Z theor BioL (1988) 131, 389-400
Competition Between Age Classes and Population Dynamics
Bo EBENMAN
Department of Biology, LinkiJping University, S-581 83 Link6ping, Sweden
(Received 3 July 1987, and in revised form 11 November 1987)
In many organisms, t.he generations overlap. It is shown, from a theoretical point
of view, that competition between the different age classes in such organisms will
have important population dynamics consequences. Strong competition between
age classes can be either a stabilizing or destabilizing force depending on the stage
of the life cycle--juvenile survival or adult survival/reproduction--at which density
dependence occurs. When juvenile survival is density dependent, competition has
a stabilizing effect. On the other hand, when reproduction is density dependent,
strong competition tends to be destabilizing, i.e. it will cause population density to
fluctuate (limit cycles, "chaos"). It is also shown that delayed effects on reproduction,
caused by strong competition as the adults grow up, tend to be destabilizing. These
findings have interesting implications for the population dynamic behaviour in
organisms with different types of life cycles, and it is discussed how these theoretical
findings might be relevant to the interpretation of patterns of population dynamic
behaviour documented in field and laboratory studies published by other ecologists.
Introduction
Studies in theoretical population ecology have shown that relatively simple deterministic models of population growth can lead to very rich dynamic behaviour
including stable equilibrium points, limit cycles and "chaos" (May, 1974, 1976;
Schafter, 1985; see Nisbet & Gurney, 1982 for a review). Somewhat more complicated
models that take into account individual variation (phenotypic structure) within
populations, have also been formulated and analysed (Oster et al., 1976). One
obvious source of variation is individual differences due to age (or more generally
physiological age) (Beddington, 1974; Guckenheimer et al., 1977; Van Sickle, 1977;
Murphy, 1983; Nisbet & Gurney, 1982; Metz & Diekmann, 1986). These structured
single species models can also result in very rich dynamic behaviour. Rich dynamic
behaviour due to internal factors has also been documented empirically (e.g.
Nicholson, 1950; Gurney et al., 1980; Murdoch & McCauley, 1985; McCauley &
Murdoch, 1987).
An important, but little studied (May et al., 1974; Tschumy, 1982; Bellows, 1982a,
b; Ebenman, 1987), process of frequent occurrence that may influence the dynamic
behaviour in populations with overlapping generations is competition between age
classes. With the help of analytical methods, I complement, generalize and extend
the results reached by Ebenman (1987). I examine how the intensity of competition
(as measured by the magnitude of the competition coefficients) between age classes
influences population stability by the aid of a general discrete-time model with two
age classes. I show that strong competition between juveniles and adults can be
either a stabilizing or destabilizing force depending on the stage of the life cycle at
389
0022-5193/88/080389+ 12 $03.00/0
O 1988 Academic Press Limited
390
a. EBENMAN
which density dependence occurs. The ecological implications o f the theoretical
results are discussed in some detail.
Analysis
The model to be analysed is a general discrete-time model with two age classes:
No( t + 1) = N~( t)Bf[ N~( t) + aN0(t)]
Nt( t + 1) = No( t)Sg[ No( t) + flN~( t) ].
(1)
Here No and N~ are the densities of age class 0 and l, respectively, B is the maximum
per capita birth rate and S is the maximum survival rate (i.e. the vital rates in the
absence of density dependence). The functions f and g give the effects of the effective
population densities, E~ = N~ + aNo and Eo = No+/3N~, respectively, on per capita
birth rate and survival rate, respectively. By definition df/dE~ < 0 and dg/dEo<O.
The competition coefficient a measures the relative depression in per capita birth
rate, caused by an individual of age class 0 compared to another of age class 1.13
measures the relative depression in per capita survival rate, caused by an individual
of age class 1 compared to one of age class 0.
An equilibrium point No*, N * satisfies
N* Bf( N*~ + a N * ) - No* = 0
(2)
N*o Sg( NO*+ fiN*) - N* = 0
DENSITY-DEPENDENT
REPRODUCTION
Now consider the case when survival is density independent (i.e. g( No + flN~) =- 1).
This gives
N*=SNO*
and
SBf[(S+a)N*]=I.
(3)
The local stability of the equilibrium point can be determined by the conventional
techniques of linearized stability analysis. When g ( N 0 + flN~)=- 1, the condition for
stability is
- N * S B a f ' [ ( S + a)No*] < - No*S2Bf'[(S + a)No*] < 2
(4)
where f ' is the derivative of the function f. The left part of the condition simplifies
to
c~<S.
(5)
- N * S 2 B f ' [ ( S + a ) N * ] = F < 2,
(6)
The right part,
is not so straightforward. From equation (3) we have it that f [ ( S + a)No*] = 1/SB.
Now, let ( S + a ) N o * = C be the solution to this equation. Then No*= C / ( S + a ) .
Condition (6) now becomes - [ C / ( S + a ) ] S : B f ' ( C ) = F < 2 , where f ' ( C ) is a negative constant. As can be seen, d F / d a < 0. Thus, increased competition between the
age classes has a stabilizing effect as long as a is smaller than S. The effect of
COMPETITION
BETWEEN
AGE
391
CLASSES
increasing a is to reduce the average time lag. (This can be seen by rewriting
equation (1) with g = 1 in the form N t ( t + 2 ) = SBNt(t)f[N,(t)+(a/S)N,(t+ 1)].)
The condition on B for stability to prevail then becomes less restrictive. (For a more
penetrating and general discussion of the effects of distributed time lags on stability
see MacDonald (1978).) Table 1 gives the stability condition (4) for some different
forms for the function f. Figure 1 shows the parameter space that corresponds to
stable, positive equilibrium points for f = exp ['-k(N, + otNo)]. It can be seen that
an increase in a increases the domain of stable B values as long as a is smaller
TABLE 1
Specific formulae for the functions f( Nl + clNo) and g( No + tiNt) and corresponding
conditions for stability of equilibrium points of equation (1)
f ( N, + aNo)
g( N o + ~N, )
[ - k ( Nt + aNo) ]
[ l + a ( N , + a N o ) ] -b
exp{-[k(N, + aNo)] °}
Stability condition
1
1
1
exp
i S [ I n (SB) - 2 ] < a < S
S{½b(SB)I/b[(SB)t/h - 1] - 1} <
S[½0 In (SB) - 1] < ~x < S
a<S
1 - c(N, + aNo)
1
S[½(SB-1)-I]<~<S
1
exp [-k'(No+tiN,) ]
½B[ln (sB) -2] < ~ < B
1
1
[1 + a'(No+ tiNt)]-""
B{½b'(SB) t / v [(SB) l / b ' - 1] - l } < t i < B
exp{ - [k'( N O+ tiN t)]"'}
l-c'(No+tiN,)
B[½O'ln (Be)- 1 ] < ~ < B
B[½(SB-I)- 1] </8 < S
1
Note: k, a, b, c, 0 and k', a', b', c', 0' are measures of the density dependent feedback on reproduction
and juvenile survival, respectively. For definitions of the other parameters see the text.
30Ia
(b)
)
45
4O
35
3O
I0
25
5
0
~5
I-0
0
0.5
1.0
FIG. 1. Stability of equilibrium points when reproduction is density dependent and of the form f = exp
[ - k ( N I +aNo)]: (a) maximum survival rate is high, S =0.9; (b) maximum survival rate is low, S = 0 . 3 .
Shaded areas correspond to stable, positive equilibrium points.
B.EBENMAN
392
than S. Thus, when survival is high, c o m p e t i t i o n b e t w e e n the age classes tends to
have a stabilizing influence, whereas if survival is low, strong c o m p e t i t i o n has a
destabilizing influence.
DENSITY-DEPENDENT SURVIVAL
NOW, c o n s i d e r
the
case
when
reproduction
is density
independent
(i.e.
f(N~ +~3No)=-1). This gives
N*o = BN*
SBg[(B+/3)N*] = 1.
and
(7)
The condition for stability n o w b e c o m e s
-N*SBg'[(B+/3)N*] < -N*SB2g'[(B +/3)N*) < 2
(8)
where g' is the derivative o f the function g. T h e left part o f the condition simplifies
to
/3 < B.
(9)
For the right part,
-N*SB2g'[(B +/3)N*] = G < 2,
(10)
it can be shown, in the s a m e way as for condition (6), that dG/d/3<O. Thus,
c o m p e t i t i o n between age classes is a stabilizing process as long a s / 3 < B, which in
reality is p r o b a b l y always the case. Table 1 gives the stability condition for s o m e
ditIerent expressions for the function g. Figure 2 shows the p a r a m e t e r space that
c o r r e s p o n d s to stable, positive equilibrium points for g = exp [ - k ' ( N o + flN~)].
W h e n both survival a n d r e p r o d u c t i o n are density d e p e n d e n t , the condition for
stability b e c o m e s very complex. M o r e o v e r , it is in most cases not possible to solve
251(a]
15
15
IO
I
Ill
I
0"5
I-0
0
I
0"5
I
I'0
FIG. 2. Stability of equilibrium points when juvenile survival is density dependent and of the form
g = exp [-k'(No+flNl)]: (a) maximum survival rate is high, S = 0.9; (b) maximum survival rate is low,
S = 0.3. Shaded areas correspond to stable, positive equilibrium points.
COMPETITION
BETWEEN
AGE
CLASSES
393
analytically for N* and N*. However, in accordance with the analytical results,
computer simulations of specified systems (f=exp[-k(Nt+aNo)]
and g =
exp [-k'(No+/3N~)]) show that strong competition between the age classes has a
stabilizing effect when the density-dependent feedback on reproduction is small
compared to that on survival (k < k'), whereas if k > k', it has a destabilizing effect
(Ebenman, 1987).
Destabilization of an equilibrium point leads to limit cycles or some other kind
of complex dynamic behaviour (aperiodic orbits, chaos) about the unstable equilibrium point (see Fig. 4 and Ebenman, 1987). The outcome may also be competitive
exclusion of one cohort by the other. This will always be the case for a > S or 13> B
(i.e. when the left side of conditions (4) and (8), respectively, are violated).
There may also exist multiple steady states. This will only occur in a rather narrow
region of parameter space. It requires combinations of parameter values that probably seldom would be found in natural populations (high B- and S-values). However,
with the functions f and g of the form f=exp{-[k(N~+aNo)] °} and g =
exp{-[k'(No+/3N~)] e'} with 0 and 0 ' > 1 , i.e. when the effect of an increase in
effective population size is small at low densities and large at higher densities,
multiple steady states may be possible (see also Tschumy, 1982).
DELAYED
FROM
EFFECT
ON
COMPETITION
REPRODUCTION
IN THE JUVENILE
ARISING
STAGE
It is well known from many studies on invertebrates, amphibians and plants that
competition among juveniles may result in retarded individual growth rates and
reduced adult size (see Barker & Podger, 1970; Collins, 1980; Prout & McChesney,
1985; Wilbur & Collins, 1973; Weiner, 1985). The consequence of this is a reduced
birth rate, since fecundity depends on body size. The population dynamic consequences of such delayed phenomenon for populations, with non-overlapping generations,
have been studied from a theoretical point of view by Prout & McChesney (1985).
(For the case of overlapping generations see also Frauenthal, 1975.)
To complement the cases investigated above, I will now analyse how competition
between age classes affect the dynamic behaviour in populations experience such
delayed effects on reproduction. It is assumed that competition does not reduce the
survival of the juveniles, only their growth rates (this seems reasonable since it has
been documented to be the case in several studies, e.g. Sutherland, 1970; Haven,
1973; Underwood, 1976; Collins, 1980; Scheiring et al., 1984). Consider the case
when there is no direct effects of density on adult reproduction. The model will
then taken the form
No(t+ 1) = N,(t)Bh[No(t- 1)4- ~/N,(t- 1)]
N~(t+ 1) = No(t)&
(11)
An equilibrium satisfies
N*, = N*oS,
BSh[(I + yS)No*] = I.
(12)
394
B. E B E N M A N
NOW, assuming that the function h is of the form h = exp { - L [ ( 1 + ),S)No]}, gives
No* = In (SB)/[L(1 + yS)]. Here y is the competition coefficient and L is a measure
of the density-dependent feedback. The local stability of the equilibrium can then
be determined by application of the S c h u r - C o h n test. The condition for stability is
In(SB)3,S
l+yS
<
In (SB)
l + yS
<2-
(ln(SB)~/S~ 2
\ l + 3,S ] "
(13)
The left part of the condition simplifies to
V < 1/S.
(14)
The right part is more complicated. Figure 3 shows the parameter space that
corresponds to stable, positive equilibrium points. It can be seen, as in the case of
direct density-dependent reproduction (Fig. 2), that strong competition between
juveniles and adults tends to have a destabilizing effect. However, the condition on
the competition coefficient is not so hard in the case of a delayed effect on reproduction (arising from competition when the adults grow up) (3,< 1/S), as in the case
of a direct density-dependent reproduction (a < S). Violation of the right part of
condition (13) leads to limit cycles or some other kind of complex dynamic
3G
20
IO
o
0.5
y
~-0
1.5
FIG. 3. Stability of equilibrium points in the case of delayed effects on reproduction arising from
competition when the adults grew up. Shaded areas correspond to stable, positive equilibrium points.
Solid line: S = 0.3 and broken line: S = 0.9.
COMPETITION
BETWEEN
AGE
CLASSES
395
behaviour, whereas violation of the left part results in extinction of one of the
cohorts through competitive exclusion.
The case of delayed effects will not be developed further in this paper. To analyse
this phenomenon it would be more suitable to formulate a model based on size,
not age, since the effect, in most cases, is due to density-dependent growth rates in
the juveniles (see Murphy, 1983; Metz & Diekmann, 1986; Ebenman, 1988; Ebenman
& Persson, 1988).
Discussion
The following discussion will consider the biological implications of the results
presented above. (For a more technical discussion and comparisons with other
theoretical studies, the reader is referred to Ebenman, 1987.) I also discuss how the
theoretical findings might be relevant to the interpretation of patterns of population
dynamic behaviour documented in field and laboratory studies, published by other
ecologists.
The results that have been derived above have interesting implications for the
population dynamic behaviour in organisms with different types of life cycles. The
strength of competition (the magnitude of the competition coefficient) between age
classes/life stages will generally depend on the form of the life cycle. In most
organisms with complex life cycles there is a discrete niche shift, concomitant with
the metamorphosis (Wilbur, 1980). As a result, there is no competition between the
different life stages (a =/3 = 0). Examples are amphibians, holometabolous insects
and many marine invertebrates. (There are exceptions, e.g. some phytophagous
insects which do not undergo a discrete niche shift in connection with the metamorphosis. As a result the larvae and adults are potential competitors.) In organisms
with "simple" life cycles (i.e. organisms not undergoing metamorphosis), like
reptiles, fishes and hemimetabolous insects, the niche differences between the age
classes are in most cases not so pronounced. (Some representatives of these groups,
e.g. piscivorous fish, do however undergo rather abrupt niche shifts as they grow.)
Instead, individuals of different age classes partly overlap in their use of resources
(or > 0,/3 > 0) (Keast, 1977; Polis, 1984; Werner & Gilliam, 1984). Among birds and
mammals (which are almost fullgrown at the age of independence) there is seldom
any significant ecological segregation between the age classes (a and/3 large) (Keast,
1977).
The present models imply that populations of organisms with relatively intense
competition between age classes (e.g. organisms with simple life cycles), and where
the density-dependent feedback on juvenile survival is much greater than that on
reproduction, should be m o r e stable than those whose reproduction is highly density
dependent, but survival weakly so (Fig. 4c, d). Table 2 gives examples of field and
laboratory populations that conform to this prediction. (Field and laboratory data,
where all relevant parameters have been specified, and against which predictions
can be tested are very scarce. Ideally there should be data on the same species for
both regimes of density dependence, since stability will also depend on other factors,
like S and B, which may vary among species. All the cases listed in Table 2 conform
396
B. E B E N M A N
(a)
(b)
50O
2000
+
1500
400
300
2OO
I00
~ooo
500
I
I
I
400 c)
3O0
+
1
I
400~ (d)
SM
200
IO0~f
IOO
I
I
I0
20
Time
I
30
"~
I0
I
20
30
Time
FIG. 4. Dynamics of populations governed by equation (1) with f = exp [ - k ( N t + aNo) ] and g = exp
[ - k ' ( No+ fiN1) ]. (a) Organisms with no competition between age classes (a =/3 = 0, e.g. organisms with
complex life cycles) and where density dependence is primarily on reproduction (k > k'). Parameter
values: B = 50, S = 0.3, k = 0.01, k ' = 0.001 and a =/3 = 0. (b) Organisms with no competition between
age classes and where density dependence is primarily on juvenile survival (k < k'). Parameter values:
B = 50, S = 0.3, k = 0.001, k'= 0.01 and ~ =/3 = 0. (c) Organisms with competition between age classes
(a > 0,/3 > 0, e.g. organisms with simple life cycles) a n d where k > k'. Parameter values: B = 10, S = 0.6,
k = 0 . 0 1 , k'=0.001 and a =/3 =0.8. (d) Organisms with competition between age classes and where
k < k ' . Parameter values: B = 10, S = 0 . 6 , k = 0 . 0 0 1 , k ' = 0 . 0 1 and a = / 3 = 0 . 8 . In all cases the starting
values are N o = NI = 100. Note the dramatic differences between organisms with no competition and
strong competition between age classes: they show completely opposite d y n a m i c responses to density
dependence.
to the prediction. This is not due to a conscious selection of data. The cases shown
in the table represent all the studies known to me, where even crude estimates of
relevant parameters can be made.)
It is relevant here to mention the findings of Murdoch and coworkers (Murdoch
et al., 1984; Reeve & Murdoch, 1986; Murdoch et al., 1987) working with the pests
red scale and olive scale and their parasitoids. According to these authors, the factor
controlling the pest populations (the parasitoids) does not act in a density-dependent
way (however, see Huttaker et al., 1986 for an alternative position concerning the
olive scale). The red scale has relatively stable populations (DeBach, 1974). Reeve
& Murdoch (1986) point out that the generations overlap and that density-dependence is restricted to survival of first instars. Adults and juveniles are potential
competitors since they exploit the same resources. The olive scale, on the other
hand, has unstable populations (Murdoch et al., 1984). In this species, the generations do not overlap and hence there should be no competition between juveniles
and adults. According to the present theoretical analysis, stability of the red scale
may be due to reduced juvenile survival caused by competition from adults.
(However, see the work of Murdoch et al. and Huffaker et al. for alternative
explanations for the dynamic behaviour of these pest populations.)
COMPETITION
BETWEEN
AGE CLASSES
397
TABLE 2
Population dynamics of field and laboratory invertebrate populations
Strength of
competition
a =/3 = 0
a>0,/3>0
Density-dependent juvenile survival
Sheep blowfly, Lucillis cuprina:
population cycles (Nicholson, 1957).
Density-dependent adult
survival/reproduction
Sheep blowfly, Lucillia cuprina*:
relatively stable population
(Nicholson, 1957).
The weevil, Callosobruchus chinensis*:
stable populations (Utida, 1967; Fuji,
1968).
The weevil, Callosobruchus maculatus:
population cycles (Utida, 1967; Fuji,
1968).
The moth, Sitotraga cerealella: population
fluctuations (Crombie, 1944).
Cinnabar moth, Gyria jacobaeae:
population fluctuations (Dempster, 1971).
Colorado potato beetle, Leptinotarsa
The California red scale, Aonidiella
decemlineatat: population fluctuations
aurantii: stable populations (DeBacb,
(Harcourt, 1971).
1974; Reeve & Murdoch, 1986).
Cladoceran species, Daphnia¢.:
The beetle, Rhizopetha dominica§, stable
population fluctuations (Pratt, 1943;
population (Crombie, 1944).
Slobodkin, 1954; Frank et al., 1957;
Murdoch & McCauley, 1985;
McCauley & Murdoch, 1987).
The scorpion, Paruroctonus mesaensis:
stable population (Polis & Farley, 1980;
Polis, 1984).
* Density-dependent feedback on larval survival also.
§ Relatively weak density-dependent feedback on fecundity.
t Density-dependent adult emigration.
~tDensity-dependent juvenile growth rate also.
McCauley & Murdoch (1987) and Murdoch & McCauley (1985), in reviewing
the demography and population dynamics of field and laboratory populations of
Daphnia species, pointed out that average per capita fecundity was relatively
constant (no density dependence) in stable populations, whereas in cyclic populations, fecundity was strongly density dependent. Generations overlap and adults
and juveniles are potential competitors. The observed patterns are thus consistent
with the above prediction (see also Table 2).
Hamrin & Persson (1986) studying the planktivorous fish Coregonus albula, found
population cycles with a periocity of two years. They suggested that competition
between age classes, with juveniles experiencing a greater abundance of available
resources (i.e. k ' < k), was responsible for the population oscillations. This fits in
well with the results derived from the present models. Although the present models
only consider semelparous organisms (see below), competition between age classes
is also a process that may be relevant to the search for an understanding of the
well-known population cycles in small mammals. For example Windberg & Keith
(1978) found that local populations of the snowshoe hare, where juvenile survival
was density dependent, cycled at a much reduced amplitude compared with nearby
populations with no pronounced density dependent juvenile survival. One possible
398
B. EBENMAN
reason for this is that competition between age classes may be stabilizing when
juvenile survival is density-dependent (see also Abramsky & Tracy, 1979). According
to Lack (1954), reproduction in birds is in most cases only weakly density dependent,
whereas in mammals there often is a heavy fall in birth rate at high densities. This,
together with a potential intense competition between age classes in these groups,
may account for the relative rarity of documented population cycles in birds as
compared to mammals.
For organisms with no or weak competition between the age clases (e.g. organisms
with complex life cycles) the situation is the converse: populations where density
dependence is primarily on juvenile survival should be less stable than populations
where the density dependence is primarily on reproduction (Fig 4a, b, see also figs
2 and 3 in Ebenman, 1987). (There is no difference in stability between the special
cases k' = 0, k > 0 and k' > 0, k = 0 when a =/3 = 0, however for 0 < a < S, the first
case is more stable than the second, all others being equal, cf. Figs 1 and 2.) This
prediction is consistent with Nicholson's (1950, 1957) observations of population
dynamics in laboratory cultures of the sheep blowfly. He found that populations
that were regulated solely by the rate of food supply to larvae, fluctuated more
(greater cycle amplitude) than those that were also regulated by the rate of food
supply to adults. In a series of studies on the population dynamics of the weevil
species Callosobruchus chinensis and C. maculatus (pests of leguminous crops),
Bellows (1982a, b 1984) pointed out differences in the dynamic behaviour between
these species. According to Bellows (1982a), there is no competition between the
mature adults and larvae in these species. Long term laboratory studies by Utida
(1967) and Fujii (1968) demonstrated population oscillations in C. maculatus and
stable populations of C. chinensis. In C. maculatus, reproduction was density
independent, whereas in C. chinensis there was density dependence in both the
adult and larval stages. Table 2 gives more examples from field and laboratory
populations that conform to the prediction.
LIMITATIONS
OF THE
MODELS
The main limitation of the models analysed in the present work is that they deal
with organisms that reproduce only once. The results derived may therefore only
apply to semelparous organisms. It is not known if these results are also valid for
iteroparous organisms (some preliminary computer simulations indicate that they
may be valid). Hence, I do not propose that the present models faithfully portray
the dynamics in all the population described in the examples above, or that we are
dealing with a universal phenomenon. The number of populations satisfying all the
assumptions of the models surely is circumscribed. However, this does not make
the above examples irrelevant to the results from present models.
The examples, discussed above do not, of course, prove the hypothesis put forward
here to account for the observed population dynamic behaviour to be correct. There
may be other causes behind such dynamic behaviour (e.g. see Gurney et al., 1980
for an alternative interpretation of the blowfly data.) However, the observations do
not contradict the view set out here. Thus, I think it is fair to conclude that the
COMPETITION
BETWEEN
AGE CLASSES
399
models point to a m e c h a n i s m - - c o m p e t i t i o n between age classes--that may be
responsible for and important in shaping the actual dynamic behaviour in natural
populations. I think that this is one important role o f theoretical work: to point out
possible causes behind observed patterns; not to establish the actual ones. That is
an empirical question.
I thank Thomas Alerstam, Karl Gustav Andersson, Sam Erlinge, Sigfrid Lundberg, Robert
May, Einar Olafsson, Lars-Erik Persson, Fredrick Schlyter, Per Weinerfelt and Uno
Wennergren for valuable help and discussions and Steffi Douwes for help with the figures.
This research was supported by the Swedish Natural Science Research Council.
REFERENCES
ABRAMSKY, Z. & TRACY, C. R. (1979). Population biology of a "noncycling" population of prairie
voles and a hypothesis on the role of migration in regulating microtine cycles. Ecology 60, 349.
BARKER, J. S. F. & PODGER, R. N. (1970). Interspecific competition between Drosophila melanogaster
and Drosophila simulans: Effects of larval density on viability, developmental period and adult body
weight. Ecology 51, 170.
BEDDINGTON, J. R. (1974). Age distribution and the stability of simple discrete time population models.
J. theol-. Biol. 47, 65.
BELLOWS, T. S. JR. (1982a). Analytical models for laboratory populations of Callosobruchus chinensis
and C. maculatus (Coleoptera, Bruchidae). J. Anita. Ecol. 51, 263.
BELLOWS, T. S. JR. (1982b). Simulation models for laboratory populations of Callosobruchus chinensis
and C. maculatus. J. Anita. Ecol. 51, 597.
BELLOWS, T. S. JR. & HASSELL, M. P. (1984). Models for interspecific competition in laboratory
populations of Callosobruchus spp. J. Anim. EcoL 53, 831.
COLLINS, N. C. (1980). Developmental responses to food limitation as indicators of environmental
conditions for Ephydra cinerea (Diptera). Ecology 61, 650.
CROMBIE, A. C. (1944). On competition between different species of gramnivorous insects. Proc. R. Soc.
LondB 132, 362.
DEBACH, P. (1974). Biological control by natural enemies. New York: Cambridge University Press.
DEMPSTER, .J.P. (1971). The population biology of the cinnabar moth, Tyriajacobaeae L. (Lepidoptera,
Arctiidae) Oecologia 7, 26.
EBENMAN, B. (1987). Niche differences between age classes and intraspecific competition in agestructured populations. J. theor. BioL 124, 25.
EBENMAN, B. (1988). Population dynamics in age- and size-structured populations: Intraspecific competition. In: Ecology and evolution in size-structured populations. (Ebenman, B. & Persson, L., eds). Berlin:
Springer-Verlag (in press).
EBENMAN, B. & PERSSON, L. (eds.) (1988). Ecology and evolution in size-structured populations. Berlin:
Springer-Verlag (in press).
FRANK, P. W., BOLL, C. D. & KELLY, R. W. (1957). Vital statistics of laboratory cultures of Daphnia
pulex DeGeer as related to density. Physiol. Zool. 30, 287.
FRAUENTHAL, J. (1975). A dynamic model for human population growth. Theor. Pop. Biol. 8, 64.
FUJll, K. (1968). Studies on interspecies competition between the azuki bean weevil and the southern
cowpea weevil. II1. Some characteristics of strains of two species. Res. Pop. Ecol. 10, 87.
GUCKENHEIMER, J., OSTER, G. & IPAKTCHI, A. (1977). The dynamics of density dependent population
models. J. Math. Biol. 4, 101.
GURNEY, W. S. C., BLYTHE, S. P. & NISBET, R. M. (1980). Nicholson's blowflies revisited. Nature 287,
17.
HAMRIN, S. F. & PERSSON, L. (1986). Asymmetrical competition between age classes as a factor causing
population oscillations in an obligate planktivorous fish species. Oikos 47, 223.
HARCOURT, D. G. (1971). Population dynamics of Leptinotarsa decemlineata (Say) in eastern Ontario.
Can. Entomol. 103, 1049.
HAVEN, S. B. (1973). Competition for food between the intertidal gastropods Acmaea scabra and A.
digitalis. Ecology 54, 143.
HUFFAKER, C. B., KENNETT, C. E. & TASSAN, R. L. (1986). Comparisons of parasitism and densities
of Parlatoria oleae (1952-1982) in relation to ecological theory. Amer. Natur. 128, 379.
400
B. E B E N M A N
KEAST, A. (1977). Mechanisms expanding niche width and minimizing intraspecific competition in two
centrarchid fishes. Evol. Biol. 10, 333.
LACK, D. (1954). The natural regulation of animal numbers. Oxford: Oxford University Press.
MACDONALD, N. (1978). Time lags in biological models. Berlin: Springer Verlag.
MAY, R. M. (1974). Biological populations with nonoverlapping generations: stable points, stable cycles,
and chaos. Science 186, 645.
MAY, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature 261,459.
MAY, R. M., CONWAY,G. R., HASSELL, M. P. & SOUTHWOOD, T. R. E. (1974). Time delays,
density-dependence and single-species oscillations. J. Anita. Ecol. 43, 747.
McCAuLEY, E. & MURDOCH, W. W. (1987). Cyclic and stable populations: Plankton as paradigm.
Amer. Natur. 129, 97.
METZ, J. A. J. • DIEKMANN, O. (1986). The dynamics of physiologically structured populations. Berlin:
Springer-Verlag.
MURDOCH, W. W. & McCAULEY, E. (1985). Three distinct types of dynamic behaviour shown by a
single planktonic system. Nature 316, 628.
MURDOCH, W. W., NISBET, R. M., BLYTHE, S. P., GURNEY, W. S. C. & REEVE, J. D. (1987). An
invulnerable age class and stability in delay-differential parsitoid-host models. Amer. Natur. 129, 263.
MURDOCH, W. W., REEVE, J. D., HUFFAKER, C. B. & KENNE'r'r, C. E. (1984). Biological control of
olive scale and its relevance to ecological theory. Amer. Natur. 123, 371.
MURPHY, L. F. (1983). A nonlinear growth mechanism in size structured population dynamics. J. theor.
Biol. 104, 493.
NICHOLSON, A. J. (1950). Population oscillations caused by competition for food. Nature 165, 476.
NICHOLSON, A. J. (1957). The self-adjustment of populations to change. Cold Spring Harbor Syrup.
Quant. Biol. 22, 153.
NISBET, R. M. & GURNEY, W. S. C. (1982). Modelling fluctuating populations. Chichester: Wiley.
OSTER, G., IPAKTCHL A. & ROCKLtN, S. (1976). Phenotypic structure and bifurcation behavior of
population models. Theor. Popul. Biol. 10, 365.
POLLS, G. A. (1984). Age structure component of niche width and intraspecific resource partitioning:
can age groups function as ecological species? Amer. Natur. 123, 541.
POLLS, G. A. & FARLEY, R. D. (1980). Population biology of a desert scorpion: survivorship, microhabitat,
and the evolution of life history strategy. Ecology 61, 620.
PRATt, D. M. (1943). Analysis of population development in Daphnia at different temperatures. BioL
Bull, 85, 116.
PROUT, T. & MCCHESNEY, F. (1985). Competition among immatures affects their adult fertility.
Population dynamics. Amer. Natur. 126, 521.
REEVE, J. D. & MURDOCH, W. W. (1986). Biological control by the parasitoid Aphyris melinus, and
population stability of the California red scale. J. Anita. Ecol. 55, 1069.
SCHAF~ER, W. M. (1985). Order and chaos in ecological systems. Ecology 66, 93.
SCHEIRING, J. F., DAVIS, D. G., RANASINGHE, A. & TEARE, C. A. (1984). Effects of larval crowding
on life history parameters in Drosophila melonogaster Meigen (Diptera: Drosophilidae). Ann. Entomol.
Soc. Am. 77, 329.
SLOBODKIN, L. B. (1954). Population dynamics in Daphnia obtusa Kurz. Ecol. Monogr. 24, 69.
SUTHERLAND, J. (1970). Dynamics of high and low populations of the limpet Acmaea scabra (Gould).
Ecol. Monogr. 40, 169.
TSCHUMY, W. O. (1982). Competition between juveniles and adults in age-structured populations. Theor.
Popul. Biol. 21,255.
UNDERWOOD, A. J. (1976). Food competition between age-classes in the intertidal neritacean Nerita
atramentosa Reeve (Gastropoda: Prosobranchia). J. exp. mar. Biol. Ecol. 23, 145.
UTIDA, S. (1967). Damped oscillations of popultion density at equilibrium. Res. Pop. Ecol. 9, 1.
VANSICKLE, J. (1977). Analysis of a distributed-parameter population model based on physiological
age. J. theor. Biol. 64, 571.
WEINER, J. (1985). Size hierarchies in experimental populations of annual plants. Ecology 66, 743.
WERNER, E. E. & GJ LLIAM, J. F. (1984). The ontogenetic niche and species interactions in size-structured
populations. Ann. Reo. Ecol. Syst. 15, 393.
WILBUR, H. M. (1980). Complex life cycles. Ann. Rev. Ecol. Syst. 11, 67.
WILBUR, H. M. & COLLINS, J. P. (1973). Ecological aspects of amphibian metamorphosis. Science 182,
1305.
WINDBERG,L. A..g KEITH, L. B. (1978). Snowshoe hare populations in woodlot habitat. Can. J. Zool.
56, 1071.