Density Dependence and Age Structure: Nonlinear Dynamics and Population
Behavior
Kevin Higgins; Alan Hastings; Louis W. Botsford
The American Naturalist, Vol. 149, No. 2. (Feb., 1997), pp. 247-269.
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Vol. 149, No. 2
The American Naturalist
February 1997
DENSITY DEPENDENCE AND AGE STRUCTURE: NONLINEAR DYNAMICS AND POPULATION BEHAVIOR KEVINHIGGINS,"~'~'*
ALANH A S T I N G S , "AND
~ ' ~ LOUIS
'~
W. BOTSFORD~'~'$
'Division of Environmental Studies, University of California, Davis, California 95616; lInstitute of Theoretical Dynamics, University of California, Davis, California 95616; 3Department of Wildlife, Fish, and Conservation Biology, University of California, Davis, California 95616; 4Center for Population Biology, University of California, Davis, California 95616 Submitted December 5, 1994; Revised April 19, 1996; Accepted April 23, 1996
Abstract.-We characterize the dynamics of age-structured density-dependent populations with
yearly reproduction. In contrast to prior studies focusing primarily on behavior near the stability
boundary, we describe the dynamics over the full range of linear and nonlinear behavior. We
describe model dynamics in terms that have direct biological interpretations. We illustrate the
use of our approach by examining the dynamics of Dungeness crab (Cancer magister) in detail.
Model dynamics are found to be very sensitive to changes in life-history parameters. Small
changes in vital rates can cause population density to suddenly jump from low to high variability
or vice versa. Nonmonotonic switching between chaotic and nonchaotic dynamics is also observed. The period (or dominant timescale) of cyclic behavior is loosely related to values of
vital rates, typically increasing with adult survivorship, but can remain constant while vital rates
change. Model dynamics are also found to be sensitive to environmental perturbations. For
example, model dynamics may be chaotic or nonchaotic for fixed parameter values with environmental perturbations switching model dynamics between these distinct behaviors (i.e., the dynamics are nonstationary). These findings illustrate one possible explanation for the variety of
dynamic behavior in Dungeness crab populations (and other natural populations) and temporal
(or spatial) shifts in behavior.
Intrinsic regulatory mechanisms can cause fluctuations in population density
(Hutchinson 1978; Kingsland 1985). In this article, we analyze a common situation
that can lead to such fluctuations: populations with density-dependent recruitment and age structure.
Early investigations into the role of density dependence in the regulation of
natural populations ignored age structure. One-dimensional, density-dependent,
difference equation models of the type investigated by May (1974a, 1976), and
May and Oster (1976) are prominent examples. Although such simple models lack
age structure, they have served the important purpose of showing that density
dependence can produce complex dynamics, including cycles and chaos.
We begin by reviewing the seminal work of Ricker (1954). Using hand-drawn
reproduction curves (i.e., not the equation n,, = n,er['-rll'kl
) and various assump-
,
* To whom correspondence should be addressed; E-mail: [email protected]. t
E-mail: [email protected]. $ E-mail: [email protected]. Am. Nat. 1997. Vol. 149, pp. 247-269. 0 1997 by The University of Chicago. W3-0147197i4902-0003$02.00. All rights reserved 248
THE AMERICAN NATURALIST
tions about age structure, Ricker's seminal work on the effects of age structure
on a population with density-dependent recruitment made the following points.
First, the amplitude of oscillations in population density tends to decrease as
more cohorts comprise the spawning stock. Second, the amplitude of oscillations
in population density increases rapidly with an increase in the number of years
(or time periods) between parental egg and the first production of filial eggs, up
to a limit imposed by the shape of the reproduction curve (i.e., egg production
as a function of adult density). Third, the period of oscillations in population
density is determined by the mean generation time from parental egg to filial eggs
and is twice this interval or close to it. Fourth, the period of oscillations in
population density is independent of the exact shape of the reproduction curve.
Fifth, the period of oscillations in population density is independent of the number
of cohorts in the spawning stock, provided more than one exists. Finally, for the
specific case in which only two adult age classes reproduce, raising the age of
reproductive maturity leads to increasingly irregular oscillations (see fig. 10,
Ricker 1954) if the non-age-structured model has regular oscillations (see fig. 7,
Ricker 1954).
Many models have recognized that age structure (i.e., repeated reproduction,
delayed reproduction, age-dependent fecundity, and age-dependent mortality) is
an important feature of natural populations (including Ricker 1954; Caswell 1989;
Kareiva 1989). More realistic age-structured density-dependent models have been
studied (Ricker 1954; Clark 1976; Guckenheimer et al. 1977; Botsford and Wickham 1978; Oster 1978; Levin and Goodyear 1980; McKelvey et al. 1980; Levin
1981; Botsford 1986, 1992; Bergh and Getz 1988; Cushing 1988). Yet the dynamics
of such models is not completely understood (Sissenwine et al. 1988; Caswell
1989; Nisbet and Onyiah 1994). Equilibrium analysis (analysis of linearized models coupled with simulation), often the only practical analytical tool available for
investigating such complex models, leads to a focus on the shift from equilibrium
to limit cycle behavior (e.g., Levin and Goodyear 1980). Beginning with Ricker,
many authors have recognized that mean generation time is an important factor
setting the period of cycles (i.e., period is roughly twice the mean generation
time). Another typical result (also noted by Ricker) is that as the slope of the
recruitment versus density relation steepens, stability is decreased (i.e., cycles
emerge). Guckenheimer et al. (1977) showed that simple age-structured models
display the full array of dynamics (i.e., equilibria, cycles, and chaos), exhibited
by non-age-structured, one-dimensional model counterparts, and a capacity for
alternate stable dynamic behaviors (Guckenheimer et al. 1977).
The basic assumptions of the model are a constant rate of adult survivorship
(except in cases of truncation), density-dependent recruitment (independent of
adult age), and a fixed developmental period (time from egg to reproductive maturity). We first characterize fluctuations in terms of variability, timescale, and
predictability, for populations whose life-history parameters are constant in time.
We then include a further complication, namely, environmental perturbations.
In particular, we look at the robustness of long cycles in age-structured populations with density-dependent recruitment to noise and the effect of noise on the
amplitude of cyclic oscillations. We also consider how alternate stable states
DENSITY DEPENDENCE AND AGE STRUCTURE
249
(i.e., dynamics) may be affected by environmental noise. Here we explore model
behavior beyond the stability boundary. We specifically look at the amplitude of
cyclic behavior, the period of cyclic dynamics, and the potential for chaotic dynamics in this region. We study the model numerically and draw inferences about
natural populations, characterizing the model's behavior as if it were data from
natural populations. We focus on three biologically important dynamic issues:
variability of population density, approximate cycle periods, and predictability of
population density. We calculate these descriptors over a wide range of important
life-history parameters (see app. B), which necessitates multiple representations
of the bifurcation structure. Each descriptor describes a different biological implication of a model. For example, a population with strongly chaotic dynamics
(i.e., a large, positive largest Lyapunov exponent) but relatively low variability
may for practical purposes be in equilibrium. Such a population would be relatively predictable even though strongly chaotic. We use these descriptors to study
fundamental changes in dynamics associated with changes in age structure and
density dependence. Of special importance are sudden changes in these descriptors for small changes in life-history parameters. Parameters that change gradually in time or space may lead to large changes in behavior, confounding predictability through time or across space.
Dungeness crab (Cancer magister) populations display some of these characteristics. We discuss the characteristics of single populations along the west coast
of North America, ignoring spatial effects (Botsford et al. 1994; Hastings and
Higgins 1994). Some Dungeness crab populations display cyclic fluctuations in
recruitment with a period of roughly 11 yr (see fig. 1). Density-dependent recruitment is a potential cause for these cycles (Botsford et al. 1989). Possible mechanisms of density-dependent recruitment include cannibalism, density-dependent
fecundity, and an egg-predator worm. Each mechanism could involve the juvenile
and adult age structure in different ways. We use this empirical system to illustrate the general relationships between life-history characteristics and population
behavior described here.
RICKER MODEL WITH AGE STRUCTURE
In many natural populations iteroparity is the rule: adults survive after reproduction and remain in the adult breeding population to reproduce again. We study
an age-structured Ricker model with biologically important reproductive delays
and adult survivorship. Adult dynamics follow
Adults, represented by n,, either survived from the previous year with a densityindependent rate given by S , or were recruits (in number R,-,) D yr ago; D is
age at reproductive maturity. The rate of juvenile survivorship, S j , is assumed
independent of juvenile age; S; is the density-independent rate of survival
through D yr to reproductive maturity. An age-structured Ricker model is immediately obtained by assuming
250
THE AMERICAN NATURALIST
FIG.1.-Time series for Dungeness crab catch at Eureka and Crescent City, California,
and Grays Harbor, Willapa Bay, and Columbia River.
as the functional form for density-dependent recruitment. Equation (2) matches
Ricker's original model but includes a delay of D yr between birth and reproductive maturity (D = 1 in Ricker's model; Clark 1976). Here r controls the
degree of density dependence, and k gauges density dependence. Model (1) becomes
n , = n,-,So
+ n , - D e r ( l - n r - ~j ' k. ) ~ D
(3)
Rescaling n to eliminate the parameters S j and k (see app. A) leads to
We also examine another important factor in the dynamics of many populations: the removal of the oldest individuals from the reproductive class by truncation (a sharp reduction in the rate of adult survivorship for older adults).
CHARACTERIZATION OF MODEL DYNAMICS
We characterized the biologically relevant behavior of the age-structured
Ricker model with the following measures.
Stability Boundary: Equilibrium
The stability boundary dividing equilibrium from nonequilibrium behavior is a
basic descriptor of dynamics. For example, as adult survivorship is increased in
DENSITY DEPENDENCE AND AGE STRUCTURE
25 1
figures 2A-E, oscillations in the model occur at progressively lower levels of r
(i.e., 6 moves to the left); in figure 2F, this trend is reversed.
Variability of Population Density: Attractor Size May's (1981) definition of population variability over a long time series is HL
highest observed density
lowest observed density '
For example, consider the Dungness crab time series (solid circles) in figure 1.
Its variability is
To apply this measure of variability to a model, we iterate the model until transients have ceased, and then we note the highest and lowest population densities
observed in a specified subsequent number of time steps. Table 1 contrasts the
coefficient of variation with measure (5) of variability for the Dungeness crab
catch data.
Period of Cyclic Oscillations
A population's oscillatory period is a directly measurable quantity that can be
estimated from observed time series (see Olsen et al. 1988; Olsen and Schaffer
1990). The approximate period of oscillation (time of return to approximately the
same density) of a time series can be formally computed using the discrete Fourier
transform (see app. B and Lichtenberg and Lieberman 1992 for details). Table 1
lists approximate oscillation periods for each of the Dungeness crab data sets.
Predictability of Population Density (Chaos)
Intrinsic regulatory mechanisms may cause chaotic behavior precluding longterm population prediction. The best measure of this lack of predictability is the
largest Lyapunov exponent (LLE; Hastings et al. 1993). The LLE can be calculated for data collected from natural populations or model runs (Ellner 1991;
Ellner et al. 1992; Hastings et al. 1993; Ellner and Turchin 1995). There is not
yet a consensus on the best approach to determine LLE for empirical data.
Systems with positive LLE are chaotic; long-term prediction may be possible
only if LLE is 0 or a negative number (i.e., not chaotic). The insets in figures
2B, E, and F display time series with negative LLEs; insets in figures 2A, C, and
D display time series with positive LLEs.
GENERALIZATIONS: DETERMINISTIC
MODELS
Given the imprecision of our models and parameter estimates, it is unwise to
draw general conclusions from a single parameter choice. We characterize the
nonlinear dynamics for the entire parameter space in terms of variability, period
of cyclic oscillations, and the largest Lyapunov exponent. This represents a non-
DENSITY DEPENDENCE AND AGE STRUCTURE
TABLE 1
DUNGENES
C RABSTATISTICS
Location
Morro and Monterey Bays
Bodega and San Francisco Bays
Fort Bragg
Eureka and Crescent City
Brookings, Gold Beach, and Port Orford
Coos and Winchester Bays
Newport and Depoe Bay
Tillamook and Garibaldi
Astoria and Warrenton
Grays Harbor. Willapa Bay, and Columbia River
Period
(yr)
Variability
(ffIL.1
CV
8
32
10.7
10.7
16
12.8
10.7
10.7
10.7
10.7
165.9
43.1
89.4
77.2
52.0
8.2
14.8
8.3
6.4
8.5
1.40
1.08
.77
.66
.85
.52
.57
.47
.49
.61
NOTE.-Summary of Dungeness crab statistics for ports from central California in the south (Morro
and Monterey Bays) to Washington in the north (Grays Harbor, Willapa Bay. and Columbia River).
The southern part of the range (Morro and Monterey Bays and Bodega and San Francisco Bays)
appears to have changed its dynamic behavior (i.e., collapsed). Behavior seems to have switched from
large-amplitude oscillations to a steady, low density. Therefore. the values for period are unreliable.
Variability for these two ports should be taken as a measure of the ratio of former stock density
compared to the present. The estimates of period were computed using the Fourier transform (see
app. B). CV = coefficient of variation.
linear extension of the well-developed sensitivity analysis for linear matrix population models (Caswell 1989). In this section, we look at adult survivorship, fecundity, and length of reproductive delay, without truncation.
Stability Boundary
The location of the stability boundary is strongly governed by the age of reproductive maturity (see, e.g., Clark 1976; Bergh and Getz 1988; table in Botsford
1992). Movement of the stability boundary due to changing the age of reproductive maturity is most pronounced at high rates of adult survivorship. An increase
in the age of reproductive maturity moves the stability boundary to lower r
FIG. 2.-Adult population density plotted against r for six rates of adult survivorship, S o
(eq. [4] with D = 3). A, S o = 0.0. The inset shows a portion of the time series for r = 2.9.
This time series is chaotic with a largest Lyapunov exponent of 0.13. B, S, = 0.05. Left
inset, r = 2.1, LEE = -0.02 (not chaotic). Right inset, r = 3.1, LLE = -0.10 (not
chaotic). C, S o = 0.1. Inset, r = 2.8. LLE = 0.06 (chaotic). D, S o = 0.12. The inset shows
a portion of the time series for r = 3.9. In this region of parameter space, the type of
dynamics (i.e.. limit cycles or chaos) and therefore the largest Lyapunov exponent depends
on the initial conditions (n3 = 1.1, n2 = 1.1. n, = 1.1). The time series in the inset is chaotic
with a largest Lyapunov exponent of 0.11. For the initial condition (n3 = 1.4, n? = 0.1, n,
= 1.4; time series not shown), the largest Lyapunov exponent is -0.06 (not chaotic). E, So
= 0.4. Inset, r = 3.75, LLE = -0.05 (not chaotic). F, S o = 0.6. Inset: r = 3.0, LLE =
-0.008 (not chaotic). The approximate period for the time series is 8.0. Variability, HIL,
is 7.4.
THE AMERICAN NATURALIST
Adult ~ u ~ i v o r s h i p
Adult survivorship
Adult su~ivorship
Adult survivorship
FIG. 3.-Population variability, HIL, plotted as a function of adult survivorship. and r .
Figure A-D correspond to I-, 2-, 3-. and 4-yr delays ( D )in reproductive maturity. Oscillating
populations are indicated by HIL > 1. The highest variability observed in the Dungeness
crab catch data is HIL = 166 for Monterey and Morro Bay.
(e.g., cf. fig. 3A with fig. 20). The range of life-history parameters that produce
equilibrium (i.e., ln[HIL] = 0) decreases with increasing age of reproductive
maturity.
Population Variability
The highly variable dynamics of the non-age-structured Ricker model are dramatically reduced by adult survivorship (see fig. 3A). Variability of the non-agestructured Ricker model is displayed in figure 3A at S, = 0. At high adult survivorship ( S , > 0.6), the model reaches equilibrium but only for short times to first
reproduction.
For reproductive delays greater than 1 yr, the age-structured model displays
variability that lacks a simple pattern in parameter space (see fig. 3B-D). Thus,
unlike the case in which the delay is 1 yr, an increase in S, does not guarantee
a reduction in variability. One pattern that does emerge is that increasing the
delay in reproductive maturity increases the range of parameter space that exhibits extreme variability (e.g., the range of dynamics with HIL > 148 increases).
DENSITY DEPENDENCE AND AGE STRUCTURE
255
Period of Cyclic Oscillations
The most important factor governing the period of cycles in the dynamics is
the age of reproductive maturity, D (fig. 5A-D later). For D = 2, 3 , 4 , the patterns
for the period of the dynamics are all roughly scaled versions of one another. At
low adult survivorship, S,, and high fecundity, r, the graphs for these cases have
cycles with a short period (=3, 2, and 2.67 yr, respectively). For other parameter
values, one finds a period that is equal to or slightly greater than twice the delay
in reproductive maturity. Overall, the period tends to increase with increased
survivorship, although at very high r and S,, the trend of increasing period shows
a slight reversal. In contrast, along the stability boundary (where most previous
studies have been done), the period increases smoothly with adult survivorship
(or mean age).
Predictability of Population Density (Chaos)
The chaotic dynamics of the non-age-structured Ricker model are eliminated
by adult survivorship (Botsford 1992). The L L E of the non-age-structured Ricker
model is displayed later in figure 7A at S, = 0. However, the dynamics of this
special case are not typical of delays in reproductive maturity of more than 1 yr.
Populations with multiyear delays in recruitment exhibit chaotic dynamics with
an apparently fractal pattern of positive largest Lyapunov exponents in parameter
space (i.e., magnification of a region of positive LLEs produces a similar irregular
pattern). There are broad regions in parameter space of chaos interspersed with
periodic (and quasi-cyclic) dynamics. Increasing the delay in first reproduction
increases the range of parameter space that exhibits a positive LLE (cf. fig.
7A-D). Of special interest are abrupt transitions from regions of regular dynamics
to strongly chaotic dynamics for a small change in parameters (the slope of L L E
near a transition from periodic dynamics is usually steep).
TRUNCATION: DETERMINISTIC
MODELS
Stability Boundary
The location of the stability boundary is strongly governed by age structure
truncation (Hastings 1984). Whether a population exhibits equilibrium depends on
the width (e.g., last reproductive age minus first reproductive age) of a breeding
population's age structure (Botsford and Wickham 1978; Levin and Goodyear
1980; Levin 1981). Removing older adults (e.g., through harvest or senescence)
beyond some age, T, switches dynamics from equilibrium to oscillatory cycles
(e.g., cf. figs. 3C and 4A; see Botsford and Wickham 1978).
Population Variability
Truncation dramatically increases variability at higher adult survivorship rates
(cf. figs. 3C and 4A for S, = 0.8). At intermediate adult survivorship rates
(S, = 0.4), the degree of variability depends on the age of truncation, T. Removal
of adults after age 7 does not affect variability very much in this region (cf. figs.
3C and 4B). In contrast, removal of adults after age 4 dramatically increases
256
THE AMERICAN NATURALIST
Adult survivorship
Adult survivorship
FIG.4.-The effect of truncation on population variability, HIL, plotted as a function of
adult survivorship and r for a population with reproductive maturity at 3 yr (D = 3). In both
figures, the harvest rate is 0.9. A, Adults are harvested after reproduction in their fourth
year ( T = 4). B, Adults are harvested after reproduction in their seventh year ( T = 7).
variability (HIL = 55 + HIL = 2,981) at intermediate adult survivorship rates
(cf. figs. 3C and 4A). Thus, adults in age classes 5-7 provide a dramatic reduction
in variability and are an example of a stabilizing tail (Hastings 1984).
Period of Cyclic Oscillations
Truncation introduces a second timescale (the first due to the reproductive
maturity delay) into the dynamics. At high adult survivorship rates, truncation
causes oscillatory dynamics in which the untruncated model produces equilibrium
(e.g., cf. figs. 5C and 6A for adult survivorship greater than 0.8). The dynamics
at high S, tend to oscillate with a period equal to the sum of the delay and the
age of truncation ( P = D + T = 3 + 4 = 7 in fig. 6A; P = 3 + 7 = 10 in fig.
6B). However, as in the untruncated case, a P = 2 0 plateau occurs at low S,.
Predictability of Population Density (Chaos)
Truncation has opposite effects depending on whether truncation occurs at an
age much older than the age of reproductive maturity or at an age only slightly
greater than the age of reproductive maturity. Truncation at an age only slightly
greater than the age of reproductive maturity increases the range of parameters
with a positive LLE dramatically (cf. figs. 7C and 8A). When the truncation age,
T, is much greater than the age of first reproduction, D, the range of parameters
exhibiting a positive LLE drops dramatically (cf, figs. 7C and 8B).
RANDOM ENVIRONMENTAL NOISE
By subjecting the model to parametric random perturbations (Kornadt et al.
1991), we test the robustness (May 1974b) of the determistic dynamics described
earlier. The sensitivity of dynamics to environmental noise depends on the degree
DENSITY DEPENDENCE AND AGE STRUCTURE
Period
(years)
,
Period
(years)
survivorship 1
survivorship
1
1
1
I
1
Period
(years)
Period (years) .9 .9
survivorship survivorship
FIG. 5.-The approximate period, P, plotted as a function of adult survivorship and r.
Figure A-D correspond to 1-, 2-, 3-, and 4-yr delays ( D ) in reproductive maturity. The range
of adult survivorship is 0.02-0.9.
of density dependence (i.e., the degree of overcompensation; Horwood and Shepherd 1981; Botsford 1986), the harvest rate (Beddington and May 1977; May et
al. 1978; Horwood and Shepherd 1981; Reed 1983; Botsford 1986), and the width
of the cohort size distribution (Botsford 1986). All three of the dynamic descriptors that we have employed can be calculated for noisy dynamics. The extension
of the definitions for period (Allen and Basasibwaki 1974) and variability are
straightforward. The largest Lyapunov exponent can be calculated for noisy dynamics (Ellner 1991; Ellner et al. 1992; Ellner and Turchin 1995) but is computationally intensive.
Following Tuljapurkar (1989), we introduce noise by making r and S , timedependent random variables (see app. C). Adding noise to r in effect includes
lognormal noise in recruitment. If r is a Gaussian random variable with a coefficient of variation (CV) of 0.05, one gets much the same picture for variability as
THE AMERICAN NATURALIST
FIG.6.-The effect of truncation on the approximate period, P, plotted as a function of
adult survivorship and r for a population with reproductive maturity at 3 yr. The harvest
rate is 0.9. The range of adult survivorship is 0.02-1.0. A, Adults are harvested after reproduction in their fourth year ( T = 4). B , Adults are harvested after reproduction in their
seventh year ( T = 7).
with r constant (cf. figs. 9A and 3 0 . This holds for much larger coefficients of
variation (at least as high as CV = 0.3; not shown). Random variation in adult
survivorship (CV = 0.05) destroys the canyon of low variability at low adult
survivorship ( S , = 0.1; cf. figs. 9B and 3 0 .
With random vital rates, regions of parameter space that were in equilibrium
show fluctuations. The period shows a relatively smooth transition across the
stability boundary (fig. 9 0 into the disturbed equilibrium. Within the region that
was oscillating before the introduction of noise, the essential features of the
period are preserved after the introduction of noise with one exception. The
canyon of period 2 dynamics (at low S, and high r) shows an erratic pattern for
the period with the introduction of noise into S,. The canyon of period 2 and
plateaus of constant period are preserved for noisy r (not shown).
Another issue associated with environmental noise is the potential for the dynamics to switch between different stable behaviors in the corresponding deterministic model (May 1977; Beddington 1984). This can happen because densitydependent models may have more than one type of stable behavior for the same
parameters (Guckenhemier et al. 1977; Hastings 1993). In the absence of environmental noise, one of the alternate stable behaviors is attained based on the initial
conditions (i.e., history). Perturbations in population density can be thought of
as providing new initial conditions at each disturbance. Thus, the model may
take on a new behavior after a disturbance. Figure 10A, B displays variability for
equation (4) for two different initial conditions. It is clear that these two pictures
show a large difference in variability over a substantial range of the parameters.
Figure 10C shows this same range of parameters with the initial condition for
each point in parameter space being chosen from a uniform random distribution.
The occurrence of high and low variability for random initial conditions indicates
DENSITY DEPENDENCE AND AGE STRUCTURE
B
Adult survivorship
Adult survivorship
0.9
-.
Adult survivorship
0.9
Adult survivorship
0.9
0.9
FIG.7.-Positive largest Lyapunov exponents plotted as a function of adult survivorship
and r. Figure A-D correspond to I-, 2-, 3-, 4-yr delays (D)in reproductive maturity. Negative
and zero exponents are graphed as zeros. All exponents have been rescaled relative to a
1-yr delay.
that dynamics with both variabilities may be likely. Over this same range of
parameters, the largest Lyapunov exponent also depends on the initial conditions
(not shown).
AN EXAMPLE: DUNGENESS CRAB
Our results characterize the behavior that can be expected from certain combinations of parameter values, particularly adult survivorship (S,) and densitydependent fecundity (r). Rather than attempt a detailed study of the Dungeness
crab, we use this species to provide an example of how these general results
might help us understand a real population. Population dynamic behavior of the
Dungeness crab depends on how the age structure of Dungeness crab affects
density-dependent recruitment (and the shape of the density-dependent recruit-
THE AMERICAN NATURALIST
Adult survivorship
0.9
Adult survivorship
0.9
FIG.8.-The effect of truncation on the largest Lyapunov exponent, LLE, plotted as a
function of adult survivorship and r for a population with reproductive maturity at 3 yr
(D = 3). In both figures, the harvest rate is 0.9. All exponents have been rescaled relative
to a 1-yr delay. A , Adults are removed after reproduction in their fourth year ( T = 4). B,
Adults are harvested after reproduction in their seventh year ( T = 4).
ment relationship; Botsford and Wickham 1978). Because only males are harvested, the age structures of males and females differ; hence, we can use the
results obtained here to determine whether each age structure can cause cycles
with the observed characteristics. In doing so, we pose the question of whether
males alone or females alone can cause the observed cycles as an example of
how our results can be used to determine causal mechanisms. This example is
not a comprehensive evaluation of the causes of cycles in the Dungeness crab
catch, as it omits several relevant dynamic aspects (e.g., economic effects and
stochasticity).
Rather than attempting to evaluate specific density-dependent recruitment
mechanisms, we will examine the consequences of whether a mechanism involves
males only or females only. Reproductive maturity for both females and males
would begin near ages 2 or 3. We assume that both males and females have
reasonably high natural survival rates (i.e., unfished survival rate ~ 0 . 8 and
) that
only males are harvested. Because it is estimated that almost all harvestable
males (i.e., carapace width greater than 159 mm, ages 4 or 5) are removed each
year (Methot and Botsford 1982), we take harvest to be a measure of both male
and female recruitment. We take biomass to be an index of the number of adult
males (i.e., catch is composed of new recruits to the fishery, all the same size).
To approximate the male and female age structures, we will assume that males
are represented by a population with maturation delay of 3 yr and truncation after
age 4, whereas females are represented by a population with maturation at age 3
and truncation (senescence) after age 7. Thus, we can use figures 4A and 6A to
determine whether a population with the male age structure could cause the
values of variability and cycle period observed in the Dungeness crab cycles and
figures 4B and 6B to determine whether the female age structure could cause
these.
DENSITY DEPENDENCE AND AGE STRUCTURE
Adult survivorship
Adult survivorship
13 12 11 10 9
8
Period 7 (years) 6
5
4
3
2
L-&.9
?-
0.3
I
Adult
survivorship
FIG.9.-The effect of stochastic variation in the parameters on population variability and
period. In all cases, reproductive maturity is at 3 yr (D = 3). A, Variability is plotted as a
function of vital rates. Here r is a random variable with a coefficient of variation equal to
0.05 (cf. fig. 3C). B, Variability is plotted as a function of vital rates. Adult survivorship is
a random variable with a coefficient of variation equal to 0.05 (cf. fig. 3C). C , The period
of cyclic oscillations is plotted as a function of vital rates. Adult survivorship is a
random variable with a coefficient of variation equal to 0.1 (cf. fig. 5C). The period is
only slightly affected by random variation in r for coefficients of variation up to 0.3 (not
shown).
The values of variability, HIL, observed for Dungeness crab range up to 166
(table 1). Figure 4B indicates that female age structure could cause this level of
variability if r were 2 3 and adult survivorship were near 0.8. Most of the estimates of period for oscillations in the Dungeness crab landing data were near
10.7 yr (table 1). Figure 6B indicates that female age structure could cause a value
of period close to 10 yr for a wide range of values of r for high adult survivorship.
Thus, the model for females and the period and variability of oscillations in the
data are in fairly good agreement.
THE AMERICAN NATURALIST
Adult survlvor~hip
22026
8103
2981
1097
403
:
H/L
20 (variability)
7.4
2.8
4
r
Adult survivorship
FIG. 10.-Alternate dynamic states (variability) due to different initial conditions (i.e,
different initial age distributions) plotted for reproductive maturity at 3 yr (eq. [4] with D =
3). A, Initial condition, n3 = 1.1, n2 = 0.5, and n , = 1.1. B , Initial condition, n3 = 0.1, n2
= 1, n2 = 1.2, and n, = 1.9. C, Initial condition, chosen for each age (1-3) from a uniform
distribution for each point on the plot.
DISCUSSION
We have sought an understanding of fluctuations in population density that
arise because of intrinsic regulatory mechanisms and how these may vary with
observable life-history characteristics (i.e., the rate of adult survival and the
degree of density dependence). The resulting dynamics cannot be described as a
simple function of life-history parameters (see table 2 for a summary of important
findings). Dramatic changes in population dynamics may result from very small
changes in life-history parameters; in fact, dramatic changes in population dynamics may even occur if parameter values are fixed. For example, exploitation and/
or environmental noise can suddenly shift a population from equilibrium to highly
variable, chaotic oscillations. This sensitivity of dynamics to small parameter
changes (or environmental perturbations) suggests even more complex phenomena in general age-structured models.
Ricker's (1954) seminal study dealt with multiple-age spawning stocks by assuming a fixed maturation period and a fixed number of years of repeated repro-
TABLE 2
Variability
Density-dependent fecundity
Adult survivorship
Age of maturity (delay)
Truncation (harvest or senescence)
Period
Variability changed erratically with The period is twice the age of reproductive maturity at low r
r.
Variability may decrease for an in(and low S,) (fig. 5C).
crease in r.
The period may lock in (i.e., not
Variability switches between exchange) as adult survivorship is
tremes suddenly (i.e., for a 1%
varied. At high r, the period
change in the rate of adult survitends to increase in sharp steps
vorship).
as adult survivorship is increased (fig. 5C).
The period is twice the age of reproductive maturity at low rates
of adult survivorship (if r is not
high) (fig. 5C).
The period shows a similar pattern
Highly variable dynamics cover a
for all delays greater that 1 yr
larger range of vital rates as the
age of reproductive maturity is
(fig. 5B-D). The scaling factor is
twice the age of reproductive maincreased (fig. 3A-D).
turity.
The period is equal to the sum of
Variability is increased most dramatically when the age of truncathe age of maturity and the age
of truncation at high rates of
tion is only slightly greater than
adult survivorship and low r
the age of reproductive maturity
(P = D + 7) (fig. 6A, B).
(fig. 4A).
The period is erratic for high r
when truncation is slightly
greater than the age of maturity
(fig. 6A, B).
Predictability (Chaos)
Chaos becomes more intense as r
is increased (fig. 7C).
Chaos displays a fractal pattern as
a function of r (fig. 7C).
Chaos becomes more intense (and
likely) as adult survivorship is
decreased (fig. 7C).
Chaos displays a fractal pattern as
a function of adult survivorship
(fig. 7C).
Chaos covers a greater range of vital rates with an increase in the
age of reproductive maturity
(fig. 7A-D).
Harvest of all but the youngest
adults dramatically increases the
likelihood of chaos (cf. figs. 7C
and 8A).
Chaos is dramatically reduced by
harvesting the very oldest adults
(cf. figs. 7C and 8B).
264
THE AMERICAN NATURALIST
duction (i.e., truncation). A comparison of our results with those of Ricker, indicating our extensions, is most appropriate in cases in which we consider
truncation.
In general, increasing the age of reproductive maturity increases the amplitude
of oscillations for fixed parameter values (Ricker 1954). For some regions of
parameters, we show, however, that increasing the age of reproductive maturity
may decrease the amplitude of oscillations (cf. fig. 3B with valleys in fig. 3C).
This result also holds in cases with age structure truncation (not shown).
In general, the period of oscillations is twice the mean age of adults (Ricker
1954). For some regions of parameters, we show, however, that the period of
oscillation does not change with mean age of adults (i.e., the period does not
change as the rate of adult survivorship changes). Furthermore, there is a range
of parameters at low adult survivorship rates in which the period is very short and
does not follow the "two times mean age rule." Also, the period may increase, for
fixed mean age (i.e., S , = constant), as r is increased.
Variability
Variability displays a complex relation to life-history parameters, with high and
low variability intermixed in a fractal-like way. A consequence of this fractal
pattern is that life-history changes on any scale may bring abrupt changes in
population variability. Although the overall trend is for variability to increase
with decreased adult survivorship, this trend is not monotonic. Furthermore, a
wider range of life-history parameters (i.e., S,, r) show high variability as the age
of maturity increases.
Oscillations with radically different levels of variability may exist for the same
parameter values. Highly variable oscillations result from truncation (by harvest
or senescence). This effect is most pronounced when adult survivorship is high
and the age of truncation only slightly exceeds the age of maturity.
Chaos
Like variability, positive largest Lyapunov exponents (chaos) occur in a fractal
pattern. Therefore, small or large changes in life-history parameters may cause
switching between chaotic and nonchaotic dynamics. For the same life-history
parameters, chaotic and nonchaotic dynamics may occur, and there may be more
than one largest Lyapunov exponent for fixed parameter values.
While his work did not explore reproduction curves with chaotic dynamics,
Ricker found that dynamics became increasingly irregular for longer reproductive
delays. This conclusion is consistent with our finding that increasing the age of
reproductive maturity increases the likelihood of chaos.
Truncation can either increase or decrease the likelihood of chaos. When the
age of truncation is only slightly greater than the age of reproductive maturity,
chaos covers a wider range of life-history parameters (when compared with the
untruncated case). Chaos almost disappears when the age of truncation is substantially greater than the age of maturity (compared to the untruncated case).
DENSITY DEPENDENCE AND AGE STRUCTURE
Period
Compared to variability and chaos, the period of cycles in population density
displays a more regular dependence on life-history parameters. The approximate
period of the dynamics does not seem to depend on whether the dynamics are
chaotic, cyclic, or a disturbed equilibrium.
For all delays in reproductive maturity greater than 1 yr, there are regions
(plateaus in the figures) in which the period is constant as life-history parameters
are varied, but along the stability boundary the period changes smoothly with
life-history parameters. The scaling factor between these figures is twice the age
of reproductive maturity.
A limited range of life-history parameters gives rise to oscillations with a period
equal to twice the age of reproductive maturity (i.e., most adults die after reproduction). The period of oscillation increases roughly with adult survivorship. This
is not surprising, because mean age of adults increases with adult survivorship
(consistent with Ricker 1954).
Environmental Noise
Environmental noise can alter the dynamics of age-structured, densitydependent populations. Relatively small perturbations in adult survivorship rates
destroyed canyons of low variability in parameter space. Variability in these low
regions increased to levels comparable to the surrounding high variability. In
contrast, variability was insensitive to large perturbations in density-dependent
fecundity (i.e., r). Regions of low variability were preserved when r was varied
randomly. The period of cyclic oscillations was insensitive to noise in either of
these parameters.
Multiple attractors for the same parameters raise the possibility that dynamic
patterns (i.e., chaos andlor variability) may switch because of environmental
perturbations. Environmental noise of the right magnitude may cause persistent
switching between high and low variability (i.e., stochastic resonance; Maddox
1994). In a noisy environment, the dynamics may not settle into either high or
low variability but switch back and forth indefinitely.
Dynamics may also switch because of the fractal structure in the occurrence
of chaos and high variability in parameter space. Small changes in the life-history
parameters may cause switches between high and low variability and chaotic and
nonchaotic dynamics.
CONCLUSIONS
Far-from-equilibrium behavior is characteristic of many, if not most, natural
and harvested populations. However, analyses of many models, even strongly
density-dependent ones, have focused on either equilibria or the onset of cyclic
behavior. Generalizations about far-from-equilibrium behavior do emerge. Although dependence on parameters is complex in general, sharp reductions in
survivorship of older individuals dramatically increase variability by any of our
266
T H E AMERICAN NATURALIST
measures. We believe that the present study should be the first of many examinations of more realistic models of population ecology that seek to find generalizations about dynamics for all biologically reasonable parameters.
ACKNOWLEDGMENTS
We thank J. Keizer and the Center for Computational Biology at the Institute
of Theoretical Dynamics for the generous use of their computational facilities.
K.H. thanks J. Wagner for computational support. We also thank B. Holt, R.
Nisbet, and an anonymous referee for comments on an earlier version of the
manuscript. K.H. also thanks J . Goldey and L. Goldey.
APPENDIX A
MODELNONDIMENSIONALIZATION
Let q = ln(Sf), ?
tildes, we have
=
r
+ q , and f i , = r n , i A . Substituting into equation (3) and dropping
n,
=
n , , S,
+ n,-Der(''rf-D~.
APPENDIX B
COMPUTATIONAL
TECHNIQUES
VARIABILITY
Population variability, H i L , was computed for each point in a two-dimensional parameter array (r and S,). For each point, the model was iterated 10,000 times. For time steps
10,001-11,000, the highest and lowest observed densities were used to compute H i L .
PERIOD
The period, P, was computed for the preceding array. For each point in the range of
parameters, the model was iterated 10,000 times. The Fourier transform was applied to
the subsequent 1,024 values. The frequency with the largest amplitude was used to compute the approximate period. The Fourier transform was applied to time series data for
Dungeness crab (padded with zeros to the next highest power of 2). The frequency with
the largest amplitude was used to compute the approximate period.
LARGEST LYAPUNOV EXPONENT
The largest Lyapunov exponent, L L E , was computed for the previous array. For each
point, the model was iterated up to 30,000 times. The Parker-Chua algorithm (Parker and
Chua 1989) was used to compute the largest Lyapunov exponent over this time interval.
APPENDIX C
+
6 , with (6,) =
We defined the parameter controlling density dependence to be p , = r
0. Since (6,) = 0, we have ( p , ) = r. The coefficient of variation of (6,) can be adjusted to
any desired noise level. Two different sequences 6 , and 6: of random perturbations (with
identical means and variances) may produce different results for variability (and period)
DENSITY DEPENDENCE AND AGE STRUCTURE
since matrices do not commute. Therefore, we used the same sequence of perturbations
for each parameter combination.
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Associate Editor: Robert D . Holt