reviews
reviews
-1
TREE
TREE vol. 8,
8, no.
no. 6,
6, June
June 1993
1993
Life
colLife history
history theory
theory and
and behavioral
Gehavioraleecol-
ogy,
ogy, two
two branchesin
branches in the
the study
study Ofadaptaof adupta-
Dynamic
DynamicModelsof
Models of Behavior:
Behavior:An
An
Extension
Extensionof
of LifeHistoryTheory
life Hook Theory
tion,
tion, have
have relied
relied extensivelyon
extensively on mathmathematical models, but have tended to
ematical models, 6ut have tended to
employ different types Ofmodels,and
employ different types of models, and different currencies
currencies Of
of fitness.
fitness. Recently,
Recently, aa
new
approachbasedon
dynamic,
In the 1970s it was observed
that
new approach based on dynamic, statestateIn the 1970s it was observed that
variablemous
variable modelshasbeenincreasinglyaphas 6een increasingly ap this life history optimization prinprinplied
dapta- this life history optimization
plied to
to the
the study
study ot
of behavioral
behavioral aadaptaciple could be understood in terms
tions_In fact, this approachamounts to a ciple could be understood in terms
tions. In fact, this approach amounts to a of dynamic programming, as forof dynamic programming, as forunificationof
unification of life
life history
history tkeorg
theory and
and be6e- mulated
mulated by
by ællmana.
Bellman4. Dynamic
Dynamic
kavioral
ecology,to
the
extent
that
the
haviorul ecology, to the extent that the programming
is
programming
is an
an algorithmic
algorithmic
line
separatingthe
two
fields
is
virtually
line separating the two fields is virtually method that allows one to determethod that allows one to deterobliterated
o6literated. Dynamic models (usually
(usually mine the optimal control strategy
mine the optimal control strategy
solvedby
computer)
tan
yield
both
Enersolvedby computer) can yield 60th gener- for
for aa dynamical
dynamical system.
system. Dynamical
Dynamical
al
al principles
principles and
and testable,quantitative
testa6le, quantitative or
or systems
are characterized by their
systems
are
characterized
by their
qualitative
to
qualitative predictions
predictions about
about specificbespecific be- states. I will use the
Colin W. Clark
Cohn W. Clark
studies, one example will
studies; one example will be dedescribed in some detail, Finally I will
scribed in some detail. Finally I will
outline
outline the
the main
main advantages
advantages of
of the
the
method,
and
discuss
its
principal
method, and discuss its principal
limitations.
limitations.
under
Foraging under risk of predation
years ago
ago itit was
was
that foraging and
that foraging and
antipredator behavior could
could not
not be
be
encompassed
by
a
single
model,
states. I will use the symbol X(t) to encompassed by a single model,
havioraland
life
history
phenomena.
fitness
havioral and rife history phenomena.
denote a system's state X at time t. because the
denote a system’s state Xat time t. because the corresponding fitness
currencies
were incommensurable.
The state variable X may have
The state variable X may have currencies were incommensurable.
Life
Life history
history theory
theory analyses
analyses the
the one
one or
or many
many components. In
In the
the Indeed,
Indeed, the
the currency
currency problem
problem was
was
sometimes
listed as a maior limibiological
setting,
state
variables
trade-offs that organisms face in
sometimes listed as a major limitrade-offs that organisms face in biological setting, state variables
making
of individual
individual organisms
organisms may
may inin- tation
tation ofof optimization
optimization theory
theory inin
making decisions
decisions that
that affect
affect their
their of
evolutionary
biol*y'_
I
clude En)dy mass, somatic energy
fecundity
and
mortality
schedules,
evolutionary
biology7. I Will
will use
use
fecundity and mortality schedules.
clude body mass, somatic energy
two very simple rncxiels to show
Most such trade-offs
can be classireserves,
other
reMost such trade-offs can be classi- reserves, other biochemical
re- two very simple models to show
how easily the currency problem
fied as
how easily the currency problem
fied as being concerned
concerned either
either with
with serves,
serves, territory
territory size,
size, number
number of
of can
resolved8. The
illusoffspring
being
cared
for,
and
other
current
versus
future
reprcKiuction,
offspring being cared for, and other
can be resolved6. The models illuscurrent versus future reproduction,
trate
dynamic
mcxieling
and
the
biologically
relevant
factors;.
or with size versus
of offor with size versus number of off- biologically
relevant
factors5. trate dynamic modeling and the
use of state variables.
These state variables change over
spring'.
But
many
important
spring’. But many important be- These state variables change over use of state variables.
First, consider
havioral
First, consider aa continuously
continuously
havioral decisions
decisions involved
involved with
with time.
time, in
in ways
ways that
that are
are affected
affected by
by
reprcxiucing
foraging,
foraging, predator
predator avoidance,
avoidance, territerri- the
the organism's
organism’s Ekhavior.
behavior. Behavior
Behavior
reproducing animal
animal that
that forages
forages
for
a
certain
of
also
affects
the
organism's
reprotorial
for a certain period of time
time and
and
torial defense,
defense, migration,
migration, social
social bebe- also affects the organism’s reprohavior,
then lays
lays aa clutch
clutch ofof eggs
eggs deterdeterductive output,
output, i.e.
i.e. its
its fitness.
fitness. then
havior, etc.,
etc., do
do not
not readily
readily fit
fit into
into ductive
mined by her foraging success. The
Finally,
the
Finally, the
the state
state XX may
may be affected
affected
mined by her foraging success. The
the classical
classical life
life history
history framework.
framework.
Yet these decisions also affect surby
extemal
environmental
events
events prcress
process isis repeated
repeated for
for aa total
total of
of TT
Yet these decisions also affect sur- by external environmental
(or
as
long
as
the
animal
not
entirely
under
the
organism's
Vival
and
reproduction,
and
have
periods (or as long as the animal
vival and reproduction, and have not entirely under the organism’s
survives).
control.
therefore
survives). In
In any
any given
given time
time peritxi
period
therefore been subiect
subject to
to the
the same
same control.
the animal can
one of two
Traditional
life
history
theory
selective forces as conventional
Traditional
life history theory
the animal can choose one of two
selective forces as conventional
does not allow for state variables.
foraging
life
does not allow for state variables.
foraging habitats.
habitats, habitat
habitat ii Iring
being
life history
history traits.
traits.
characterized by two parameters:
By
using
state
variables
in
dynamic
The
concept
of
reprcxiuctive
of reproductive
By using state variables in dynamic
characterized by two parameters: qj
The concept
value
decision models,
models, itit is
is possible
possible to
to (average
(average fecundity
fecundity per
per foraging
foraging
value isis central
central to
to life
life history
history thethe- decision
achieve an extension
and unificabout) and
(predation risk per
ory23.
Reproductive
value
is
an
orbout) and pj (predation risk per
ory2m3.Reproductive value is an or- achieve an extension and unification
ganism's
tion of
of both life
life history
history theory
theory and
and foraging
foraging bout).
bout).
ganism’s expected future
future reprtxiucreproduck*havioral
This
Assume
tive
success
(rx»ssibly
discounted,
behavioral ecology. This approach
Assume that
that @,<< & and
and g,
p, << gr
p2,
tive success (possibly discounted,
i.e. habitat
2 is more
is
now
increasingly
used
in
in
the
case
of
a
growing
popuis now being increasingly used in i.e. habitat 2 is more productive
in the case of a growing populationl
the study
study of
of behavioral
behavioral adaptadapt- and
and more
more risky
risky than
than habitat
habitat l.I.
lation) not
not including
including current
current reprorepro- the
Which is the optimal habitat?
ations.
Behavioral
ecologists
and
duction.
An
optimal
life
history
ations. Behavioral ecologists and Which is the optimal habitat?
duction. An optimal life history
To
profile
other evolutionary
evolutionary biologists
biologists nc»w
now
To Ergin
begin with,
with, consider
consider the
the optioptiprofile is
is one
one that
that at
at each
each age
age maxmax- other
imizes
need toto be as
as familiar
familiar with
with the
the mization
mization problem
problem inin the
the final
final
imizes the
the sum
sum ofof current
current reprorepro- need
duction
elements of
of dynamic
dynamic modeling as
as v*riod
period T.
T. Here
Here only
only aa single
single dededuction plus
plus reprcxductive
reproductive value.
value. elements
The
The central
central idea
idea of
of life
life history
history thethe- they
they have
have been
been with
with life
life history
history Cision
cision isis involved.
involved. Expected reproreproductive
ory,
ory, then,
then, isis that
that there
there must
must exist
exist aa theory.
theory. Fortunately,
Fortunately, the
the technique
technique
ductive Output
output from
from habitat
habitat ii isis
trade-off
trade-off in
in the
the sense
sense that
that aa high
high of
of dynamic
dynamic programming
programming isis easy
easy equal
equal to
to (I —g)
- p> "and
qjand the
the optimal
optimal
level
level of
of current
current repnxiuctive
reproductive effort
effort to
to learns;
learn6; graduate
graduate students
students of
of bibi- decision
decision maximizes
maximizes this
this quantity
quantity.
Let
Ology
results
ology With
with limited
limited mathematical
mathematical
Let us
us denote
denote this
this maximum
maximum exexresults in
in aa decrease
decrease in
in reproducreproducpected
in
background
are
now
routinely
tive
value
—
the
•cost
of
reproducbackground
are now routinely
pected reproduction in pericxå
period rby
T by
tive value - the ‘cost of reproduclearning
tion•.
tion’. Optimization
Optimization of
of this
this trade-off,
trade-off,
learning to
to construct
construct and
and analyse
analyse
FIT):
through
through natural
natural selection,
selection, is
is what
what dynamic
dynamic behavioral
behavioral mcxiels.
models.
II will
F(T)=~=~(I-Pj)@j
determines
(1)
determines the
the variety
variety of
of life
life history
history
will first
first explain
explain the
the logic
logic of
of dy
namic
programming,
using
two
strategies
observed
in
nature.
namic programming,
using two
strategies observed in nature.
simple
simple models of
of foraging
foraging under
under Since
Since we
we are
are assuming
assuming that
that the
the nunumerical values of the parameters
risk
of
predation.
II will
then
briefly
Colin
Clark
is
at
Irstitute
of
risk
of
predation.
will
then
briefly
merical
values
of
the
parameters
$,,
Cohn Clark is at the Institute
of Applied
discuss
Mathematics.University
of
discuss recent
recent applications
applications ofof the
the pj are
are given,
given, Eqn
Eqn I1 specifies
specifies the
the
Mathematics,
University
of British
British Colwmbia,
Columbia,
numerical value of FIT). This value
methcxi
to
Vancouver.
method
to SEECific
specific behavioral
numerical value of F(T). This value
Vancouver, K,
BC, CanadaVET
Canada V6T IZ2
122.
0 1993, Elsevier
Science Publishers Ltd
Ltd •;u'Kl
(UK)
Not
Not
many
many
years
widely thought
widely thought
antipredator
205
reviews
reviews
TREEvol.
TREE vol. e8, no.
no. 6,
6, June
June 1993
1993
represents
represents expected
expected reprcxduction
reproduction
at
at the
the start
start of
of period
period T,
T, assuming
assuming
that
the
optimal
habitat
that the optimal habitat is
is choæn.
chosen.
Let
Let the
the optimal
optimal habitat
habitat in
in period
period TT
T).
be denoted
denoted by
by i"i*(T).
Next,
Next, what
what is
is the
the optimal
optimal habitat
habitat
for
for earlier
earlier time
time periods t?t? For
For
t:t = T—
T- II the analog to Eqn 1 isis
FIT-I)
F(T-1)
=ya;(l-~J$j+F(T)l
(2)
Understanding
Understanding Eqn
Eqn 22 is
is the
the crux
crux
of
dynamic
programming.
The
of dynamic programming. The arguargument
ment is
is simplicity
simplicity itself.
itself. Consider
Consider
the
the decision
decision in
in period
period TT —
- l.1. IfIf
habitat
habitat
ability
ability
ii is
is chosen,
chosen, then
then the
the probprob-
of
of surviving
surviving the
the period
period is
is
(l( 1 —g),
- pi), and
and total
total expected
expected reprorepro(conditional
(conditional on
on surviving
surviving
period T—
T- l)1) isis then
then (qi + exsrcted
expected
reproduction inin period T) = & ++ FIF(T).
T).
Expected total
total reproduction
reproduction at
at the
the
start
start of
of period
period T—
T - II is
is therefore
therefore
equal
to
'1
+
equal to (1 - ,u.)[$~ + F(T)]. The
The
Optimal
optimal habitat
habitat ilT—
/*(T - l)1) in
in grricxi
period
T—
T- I1 maximizes
maximizes this
this expression,
expression, as
as
stated
stated in
in Eqn
Eqn 2.
2. The
The value
value F( T - l)1)
represents
represents maximum
maximum expected
expected total
total
reproduction
over
the
reproduction
over the last
last two
two
periods.
Exactly
Exactly the
the same
same argument
argument
shows
shows that,
that, for
for any
any tt << TT
duction
duction
F(t)=m;(Wj)[4j
++F(t+l)l
13)
(3)
reservesat
reserves at the
the Erginning
beginning of
of period
period
t.t. AIM)
let
denote
Also let R[X(T)] denote reproreproof
ductive output
output inin the
the final
final period
of its
its simplicity.
simplicity, the
the model
model can
can be
be ductive
T. We assume that R is an increasexpressed
T. We assume that R is an increasexpressed in
in the
the standard
standard life
life
ing
history
ing function,
function, with
with aa threshold
threshold XXcrit
history notation.
notation. For
For pedagogical
such that
purvx:ses
I
prefer
not
to
do
so
purposes I prefer not to do so such that
here.)
here.)
AA numerical
)
R[X(T)] = 0 ifif X(T) <xe„t
X,,
numerical example
example (see
(see Box
Box I1)
illustrates
the iterative
solution of
illustrates the iterative solution of
Eqns
The optimal
optimal foraging
foraging strategy
strategy isis
Eqns I1 and
and 3.
3. For
For tt near
near the
the terter- The
that
which
maximizes
mina!
time
T.
the
risky
habitat
that which maximizes expected
expected
minal time T, the risky habitat
terminal
2)
is
optimal.
but
for
ts
T—5
reproduction.
(i* = 2) is optimal, but for t I T - 5 terminal reproduction
the
Reproductive value
value now
now Ercomes
becomes
the safer
safer habitat
habitat (i•
(i” == I)I) becomes
becomes
state
dependent:
optimal.
The
prediction
is
that
optimal. The prediction
is that state dependent:
older
older animals
animals will
will accept
accept greater
greater
ftx.n
F(x,O
predation
predation risks
risks than
than younger
younger aniani—max
EIRIX(T//
x)
(4)
=maxE{RIX(T)]/X(t)=x)
(4)
mals.
The
explanation
for
this
lies
mals. The explanation for this lies
Here E denotes mathematical
ex.
in
value
In
subin reproductive value F(t). In sub- Here E denotes mathematical exjecting
jecting itself
itself to
to predation
predation risk,
risk, the
the grctation
pectation (relative
(relative to
to the
the random
random
animal
tx)th
its
current
and
predation
event)
and
animal places both its current and predation event) and the
the prefix
prefix
future
future reprcxduction
reproduction at
at risk.
risk, while
while •max'
‘max’ refers
refers toto the
the use
use ofof the
the optioptigaining
gaining only
only aa m.tential
potential increase
increase inin mal
mal strategy.
strategy. Equation
Equation 44 leads
leads imimcurrent
The
current reproduction.
The larger
larger mediately
mediately toto the
the dynamic
dynamic programprogramming
the
the future
future reproduction (reproduce
(reproducming algorithm•
algorithm:
tive
value).
the
more
important
it
is
tive value), the more important it is
to
to protect
protect this
this 'asset'.
‘asset’. Since
Since reproreproF(x,T)= R(x)
(5)
ductive
ductive value
value isis aa decreasing
decreasing funcfunction
ftx,t)
tion ofof time
time tt Ifewer
(fewer reproductive
episodes
episodes remain
remain as
as tt increases
increases totowards
wards T).
T), older
older animals
animals can
can afford
afford
greater
risks.
greater risks.
Similar
The
The first
first iteration
iteration t= T—
T- 1gives
gives
Similar effects
effects ofof temporal
temporal reducreductions
tions in
in reprcxfuctive
reproductive value
value have
have
F(x,T -1)
been discussed,
discussed, for
for example.
example, reregarding
parental
investment•
garding parental investment9.
=max(l-pi)R(x+g,)
(7)
the
the standard
standard life
life history
history optimizaoptimization
principle.
(In
tion principle.
(In fact.
fact, because
i=1,2
Equation
Equation
33 is
is the
the dynamic
dynamic
propro-
Gro.th
Growth
model still
still lacks aa
mcxiel.
model. This
This equation
equation has
has three
three imim- state
state variable.
variable, and
and may
may be mismisgxytant
portant features:
features: first.
first, itit isis clcxely
closely leadingly
leadingly simplistic
simplistic. Let
Let us
us therethererelated
related to
to traditional
traditional life
life history
history fore
fore consider
consider aa different
different situation.
situation,
still
involving
foraging
theory;
second,
it
provides
a
practheory; second, it provides a prac- still involving foraging under
under risk
risk Of
of
predation.
Assume
now
that
tical
algorithm
for
computing
the
tical algorithm for computing the predation. Assume now that the
the
optimal
optimal decision
decision ilo
i*(t) for
for all
all times
times forager
forager reprcxiuces
reproduces once
once only,
only, atat
t,r; and
and third.
third, similar
similar equations
equations can
can the
the end
end TT ofof the
the foraging
foraging season.
output
be
be obtained,
obtained, by
by analogous
analogous arguargu- Reprcxiuctive
Reproductive
output isis deterdetermined
by
somatic
ments,
for
much
more
complex
ments,
much more complex be- mined by somatic reserves
reserves atat time
time
TT. Somatic
havioral
havioral models.
Somatic reserves
reserves are
are increased
increased
To
To see
see the
the connection
connection with
with life as
as aa result
result of
of foraging.
foraging, and
and are
are also
also
history
required for
for metaix»lism
metabolism throughthroughhistory theory,
theory, observe
observe that
that at)
F(t) required
represents the
the animal's
animal’s total
total exex- out
out the
the season.
season. By
By foraging
foraging inin the
the
more
productive
habitat,
the
reprcxiuction
from
time
more
productive
habitat,
the ani•
anipected reproduction
from time
period tt through
through to
to the
the final
final mal
mal can
can increase
increase itsits rate
rate Of
of somatic
period r,T, assuming
assuming that
that the
the animal
animal growth,
growth, but
but at
at an
an increased
increased risk
risk of
of
employs
employs the
the optimal
optimal habitat
habitat at
at predation.
predation. We
We imagine
imagine the
the same
same
all
all times.
times. This
This isis exactly
exactly how
how two
two habitats
habitats as
as before.
before, but
but the
the par•
parare
now
Fisher
Fisher defined
defined reproductive
reproductive value.
value. ameters
ameters are now
Equation
Equation 33 therefore
therefore asserts that
that
g,
gi = net
net increase inin rærves
reserves per
per
the
the optimal
optimal behavioral
behavioral decision
decision atat
period
time
time tt isis that
that which
which maximizes
maximizes the
the
risk
predation
risk
per
Pi =
sum
sum Of
of expected current
current reproreproperiod
duction
duction (l(1 —
- pi)+i plus
plus exsrcted
expected
Clearly
future
i.
Clearly We
we now
now need
need aa State
state
future reprcxiuction
reproduction II(1 —
- &)flt + I1).
Again,
Again, this
this agrees
agrees exactly
exactly with
with variable.
variable. Let
Let Xin
X(t) denote
denote somatic
gram
ming equation
gramming
equation for
for the
the present
present
206
The
The forqoing
foregoing
Here
Here
the
the
trade-off
growth
growth and
and mortality
mortality
between
between
is
is explicit.
explicit.
From
From the
the conditions
conditions
g,< g, and
itit follows
follows from
from Eqn
Eqn 77that
that there
there isisaa
reserves
reserves level
level rXT-, >> with
with the
the propprop-
erty
erty that
that
.*
I T-l
2ifx<x
2 if x<x*
=
lifx>x
1 if x
T-l
(8)
> xi-,
Animals
Animals with
with low
low reserves
reserves atat time
time
T—
T- I1 should
should
employ
employ
the
the
risky.
risky,
prtxiuctive
habitat,
but
productive
habitat,
but those
those
with
with high
high reserves
reserves should
should use
use the
the
safer,
habitat.
safer, less-productive
less-productive
habitat. AA
similar
similar situation
situation applies
applies for
for earlier
earlier
time
time
periods
t.t. The
The level
level
which
which the
the optimal
optimal
isis an
an increasing
increasing
x’r atat
habitat
habitat switches
switches
function
function
of
of time
time
t(Fig_
t (Fig. 1).
I).
In
In this
this model reprtx*uctive
reproductive value
value
nx,0
flx,t) isis kx)th
both time
time and
and state
state dede-
pendent. The
The terms
terms Of
of the
the trade-off
growth
and
between growth and predation
predation risk
risk
change
change over
over time,
time, and
and depend on
on
reviews
reviews
TREE vol. 8, no. 6, June '993
TREE vol. 8, no. 6, June 1993
the animal's current state. The
the animal’s current state. The
same asset-protection
principle
same asset-protection
principle
used to explain the predictions of
used to explain the predictions of
also applies
also applies
for constant
for constant
in t (because
x, F(x,t) is decreasing in t (because
older animals have less time left in
older animals have less time left in
which to grow); as the asset value
which to grow); as the asset value
Flx,n decreases, optimal behavior
flx,t) decreases, optimal behavior
favors more risky choices. Similarly
favors more risky choices. Similarly
is an increasing function of x,
f+f, t) is an increasing function of X,
for fixed t, implying that larger
for fixed t, implying that larger
animals should adopt less risky
animals
should adopt less risky
choices. These two considerations
choices. These two considerations
showthat
the switchingline x; has
show that the switching line xi has
gxysitive slogr (Fig. l).
positive slope (Fig. I).
the previous model
the previous model
here. For example,
here. For example,
x,
is decreasing
Appncatims
Applications
The simple
The simple
described
theoretical
theoretical models
models
were to acquaint
described
above
were to acquaintof
reader
with the mechanics
the reader with the mechanics of
state-variable modeling of behavior
state-variable modeling of behavior
and life history. In recent years
and life history. In recent years
this technique has been used in
this technique has been used in
many empirical studies. Examples
many empirical studies. Examples
include
include:
the
• oviposition behavior of the para-
l oviposition
behavior of the parasitic wasp Nasonia vitripennis1°
•
allocation in breeding tree
l food allocation
in breeding tree
swallows (Tachycineta bicolon (D.
swallows (Tachycineta bicolorj (D.
Winkler and F. Adler, unpublished)
Winkler and F. Adler, unpublished)
•
caching in Carolina chickl food
caching in Carolina chickadees (Parus carolinensis) I
adees (Parus carolinensis) ’ ’
• diel vertical migration of juvenile
l die) vertical migration
of juvenile
sockeye salmon (Oncorhynchus
sockeye
salmon
( Oncorhynchus
nerka) 12
sitic wasp Nasonia vitripennis10
eling
* 300
t
greatly extends our
eling approach greatly extends our
ability
ability to
to understand
understand evolutionary
evolutionary
which
G.C. Williams
adaptation,
which G.C. Williams
has called •a phenomenon of
has called ‘a phenomenon of pervasive
vasive importance
importance in
in biology•20_
biology’20.
Xcrit
Example: termral decHneIn
Example: temporal decline in the mass
mass
at fledging of seabirds
at fledging of seabirds
A common phenomenon in the
0
A common phenomenon
in the
life history of birds Erlife history of birds beto the family Alcidæ
longing to the family Alcidae
(murres, puffins, auks, etc.) is a de(murres, puffins, auks, etc.) is a decline,
cline, as
as the
the breeding
breeding season
season proprogresses, in the average mass at
gresses, in the average mass at
which nestlings fledge. Considerwhich nestlings fledge. Considerable
variation
cxcurs
within
able
variation
occurs
within
species,
in
the
age
and
mass
species, in the age and mass at
at
fledging,
fledging, and
and even
even greater
greater variavariation
different
tion occurs between
different
sgkcies14,
but
the
pattern
species r4, but the pattern of
of temtemporal decline of average mass at
poral decline of average mass at
fledging
fledging is
is widespread within
within many
many
populations.
A standard explanation attributes
A standard explanation attributes
this
this decline
decline in
in fledging
fledging mass
mass to
to aa
corresgXJnding decline in the availcorresponding decline in the availability
ability of
of food,
food, but
but little
little ifif any
any
evidence suEmrts this claim. An
evidence supports this claim. An
alternative
explanation 14 hygfthalternative
explanationI
hypothesizes that the phenomenon conesizes that the phenomenon constitutes
a
to
stitutes a behavioral adaptation to
iuvenile
juvenile
longing
For the
I
20
40
60
Time, t
60
Time, t
Fig.
Optimal
habitat
choice for a
Fig. 1. Optimal
habitat
choice for a model of foraging
risk Ofpredation. The Shadedarea is a region Of
under risk of predation.
The shaded
area is a region of
zero fitness: fora«rs whose reserves x at time r lie in
zero fitness:
foragers
whose reserves
x at time t lie in
this
this regionare
region are unableto
unable to growlargeenotÆhto
grow large enough
to repmreproduce
duce. Otherwise,
foragers ShoulduseHa
should use H2 (risky habitat'
habitat)
H, Isafer habitat' if their
x at time t lie in
or H, (safer habitat)
if their reserves
x at time t lie in
indicated region. Typical growth Curvesare Shown
the indicated
region. Typical
growth curves are shown
lines.
parameters;
g,
as dotted
lines. IModel
(Model
parameters:
p, = 0.02,
0.02,
& = 0.04, g, = 2, g2 = 5, X&, = 100)
the
the trade-off Ertween
between
growth rates and mortality
relative
relative
risks in
growth rates and mortality risks in
the
the nest
nest and
and at
at sea.
sea. Typically
Typically the
the
nest
is a safer
environment
for
nest is a safer environment
for
young birds, but growth rates are
young birds, but growth rates are
higher at sea.
higher at sea.
A
A simple
simple dynamic
dynamic programming
programming
(Box 2) was developed to
model (Box 2) was developed to
test whether this hw)thesis
test whether this hypothesis was
was
reasonable.
The model is similar to
reasonable. The model is similar to
the
the growth
growth model of
of Eqns
Eqns 4-7
above,
except
that
above, except that the
the nestling
nestling
growth
growth rate
rate isis assumed
assumed to
to be on
on
Of
nerka) ’ 2
• territorial
bequeathal in Ameril territorial
bequeathal in American red squirrels
(Tamiasciurus
can red squirrels ( Tamiasciurus
h udsonicus)
hudsonicus) I3
• fledging behavior
of common
l fledging
behavior of common
murres
aalge)14
murres (Uria
( CJriaaalge)14
•0 parent—offspring
conflict
and
parent-offspring
conflict
and
nestling mass decline in dovekies
nestling mass decline in dovekies
(Alle alle'15
(Ale alle) I5
• anaerobic
diving
l anaerobic
diving of
of Westem
Western
grebes
(Aechmorphus
cxcidengrebes
(Aechmorphus
occidentalis 16
talis) ’ 6
• host seeking in mc%quitoeS17
l host seeking in mosquitoesI
• vegetative
versus reprcxiuctive
l vegetative
versus reproductive
growth in plants' 819
growth in plants’a,‘9
and flexibility of the
The generality method
and flexibility
of the
are attested
state-variable method are attested
by the variety of species and types
by the variety of species and types
of behavior encompassed by these
of behavior encompassed by these
publications.
These studies
have
publications.
These studies have
led
to
new
insights
and
hypotheses
led to new insights and hypotheses
regarding the adaptive significance
regarding the adaptive significance
of many Erhavioral and life history
of many behavioral and life history
traits. Indeed, the distinction
traits. Indeed, the distinction between life history and kkhavioral
tween life history and behavioral
decisions becomes largely blurred
decisions
becomes largely
blurredIn
in the state-variable
framework.
in the state-variable framework. In
my opinion, the state-variable
my opinion, the state-variable mod-
A1-gag
2
The generality
state-variable
2.7
2
11.0
025,
207
s
review
TREE
vol. 6,
8, no.
no. 6,
6. June
June 1993
TREE vol.
1993
current
with
current body mass:
mass: g,
g, = gl(x),
g,(x), with
g,(x) declining
at high
(parents
g,(x)
declining at
high x (parents
are
are unable
unable to
to provision
provision nestlings
nestlings at
at
a
rate
sufficient
to
overcome
ina rate sufficient to overcome increasing
costs).
creasing metabolic
metabolic
costs). Also,
Also,
only
habitat
only a
a single
single ontogenetic
ontogenetic habitat
switch (fledging)
switch
(fledging) is
is allowed.
allowed.
The
rncxiel indeed
The model
indeed predicts
predicts aa
ternEX)ral
decline
in
fledging
mass",
temporal decline in fledging massr4,
a
result
that
can
explained
a result that can be explained in
in
terms
terms of
of the
the asset-protection
asset-protection prinprinciple referred
referred to
light
ciple
to earlier.
earlier. First,
First, light
birds should
should remain
remain in
birds
in the
the nest,
nest,
where growth
growth rates
where
rates are
are higher,
higher, and
and
mortality risks lower,
at sea.
mortality
lower, than
than at
sea.
As nestlings
As
nestlings grow
grow larger,
larger, however,
however,
their
growth
rate
in
their growth rate in the
the nest
nest dedecreases and
and eventually
creases
eventually falls
falls below
the potential
at sea.
the
potential growth
growth rate
rate at
sea.
The trade-off
and
The
trade-off between
between safety
safety and
low growth
low
growth rate
rate in
in the
the nest
nest then
then
becomes crucial.
becomes
crucial.
given
A nestling that
that grows
grows to
to aa given
mass x earlier has
has greater
greater reproreproductive value
value than
ductive
than another
another nestling
nestling
that reaches
reaches xx later
later on,
that
on, because
because
the former
former nestling
nestling has
the
has greater
greater
for growth
to the
potential for
growth prior
prior to
the
end of the
the season.
Having a greater
end
season. Having
greater
reproductive
asset to
to protect,
protect, the
reproductive
early grower
grower will be more
more careful
predation risk, and
about predation
and will thus
leave the nest
nest than
be less apt to leave
than
later-growing nestling.
the later-growing
nestling. Given
that the
the later bird fledges
that
fledges on a certain date
date at
at a
a certain
mass x,
tain
certain mass
x, the
the
bird
that
reaches
x
at
an
earlier
bird that reaches x at an earlier
date will
Will benefit
benefit by remaining
remaining in
date
has grown
larger
the nest until it has
grown larger
than x. In other
Other words, nestlings
nestlings
than
that fledge
fledge early will
Will do so at
at larger
larger
that
mass levels
levels than
mass
than later
later fledgers.
fledgers.
Among different
different nestlings
nestlings in
Among
in aa
given colony,
colony, variation
variation in mass
mass may
may
given
result
result from
from variation
variation
'Of
not
particular
not very
very sensitive
sensitive to
to a
a particular
that variabehavioral decision,
decision, so
so that
variain
behavior
would
in behavior would be selecselectively
neutral.
tively neutral.
The
The evolutionary
evolutionary implications
implications of
of
are
space.
In practice,
stochastic environments
environments
are exexspace. In
practice, the
the data
data show
show stochastic
considerable
a
tremely subtle,
subtle, however
however. ReproReproconsiderable
spread around
around
a tremely
ductive
fitness, is
mean,
negatively
fledging
mean, negatively sloped, fledging
ductive value,
value, or
or fitness,
is usually
usually
defined
boundary. AA more
more complex
complex model
defined in
in terms
terms of
of expected
expected (aver(average)
success,
age
and
maturation
incorporating age and maturation
age) reproductive success, but
but this
this
is
in the
effects
account
effects could
could perhaps
account
is known
known to
to be
be incorrect
incorrect in
the case
case
for some
environments,
for
some of
of this
this variation,
variation, but
but of
of •coarse-grained'
‘coarse-grained’
environments,
in
such
such aa model has
has not
not yet
yet been
in which
which the
the timescale
timescale of
of fluctufluctuations
is
comparable
to
generation
developed.
ations is comparable to generation
An
extension of
fledging times21•22.Although
An extension
of the
the fledging
times2im22. Although state-variable
state-variable
rncxiel
has
used
to
study
the
readily include
model has been used to study the models readily
include random
random enenphenomenon
of
nestling-mass
devironmental
variations,
phenomenon of nestling-mass de- vironmental
variations, they
they still
still
cline in
rely
cline
in seabirds".
seabirds15. This
This extended
extended
rely on
on expected reproduction as
as
conmodel treats
treats parent—offspring
parent-offspring
con- the
the criterion
criterion Of
of fitrBs.
fitness. These
These
terms of
dynamic evolutiontherefore appropriate
flict in terms
of dynamic
evolutionmodels are
are therefore
appropriate
to
fluctuations,
i.e.
ary games
ary
games.
to fine-grained
fine-grained
fluctuations,
i.e.
fluctuations
that
take
place
on
fluctuations that take place on aa
that is
Advantages and
and nmIUti0%
limit&Ions
timescale that
is short
short relative
relative to
to
The
The state-variable
state-variable
approach to
to an
an individual's
individual’s life span.
span, How
How to
to
modeling
life history,
with
modeling tRhavior
behavior and
and life
history,
modify the
the approach
approach to
to deal
deal with
like
has coarse-grained
like any
any modeling
modeling framework,
framework, has
coarse-grained environments
environments is
is not
not
both
advantages
and
limitations.
yet
clear.
Indeed,
evolution
cannot
both advantages and limitations.
yet clear. Indeed, evolution cannot
Its
most important
fully maximizing
Its most
advantages are
are be a
a fully
maximizing
process
under these
the
followings:
the following?
under
these circumstances'
circumstances7f2’.
Nature is complex.
Nature
complex. ItIt is therefore
therefore
•l Model parameters
have direct
parameters have
sometimes
mistakenly
thought
that
sometimes
mistakenly
thought
that
biological meaning,
meaning, and
and can often
often
of
nature
must
be
equally
models
of
nature
must
be
equally
be measured
measured experimentally.
complex.
knew betcomplex. Einstein, who
who knew
•l Model predictions
quantitatpredictions are
are quantitatter,
said
that
theories
should
be as
ter, said that theories should be
as
ive
ive and
and subje•ct
subject to
to experimental
simple
as
but
not
simple
as
possible,
but
not
too
testing.
However,
qualitative
pretesting. However, qualitative pre- simple. What is meant by simple'
simple. What is meant by ‘simple’
dictions
dictions and
and general
general principles
principles can
can depends
in
part on
in part
on current
current techalso
be useful
also be
useful in
in broadening Our
our depends
nology_ Life history theory
theory was denology.
understanding of adaptation.
understanding
adaptation.
veloped
veloped long
long before
before the
the advent
advent of
of
•l Multiple
choices, and
Multiple behavioral choices,
and automatic
computation.
The
automatic
computation.
modthe associated
ass.xiated
trade-offs,
can
the
trade-offs,
can be em extension
to
em extension
to state-variable
state-variable
studied in a single
studied
single model.
is
a
consequence
of the
models
consequence
•l Constraints
Constraints on
on behavior,
behavior, and
and on
on computer
revolution.
(Dynamic
computer
revolution.
(Dynamic
state
variables,
are
a
natural
comstate variables, are a natural com- programming
programming is itself a computercomputerponent Of
of the
the model.
intensive
technique.)
But
it is well
intensive
technique.)
But
well
•l The
implications Of
The implications
of environmenenvironmenknown
that
dynamic
optimization
known
that
dynamic
optimization
tal fluctuations
tal
fluctuations (both deterministic
deterministic
models with
with more
more than
than aa few
few state
state
and stochastic) can be analysed
analysed.
variables
variables quickly
quickly encounter
encounter the
the
State-variable
mcxiels
realistic
'curse
State-variable
models with
with realistic
‘curse of dimensionality•«,
dimensionality’4*6, which
stcxhastic
not precan overwhelm
latest cornput•
stochastic components do not
overwhelm the latest
computdict a unique behavioral repering capacity.
A similar
toire, but rather predict behavioral
similar problem arises
arises in senvariation
within a given population,
sitivity testing,
which can become
variation within
population,
testing, which
become
resulting
from environmentally
indaunting
resulting from
environmentally
daunting for a complex
complex model induced
duced variation
variation in
in individual
individual states.
states. volving numerous
numerous parameters.
parameters. But
From this point
view, at least
as I hope
point of view,
hope to have
have demonstrated,
demonstrated,
part Of
inevitably endynamic
of the variability inevitably
dynamic models Of
of behavior
behavior need
countered in field studies can be not be highly complex in order to
seen
to have
signifimake useful,
predictions.
seen to
have adaptive
adaptive
make
useful, testable
testable predictions.
cance,
rather than
simply being
Simple
cance, rather
than simply
being
Simple dynamic
dynamic models often gendisregarded
disregarded as meaningless
meaningless statis- erate
erate predictions
predictions that
that differ radtical
noise
in
observations.
On
ically from
tical noise
in the
the observations.
On ically
from equilibrium
equilibrium models. In
In
the other
Other hand,
model may
may
Other
hand, a model
other cases no nondynamic
nondynamic analog
analog
sometimes
indicate
is
may be possible at
sometimes
indicate that
that fitness
fitness is
at all.
e ab
?oiidr
uutnx:r
in the
in
in hatch
hatch date,
date,
or
or in
in nest
nest growth
growth rate.
rate. Our
Our simple
simple
model
predicts
that
all
nestlings
in
model predicts that all nestlings in
a
given
colony
will
fledge
along
a given colony will fledge along a
a
single
single 'fledging
‘fledging boundary
boundary’ in
in t—x
t-x
tions
tions
ing
TREE
TREE vol. 8.8,no.
no. 6,6,June
June
reviews
reviews
1993
The
The models described inin this
this productively, about the
the meaning
meaning Of
of
brief
brief survey
survey have
have all
all concemed
concerned inin- adaptation
adaptation.
dividual
dividual optimization.
optimization, and
and itit might
might
thought
that
the
technique
be thought that the technique isis
References
limited
limited to
to this
this case.
case. Fortunately.
Fortunately,
I I Leswls.
Lessels, CM.
C.M. (1991) in Behavioral
this
this isis not
not so;
so; aa variety
variety ofof dynamic
dynamic
Ecology:
An Evolutionary
Approach
game-theoretic,
or
game-theoretic,
or ES
ESS, models
(Krebs, 1.R. and Davies,
N.B., eds), pp.
32-68, Blackwell
have
have tren
been developed15f24. Dynamic
Dynamic
2 Fisher. RA
R.A. ( 1930) The Genetic&
Theory
ESSmodels
ESS models can
can be
be extremely
extremely comcom- 2of Fisher,
Natural Selection,
Clarendon
plex
unless
ingenuity
is
applied
n
plex unless ingenuity is applied n 3 Steams
Steams, sc
S.C. (I 992) The
The Evolutionon.ife
Evolution
of Life
the
the choice
choice ofof simplifying
simplifying assumpassumpHistories,
Oxford University
University
Press
44 eellman.
tions.
Bellman, RR. (1957) Dynamic
Programming,
tions. For
For example.
example, inin aa two-particitwo-participant
pant game
game between dominant
dominant and
and Princeton University Press
subordinate, the
the priority
priority Of
of decision
cision can
can be
be asumed
asumed toto trlong
belong toto
the
the dominant
dominant15.
In
In aa sense.
sense, the
the limitations
limitations ofof dp
dy
namic
may
namic modeling may be
be aa blessing
blessing
in
in disguise,
disguise, ifif they
they cause
cause scientists
scientists
to
to think
think profoundly
profoundly atX)ut
about their
their syssysterns,
tems, rather
rather than
than relying
relying on
on incomincomprehensible
prehensible computer
computer simulations
simulations.
I 1 believe that
that state-variable
state-variable modeling isis helping
helping evolutionary
evolutionary biolobioloeling
gists
gists to
to think
think differently.
differently, and
and more
more
Virugs
Viruses have
have
DD. and
5 McFarland.
McFarland,
and Houston.
Houston, KI.
A.1. ( 198 I )
Quantitative
Ethology:
The State Space
Approach,
Pitman
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Mangel, M. andCIaå.C.W.
and Clark, C.W. '1988'
( 1988)
Dynamic
Modeling
in Behavioral
Ecology,
Princeton
Press
Princeton University
University
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7 Lewontin,
R.C. (1987) in The Latest on the
Best: Essays on the Evolution
of Optimality
(Dupti,
J., ed.), pp. 151-150.
151-159, MIT
MIT Press
Press
'H,
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McNamara,
J.M. and
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A.I. (1986)
Am
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R.L. (1974) Am. Zoo/. 14,
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10
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1-22
6een assumed
assumed toto play aa
uptake2 and
and methcxis
methods for
for measuremeasureuptake2
ment
of
grazing3,
were
developed.
ment of grazing'. were
In this
this
In
perspective,
recent work
work
recent
showing that
that viruses
viruses may
may be ix»th
both
showing
research inin marine
marine microbial
microbial ecolecolresearch
active inin cell
cell lysis'.
lysis5, represents
represents
active
discovery
of
yet
another
piece
discovery of yet another piece
F. Thingstad, M.Heu
Heldal, G.
G. Bratbak and I. Dundas are
at the Dept of Microbiology
and Plant Physiology,
University of Bergen, jahnebakken 5, N-5020 Bergen,
l_meßity
lahneba"en 5.
Norway.
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Q 1993, Elsevier Science Publishers
C
seiÄ
Ltd (UK)
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131,271-290
13 Price,
Price, K.K. (1992) Bull. Math. Biol. 54,
335-354
14 Ydent*rg.R.C.
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R.C. (1989) Ecology70,70.
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15 Clan.
Clark, C.W..ndYdenkx•rg.
C.W. and Ydenberg,
R.C. (1990)
Evol. Ecol. 4, 3 12-325
16
R.C,
16 Ydenberg,
R.C. and
and Clark.
Clark, C.W.
C.W. (1989)
I. Theor
Theor. Biol
Bioi. 139,437-449
17
17 Roitberg,
B.D. and
and Friend,
Friend, W.G. (1991)
Bull. Math
Math. Biol. 54,40 I-4 I 2
l.I.and
I8 Kozlowski,
Kozlowski,
and Wiegert,
Wiegert, R.G.
R.G. 11987)
(1987)
EVO/
Evol. Ecol. I 1,23 l-244
l.1.and
19 Kozlowski.
Kozlowski,
and Ziölko.
Ziblko, M.
M. (19881
Theor. pop.
fop. Biol. 34,
34, II&l29
20
and
20 Williams.
Williams, G.C
G.C. (1966) Adaptation
Adaptation
and
Natural
*lection, Princeton
Naturalselection,
Princeton University
University
Press
2 1 Levins, RR. (I 1968) Evolution
Evolution inin Changing
Changing
Environments,
Princeton University
University
Press
Environments.
Princeton
Press
22 Yoshimura,
J. and Clark, C.W. ( 199 I )
Evat
Evol. Ecoi. 5,5, 173-192
23
23 Yoshimura.
Yoshimura,
).and
and Clark.
Clark, C.W
C.W., eds
Adaptation
Stochastic
Environments,
Adaptation
inin Sttxhastic
Environments,
Springer un
(in press
press) I
Sprin«r
24
24 Houston.
Houston, A1.
A.I. and
and McNamara.
McNamara,
1.M. (1987)
j. Theor. Biol. 129:37-68
129: 57-68
1.
T.F.
hingstad, M.
T.F.TThingstad,
M.Heldal,G.
Heldal,G.Bratbak
Bratbakand
andl.I.Dundas
Dundas
one uses
uses the
the allegory
allegory ofof aa iigjigIfIf one
saw
puzzle,
the
last
two
decades
saw puzzle, the last two
ofof
ogy have
have been
been charKterized
characterized by
by atattempts toto fitfit together
together emerging
emerging
tempts
pieces ofof knowledge
knowledge toto form
form aa pie
picpieces
ture
of
how
elements
are
cycled
in
ture of how elements are cycled in
the
microbial
food
web:
who
are
the microbial
web: who are
the trophic
trophic partners
partners inin the
the food
the
web, how
how active
active are
are they
they and
and how
how
web.
this ecosystem
ecosystem controlled
controlled by
by exexisis this
ternal and
and by
by intemal
internal fm-tors?
factors? The
The
ternal
very existence
existence ofof many
many ofof its
its
very
pieces
remained
unknown
until
pieces remained
unknown until
new techniques
techniques such
such as
as fluoresfluoresnew
cence microscopy’, , measurement
measurement
cence
of DNA
DNA synthesis
synthesis by
by thymidine
thymidine
Of
Nar.
Nat.
AreViruses
annersin
Are VirusesImportant
ImportantP
Partners
in
PelagicFood
Pelagic F Webs?
rather negligiblerok
negligible roleas
aspartnersin
partners in micromicrorather
6iaI foodweb
we6dynamics.
dynamics. However, recent
bial
discoveries
suggest that tke
the rate
rateofofvirally
virally
discoveries
suggestthat
induced lgsis of marine
marine mkrobia/
microbial popuinducedlysis
lations may be
6esignifiant.
significant. 7%is,
Tclis,inin turn.
turn,
lationsmay
may
have
important
consequences
for
the
may haveimportant'onsequen«s tke
developing conceptual frameworkof
framework of tke
the
devebpingconceptual
microbial foodVeb
we6.
il1 I Lueas,
Lucas, J.R. and
and waiter.
Walter, L.R.
L.R. ( I99 1) Anim.
Behav
Behav. 41.579-601
4 1,579-60 I
12
12 Clark.
Clark, C.W
C.W. and Levy, D.A. (1988) Am.
numerous
numerous
the water
water mass',
mass4, and
and
inin the
has toto be
be fitted
fitted
has
the
the
that
that
somewhere inin
inin somewhere
the
very
center
of
this
puzzle. InIn
the very center of this puzzle.
addition to
to •classical'
‘classical’ questions
questions rereaddition
lated toto element
element cycling
cycling and
and food
lated
web dynamics.
dynamics, the
the virus
virus story
story adds
adds
web
new
aspects
to
the
puzzle,
such
as
new
to the puzzle, such as
the
potential
exchange
of
genetic
the EX'tential exchange Of genetic
information
between
microbial
information
microbial
species and
and the
the potential impact
impact ofof
species
viruses
on
the
diversity
of
the mimiviruses on the diversity of the
crobial
population.
With
this
develcrobial 'xvulation.
With this development, •phage
‘phage ecology
ecology’ isischanging
changing
opment.
from aa field
field mainly
mainly concerned
concerned With
with
from
the sanitary
sanitary aspects Of
of survival
survival Of
of
the
enteric
viruses
in
natural
environenteric
viruses
in natural
environments? toto aa discipline
discipline inin the
the mainmainments6
stream Ofof research
research inin the
the field
field
of general
general
Of
microbial ecology
ecology Of
of
microbial
aquatic
ecosystems.
aquatic ecosystems _
The existence
existence Of
of Viral
viral activity
activity and
and
phage-host
systems inin aquatic
aquatic
phage—host
systems
ecosystems where
where the
the hosts
hosts are
are ininecosystems
digenous has
has been
been known
known for
for more
more
digenous
than 30
30 years',
years7. Until
Until recently,
recently, how
howthan
ever, attention
attention toto the
the role
role ofof virusvirusever,
es inin the
the marine
marine environment
environment was
was
es
hampered by
by aageneral
general acceptance
acceptance
hampered
of arguments
arguments for
for aa low
low probability
probability
Of
of infection.
infection. Viruses
Viruses were
were known
known
Of
from
laboratory
studies
to
be
host
from latX)ratory studies to
host
specific, which,
which, combined
combined with
with relarelatively sparse
sparse populations
populations Of
of hosts
hosts
tively
(c.10”
ml-’ for
for bacteria;
bacteria; 103—104
103-IO4ml-l
ml-’
(c.
100 ml-'
for
small
eukaryotes)
and
the
asfor small eukaryotes) and the as.
sumption
of
generally
very
diverse
sumption of generally very diverse
communities, seemed
seemed toto make
make the
the
communities,
probability ofof virus—host
virus-host encounter
encounter
probability
very small.
small. Thus,
Thus, there
there was
was aa relarelavery
tive
lack
of
interest
in
early
work
tive lack Of interest in early work
suggesting a high number of free
suggesting a high number of free
virusesa.
viruses'.
209