occur
frequently
because
the
prey
have
low
costs
of
predation
when
following
either
strategy.
The
selective
pressure
from
a
decreased
birth
rate
is
easily
offset
by
the
dual
benefits
of
optimizing
birth
rate
and
avoiding
predation
when
switching
to
the
alternate
strategy.
This
is
not
true
in
the
unidirectional
model
because
switching
to
a
strategy
that
optimizes
birth
rate
in
the
prey
incurs
high
costs
of
predation,
and
so
switching
occurs
less
frequently.
However,
it
is
impossible
to
directly
compare
the
tendency
toward
oscillation
between
the
two
models
because
the
attack
rate
functions
utilize
different
shape
parameters.
38
Comparison
to
Previous
Coveolutionary
Models
These
models
largely
support
the
findings
of
previous
coevolutionary
models.
Stabilizing
selection
is
a
necessary
condition
for
oscillations
in
both
models,
but
strong
stabilizing
selection
contributes
to
stable
system
dynamics.
This
is
because
oscillations
require
that
two
evolutionary
strategies
exist,
and
that
the
prey’s
better
strategy
changes
depending
on
the
trait
value
of
the
predator.
The
presence
of
stabilizing
selection
creates
a
tradeoff
between
birth
rate
and
predator
evasion
in
the
prey,
which
generates
the
two
evolutionary
strategies.
However,
when
the
strength
of
this
selection
is
strong,
the
prey
species
is
confined
to
follow
a
strategy
of
low
phenotypic
evolution,
which
prevents
oscillations
from
occurring.
Additionally,
when
the
parameters
governing
the
evolutionary
abilities
of
the
two
species
( γ ,
Θ ,
and
the
additive
genetic
variances
of
the
species)
are
comparable
in
the
predator
and
the
prey,
oscillations
are
most
likely
to
occur
and
the
amplitudes
of
the
oscillations
are
large.
The
results
of
this
coevolutionary
model
are
different
from
the
outcomes
of
similar
coevolutionary
models
under
some
circumstances.
Firstly,
previous
analyses
have
found
that
the
maximum
growth
rate
of
the
prey
species
has
no
effect
on
the
stability
of
a
bidirectional
system
(Mougi
2012).
However,
this
model
demonstrated
multiple
instances
where
increasing
the
value
of
c
altered
the
behavior
of
the
system
from
reaching
a
stable
equilibrium
to
exhibiting
persisting
oscillations.
In
fact,
every
parameter
could
potentially
induce
oscillations
in
the
system.
Additionally,
a
stability
analysis
performed
in
a
previous
study
of
a
bidirectional
system
suggested
that
predator
evolution
is
stabilizing
because
models
where
only
the
predator
evolves
are
always
stable
(Mougi
2012).
In
the
model
presented
here,
when
the
39
predator
and
the
prey
are
both
capable
of
evolution,
decreasing
the
predator’s
cost
of
evolution
can
induce
oscillatory
dynamics.
In
this
bidirectional
model,
decreasing
values
of
Θ causes
oscillations
in
the
trait
values
and
abundances
of
both
species.
Reducing
the
predator’s
cost
to
phenotype
evolution
allows
the
predator
to
more
closely
track
the
prey’s
evolving
phenotype,
which
results
in
a
higher
attack
rate.
The
increased
pressure
from
predation,
even
at
high
trait
values
of
the
prey,
causes
the
prey
to
switch
evolutionary
strategies
more
frequently,
resulting
in
oscillations
in
the
trait
values
and
abundances
of
both
species.
One
notable
difference
in
the
model
presented
here
and
the
model
used
in
Mougi,
2012,
where
the
analysis
was
performed,
is
the
death
rate
function
of
the
predator.
This
model
utilizes
a
quadratic
function
to
describe
the
increase
in
the
predator’s
death
rate
due
to
phenotypic
evolution,
whereas
the
previous
model
uses
a
function
from
the
Gaussian
family.
The
difference
in
the
shape
of
the
curves
may
be
responsible
for
some
of
the
discrepancies
in
the
outcomes
of
the
two
bidirectional
models.
Applications
to
Empirical
Systems
These
models
predict
that
predator‐prey
systems
where
the
predator
attains
its
maximal
attack
rate
by
matching
the
phenotype
of
the
prey
should
be
more
prone
to
oscillations
in
abundance
and
phenotypic
traits
than
systems
where
the
predator
must
increase
its
trait
value
over
that
of
the
prey.
Traits
of
the
prey
that
generate
the
corresponding
bidirectional
attack
rate
function
include
body
size,
coloration,
and
chemical
signals.
Previous
studies
of
predator‐prey
systems
provide
examples
of
oscillatory
behavior
in
systems
mediated
by
these
kinds
of
traits.
Phenotypic
cycling
has
been
demonstrated
in
plant‐herbivore
interactions
where
the
susceptibility
of
a
plant
to
predation
by
an
insect
depends
upon
the
insect’s
ability
to
metabolize
the
suite
of
40
compounds
produced
by
the
plant
(Berenbaum
and
Zangrel
1998).
This
study,
and
others,
have
demonstrated
that
rapid
evolution
occurs
on
time
scales
relevant
to
ecological
interactions,
and
can
strongly
impact
the
abundances
of
predator
and
prey
species
(Hairston
et
al.
2005,
Strauss
et
al.
2008,
terHorst
et
al.
2010).
However,
because
many
evolutionary
changes
occur
on
a
timescale
too
long
for
any
individual
to
directly
observe,
the
fossil
record
may
serve
as
a
way
to
test
the
predictions
of
these
models.
One
specific
hypothesis
is
that
species
known
to
suffer
from
heavy
predation
may
experience
more
frequent
changes
in
body
size
than
more
resistant
prey.
The
likelihood
of
oscillation
in
a
unidirectional
system
is
strongly
dependent
on
the
steepness
of
the
attack
rate
curve
as
a
function
of
the
relative
phenotypes
of
the
predator
and
the
prey.
When
the
curve
is
steep,
oscillations
are
more
likely
to
occur.
An
empirical
example
of
a
unidirectional
system
where
the
attack
rate
responds
strongly
to
changes
in
the
relative
phenotypes
of
the
two
species
is
the
algae‐rotifer
system
comprised
of
Chlorella
vulgaris
and
Brachionus
calyciflorus.
Some
phenotypes
of
the
algae
produce
a
compound
that
is
highly
toxic
to
the
rotifer,
which
causes
a
decrease
in
pressure
from
predation
as
the
rotifers
are
unable
to
consume
the
toxic
resource
(Yoshida
et
al.
2003).
However,
when
the
predator
population
is
low
and
there
is
little
pressure
from
predation,
the
non‐toxic
phenotype
is
more
competitively
successful
because
its
growth
rate
is
faster
than
that
of
the
toxic
phenotype
(Yoshida
et
al.
2003).
Unidirectional
systems
where
the
attack
curve
is
less
steep
should
equilibriate
with
the
predator
having
a
more
extreme
phenotype
than
the
prey.
In
addition
to
toxicity,
traits
of
the
prey
that
might
generate
this
attack
rate
curve
include
speed
and
strength.
In
systems
where
the
attack
rate
is
mediated
by
these
traits,
the
phenotype
of
the
prey
over
41
time
is
indicative
of
the
amount
of
predation
inflicted
upon
the
prey
species.
Under
circumstances
where
there
is
low
predation,
the
prey’s
trait
value
is
relatively
close
to
its
trait
value
of
minimum
cost,
but
increases
as
the
pressure
from
the
predator
increases.
The
prey’s
trait
value
serves
as
an
indicator
of
predation
pressure
in
the
unidirectional
system
but
not
in
the
bidirectional
system
because
the
prey
can
only
avoid
predation
in
this
system
only
by
increasing
their
phenotype.
The
unidirectional
system
equilibriates
with
Δ > 0 ,
which
is
a
region
of
the
graph
where
the
attack
rate
is
saturating
as
a
function
of
Δ .
Thus,
unlike
the
bidirectional
system,
the
degree
of
phenotypic
evolution
at
equilibrium
reflects
the
pressure
of
evolution
due
to
the
attack
rate
of
the
predator.
This
pattern
of
net
directional
selection
due
to
predation
in
unidirectional
systems
has
been
demonstrated
using
the
burst
speed
of
tadpoles
as
the
prey’s
phenotypic
trait
of
interest.
Tadpoles
with
high
burst
speeds
evade
predation
much
more
than
tadpoles
with
an
average
burst
speed,
indicating
strong
selection
for
an
increasing
phenotype
(Watkins
1995).
At
high
levels
of
predation,
when
this
directional
selection
is
strong,
the
phenotype
of
the
prey
should
increase
correspondingly.
These
models
also
indicate
that
predator‐prey
systems
are
stable
when
the
prey
species
has
a
much
faster
rate
of
evolution
than
the
predator
species.
This
can
occur
if
the
prey’s
generation
time
is
much
shorter
than
that
of
the
predator
or
if
the
prey
have
a
lower
cost
than
the
predator
of
evolving
its
phenotype.
In
the
case
of
mosquito
larvae
preying
upon
protozoa,
the
generation
time
of
the
predator
is
more
than
two
orders
of
magnitude
longer
than
that
of
the
prey
(terHorst
et
al.
2010).
In
accordance
with
the
predictions
of
these
models,
the
system
shows
convergence
upon
a
stable
equilibrium
in
the
predation
experiments
carried
out
in
terHorst
et
al.
2010,
and
the
negative
impact
of
the
predator
on
42
the
prey
population
diminishes
strongly
over
the
course
of
the
experiments.
These
models
suggest
this
stability
is
due
to
rapid
evolution
in
the
prey
leading
to
a
decreased
attack
rate
of
the
predator
and
the
lack
of
corresponding
rapid
evolution
in
the
predator.
Both
the
unidirectional
and
bidirectional
models
predict
that
systems
exhibiting
damped
oscillations
will
equilibriate
with
the
prey
at
a
stable
fitness
minimum,
providing
the
assumptions
of
the
model
are
met.
However,
some
of
these
assumptions,
such
as
random
mating
in
the
prey
and
a
lack
of
stochasticity,
are
unlikely
to
be
upheld
in
empirical
systems.
When
these
conditions
are
relaxed,
these
models
provide
a
starting
point
for
conceptualizing
the
evolution
of
polymorphisms
in
prey
species.
The
fitness
surface
of
the
prey
at
equilibrium
after
a
period
of
damped
oscillations
is
analogous
to
the
fitness
surface
of
a
population
under
disruptive
selection.
In
both
cases,
the
individuals
with
the
mean
phenotype
of
the
population
have
the
lowest
fitness
(Rueffler
et
al.
2006).
Previous
studies
have
demonstrated
that
conditions
of
disruptive
selection
favor
assortative
mating
between
the
prey,
as
fitness
increases
as
the
phenotype
of
the
individuals
deviates
further
from
the
mean
phenotype
(Benkman
1999,
Kopp
and
Hermisson
2006).
Thus,
future
coevolutionary
models
may
find
it
of
interest
to
allow
for
a
degree
of
non‐random
mating
in
the
prey
species
as
to
observe
the
trajectories
of
the
system
under
the
conditions
where
these
models
show
damped
oscillations.
Additionally,
both
empirical
(Rundle
et
al
2003,
Nosil
and
Crespi
2006)
and
theoretical
studies
(Meyer
and
Kassen
2007,
Reed
and
Janzen
1998)
have
indicated
that
a
predator
can
impose
disruptive
selection
upon
a
prey
species,
and
that
this
selection
can
lead
to
polymorphisms
in
the
prey
species.
However,
it
is
unclear
how
the
frequency‐dependent
nature
of
the
fitness
surface
of
the
prey
would
affect
43
the
evolution
of
prey
populations
with
diverging
phenotypes,
even
after
accounting
for
stochasticity
and
non‐random
mating.
Limitations
of
the
Models
Although
models
can
be
useful
for
offering
general
insights
into
complex
biological
systems,
there
are
limitations
on
the
generalizability
of
such
models,
including
those
presented
here.
These
models
pertain
to
a
predator‐prey
system
where
the
predator
is
a
specialist
and
the
prey
suffers
predation
from
only
one
predator.
Specific
assumptions
of
this
model
include
treating
the
predator
and
prey
traits
as
continuous
variables,
a
lack
of
inducible
defenses
and
phenotypic
plasticity,
constraining
the
additive
genetic
variances
to
be
constant
throughout
the
model,
and
using
the
mean
trait
value
of
the
predator
and
prey
species
to
represent
the
phenotypes
of
each
population.
Additionally,
this
model
assumes
that
the
phenotypes
of
the
predator
and
the
prey
are
uncorrelated
with
the
fitnesses
of
the
species,
except
for
imposing
a
higher
mortality
rate
in
the
predator
and
a
lower
birth
rate
in
the
prey.
This
assumption
is
invalid
in
cases
where
the
phenotype
otherwise
affects
birth
rate
in
the
prey,
such
as
if
larger
body
size
increases
fecundity.
Lastly,
these
models
imply
spatial
homogeneity
and
a
lack
of
refuges
for
the
prey,
as
neither
of
these
aspects
is
explicitly
addressed
in
the
equations.
Despite
these
limitations,
there
are
many
emergent
patterns
from
the
two
models
that
hold
up
under
a
broad
range
of
parameter
space.
It
is
clear
from
the
exploration
of
these
coevolutionary
models
that
allowing
for
evolution
of
the
predator
and
the
prey
fundamentally
alters
some
of
the
behaviors
of
predator‐prey
systems.
Additionally,
the
shape
of
the
attack
rate
function,
based
on
the
phenotypes
of
the
predator
and
the
prey,
contributes
to
different
community
and
evolutionary
dynamics
in
predator‐prey
systems.
44
Thus,
the
rate
of
evolution
of
the
two
species
and
the
characteristics
of
the
traits
that
mediate
predator‐prey
interactions
can
have
profound
influences
on
the
location
and
stability
of
equilibria
in
these
systems.
45
LITERATURE
CITED:
Abrams, P. A. 1986. Is Predator-Prey Coevolution an Arms Race? Trends in Ecology and
Evolution 1: 108-110.
Abrams, P. A. 1990. The Evolution of Anti-Predator Traits in Prey in Response to Evolutionary
Change in Predators. Oikos 59: 147-156.
Abrams, P. A. 1995. Implications of dynamically variable traits for identifying, classifying and
measuring direct and indirect effects in ecological communities. American Naturalist
146:112–134.
Abrams,
P.A.
2000.
The
Evolution
of
Predator‐Prey
Interactions:
Theory
and
Evidence.
Annual
Review
of
Ecology
and
Systematics
31:
79‐105.
Abrams,
P.
A.,
H.
Matsuda,
and
Y.
Harada.
1993.
Evolutionarily
Unstable
Fitness
Maxima
and
Stable
Fitness
Minima
of
Continuous
Traits.
Evolutionary
Ecology
7:
465‐87.
Agrawal,
A.
A.,
2001.
Phenotypic
Plasticity
in
the
Interactions
and
Evolution
of
Species.
Science
294:
321‐326.
Arnold,
S.
J.
1992.
Constraints
on
Phenotypic
Evolution.
The
American
Naturalist
140:
85‐
107.
Benkman,
C.
W.
1999.
The
Selection
Mosaic
and
Diversifying
Coevolution
between
Crossbills
and
Lodgepole
Pine.
The
American
Naturalist
153:
75‐91.
Benkman,
C.
W.,
W.
C.
Holimon,
and
J.W.
Smith.
2001.
The
Influence
Of
A
Competitor
On
The
Geographic
Mosaic
Of
Coevolution
Between
Crossbills
And
Lodgepole
Pine.
Evolution
55:
282‐294.
Berenbaum,
M.
R.
and
A.
R.
Zengerl.
1998.
Chemical phenotype matching between a plant and
its insect herbivore. Proceedings of the National Academy of Sciences 95: 13743-13748.
Berryman, A. A. 1992. The origins and evolution of predator–prey theory. Ecology 73:1530–
46
1535.
Bolker, B., M. Holyoak, V. Krivan, L. Rowe, and O. Schmitz. 2003. Connecting theoretical and
empirical studies of trait mediated interactions. Ecology 84:1101–1114.
Brodie, E. D., III, and E. D. Brodie Jr. 1991. Evolutionary Response of Predators to Dangerous
Prey: Reduction of Toxicity of Newts and Resistance of Garter Snakes in Island
Populations. Evolution. 45:221-224
Brodie, E. D., III, and E. D. Brodie Jr. 1999. Costs of exploiting poisonous prey: evolutionary
trade-offs in a predator-prey arms race. Evolution 53:626–631.
Dawkins, R. and J. R. Krebs. 1979. Arms races between and within Species. Proceedings of the
Royal Society B: Biological Sciences. 205: 489-511.
De
Mazancourt,
C.,
and
U.
Dieckmann.
2004.
Trade‐Off
Geometries
and
Frequency‐Dependent
Selection.
The
American
Naturalist
164:
765‐78.
Dieckmann,
U.,
and
R.
Law.
1996.
The
Dynamical
Theory
of
Coevolution:
A
Derivation
from
Stochastic
Ecological
Processes.
Journal
of
Mathematical
Biology
34.5:
579‐612.
Dieckmann,
U.,
P.
Marrow,
and
R.
Law.
1995.
Evolutionary
Cycling
in
Predator‐prey
Interactions:
Population
Dynamics
and
the
Red
Queen.
Journal
of
Theoretical
Biology
176:
91‐102.
Fussman,
G.
F.,
S.
P.
Ellner,
K.
W.
Shertzer,
N.
G.
Hairston
Jr.
2000.
Crossing
the
Hopf
Bifurcation
in
a
Live
Predator‐Prey
System.
Science
290:
1358‐1360.
Garland
Jr.,
T.
1983.
The
relation
between
maximal
running
speed
and
body
mass
in
terrestrial
mammals.
Journal
of
Zoology.
199:157‐170.
Gavrilets,
S.
1997.
Coevolutionary
Chase
in
Exploiter–Victim
Systems
with
Polygenic
Characters.
Journal
of
Theoretical
Biology
186:
527‐34.
Ginzburg,
L.
R.
1986.
The
Theory
of
Population
Dynamics:
I.
Back
to
First
Principles.
Journal
47
of
Theoretical
Biology.
122:
385‐399.
Gotelli,
Nicholas
J.
A
Primer
of
Ecology.
Sunderland,
MA:
Sinauer
Associates,
2001.
Hairston Jr., N. G., Ellner, S. P., Geber, M. A., Yoshida, T. & Fox, J. A. 2005. Rapid evolution
and the convergence of ecological and evolutionary time. Ecology Letters 8: 1114–1127.
Hairston
Jr.,
N.G.
Jr,
Lampert,
W.,
Ca´ceres,
C.E.,
Holtmeier,
C.L.,
Weider,
L.J.,
Gaedke,
U.
et
al.
1999.
Rapid
evolution
revealed
by
dormant
eggs.
Nature
401:
446.
Hanifin,
C.
T.,
M.
Yotsu‐Yamashita,
T.
Yasumoto,
E.
D.
Brodie
III,
E.
D.
Brodie
Jr.
1999.
Toxicity
of
Dangerous
Prey:
Variation
of
Tetrodotoxin
Levels
Within
and
Among
Populations
of
the
Newt
Taricha
granulosa.
Journal
of
Chemical
Ecology.
25:2161‐
2175.
Holling, C. S. 1959. The components of predation as revealed by a study of small mammal
predation of the European pine sawfly. Canadian Entomologist 91: 293-320
Holt, Robert D. “Natural Enemy-Victim Interactions: Do We Have A Unified Theory Yet?" The
Theory of Ecology. Ed. Samuel M. Scheiner and Michael R. Willig. Chicago: University
of Chicago, 2011. 125-63.
Huston,
M.
1979.
A
General
Hypothesis
of
Species
Diversity.
The
American
Naturalist
113:
81‐101.
Ioannou,
C.
C.,
and
J.
Krause.
2009.
Interactions
between
Background
Matching
and
Motion
during
Visual
Detection
Can
Explain
Why
Cryptic
Animals
Keep
Still.
Biology
Letters
5:
191‐93.
Kopp,
M.
and
J.
Mermisson.
2006.
The
evolution
of
genetic
architecture
under
frequency‐
dependent
disruptive
selection.
Evolution
60:
1537‐1550.
Křivan,
V.,
and
O.
J.
Schmitz.
2004.
Trait
and
Density
Mediated
Indirect
Interactions
in
48
Simple
Food
Webs.
Oikos
107:
239‐50.
Lande,
R.
1982.
A
quantitative
genetic
theory
of
life
history
evolution.
Ecology
63:607‐15
Maynard
Smith,
J.
and
Slatkin,
M.
1973.
The
Stability
of
Predator‐Prey
Systems.
Ecology.
54:
384‐391.
McCollum, S. A., and J. D. Leimberger. 1997. Predator-induced morphological changes in an
amphibian: predation by dragonflies affects tadpole shape and color. Oecologia 109:615–
621.
McCollum, S. A., and J. VanBuskirk. 1996. Costs and benefits of a predator-induced
polyphenism in the gray treefrog Hyla chrysoscelis. Evolution 50:583–593.
McPeek, M. A., A. K. Schrot, J. M. Brown. 1996. Adaptation to Predators in a New Community:
Swimming Performance and Predator Avoidance in Damselflies. Ecology 77: 617-629.
Merilaita, S., and J. Lind. 2005. Background-matching and Disruptive Coloration, and the
Evolution of Cryptic Coloration. Proceedings of the Royal Society B: Biological Sciences
272: 665-70.
Meyer, J. R. and R. Kassen. 2007. The effects of competition and predation on diversification in
a model adaptive radiation. Nature 446: 432-435.
Moitoza
D.
J.
and
D.
W.
Phillips.
1979.
Preference,
and
Nonrandom
Diet:
The
Interactions
Between
Pycnopodia
helianthoides
and
Two
Species
of
Sea
Urchins.
Marine
Biology
53:
299‐304.
Morran,
L.
T.,
O.
G.
Schmidt,
I.
A.
Gelarden,
R.
C.
Parrish,
and
C.
M.
Lively.
2011.
Running
with
the
Red
Queen:
Host‐Parasite
Coevolution
Selects
for
Biparental
Sex.
Science
333:
216‐218
Mougi,
A.
2012
Predator‐prey
coevolution
driven
by
size
selective
predation
can
cause
anti‐
synchronized
and
cryptic
population
dynamics.
Journal
of
Theoretical
Biology
81:
49
113‐118.
Mougi, A., and Y. Iwasa. 2010. Evolution towards oscillation or stability in a predator–prey
system. Proceedings of the Royal Society B. 277:3163–3171.
Mougi,
A.,
and
Y
Iwasa.
2011.
Unique
Coevolutionary
Dynamics
in
a
Predator–prey
System.
Journal
of
Theoretical
Biology
277:
83‐89.
Murdoch
W.
W.
1973.
The
Functional
Response
of
Predators.
Journal
of
Applied
Ecology.
10:335–342.
Nosil,
P.
and
B.
J.
Crespi.
2006.
Experimental
evidence
that
predation
promotes
divergence
in
adaptive
radiation.
Proceedings
of
the
National
Academy
of
Sciences
24:
9090‐
9095.
Nuismer,
S.
L.,
M.
Doebeli,
and
D.
Browning.
2005.
The
Coevolutionary
Dynamics
Of
Antagonistic
Interactions
Mediated
By
Quantitative
Traits
With
Evolving
Variances.
Evolution
59:
2073‐2082.
Nuismer,
S.L.,
B.J.
Ridenhour,
and
B.
P.
Oswald.
2007.
Antagonistic
Coevolution
Mediated
By
Phenotypic
Differences
Between
Quantitative
Traits.
Evolution
61.8:
1823‐834.
Pratt
D.
M.1974.
Behavioral Defenses of Crepidulafornicata against Attack by
Urosalpinx cinema. Marine Biology 27:47-49.
Preisser, E. L., D. L. Bolnick, and M. F. Benard. 2005. Scared to death? The effects of
intimidation and consumption in predator–prey interactions. Ecology 86:501–509.
Ramos-Jiliberto, R. 2003. Population dynamics of prey exhibiting inducible defenses: the role of
associated costs and density-dependence. Theoretical Population Biology 64: 221–231
Reed, W. L. and F. J. Janzen. 1999. Natural selection by avian predators on size and colour of a
freshwater snail (Ponacae flagellata). Biological Journal of the Linnean Society 67: 331-
50
342.
Reznick,
D.
A.,
H.
Bryga,
and
J.
A.
Endler.
1990.
Experimentally
Induced
Life‐history
Evolution
in
a
Natural
Population.
Nature
346:
357‐59.
Rosenzweig,
M.
L.
1971.
Paradox
of
Enrichment:
Destabilization
of
Exploitative
Ecosystems
in
Ecological
Time.
Science
171:
385‐387.
Rosenzweig,
M.
L.
1973.
Evolution
of
the
Predator
Isocline.
Evolution
27:
84‐94.
Roy,
S.,
and
J.
Chattopadhyay.
2007.
The
Stability
of
Ecosystems:
A
Brief
Overview
of
the
Paradox
of
Enrichment.
Journal
of
Biosciences
32.2:
421‐28.
Rueffler,
C.,
Van
Dooren,
T.
J.
M.,
O.
Leimar,
and
P.
A.
Abrams.
2006.
Disruptive
selection
and
then
what?
Trends
in
Ecology
and
Evolution.
21:
238‐245.
Rundle,
H.
D.,
S.
M.
Vamosi,
and
D.
Schluter.
2003.
Experimental
test
of
predator’s
effect
on
divergent
selection
during
character
displacement
in
sticklebacks.
Proceedings
of
the
National
Academy
of
Sciences.
100:
14943–14948
Saloniemi,
I.
1993.
A
Coevolutionary
Predator‐Prey
Model
with
Quantitative
Characters.
The
American
Naturalist
141:
880.
Schaffer,
W.,
and
M.
Rosenzweig.
1978.
Homage
to
the
Red
Queen.
I.
Coevolution
of
Predators
and
Their
Victims.
Theoretical
Population
Biology
14:
135‐57.
Shertzer,
K.W.,
S.
P.
Ellner,
G.
F.
Fussmann,
and
N.
G.
Hairston
Jr.
2002.
Predator‐prey
Cycles
in
an
Aquatic
Microcosm:
Testing
Hypotheses
of
Mechanism.
Journal
of
Animal
Ecology
71:
802‐15
Slobodkin,
L.
B.
1974.
Prudent
Predation
Does
Not
Require
Group
Selection.
The
American
Naturalist
108:
665‐678.
Smith,
T.,
and
M.
Huston.
1989.
A
Theory
of
the
Spatial
and
Temporal
Dynamics
of
Plant
51
Communities.
Plant
Ecology.
83:
49‐69.
Solomon M. E., 1949. The Natural Control of Animal Populations. Journal of Animal Ecology.
18:1-35.
Strauss, S. Y., J. A. Lau, T. W. Schoener, and P. Tiffin. 2008. Evolution in ecological field
experiments: implications for effect size. Ecology Letters 11: 199-207.
terHorst,
C.
P.,
T.
E.
Miller,
and
D.
R.
Levitan.
2010. Evolution of prey in ecological time
reduces the effect size of predators in experimental microcosms. Ecology 91: 629-636.
Tilman
D.
1990.
Constraints
and
trade‐offs:
toward
a
predictive
theory
of
com‐petition
and
succession.
Oikos
58:3‐15
Turchin,
P.
2001.
Does
population
ecology
have
general
laws?
Oikos
94:17‐26.
Van
Buskirk
J,
Schmidt
B.
R.
2000.
Predator‐induced
Predator‐induced
phenotypic
plasticity
in
larval
newts:
trade‐offs,
selection,
and
variation
in
nature.
Ecology
81:3009‐28
Van Buskirk, J., and S. A. McCollum. 1999. Plasticity and selection explain variation in tadpole
phenotype between ponds with different predator composition. Oikos 85:31– 39.
Van Valen, L. 1973. A new evolutionary law. Evolutionary Theory 1: 1-30.
Watanabe, J. M. 1983. Anti-predator defenses of three kelp forest gastropods: contrasting
adaptations of closely related prey species. Journal of Experimental Marine Biology and
Ecology 71:257-270.
Watkins, T. B. 1995. Predator-Mediated Selection on Burst Swimming Performance in Tadpoles
of the Pacific Tree Frog, Pseudacris regilla. Physiological Zoology 69:154-167.
Werner, E., and S. Peacor. 2003. A review of trait-mediated indirect interactions in ecological
communities. Ecology 84:1083–1100.
West
K,
Cohen
A,
Baron
M.
1991.
Morphology
and
behavior
of
crabs
and
gastropods
from
52
Lake
Tanganyika,
Africa:
implications
for
lacustrine
predator‐prey
coevolution.
Evolution
45:
589‐607
Yoshida,
T.,
S.
P.
Ellner,
L.
E.
Jones,
B.
J.
M.
Bohannan,
R.
E.
Lenski,
and
N.
G.
Hairston.
2007.
Cryptic
Population
Dynamics:
Rapid
Evolution
Masks
Trophic
Interactions.
PLoS
Biology
5:
E235.
Yoshida, T., L.E. Jones, S. P. Ellner, G. F. Fussman, & N. G. Hairston Jr., 2003. Rapid evolution
drives ecological dynamics in a predator-prey system. Nature 424: 303–306.
53