Journal of Animal
Ecology 0887\
56\ 079Ð083
Small mammals cycles in northern Europe] patterns and
evidence for a maternal effect hypothesis
PABLO INCHAUSTI and LEV R[ GINZBURG
Department of Ecology and Evolution\ State University of New York\ Stony Brook\ NY 00683!4134\ USA
Summary
0[ Voles undergo pronounced oscillations over periods of 2Ð4 years in northern
Europe[ A latitudinal gradient of cycle periods and amplitudes has been reported for
Fennoscandia\ with periods and amplitudes increasing towards northern latitudes[
1[ This study formulates a discrete time model based on maternal e}ects to explain
the density ~uctuation patterns of microtine rodents[ The phenotypic transmission of
quality from mothers to o}spring generates delayed density dependence\ which pro!
duces cyclic behaviour in the model[
2[ The dynamic patterns predicted by the maternal e}ect model agree with data[ We
conclude that the maternal e}ect hypothesis is a plausible\ parsimonious explanation
for vole!density cycles in northern Europe[
Key!words] individual quality\ microtines\ models\ oscillations\ population dynamics\
rodents\ voles
Journal of Animal Ecology "0887#\ 56\ 079Ð083
Introduction
Þ 0887 British
Ecological Society
079
The small mammal cycles comprise one of the long!
standing\ unsolved problems of population ecology[
Seemingly similar species of microtine rodents "voles
and lemmings# are known to exhibit a diverse array of
dynamic behaviours across their distribution ranges[
Time!series analysis of northern Fennoscandian voles
has established a geographic trend in cycle length with
periods ranging from 2 to 4 years "Henttonen\
McGuire + Hansson 0874^ Turchin 0882^ Bjo
rnstad\
Falk + Stenseth 0884#[ Southern vole populations
"south of 59>N# are thought to be semi!stable with
irregular ~uctuations "Henttonen et al[ 0874#[ This
dichotomy between northern and southern Fenno!
scandian voles is\ at best\ tentative considering the
shortness of the time series of southern Fenno!
scandian voles analysed so far "Marcstrom\ Hoglund
+ Krebs 0889^ Krebs 0885#[ Longer time series from
Russia\ at latitudes similar to southern Fennoscandia\
exhibit a 2!year cycle "data in Falk\ Bjo
rnstad + Sten!
seth 0884^ Zhigalski 0881#[ Toussaint "0889#\ Froschle
"0880# and Saucy "0883# provide evidence of cyclicity
for some southern and central European microtine
populations[
Consistent qualitative changes in individual attri!
butes\ such as body size or behaviour\ have often
been associated with density ~uctuations in most vole
species "reviews in Krebs + Myers 0863^ Stenseth +
Ims 0882#[ Whether these qualitative changes are
causes or consequences of vole cycles remains to be
empirically demonstrated[
EXPLAINING THE PATTERNS
Despite the enormous amount of research e}ort
devoted to the problem in the last 69 years "Stenseth
+ Ims 0882 quoted over 149 references in their review
of microtine research#\ there is no consensus regarding
the proximal factors or processes that cause vole cycles
"Batzli 0881^ Krebs 0885#[ The hypotheses proposed
to explain them can broadly be subdivided into two
main classes\ extrinsic and intrinsic[
The extrinsic hypotheses encompass those referring
to the interactions between voles and their food
resources or their predators[ The resource!related
hypothesis asserts that ~uctuations in abundance "e[g[
Lack 0843# or quality "e[g[ Agrell et al[ 0884# of food
resources produce changes in demographic rates
which then cause density ~uctuations[ Many
researchers "e[g[ see Stenseth + Ims 0882\ p[ 63# have
documented the decline\ or even destruction\ of veg!
etation before or during vole peaks[ Experimental
food addition has been shown to accelerate sexual
maturation "Bujalska 0864#\ to prolong the main
breeding season "Saitoh 0878# and to increase survival
as measured by persistence "Schweiger + Boutin
0884#\ percentage of individuals in reproductive con!
dition "Desy + Thompson 0872#\ recruitment "Saitoh
0878# and population density "Desy\ Batzli + Liu
070
P[ Inchausti + L[R[
Ginzburg
Þ 0887 British
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Journal of Animal
Ecology\ 56\ 079Ð083
0889# for several vole species[ While experimental
results and _eld observations indicate the importance
of resource abundance on vole dynamics\ food
additions have not prevented the decline of vole popu!
lations "Stenseth + Ims 0882^ Krebs 0885#[ However\
most food!addition experiments have been under!
taken with North American voles that do not exhibit
multi!annual cycles[ In Fennoscandia\ the authors are
aware of one multi!annual experiment with cyclic
voles in which Henttonen et al[ "0876# used a pulse!
type addition of a constant amount of food at regular
intervals[ The _nding was that the addition of food
produced increased population density and growth
rate\ but did not prevent the occurrence of cycles[ The
present study hypothesizes that this might have been
so because the positive e}ects of food addition on
reproduction are likely to be overridden by negative
e}ects from decreasing per capita resource supply as
population density increases "in the Discussion we
propose a food!addition experiment that could pre!
vent voles from cycling if the oscillations are caused
by maternal e}ects#[ The authors agree with Stenseth
+ Ims "0882\ p[ 63# in that few replicated\ food!
addition experiments of su.cient duration have been
carried out with cyclic voles to allow conclusive state!
ments about the role of food abundance on vole
dynamics[
Predation has recently received renewed attention[
Hansson + Henttonen "0877# and Hanski + Kor!
pimaki "0884# proposed that the latitudinal changes
in the abundance of vole predators may explain the
apparent gradient of cyclicityÐstability in Fenno!
scandia[ They noted that the abundance of resident
specialist predators "e[g[ small mustelids# that are cap!
able of producing cycles\ increases toward northern
latitudes\ whereas resident generalist "e[g[ red foxes#
and nomadic generalist predators "e[g[ owls#\ which
produce additional density!dependent mortality that
is thought to stabilize vole abundance\ are pre!
ponderant in southern Fennoscandia[ Predator
removals in this region do not indicate that predation
is necessary to cause vole cycles[ Marcstrom\ Kenward
+ Engren "0877# removed all martens and foxes in 7
years from two islands in northern Sweden "55>N#
and voles had 3!year cycles\ similar to unmanipulated\
mainland populations[ Norrdahl + Korpimaki "0884#
removed birds of prey from a large area in Finland in
2 years with no e}ects on vole cycles[
The intrinsic hypotheses assume the operation of
some intraspeci_c mechanism of self!regulation based
on spacing behaviour "Krebs 0867#[ In their original
versions\ these hypotheses postulated the occurrence
of density!dependent changes in individual quality
based on physiological "Christian 0849#\ genetic
"Chitty 0859\ 0856# or behavioural "Charnov + Fin!
nerty 0879# mechanisms[ Those changes in individual
quality\ in turn\ a}ect population growth\ thereby
preventing population explosions and generating
multi!annual density ~uctuations[ Boonstra "0883#
recently proposed a di}erent version of Christian|s
"0849# stress hypothesis based on changes in age struc!
ture and on the e}ects of senescence on reproductive
e}ort during density cycles[ The evidence gathered for
the three intrinsic mechanisms mentioned above tends
to be negative\ but in fairness\ these intrinsic hypoth!
eses are still viable and more research is needed before
ruling them out conclusively "Krebs 0885#[ Admit!
tedly\ there are empirical di.culties in de_ning and:or
measuring in the _eld attributes such as body
condition\ stress\ aggression and kin structure[ In the
authors view\ these di.culties should not prevent us
from recognizing the recurrent changes in individual
quality observed in most oscillating vole populations
"Krebs + Myers 0863^ Hansson 0877^ Chitty 0859\
0856\ 0885^ Krebs 0885#[
In this paper a simple dynamic model is formulated\
based on the transmission of average individual qual!
ity between consecutive generations[ This model
di}ers from previous intrinsic hypotheses in that the
phenotypic transmission of individual quality is via
maternal inheritance[ Maternal e}ects may be con!
sidered a special case of cohort e}ects mechanisms\
in which the current cohort quality depends on the
environmental conditions experienced several cohorts
before\ as it has been proposed for long!lived organ!
isms "e[g[ deer] Albon\ Clutton!Brock + Guiness 0876^
_shes] Nikolskii 0858#[ The term maternal rather than
cohort e}ects is preferred because the former has been
more readily associated with the short!term trans!
mission of individual quality "e[g[ Bernardo 0885^
Rossiter 0885#[ It is stressed that other intrinsic mech!
anisms of change and transmission of individual qual!
ity can be expressed through\ and are compatible with\
the general structure of the model as long as the chan!
ges in individual quality a}ect population growth rate[
The maternal!e}ect model essentially re~ects the
notion that the environment in which the parental
generation lived can a}ect the demographic rates of
its o}spring[ In turn\ the average quality depends on
the per capita share of resources an individual
acquires[ The combination of these two dependencies
produces delayed density dependence capable of gen!
erating cycles similar to those observed for northern
European voles[
The maternal!effect model
MODEL STRUCTURE
The maternal!e}ect model describes vole dynamics as
occurring in two non!overlapping breeding periods[
These breeding periods are neither even fractions of
the year\ nor are they governed by the same rules] the
main breeding period "spring:autumn# is generally
shorter and occurs under more benign environmental
conditions than the minor breeding period "autumn:
spring#[ Voles can breed within 1 months of birth in
some phases of the density cycle and thus they have
071
Small mammals
cycles
Þ 0887 British
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Journal of Animal
Ecology\ 56\ 079Ð083
overlapping generations with a variable number of
cohorts^ rarely do they live more than 0 year[ Also\
the length of the main breeding period changes during
the cycle\ being shorter at peak densities "Krebs +
Myers 0863#[ Voles can certainly undergo several gen!
erations and produce several litters during the main
breeding period[ We considered variants of our model
in which voles had three or four non!overlapping gen!
erations in the spring:autumn period and only one
during the autumn:spring period[ Similar to the orig!
inal model\ all these variant models can generate 2 to
4 year cycles for a set of population growth rates
corresponding to the shorter transitions during the
main breeding period[ We also consider an age!struc!
tured variant of the maternal e}ects model in Appen!
dix 0[ By treating the two breeding periods as discrete
periods\ we assume that the dynamics of the over!
lapping generations occurring during each breeding
period may be mathematically summarised as an over!
all single demographic transition characterised by the
growth rate of the breeding season[ We feel that this
characterisation is commensurate with the frequency
at which most species and locations are censused in
Fennoscandia\ where long!term censuses have been
performed annually or\ at best\ twice a year[ It is the
data availability and our strong distaste for over!
parameterised models that sway us towards the
abstraction of two breeding periods[
Winter breeding is thought to be important for the
occurrence of vole cycles "Krebs + Myers 0863#[ It
tends to be associated with the increased phase of the
density cycles "Krebs + Myers 0863^ Stenseth + Ims
0882#\ when both the spring population density and
the growth rate of the preceding autumnÐspring per!
iod are low^ a type of dependence that is re~ected in
the time!lagged version of the model "see Appendix
1#[ Direct evidence of winter breeding for some vole
species "e[g[ Clethrionomys glareolus Schreber in the
northern part of its cyclic range# is sketchy\ mostly
because of the obvious di.culties of collecting such
data[ The indirect detection of winter breeding by
observing placental scars and:or by trapping very
young animals\ is not without di.culties[ This is
because the generally low intensity of winter breeding
can be missed if spring vole abundance during the _rst
increase year is low to moderate[ Hansson "0873a# and
Kaikusalo + Tast "0873# provide evidence of winter
breeding for C[ rufocanus Sundevall\ Microtus agrestis
L[ and C[ rutilus Pallas in Fennoscandia[ Winter
breeding of C[ glareolus has been reported by Zhi!
galski "0881^ pp[ 10\ 17 and 28# in Russia\ by Ylonen\
Mappes + Viitala "0882# in central Finland\ and by
Hansson "0873a# in southern Fennoscandia[ It must
be pointed out that the maternal!e}ect model does
not require the yearly occurrence of winter breeding\
but only that it can occur in some years of the density
cycle[
While the present study|s de_nition of two non!
overlapping transitions corresponding to the two
breeding periods is certainly an abstraction\ it seems
to be a step in the alternative direction compared
with the use of continuous di}erential equations[ In
continuous models "dN:dt#\ the very language of the
theory forces the parameter estimation to be related
to a _xed\ chronological time scale[ Consequently\ one
loses the opportunity to distinguish between the two
distinct breeding periods\ which are di}erent in quality
and variable in duration[ Making the transitions
between parameter values and functions describing
contrasting portions of the yearly dynamics is prob!
lematic for continuous di}erential models[ Problems
arise because\ by de_nition\ these models assume a
smooth\ continuous transition between the two sets
of equations which describe the dynamics of each
breeding period[ This smoothing often requires speci!
fying an additional ad hoc function to transfer the
values of the state variables between breeding periods
"e[g[ Hanski + Korpimaki 0884#[ In contrast\ discrete
time models can consider breeding periods of con!
trasting intensity and variable duration\ and thus they
do not have the constraints on parameter estimation
that exist for continuous time models[ Discrete time
models do not require additional smoothing junctions
to connect the two distinct breeding periods[ The sep!
aration of the annual dynamics into two breeding
periods in this model makes errors by assuming that
all mothers reproducing in a given breeding period
are born in the previous one\ and by considering
demographic rates to be constant during a breeding
period[ These errors seem to us to be less egregious
than the ones discussed for a continuous time model
formulation[
The two sets of equations describing the annual
dynamics have the form]
7
eqn 0a
7
eqn 0b
NA NSGS:A"XS#
XA HS:A"NA\ XS#
and
NS NAGA:S"XA#
XS HA:S"NS\ XA#
where NS and NA are the vole population abundance
at the end of the spring and autumn periods "i[e[
assuming post!breeding censuses#\ XS and XA are the
average quality of the mothers at the end of the spring
and autumn\ respectively[ Equation 0a describes the
changes in population abundance and average quality
occurring over the main breeding season "spring:
autumn#\ whereas eqn 0b corresponds to those
changes occurring over the minor breeding season
"autumn:spring#[
The population growth rate functions G are mono!
tonic\ decelerating functions of the average individual
quality "Fig[ 0#[ The average quality X is related to
the amount of energy a female can gather\ or\ in other
words\ to her per capita share of the available
resources[ Function G then describes the allocation of
072
P[ Inchausti + L[R[
Ginzburg
Fig[ 0[ Population growth rate functions GSpring:Autumn and
GAutumn:Spring "eqns 0a and 0b# as a function of average indi!
vidual quality in the previous season "continuous lines#[ The
dotted line represents the modi_ed GAutumn:Spring function by
assuming that there exists a minimum average individual
quality below which reproduction ceases[
assimilated energy\ re~ected in individual quality\ into
population growth[ The higher the quality of females\
the higher the endowment they can provide to a more
numerous "function G#\ healthier progeny "function
H#\ and\ by implication\ the higher the population
growth rate[
The functions H that describes the changes in indi!
vidual quality across breeding seasons are mono!
tonically decelerating functions of the per capita share
of resources at the current generation "Fig[ 1#[ The
functions H describe the processes of acquisition of
resources "foraging behaviour# and their conversion
"trophic e.ciency# into energy devoted to repro!
duction depending on the current population abun!
dance[ A high population density will reduce the per
capita share of resources and the average individual
quality in the current breeding season which\ in turn\
will render a lower population growth rate in the next
breeding season[
The phenotypic transmission of individual quality
across breeding periods is the type of maternal e}ect
that the model refers to[ The requirement is simply
that motherÐo}spring relations can be made and that
the population growth rate can be expressed as a func!
tion of the average individual quality[ Therefore\ the
maternal!e}ect model is based on the interaction
between resource and population densities in a breed!
ing season\ with the important addition that the e}ects
of such an interaction on the population growth rate
are mediated through changes in individual quality
transmitted from mother to o}spring in the next
breeding period[
Two structural features critical to the behaviour of
the model "eqns 0a and 0b# need to be stressed[ First\
in the absence of maternal e}ects\ functions HA:S and
HS:A depend only on the population density of the
current generation[ After substituting the second
equation of the pair into the _rst one "eqns 0a and
0b#\ one obtains]
7
NA N
SGS:A"NS#
NS N
AGA:S"NA#
eqn 1
The model then becomes one!dimensional "i[e[ re~ect!
ing direct density dependence#[ If the functions G rep!
resent overcompensation\ eqn 1 can\ in principle\ pro!
duce limit cycles whose periods are 1\ 3\ 7\ etc[ "May
0865#[ However\ the direct "non!delayed# density!
dependence model represented in eqn 1 cannot pro!
duce periods and other qualitative features "such as
cycle shapes# of vole cycles for biologically realistic
parameter values[ As made abundantly clear by Tur!
chin "0882#\ Hornfeldt "0883# and others\ delayed den!
sity dependence is essential for any model to produce
the dynamic patterns observed for northern European
voles[ The transmission of individual quality is one
possible mechanism that can generate delayed density
dependence[
Second\ the argument N of the function H is evalu!
ated at the current generation "eqns 0a and 0b#\ imply!
ing that the average individual quality responds
quickly to changes in the assimilated energy at the
current generation[ Ginzburg + Taneyhill "0883 and
references therein# discuss why this is a critical math!
ematical assumption of the maternal!e}ect model[
EMPIRICAL EVIDENCE OF MATERNAL EFFECT IN
MICROTINES
Þ 0887 British
Ecological Society
Journal of Animal
Ecology\ 56\ 079Ð083
Fig[ 1[ Changes in the average individual quality between
seasons HSpring:Autumn and HAutumn:Spring "eqns 0a and 0b# as a
function of current per capita resource share "continuous
lines#[ The dotted line represents the modi_ed HAutumn:Spring
function to generate true limit cycles\ assuming that repro!
duction will occur only after the average individual quality
is higher than a threshold "Xmin#[
Consideration of di}erent types of maternally induced
e}ects has a relatively long tradition in microtine
research "Christian 0849^ Chitty 0841#[ Bernardo
"0885# and Rossiter "0885# have recently reviewed the
occurrence and the ecological and evolutionary
importance of maternal e}ects in a wide variety of
taxa[ Most of the evidence of maternal e}ects in voles
reviewed below comes from studies in the laboratory
073
Small mammals
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Þ 0887 British
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Journal of Animal
Ecology\ 56\ 079Ð083
which\ in itself\ essentially points to the potential
importance of maternal e}ects under _eld conditions[
The environment in which parents lived can have
overriding e}ects on the o}spring demography\ as
Hansson "0876\ 0878# demonstrated with wild!caught
voles C[ glareolus and M[ agrestis\ which were bred in
the laboratory for two generations under identical
conditions[ Laboratory!born females with mothers
having good food conditions grew faster\ matured and
bred earlier than their wild!born counterparts[ For
C[ glareolus\ Hansson "0877\ 0881# found negative
correlations in both body weight and reproductive
parameters "number of litters\ litter size\ time of _rst
parturition and total number of young weaned#
between mothers that descended from _eld!born par!
ents and their laboratory!born o}spring when both
generations were kept under similar nutritional con!
ditions[ Hansson "0877# interpreted these correlations
as {negative maternal e}ects|[ Well!fed mothers began
reproducing early and had relatively large repro!
ductive output\ with each young having low body
weight^ these female young presumably retained a low
body weight into adulthood\ began reproducing late
and had a relatively low reproductive output[ Boon!
stra + Boag "0876# compared the performance of M[
pennsylvanicus Ord o}spring of parents that _eld!bred
from increase and peak years under laboratory and
_eld conditions[ They found that mothers were sig!
ni_cantly larger in the peak densities\ litter size did
not di}er between increase and peak densities\ but the
progeny from mothers of the increasing populations
reared in the _eld grew faster and reached sexual
maturity earlier than that from peak populations\
while the reverse occurred for progeny reared in the
laboratory[ Boonstra + Boag|s "0876# experiment
showed that most of the phenotypic variation of o}!
spring performance was non!genetic and that
maternal and perhaps other environmental e}ects had
overriding importance[ Mihok + Boonstra "0881#
found that the breeding performance of M[ pennsyl!
vanicus changed during the density cycle such that
the progeny of decline!phase females produced fewer
litters that grew slower than those of increase!phase
females under similar conditions[ Both Hansson
"0877# and Boonstra + Boag "0876# think that the
manifestation of maternal e}ects should be more obvi!
ous under _eld conditions[ These negative maternal
e}ects may explain the typical variation in repro!
ductive output observed in vole cycles "Krebs + Myers
0863^ Stenseth + Ims 0882#] high body weight and low
reproductive output at peak densities\ and low body
weight and high reproductive output during the early
increase phase of cycles[ Rossiter "0881# argued that
negative maternal e}ects may represent an adaptive
strategy in herbivores that allows the regulation of
o}spring quality in response to changes of the quan!
tity and quality of food resources[
The age\ and presumably weight\ at mating has been
related to parental investment and o}spring per!
formance[ Hagen + Forslund "0868# showed that
females of M[ canicaudus Miller that mated younger
produced fewer but larger litters and their o}spring
were smaller at birth and had lower survival rates than
those of females that were older at mating[ Solomon
"0883# obtained very similar results for M[ ochrogaster
Wagner[ The parental investment of M[ guentheri
Danford + Alston a}ects size at weaning\ which\ in
turn\ a}ects adult weight\ rate of sexual maturation
of females and size of the _rst litters\ even under con!
ditions of constant\ unlimited food supply "German
0882#[ Juvenile survival of M[ agrestis "Godfrey 0844#
in England and of M[ montanus Peale in California
"Ho}man 0847# after peak densities was very low\
with most o}spring dying during lactating even
though their parents showed no evidence of food
shortage\ starvation\ or low body weight[
Individual quality has been shown to be transmitted
between parents and their o}spring[ The o}spring of
M[ montanus receiving 5MBOA "a compound that
stimulates the onset of reproduction# through the pla!
centa showed increased gonadal weight and body size
compared with control animals under similar con!
ditions "Frandsen\ Boyd + Berger 0882#[ Horton
"0874# and Meek\ Lee + Gallon "0882# showed that
the maternal photoperiod of M[ montanus and M[
pennsylvanicus\ respectively\ a}ected the growth and
reproduction of their progeny[ A reduced o}spring
viability in M[ townsendii Bachman "Beacham 0879#
and M[ montanus "Horton 0873^ Pinter 0877# was
attributed to poor adult health[ Maternal condition
and the quality of lactation of M[ pennsylvanicus
"Boonstra 0874# and M[ agrestis "Chitty + Phipps
0855# presumably deteriorated during the decline
phase of the cycle\ a}ecting the age of weaning and
juvenile viability[
THE MEASUREMENT OF INDIVIDUAL QUALITY IN
VOLES
Individual quality can be de_ned by biochemical\
behavioural\ physiological or developmental com!
ponents of _tness\ such as body size\ body condition\
disease resistance\ tendency to disperse\ o}spring
viability\ etc[ "Rossiter 0883#[ Thus\ population qual!
ity can be described by the mean and the variance of
the frequency distribution of the pertinent trait that
re~ects individual quality[ In the model\ individual
quality should be ideally measured as the average
amount of energy an individual devotes to repro!
ductive activities\ but direct measurements are obvi!
ously impossible in most cases[
Body weight in voles may be a possible surrogate
for individual quality based on its association with life
history traits that ultimately determine the population
growth rate\ such as age of sexual maturation\ length
of the breeding season\ litter size and survivorship[
Moreover\ the occurrence of heavy individuals at peak
densities "the Chitty e}ect# has been suggested to be a
074
P[ Inchausti + L[R[
Ginzburg
distinctive feature of vole cycles "e[g[ Krebs + Myers
0863^ Stenseth + Ims 0882#[ Attempts to use body
weight as a surrogate for individual quality were hin!
dered by the acute shortage of published time series
of body weights[ It must be pointed out that the esti!
mation of parameters in the maternal!e}ect model
would require simultaneous data of population abun!
dance and body weight[ Published time series of body
weight and abundance for northern European voles
are too short to allow parameter estimation[ There!
fore\ it was decided to keep the average quality vari!
able implicit in the model\ bearing in mind the e}ects
of average individual quality on the population
growth[
Behaviour of the maternal!effect model
The annual dynamics of the maternal!e}ect model is
de_ned by substituting speci_c expressions for func!
tions G and H in each breeding period[ The following
expressions portray the shapes of the functions G and
H "Figs 0 and 1# for the spring:autumn transition]
GS:A RSXS
aS¦XS
HS:A XS
$
eqn 2a
%
MS"S:NA#
bS¦"S:NA#
eqn 2b
Functions GA:S and HA:S are analogous to eqns 2a
and 2b but with di}erent parameter values
"RA\ aA\ bA#[ RS is the maximum reproductive rate in
the springÐautumn transition\ and MS is the maximum
rate of increase of the average individual quality in
the same time period[ Lacking knowledge of the values
of M in di}erent seasons\ it is assumed that
MS MA M[ Parameters aS and bS control the rate
of increase of the functions G and H to their asymp!
totes RS and M respectively "Figs 0 and 1#^ low values
describe steep increases of functions G and H[ The
total amount of resources available in the environ!
ment is denoted as S and is assumed to be constant in
each transition[ Therefore\ S:NA is the per capita share
of resources in the spring:autumn transition[
Variables and parameters of eqns 2a and 2b can be
re!scaled in order to simplify the analysis in terms of
the dimensionless parameters[ Parameters aS\ bS and
S can be eliminated by dividing the numerator and
denominator of eqns 2a and 2b by aS and NAS and re!
expressing variables X and N and parameter M in the
appropriate units[ A similar procedure is adopted for
the equations for the autumn:spring transition[ The
dimensionless equations for the spring "year t#:aut!
umn "year t# transition are]
0
0
Þ 0887 British
Ecological Society
Journal of Animal
Ecology\ 56\ 079Ð083
1
1
RSXS\t J
0¦XS\t G
f
F
M
XA\t pXS\t
G
0¦NA\t j
NA\t NS\t
eqn 3a
and for the autumn "year t#Ðspring "year t¦0#]
0
0
1
1
RFXA\t J
0¦XA\t G
f
F
M
XS\t¦0 XA\t
G
0¦NS\t¦0 j
NS\t¦0 NA\t
eqn 3b
Equations 3a and 3b generate neutral cycles\ that
is\ oscillations whose amplitudes and\ to a lesser
extent\ the periods depend on the initial conditions[
Ginzburg + Taneyhill "0883# obtained similar results
for the model of forest Lepidoptera[ As a consequence
of its neutral behaviour\ the maternal!e}ect model
cannot predict the actual value of cycle amplitudes[
Equations 3a and 3b can be modi_ed in a number of
ways to generate true limit cycles[ Of course\ these
modi_cations would require adding at least one extra
parameter to the model[ The paucity of data on indi!
vidual quality "or surrogates such as body size# does
not allow choice of the single modi_cation of the
model structure that best achieves the goal[ It may\
for instance\ be assumed that a minimum quality
"Xmin# exists for the winter period such that repro!
duction ceases when the average individual quality\
X\ falls below Xmin "Figs 0 and 1#[ This seemingly
minor modi_cation of the model will transform its
behaviour from neutrality to limit cycles[ The present
study chose to sacri_ce the comparisons of amplitudes
and focus on cycle periods that are much better char!
acterized for northern European voles "Henttonen et
al[ 0874^ Turchin 0882#[
PARAMETER ESTIMATION
The maximum rates of population growth\ RS and
RA\ were estimated from time series of population
abundance of northern Europe "between 43>N and
69>N# where voles are known to exhibit regular oscil!
lations[ Only studies in which censuses were carried
out both in spring and autumn for at least 09 years
and which had clear periodicity in an autocorrelation
analysis\ were used for parameter estimation[ Par!
ameters RS and RA for each locality and vole species
were the maximum of the realized growth rates in each
breeding period as calculated by the ratios "NAutumn:
NSpring# and "NSpring:NAutumn#\ respectively "Table 0#[
Two caveats should be mentioned about the esti!
mation of RS and RA[ First\ the trap index\ usually
total capture per 099 trap nights\ was assumed to be
proportional to the actual population abundance\ as
evidence from saturation trapping suggests "e[g[
Hanski\ Henttonen + Hansson 0883^ Jedrzejewski et
al[ 0885#[ Second\ those censuses for which the authors
gave an arbitrary low value of the trap index\ usually
during population crashes\ were discarded and only
the remaining censuses were used for parameter esti!
mation[
The maximal annual population growth rates "cal!
culated as the geometric average of RA and RS for
each species and locality# tend to decrease weakly with
075
Small mammals
cycles
Table 0[ Observed dynamics of 02 time series of northern European voles[ Latitudes are given as reported in the original
papers "latitudes for Mari and Udmurt were not provided and hence ranges are indicated#[ The maximum population growth
rates for the spring:autumn "RS# and autumn:spring "RA# were calculated as the maximum of the realized growth rates
"NAutumn:NSpring# and "NSpring:NAutumn#\ respectively[ RAVG is the geometric average of RS and RA[ The observed periods "Tobs#
were determined as the dominant lag of the autocorrelogram of the ln!transformed autumn "or the trapping date closer to the
autumn# trap index\ as suggested by Henttonen et al[ "0874#[ refers to the approximate latitude of a location for which the
actual latitude was not provided in the original reference
Species
0[ Clethrionomys glareolus
1[ Microtus agrestis
2[ Clethrionomys rufocanus
3[ Clethrionomys glareolus
4[ Microtus agrestis
5[ Clethrionomys rufocanus
6[ Clethrionomys rutilus
7[ Clethrionomys glareolus
8[ Clethrionomys glareolus
09[ Microtus agrestis
00[ Clethrionomys glareolus
01[ Clethrionomys glareolus
02[ Clethrionomys glareolus
Þ 0887 British
Ecological Society
Journal of Animal
Ecology\ 56\ 079Ð083
Location and
latitude ">N#
Time series
length "yrs#
RS
RA
Tobs
RAVG "yrs#
Pallasjarvi\
Finland "57#
Pallasjarvi\
Finland "57#
Pallasjarvi\
Finland "57#
Umea\ Sweden
"53#
Umea\ Sweden
"53#
Umea\ Sweden
"53#
Sotkamo\
Finland "53#
Sotkamo\
Finland "53#
Boda\ Sweden "50#
05
15[7
2[4
8[5
3
05
6[3
2[0
3[7
3
05
4[0
0[0
1[2
4
07
10[7
0[0
3[8
3
Henttonen et al[
"0876#
Henttonen et al[
"0876#
Henttonen et al[
"0876#
Hornfeldt "0883#
07
7[7
1[6
3[8
3
Hornfeldt "0883#
07
00[0
0[2
2[6
3
Hornfeldt "0883#
09
3[8
2[0
2[8
3
09
7[6
0[8
3[0
3
17
6[4
2[3
4[0
3
08
8[8
1[8
4[3
3
Henttonen et al[
"0866#
Henttonen et al[
"0866#
Marcstrom et al[
"0889#
Lindstrom "0883#
03
07[3
1[6
6[0
2
Zhigalski "0881#
04
6[7
1[2
3[1
2
Zhigalski "0881#
05
09[4
2[4
5[1
3
Zhigalski "0881#
Grimso\ Sweden
"48#
Udmurt\ Russia
"45Ð48#
Mari\ Russia
"45Ð46#
Tula\ Russia
"43#
latitude "Fig[ 2#\ except for C[ glareolus at Pallasjarvi\
Finland "57>N#[ This pattern may seem inconsistent
with the observed latitudinal increase of litter size
"Stenseth et al[ 0874^ Innes + Millar 0883#\ a par!
ameter sometimes used as an index of population
growth rate[ This contradiction disappears when one
notices that the length of the main breeding season
and the rate of sexual maturation\ two parameters that
ultimately determine the number of litters\ decrease
toward northern latitudes "Stenseth et al[ 0874^ Hans!
son + Henttonen 0874#[ In a recent review of micro!
tine life history\ Innes + Millar "0883# found a nega!
tive correlation between the length of the main
breeding period and the litter size[ All other things
being equal\ a moderate decrease in the number of
litters may more than o}set a concurrent increase in
litter size and result in a lower population growth rate[
The apparent trade!o} between the number and size
of litters across Fennoscandia may explain the lati!
tudinal trend in the maximum annual realized popu!
lation growth rate obtained for northern European
voles[
Source
Fig[ 2[ Annual maximum population growth rates as a func!
tion of latitude[ The annual growth rates were calculated as
the geometric average of RS\ the maximum population
growth rate in spring:autumn and RA\ the maximum popu!
lation growth rate in autumn:spring[ The growth rates RS
and RA were calculated as the maximum of the realized
growth rates "NAutumn:NSpring# and "NSpring:NAutumn#\ respec!
tively\ for the time series of vole abundance listed in Table
0[ The dotted line indicates the tendency of the growth rates
to decrease with latitude[
076
P[ Inchausti + L[R[
Ginzburg
Þ 0887 British
Ecological Society
Journal of Animal
Ecology\ 56\ 079Ð083
Results
The predicted dynamics obtained by sequentially
applying eqns 3a and 3b is oscillatory\ often being
quasi!periodic^ that is\ the cycles have fractional
values for their periods[ Cycle periods decrease
towards a minimum of 2 years as the average maximal
growth rates increase "Fig[ 3a#\ covering the range of
periods "2Ð4 years# observed for northern European
voles[ This intuitively obvious behaviour of the
maternal!e}ect model\ that species with higher growth
rates should pass through their cycles faster\ has
recently provoked some controversy "Berryman 0884^
Ginzburg + Taneyhill 0884#[ The periods of oscil!
lation predicted by the maternal!e}ect model agree
with those of most vole time series across northern
Europe "Fig[ 3a and b#[ Since the maximum popu!
lation growth rates roughly tend to decrease with lati!
tude "Fig[ 2#\ the maternal!e}ect model qualitatively
predicts the latitudinal gradient of periodicity of
northern European voles "Table 0#[
The strength of the inverse relation between cycle
period and the average maximal growth rate is modu!
lated by M\ the maximum increase in individual qual!
Fig[ 3[ "a# Predicted cycle periods by the maternal!e}ect model "eqns 3a and 3b# as a function of the average maximum growth
rate "calculated as the geometric average of RS\ the maximum population growth rate in spring:autumn and RA\ the maximum
population growth rate in autumn:spring#[ The curves are for di}erent values of the maximum increase in average individual
quality\ M[ "b# Observed cycle periods of the northern European vole time series as a function of average maximum growth
rate "average R#[ The dotted line indicates the tendency of vole cycle periods to decrease with R[ The numbers in the plot
correspond to the entries of Table 0[ "c# Observed cycle periods of the northern European vole time series as a function of the
scaled equilibrium population abundance\ estimated as the median of the autumn and spring trap indices for each series[
Median population abundance is roughly proportional to the maximum rate of increase of individual quality\ M "see Appendix
0#[ The dotted line indicates the tendency of vole cycle periods to decrease with M[
077
Small mammals
cycles
ity] higher M produces greater sensitivity of cycle per!
iods to changes in average maximal growth rate "Fig[
3a#[ The maternal!e}ect model predicts that\ for any
population growth rate\ the cycle periods should
decrease as the average value of M increases "Fig[
3a#[ As explained above\ there is no direct way of
estimating the maximal rate of increase of average
individual quality[ The scaled equilibrium population
size\ N\ is predicted to be proportional to M "see
Appendix 1#[ Assuming that the median population
trap index "including both spring and autumn cen!
suses# for each locality and vole species crudely esti!
mates N\ cycle periods are\ in fact\ negatively related
to the value of M "Fig[ 3c#\ as predicted by the model[
The exact value of the maximum rate of increase of
average individual quality "M# does not a}ect cycle
periods but does a}ect the amplitudes when M was
varied between 4 and 04 "roughly coinciding with the
range of values of M estimated by Ginzburg + Taney!
hill 0883# for any given pair of values of RS and RA[
In biological terms\ high values of M imply stronger
maternal e}ects\ which translate into stronger\
delayed density dependence and shorter cycles for a
given growth rate "Fig[ 3a#[ Bjo
rstad et al[ "0884#
obtained similar results when they used a linear auto!
regressive model to describe vole dynamics in Fenno!
scandia[
The typical oscillations generated by the maternal
e}ect model "Fig[ 4# are often asymmetric in time] the
increase phase lasts longer than the decline to low
abundance "see Appendix 2 for details#[ This quali!
tative feature of vole cycles has been observed in
northern Europe "Fig[ 5^ see Ginzburg + Inchansti\
0887#[
Discussion
Þ 0887 British
Ecological Society
Journal of Animal
Ecology\ 56\ 079Ð083
A key feature for understanding the causes of cyclicity
of northern European voles is the identi_cation of the
factor"s# causing delayed density dependence[ These
factors can rarely be deduced from theoretical con!
siderations\ leaving careful\ long!term experimental
manipulations of vole environments as the most
promising approach to identifying the causes of
delayed density dependence[ Thus far\ the exper!
imental manipulations involving food addition and
predation have not prevented the occurrence of cycles\
suggesting that extrinsic factors may not be necessary
for the occurrence of vole cycles[ Extrinsic factors
may\ however\ act as modi_ers of the delayed density!
dependent process by primarily a}ecting the cycle
amplitudes "Henttonen et al[ 0876^ Desy et al[ 0889#
and\ marginally\ their periods[
Intrinsic factors involving changes in individual
di}erences or quality are likely to be the primary cause
of delayed density dependence "Chitty 0859^ Hansson
0873b^ Krebs 0885#[ There is ample evidence indi!
cating that changes in individual quality occur during
vole cycles[ During density peaks\ the population age
and size structure shift toward older\ heavier animals
"Hansson 0873b#[ In contrast\ animals are lighter "Lid!
icker + Ostfeld 0880#\ less aggressive "Krebs 0874#\
grow more slowly\ and have lower juvenile viability
and reproductive output "Krebs + Myers 0863# during
declines[ The concomitant changes of individual qual!
ity and population density observed during the cycles
point to the existence of a dynamic interaction
between these variables that\ in the authors| judge!
ment\ is crucial for the understanding of vole cycles[
It is proposed in the present study that the pheno!
typic transmission of individual quality is an intrinsic
mechanism capable of causing vole cycles[ According
to our interpretation\ average individual quality refers
to the energetic value of individuals as a function of
their resource share in the current breeding period[
Although the de_nition is based on energy\ it should
be clear that individual quality also encompasses other
features that parents can provide for their o}spring
such as nutrients\ hormones\ enzymes\ immune resist!
ance\ symbionts\ pathogens\ toxins\ etc[ "see examples
in Rossiter 0885#[ Because the transmission of quality
takes place at the time scale of reproduction\ maternal
e}ects produce delayed density dependence\ which
may cause vole cycles[
The maternal!e}ect hypothesis may be exper!
imentally falsi_ed by observing the response in vole
dynamics to the experimental manipulation of indi!
vidual quality[ The spatial scale at which such manipu!
lative _eld experiments should be performed might
seem daunting\ but this should not deter their con!
ceptual formulation[ The obvious _rst requirement of
such a research programme is to _nd a variable that
adequately characterizes individual quality "see Chitty
0885 and Rossiter 0881 for a discussion#[ At present
it is unclear whether body weight is an adequate
descriptor of vole individual quality[ One experiment
could involve altering the distribution of "and thus
the average# individual quality through the long!term
selective removal of high!quality individuals[
Recently\ Moss\ Watson + Parr "0885# experimentally
prevented a red grouse population from cycling by
changing the age structure and lowering the total
population density through the long!term removal of
territorial cocks[ In terms of the maternal!e}ect
model\ these selective removals that rendered a youn!
ger age distribution can be interpreted as an increase
in the average individual quality\ since the remaining
individuals maintained a relatively high and constant
recruitment rate throughout the duration of the
removals "see Moss et al[ "0885#\ pp[ 0415Ð0416#[
Cycles could also be prevented by minimizing the
changes in average individual quality[ This might be
achieved by adding increasing amounts of food in
order to maintain a high\ nearly constant resource
ratio "S:N# as population density increases during the
cycle[ In the experimental population\ a high "S:N#
ratio would translate into a high individual quality\
which would allow it to expand until some limiting
078
P[ Inchausti + L[R[
Ginzburg
Fig[ 4[ Typical cycles produced by the maternal!e}ect model "eqns 3a and 3b# as time plots[ Initial spring population abundance
and average individual quality\ NS"9# and XS"9#\ are expressed in relation to their equilibrium values "N and X^ see Appendix
0#[ Five hundred iterates were simulated but only the last 64 are shown[ "a# 2!year cycles] RS 07\ RA 1 "avg[ R 6[23#\
M 04\ NS"9# 1N\ XS"9# X[ "b# 3!year cycles] RS 01\ RA 1 "avg[ R 3[28#\ M 09\ NS"9# 1N\ XS"9# 09X[
"c# 4!year cycles] RS 5\ RA 1 "avg[ R 2[35#\ M 4\ NS"9# 1N\ XS"9# 09X[
Þ 0887 British
Ecological Society
Journal of Animal
Ecology\ 56\ 079Ð083
factor other than food "e[g[ nesting sites# curbs the
increase[ In this population\ population growth rate
would be independent of changes in individual quality
and regulated in a direct density!dependent manner[
As it was mentioned in the introduction\ a _xed
amount of food was generally added to food!addition
experiments[ This type of manipulation cannot be
expected to alter individual quality during the density
cycle[ The reason is that the positive e}ects of food
addition on reproduction are likely to be overridden
by negative e}ects from decreasing per capita resource
supply as population density increases[ Assuming a 3!
year cycle\ the best time to begin the proposed experi!
ment would be in the spring of the second year of the
cycle and to continue it for 2 years[ If the maternal!
e}ects hypothesis is correct\ the population would
increase at a high rate for 1 years and would {miss|
the decline expected to occur during the second year
of the experiment\ the fourth year of the cycle[
Another experiment may consist of the reciprocal
translocation of high and low quality individuals of
cyclic populations into di}erent enclosures of a com!
mon habitat[ The goal is to reveal whether vole
dynamics and the individual quality have an immedi!
ate or a delayed response to the new environmental
conditions\ depending on the original individual qual!
ity[ If maternal e}ects are unimportant\ the new
environment is expected to have an overriding in~u!
089
Small mammals
cycles
Fig[ 5[ Selected time series plots and correlograms of Ln!transformed autumn population abundance "voles:099 trap nights#
for northern European voles to illustrate the increase in cycle periods with latitude and the asymmetric nature of high!
amplitude cycles[ Mari data from Zhigalski "0881#\ Umea data from Hornfeldt "0883#\ and Pallasjarvi data from Henttonen
et al[ "0876#[
Þ 0887 British
Ecological Society
Journal of Animal
Ecology\ 56\ 079Ð083
ence on individual quality and thus the dynamics of
the translocated voles would di}er from those exhi!
bited by resident voles[ Even though this experiment
would ideally require out!of!phase populations "rarely
found in nature^ Marcstrom et al[ 0889#\ it could be
nonetheless performed by transferring and following
the dynamics of voles of low and high quality in
enclosures of the new habitat[
A detail worth mentioning in the time!lagged ver!
sion of the maternal!e}ect model "Appendix 2# is the
length of delays involved[ Assuming that the spring:
autumn period approximately lasts 4 months in
Fennoscandia "Stenseth et al[ 0874#\ the autumn abun!
dance depends on the population abundance 4 and
01 months before\ and the spring abundance on the
population abundance 6 and 01 months before[ May
"0865# showed that a mechanism!independent model
based on a delay!di}erential equation with a _tted
timelag of 8 months can produce cycles similar to
those of northern European voles[ Using auto!
correlation analysis\ Hornfeldt "0883# detected a sig!
ni_cant time!lag of 8 months in the population growth
rate of three vole species in northern Sweden[ The
transmission of maternal e}ects may be a mechanism
that can account for time!lags of about 8 months
"Hansson 0876\ p[ 205#\ as depicted in the time!lagged
version of the model "Appendix 2#[
The mathematical structure of the maternal!e}ect
model can also be used to represent other intrinsic
mechanisms or other ecological interactions\ as long
as they a}ect the population growth rate similarly^ for
instance\ the variable X may represent the average
080
P[ Inchausti + L[R[
Ginzburg
resource quality that changes as a function of the
concurrent vole abundance[ Resource quality has
been shown to be transmitted over time and to a}ect
vole population growth rates "Agrell et al[ 0884#[ Para!
sitism and the indirect e}ects of predation on foraging
and reproductive behaviour may be similarly rep!
resented[ These interactions may a}ect population
growth rate via qualitative changes in individual attri!
butes without directly causing a population decline[
The ultimate resolution of the {small mammals enig!
ma| can only come from manipulative experiments
involving both intrinsic and extrinsic factors[ While
unable to prove beyond reasonable doubt that vole
cycles are primarily caused by intrinsic factors\ we feel
that the weight of the evidence points toward them
as essential\ making the maternal!e}ect hypothesis a
strong contender as an explanation of microtine
cycles[
Acknowledgements
We thank Peter Turchin for generously providing us
with data and for useful discussions\ Karen Goodell
and Charles Janson for reading and improving earlier
drafts\ and Karen Kernan for her careful editing of
the manuscript[ This is the contribution No[ 888 from
the Department of Ecology and Evolution\ SUNY at
Stony Brook[
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Þ 0887 British
Ecological Society
Journal of Animal
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minant of vole cycles[ Ecology\ 64\ 680Ð795[
Horton\ T[ "0873# Growth and reproductive development
of male Microtus montanus is a}ected by the prenatal
photoperiod[ Biology and Reproduction\ 20\ 388Ð493[
Horton\ T[ "0874# Cross!fostering of voles demonstrates in
utero e}ect of photoperiod[ Biology and Reproduction\ 22\
823Ð828[
Innes\ D[ + Millar\ J[ "0883# Life histories of Clethrionomys
and Microtus[ Mammalian Review\ 13\ 068Ð196[
Jedrzejewski\ W[\ Jedrzejewski\ B[\ Szymura\ A[ + Zub\ K[
"0885# Tawny owl "Strix aluco# predation in a pristine
deciduous forest "Bialowieza National Park\ Poland#[
Journal of Animal Ecology\ 54\ 094Ð019[
Kaikusalo\ A[ + Tast\ J[ "0873# Winter breeding of microtine
rodents at Kilpisjarvi\ Finish Lapland[ Winter Ecology of
Small Mammals "ed[ R[ Merrit#\ No[ 09\ pp[ 132Ð141[
Carnegie Museum of Natural History\ Pittsburg[ PA[
Krebs\ C[ "0867# A review of the Chitty hypothesis of popu!
lation regulation[ Canadian Journal of Zoology\ 45\ 1352Ð
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Krebs\ C[ "0874# Do changes in spacing behaviour drive
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Krebs\ C[ "0885# Population cycles revisited[ Journal of Mam!
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Krebs\ C[ + Myers\ J[ "0863# Population cycles in small
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Lack\ D[ "0843# The Natural Regulation of Animal Numbers[
Oxford University Press\ Oxford[
Lidicker\ W[ + Ostfeld\ R[ "0880# Extra!large body size in
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50\ 097Ð010[
Lindstrom\ E[ "0883# Voles cycles\ snow depth and predation[
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Marcstrom\ V[\ Hoglund\ N[ + Krebs\ C[ "0889# Population
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082
P[ Inchausti + L[R[
Ginzburg
Appendix 0
The maternal!e}ect model postulates that changes in
individual quality a}ect the population growth rate
without distinguishing whether those e}ects resulted
in changes in survival and:or fecundity[ By modelling
vole population dynamics as an age!structured
process\ one could consider the occurrence of\ and
possible trade!o}s between\ age!speci_c maternal
e}ects[ Assuming that the spring:autumn period
lasts about 4 months and that four cohorts can be
produced during this period\ the time step for each
transition matrix of the main breeding period is
approximately 4 weeks[ It is assumed that only one
cohort can be produced during the autumnÐspring
transition\ for which there is a matrix similar to that of
the main breeding period\ but with di}erent parameter
values[ Thus\ the annual dynamics consisted of apply!
ing four consecutive times the spring:autumn
matrix\ followed by the autumn:spring matrix once[
From the annual dynamics thus obtained "Fig[ A0#\
the autumn population abundance is shown after the
main breeding season\ as is customarily done for Fen!
noscandian voles[
The matrices governing these demographic tran!
sitions have the following structure]
0
9
F"X#
Sjuv"X#
Sadult"X#
1
The actual values of juvenile survival "Sjuv#\ adult sur!
vival "Sadult# and fecundity "F# depend on individual
quality[ Fecundity changes according to eqn 2a\ after
substituting F for the maximum growth rate R[ Juv!
enile and adult survival are assumed to monotonically
increase with individual quality from a minimum
value\ according to]
S"X# S¦a
0 1
X
0¦X
The multiplier a "0−S# ensures that survival can
Þ 0887 British
Ecological Society
Journal of Animal
Ecology\ 56\ 079Ð083
never become greater than 0 in the model[ Note that
the overlap of generations in this age!structured mode
occurs because adults can survive longer than one
demographic transition "i[e[ Sadult × 9# in each breed!
ing period[ The dynamics of individual quality X dur!
ing each demographic transition "i[e[ four in the main
and one in the minor breeding period# remains as
in eqns 3a and 3b with Ntot Nadult¦Njuv[ The fully
~edged\ age!structured model for the annual dynamics
has a minimum of eight parameters] six demographic
rates and the maximum rates of increase in individual
quality "M# for each breeding period[ Trade!o}s
between quality!mediated changes in the demographic
rates can\ of course\ exist but they are not considered
here[ There are several di.culties in analysing such a
complex model[ First\ some demographic parameters
such as juvenile survivals and the number of litters
"e[g[ Krebs + Myers 0863^ Boonstra 0874# are very
di.cult to estimate in the _eld\ and especially during
the winter months[ Thus\ unavoidable guesses about
the magnitudes and interdependence of parameters
need to be made[ Second\ as the number of parameters
increases it becomes more di.cult to understand how
their variation a}ects the model dynamics[ Adequate
exploration of the parameter space is cumbersome for
high dimensional models[ Within the level of uncer!
tainty of the parameter estimates of the age!structured
model\ it is possible to select combinations of reason!
able parameter values that can produce 3!year cycles
similar to those of the unstructured model "eqns 3a
and 3b#[
We regard this example to be just an indication
of plausibility that an age!structured version of the
maternal!e}ect model can indeed produce cyclic
dynamics[ The complexity of models should be con!
strained by the availability of _eld information that
can yield reasonably accurate parameter estimates to
minimize the risk of over_tting the data[ We believe
that the simpler version of the maternal!e}ect model
"eqns 3a and 3b# captures the essence of the proposed
Fig[ A0[ Four!year cycles produced by the age!structured version of the maternal!e}ect model[ The actual values of juvenile
survival "Sjuv# and adult survival "Sadult# and fecundity "F# depend on the individual quality "see text for details#[ The dynamics
of individual quality X during each breeding period as in eqns 3a and 3b[ Parameter values used were FS:A 2[4^ Sadult\S:A 9[5^ Sjuv\S:SA 9[5^ FA:S 1^ Sadult\A:S 9[2\ Sjuv\A:S 9[01\ and M 09 for both breeding periods[
083
Small mammals
cycles
mechanism "i[e[ the population growth rate changes
as a function of the average individual quality#\ while
keeping the model complexity at a level that is com!
mensurate with our ability to estimate parameter
values[
average individual quality# on vole population
dynamics[ After substituting the righthand side of the
_rst into the second equation of each pair "eqns 3a
and 3b#\ one obtains for the spring "year t#:autumn
"year t# transition]
0 1
M
Appendix 1
NA\t The equilibrium population size\ N\ of the maternal
e}ect model can be found by substituting the functions
HS:A and HA:S "eqn 2b# into the second equation of
eqn 0a and 0b\ obtaining]
0
XS\t¦0 XS\t
MS"S:NS\t¦0#
bS¦"S:NS\t¦0#
10
MA"S:NA\t#
bA¦"S:NA\t#
1
$ 0 1%
RA−
F0 NS\t?
N X0
1 0
1 0
1
eqn A1
Assuming that MS MA M "see Behaviour of
the maternal e}ect model# and using a dimensionless
model "eqns 3a and 3b#\ the scaled equilibrium popu!
lation abundance is then N M−0[
Similarly\ the equilibrium average individual qual!
ity\ X\ can be found by substituting the functions
GS:A and GA:S "eqn 2a# into the second equation of
eqns 0a and 0b\ obtaining]
X X
"RSRA#¦
aSaA 1
aS¦aA
−
1
1
0 1 0
1
eqn A2
which\ in the dimensionless version of the model "eqns
3a and 3b# becomes X z"R
SRA#−0[
Appendix 2
To elaborate further on the delayed density depen!
dence produced by the transmission of individual
quality\ eqns 3a and 3b can be expressed in the more
traditional timelagged form[ This alternative form of
eqns 3a and 3b can be used to represent the e}ect
of the unobserved or {hidden| variables "such as the
Þ 0887 British
Ecological Society
Journal of Animal
Ecology\ 56\ 079Ð083
NS\t
NA\t−0
= RSNS\t
eqn A3
NS\t¦0 After multiplying by S:NS\t¦0 and S:NA\t inside the
parentheses and solving for N]
0 0
0
0 0
0
MS MA
¦
−
−
¦
bS bA
3 bS bA
1 bA bS
0 1
0
1
NS\t
NS\t
"0¦NS\t#¦M
NA\t−0
NA\t−0
and for the autumn "year t#:spring "year t¦0# tran!
sition]
eqn A0
1
NS\t
NA\t−0
M
$ 0 1%
NA\t
RS−
NS\t−0
0 1
NA\t
NS\t−0
"0¦NA\t#¦M
0
0 1
NA\t
NS\t−0
F1 NA\t?
= RANA\t
1
NA\t
NS\t−0
eqn A4
Notice that growth rate functions F0 and F1 governing
the transitions between successive seasons depend on
both the population density and the realized popu!
lation growth rate "i[e[ the ratio of population den!
sities of the two previous seasons\ see Ginzburg\ 0887#[
The type of delayed density dependence produced
by the maternal!e}ect model is slightly di}erent from
traditional discrete delayed density!dependent models
of the form Nt¦0 NtG"Nt\Nt−0#[ One consequence
of such di}erences is the ability of the maternal!e}ect
model to generate asymmetric cycles "Fig[ 4#[ This
asymmetry arises because in the maternal!e}ect model
the realized population growth rates in eqns 3a and
3b are only constrained for positive values "i[e[ when
population is increasing# to be smaller than the maxi!
mal rate R but are unconstrained for negative values
"i[e[ when population declines#[ As a result\ population
declines often take less time than the previous
increases[ This asymmetric property of cycles is par!
ticularly evident for large!amplitude oscillations\ as
can be observed for several species of vole in northern
Europe "Fig[ 5#[