Download A simple predator–prey model of exploited

ICES Journal of Marine Science, 53: 615–628. 1995
A simple predator–prey model of exploited marine fish
populations incorporating alternative prey
Paul D. Spencer and Jeremy S. Collie
Spencer, P. D. and Collie, J. S. 1996. A simple predator–prey model of exploited
marine fish populations incorporating alternative prey. – ICES Journal of Marine
Science, 53: 615–628.
A simple two-species population model in which the predator is partially coupled to
the prey is developed. The model is an extension of traditional two-species models but
less complex than a three-species system. The growth rate of the predator depends
upon predation on the modeled and alternate prey; this formulation provides greater
realism in describing marine piscivores, such as spiny dogfish (Squalus acanthias), than
two-species predator–prey models. Two stable equilibria separated by a saddle point
potentially exist for the predator–prey system, and stochastic variability can lead to
movement between equilibrium abundance levels. In addition, endogenous limit cycles
may exist in the presence of predator fishing mortality. The model is applied to the
predation of spiny dogfish on a representative groundfish species, the Georges Bank
haddock (Melanogrammus aeglefinus). Stochastic variability is input to the model in
the form of ‘‘red noise’’ (variance is a decreasing function of frequency), a feature
observed in marine environments. Predator abundance can increase when the modeled
prey abundance is low, due to consumption of alternate prey, consistent with the
pattern observed in the spiny dogfish–haddock abundances. Increased harvesting on
the predator species allows the prey species to spend a greater proportion of time at the
high equilibrium. The model presented here poses the interesting management problem
of finding the optimal combination of fishing mortality rates for the two species.
? 1996 International Council for the Exploration of the Sea
Key words: predator–prey model, autocorrelated variability, functional response,
marine fisheries, exploitation, management.
Received 15 February 1995; accepted 20 October 1995
P. D. Spencer and J. S. Collie: University of Rhode Island, Graduate School of
Oceanography, Narragansett, Rhode Island, USA, 02882.
Introduction
A characteristic feature of marine fish populations is
dramatic changes in abundance over decadal time scales.
A review by Caddy and Gulland (1983) indicated that
exploited populations can be classified as ‘‘steady, cyclic,
irregular or spasmodic’’, with no exact criteria for the
boundaries between groups. Because the management of
marine fisheries is complicated greatly by population
fluctuations, the identification of causes of nonequilibrium behavior is of significant practical importance. Early fisheries models assumed a single-species
equilibrium view, in which temporal changes in population size are caused by corresponding changes in
exploitation (Schaefer, 1954). (‘‘Equilibrium’’ is defined
here as a fixed point which a population is very closely
associated with; any slight perturbation in population
size is countered by compensatory biotic interactions. In
contrast, ‘‘non-equilibrium’’ refers to systems which
1054–3139/96/030615+14 $18.00/0
show little association with specific fixed points.) Data
from scale deposits in anaerobic sediments, however,
indicate that population fluctuations can occur in the
absence of exploitation (Soutar and Isaacs, 1974), and
an ongoing debate in fisheries science is ascertaining the
relative importance of exploitation and environmental
factors in determining population levels.
The effect of environmental variability on populations
depends on its time scale (ô) relative to the characteristic
response time of the population (Tr, the ‘‘speed’’ at
which a population returns to equilibrium once disturbed), and in marine systems it is reasonable to expect
environmental variability to produce non-equilibrium
fluctuations. Relatively short-term changes in environmental conditions can be viewed as affecting populations
primarily through changes in growth rate, whereas
persistent, long-term changes may affect equilibrium
population levels (Pimm, 1984, Ch. 3). The latter case is
pertinent to marine systems, where variability is a
? 1996 International Council for the Exploration of the Sea
616
P. D. Spencer and J. S. Collie
decreasing function of frequency (‘‘red noise’’) and thus
has important low frequency components (Steele, 1985).
The effects of persistent changes in equilibria have been
illustrated by introducing cyclic change to the carrying
capacity, K, of the logistic model; when ô°Tr the
population dynamics will tend to smooth out the effects
of variability whereas when ô±Tr, the population
dynamics will be driven by environmental variability
(May, 1976). Environmental variability can thus substantially affect marine ecosystems where the Tr of
many organisms is less than ô (Steele, 1985). Steele and
Henderson (1984) showed that a commonly used singlespecies model can ‘‘flip’’ between multiple stable
equilibria when forced with red noise. Similarly, a
two-species, predator–prey model forced with red noise
was used to describe the dynamics of Pacific herring
(Clupea pallasi) preyed on by Pacific hake (Merluccius
productus) (Collie and Spencer, 1994).
Compensatory biotic interactions are likely to exist in
marine communities, although they are often obscured
by environmental variability. In particular, predation is
believed to play a major role in the community structure
of marine fish; for example, in the north-west Atlantic
the importance of predation by spiny dogfish (Squalus
acanthias) in structuring the Georges Bank fish community is becoming increasingly recognized (Overholtz
et al., 1991). Thus, harvesting may affect entire communities via species interactions, although the models
currently being used for management purposes are
predominantly single-species approaches. In addition,
many marine piscivores, including spiny dogfish in the
north-west Atlantic and Pacific hake, undertake extensive seasonal migrations on a spatial scale much larger
than the habitat occupied by some of their prey, and the
development of realistic models for these predator–prey
systems will require consideration of alternate prey.
Early predator–prey models have focused on onepredator, one-prey systems, and the dynamics of these
models are well known. More recent theoretical models
have considered additional trophic levels (Abrams and
Roth, 1994) and alternate prey (Vance, 1978). Additionally, many investigators have moved from traditional
models with clear equilibria to non-equilibrium models
that focus on the persistence of communities; populations may be unstable at small, ‘‘patch’’ scales, but
movement between ‘‘patches’’ may provide persistence at larger, ‘‘landscape’’ scales (DeAngelis and
Waterhouse, 1987). Unfortunately, these developments
have not provided a rigorous correspondence between
theory and observations of real ecological systems
(Kareiva, 1989). Determining the appropriate level of
detail is critical for applied work; as one considers three
(or more) species, the inherent complexity may preclude
any rigorous application to field data (May, 1977).
Spatially explicit models are of interest in a variety of
resource management applications (Dunning et al.,
1995), but the required additional knowledge of dispersal and movement rates makes their validation difficult
(Conroy et al., 1995). On large spatial scales, which
are usually of interest in marine fisheries management,
simplified, traditional models may serve as useful
approximations to system dynamics (DeAngelis and
Waterhouse, 1987).
Most applications of predator–prey models to marine
fisheries have been age-structured because predation is
generally age- or size-dependent. Andersen and Ursin’s
(1977) ecosystem model of the North Sea has evolved
into what is known as Multispecies Virtual Population
Analysis (MSVPA, Sparre, 1991). This multispecies
analysis reconstructs the abundance of each cohort,
accounting for the observed feeding interactions
between species. The goal of MSVPA has been primarily
to improve stock assessment and to estimate predation
rates and feeding preference functions. Combined with
assumptions about recruitment, the feeding preference
model can be simulated forward in time to provide
medium-term management advice (Gislason, 1991). Up
to 11 species have been included in MSVPA, but endogenous dynamics (cycles, chaos) generally do not occur
when predation and recruitment are not modeled with
strongly nonlinear functions. Attempts to link agestructured and age-aggregated models include introducing density-dependent recruitment into age-structured
models (Overholtz et al., 1991), or constructing production models with the output of MSVPA (Blinov,
1991).
In this study, we develop a simple, two-species
predator–prey model that incorporates the effect of
alternative prey. Because our intent is to understand the
dynamics of specific predator–prey systems, the model is
deliberately kept simple; it is an extension of traditional
two-species models but less complex than a three-species
system. First, the theoretical development of the model
will be discussed, and the effects of harvesting either or
both of the predator–prey species will be examined.
Second, realistic parameter values will be used to apply
the model to the spiny dogfish–Georges Bank groundfish interaction, and stochastic simulations will be run by
forcing the model with a range of variability typical of
marine environments.
Predator–prey model development
A class of models describing populations experiencing
predation, reviewed by May (1977), is helpful in describing dramatic fluctuations in abundance. A common
example is a logistic production function combined with
a sigmoidal type III predator functional response
(Holling, 1965), leading to the presence of multiple
equilibrium population levels. Continuous changes in
the intrinsic growth rate, carrying capacity, or rate of
predation can lead to discontinuous bifurcations in
A predator–prey model incorporating alternative prey
where e is the conversion of prey biomass to predator
biomass (thus, ec is the intrinsic growth rate of the
predator) and D is a quadratic closure coefficient. It
has been shown that a linear form (DH) leads to unstable equilibria and limit cycles (Rosenzweig, 1971)
and is inappropriate for marine plankton (Steele and
Henderson, 1992). Of several possible alternatives, the
form DH2 is used here as the most economical in
parameters. It can be regarded, formally, as applying a
logistic closure for H (Getz, 1984). In contrast to system
(1), the predator isocline is now constrained to pass
through the origin of the predator–prey phase space;
when prey abundance is zero, the equilibrium predator
abundance is also zero (Fig. 1b). Note that different
(a)
Predator abundance
Predator abundance
where P is the prey population size, r is an intrinsic
growth rate, K is carrying capacity, C is a consumption
rate reflecting predator abundance (i.e. C=cH, where c
is the per capita consumption rate and H is predator
abundance), and A is the prey abundance where predator satiation begins. The location of equilibria can be
seen graphically by plotting the prey and predator
isoclines (defined where dP/dt=0 and dH/dt=0, respectively) in the predator–prey phase space. The system
equilibria exist where the predator and prey isoclines
intersect; certain combinations of parameter values
result in three equilibria (referred to as the triple-value
region (TVR)), with the outermost equilibria being
stable and the central equilibrium point being unstable
(Fig. 1a). Either a single high or low equilibrium can
occur with different combinations of parameter values.
The predator dynamics are not explicitly considered in
this system as H is assumed constant, leading to a
horizontal predator isocline. This situation occurs if the
prey are an insignificant portion of the predators’ diet
and thus the predator abundance is independent of prey
abundance, although predation is clearly important in
prey dynamics.
The opposite limiting case describes situations in
which the predator consume only the modeled prey.
Steele and Henderson (1981) extended system (1) by
explicitly including the predator dynamics:
Predator abundance
population size. Models of this form have been used to
describe the dynamics of grazed vegetation (Noy-Meir,
1975) and the spruce budworm (Ludwig et al., 1978).
Steele and Henderson (1984) have shown these models
to be relevant to marine fishery systems. The model
formulation is
617
Low C
High C
(b)
Low D
High D
Low D
(c)
High D
Prey abundance
Figure 1. Predator (dotted lines) and prey (solid lines) isoclines
for (a) single-species model of Steele and Henderson (1984),
(b) two-species model of Steele and Henderson (1981), and
(c) two-species model developed in this study with F1 =F2 =0
and 0<B<1. Stable (,) or unstable (#) equilibria occur where
the isoclines intersect.
values of the predator closure term D can also lead to
single-equilibrium solutions, with high (or low) values of
D resulting in high (or low) equilibrium prey abundance.
Local stability analysis (May, 1974) was used to corroborate the graphical approach (Fig. 1b) and to verify
that the unstable equilibrium of the TVR is a saddle
point. Because the predator isocline is forced to pass
through the origin, extreme reductions in D do not result
in extreme increases in predator abundance (Fig. 1b).
Instead, the lower equilibrium point is determined
largely by the prey abundance at which satiation occurs,
and reductions in D have relatively minor effects at low
prey abundances. This system was used to describe
closed experimental enclosures in which an aggregate
group of zooplankton fed upon an aggregate group of
phytoplankton (Steele and Henderson, 1981). Because
the type III functional response is common among
vertebrates (Holling, 1965), this general predator–prey
interaction can be expected to occur at higher trophic
levels, as in the Pacific hake–Pacific herring interaction
(Collie and Spencer, 1994).
P. D. Spencer and J. S. Collie
The fishing mortality rates F1 and F2 are added to allow
the model to correspond to exploited fish populations;
the effect of harvesting will be considered in the following section. The dimensionless parameter B represents
the degree of coupling between the predator and
modeled prey species. If B=1 the predator feeds only
upon the modeled prey species and thus it is clear that
system (2) is a special case of system (3). Setting B=0
corresponds to the two populations not interacting and
the growth rate of the predator is determined by prey
outside the two-species system. For simplicity, it is
assumed that the alternate prey have a high abundance
so that the predator consumes them at the maximum
rate. A predator which alternates between two sources
of prey can be represented with 0<B<1. Although
the degree of coupling may change over short time
scales, our interest here is annual changes of abundance
over decadal time scales and allowing B to take on a
constant, annual value averages over any seasonal
patterns.
When 0<B<1, system (3) has unique properties not
observed in system (2). Again, certain combinations of
parameter values can result in a TVR solution, as seen in
Figure 1c, and changes in D can result in single equilibrium solutions. Because of the partial dependence of the
predator on the modeled prey, the predator isocline
increases with prey abundance at low prey abundances
but does not pass through the origin of the phase space.
Lower levels of D now lead to noticeable increases of the
predator isocline over the entire range of prey abundances, not simply those prey abundances larger than
the satiation level. Thus, when the system is in the low
Predator abundance
(a)
Low F1
High F1
Predator abundance
Intermediate to the two limiting cases above is the
situation in which the predator dynamics are partially
coupled to any particular prey species. The sigmoidal
predator response was derived by Holling (1965) in
experiments where the animals (mice preying on sawfly
larvae) had an alternative but less preferred food source
(broken biscuits). Feeding upon multiple prey seems
appropriate for fish which are predators on the larvae
and juveniles of numerous fish species and have alternative sources of relatively abundant food. Thus, the
growth rate of the predator should be maximal when P is
abundant but non-zero when P]0. Consumption of
multiple prey could occur over short time and space
scales or could be the result of seasonal migrations, or
some combination.
System (2) can be modified to describe a partially
coupled predator–prey interaction:
Predator abundance
618
Low F2
(b)
High F2
Low D
(c)
High D
0
Prey abundance
Figure 2. Predator (dotted lines) and prey (solid lines) isoclines;
stable (,) or unstable (#) equilibria occur where the isoclines
intersect. (a) and (b) Effect of fishing mortality rates F1 and F2
on the prey and predator isoclines, respectively. (c) Effect of the
predator closure term D when high levels of F2 result in a
negative y-intercept of the predator isocline.
prey equilibrium, very low values of D can result in high
equilibrium predator abundance. This pattern clearly
depends on an alternate prey and would not occur in a
completely coupled two-species system.
Effects of harvesting
The analysis above pertains to unexploited predator–
prey systems, and the two-species system shown in
Figure 1c can also show single or multiple equilibria as
a result of intensity of harvest on the prey and/or
predator. Each of these cases is discussed below.
Prey is harvested, predator is not harvested. This situation was analyzed by Collie and Spencer (1994) for the
case where B=1; allowing 0<B<1 does not change the
essential features of the system. Harvesting of the prey
species lowers the prey isocline in a manner shown in
Figure 2a. Here, increasing F1 to high (or low) values
results in a single low (or high) prey equilibrium. Note
that as P]0 the prey isocline approaches £ so that
A predator–prey model incorporating alternative prey
H=
and negative values of the intercept will occur when the
fishing mortality exceeds the population increase from
the alternative prey (i.e. F2 >ec(1"B)). This illustrates
the significance, in the model, of the alternative food
source. Further, it can be shown that real-valued solutions for the x-intercept exist only for ec(1"B)<F2 <ec
and are independent of D. If a positive x-intercept exists,
then as D]0 the y-intercept approaches "£ and the
location of the lower equilibrium is constrained as the
predator isocline approaches a vertical line at low prey
abundances (Fig. 2c). Thus, the dramatic increases in
equilibrium predator abundance with decreased D are
not necessarily seen with predator harvesting.
Predator fishing mortality can result in a single,
unstable equilibrium, depending on the location of
the x-intercept of the predator isocline, analogous to
Rosenzweig’s (1971) ‘‘paradox of enrichment’’. In the
absence of predator fishing mortality, single high or
low stable equilibria generally occur on the downward
sloping portions of the prey isocline (Fig. 1). With a
positive level of F2, a positive x-intercept of the predator
isocline can occur such that as D]0 a single, unstable
equilibrium occurs in the upward sloping portion of the
prey isocline. The behavior of the simple two-species
system is now characterized by the familiar limit cycle,
as illustrated by the simulation in Figure 3 with parameter values corresponding to the spiny dogfish–Georges
Bank groundfish application discussed below. Unstable
behavior is not observed at high levels of D, for which a
single high prey equilibrium occurs regardless of the
position of the x-intercept (Fig. 2c). Thus, at low levels
of D the presence of predator fishing can be potentially
destabilizing as the ability of the predator to keep the
prey at low levels is reduced.
Predator abundance
Predator is harvested, prey is not harvested. Harvesting
the predator lowers the predator isocline but, in contrast
to harvesting prey, the predator isocline is lowered
uniformly over the entire range of prey abundances. As
an analogy to prey harvesting, high (or low) levels of
F2 result in high (or low) prey equilibria (Fig. 2b).
Changes in F2 and D have similar effects on the predator
isocline, but the latter acts in a nonlinear manner due to
the quadratic closure term.
At high levels of predator harvest, the predator
isocline has a negative y-intercept in the phase plane, as
seen in Figure 2b. The value of the intercept is
(a)
Prey abundance
(b)
Population abundance
changing F1 does not significantly affect this isocline at
low prey abundances, although it clearly affects the
existence of the lower equilibrium. Further, the rate of
fishing can be increased such that the hump in the
isocline disappears.
619
0
200
400
600
800
1000
Time
Figure 3. (a) With a high rate of predator fishing F2 and low
level of D, the species isoclines (dotted lines) may intersect in
the unstable upward sloping portion of the prey isocline and
result in a limit cycle (solid line). (b) The limit cycle of (a) shown
as a time series of predator (solid line) and prey (dashed line)
abundances. Parameter values were r=0.9, K=850, c=3,
e=0.06, B=0.42, A=48, D=0.0001, F1 =0, and F2 =0.16.
Predator and prey are harvested. In the previous two
situations the presence of fishing mortality on either
species could potentially result in single or multiple
equilibria, depending, in part, on the level of D. With
fishing on both species, the location of the TVR in the
K–D parameter space varies as a function of both F1
and F2. Each of the wedge shaped areas in Figure 4
indicates the portion of the K–D parameter space where
a TVR solution exists for a given combination of fishing
mortality rates. Parameter combinations above (or
below) any individual TVR result in a high (or low) prey
equilibrium; thus, a given combination of K and D may
exist in either the high prey equilibrium, TVR, or low
prey equilibrium with different levels of fishing mortality. For example, the asterisk in Figure 4 represents a
single low prey equilibrium when F1 =0.6, F2 =0.05 but a
single high prey equilibrium when F1 =0.0, F2 =0.05.
Increasing F1 alone allows the TVR to exist at higher
values of D but not at lower levels of K, thus shifting
this region upward; similarly, reducing F2 alone also
shifts the TVR upward. The most dramatic shifts in the
location of the TVR result when increases (or decreases)
in F1 occur simultaneously with decreases (or increases)
in F2.
620
P. D. Spencer and J. S. Collie
F1 = 0.6, F2 = 0.00
0.008
F1 = 0.6, F2 = 0.05
D
0.006
0.004
*
F1 = 0.0, F2 = 0.00
0.002
F1 = 0.0, F2 = 0.05
0.0
500
1500
1000
2000
2500
K
Figure 4. Location of the TVR in the K-D parameter space as a function of F1 and F2. Parameter values were r=0.9, c=3, e=0.06,
B=0.42, and A=48. A system at the asterisk (K=850, D=0.0035) will exist in the low prey equilibrium when F1 =0.6, F2 =0.05 but
the high prey equilibrium when F1 =0.0, F2 =0.05.
The two stable equilibria for the system within the
TVR result in two separate equilibrium yield curves for
each of the prey and predator populations; for the
combinations of F1 and F2 within the TVR, only the
higher curve is shown in Figure 5. Recalling the effect of
fishing pressure on system equilibria (Fig. 2) will assist in
interpreting the equilibrium yield plots. In the absence of
predator fishing with the parameter values as in the
asterisk in Figure 4, the equilibrium prey yield will
increase rapidly with F1 as the system is in the single
high equilibrium (Fig. 5a). Further increases in F1 will
move the system into the TVR, where a separate yield
curve associated with the lower stable equilibrium exists
(not shown). Increases in F1 past a critical value can
cause the system to ‘‘flip’’ dramatically to a single low
equilibrium. Allowing positive levels of F2 results in a
higher maximum yield because the intensity of predation
is relaxed. As F2 increases, the boundaries of the TVR
occur at larger values of F1 and the range of the TVR
becomes diminished. Beyond a maximum value of F2 the
TVR does not exist for this parameter set, and the yield
curve converges to the familiar parabolic function as
the equilibrium predator size goes to zero (i.e. when
F2 =ec).
The patterns observed in the predator equilibrium
yield curve can be explained in a similar manner (Fig.
5b). The system exists in the high equilibrium at low
levels of F1 (0.00–0.29) and predator yield is virtually
independent of F1 due to the horizontal predator isocline at high prey abundances. At intermediate levels of
F1 (0.30–0.39), the system exists within the TVR and, at
high prey equilibrium, increasing F2 does not move the
system to the low equilibrium. In either case, the predator yield closely approximates a parabolic function of
F2. However, at higher levels of F1 (0.40–0.50) the
system exists in the low prey equilibrium at low F2, and
increasing F2 will move the system through the TVR to
the high prey equilibrium. Now, the yields at low levels
of F2 are reduced because of prey harvesting. Finally, at
the highest levels of F1 (>0.50) the system has a single
equilibrium with any level of F2. With recognition of
alternate prey (i.e. 0<B<1), a non-zero predator equilibrium yield curve exists even when the modeled prey
equilibrium abundance is zero (i.e. when F1 >r).
Application to the spiny dogfish–Georges
Bank haddock interaction
The final step of the analysis is to use the model to gain
insights on the interactions occurring in real marine
fisheries, such as the spiny dogfish–Georges Bank
groundfish interactions. Haddock (Melanogrammus
aeglefinus) will be used as a representative prey species
for the purpose of illustrating the model. Firstly, the
stock fluctuations of spiny dogfish and haddock will be
described. Secondly, realistic parameter values for the
spiny dogfish–Georges Bank haddock interaction will be
developed. Finally, stochastic simulations will be run
and qualitatively compared with observations.
Increased fishing has been the dominant cause of
precipitous declines in groundfish stocks on Georges
Bank (Sissenwine, 1986). The strong 1963 year-class of
haddock attracted effort to this species in 1965–1966,
resulting in record catches, and the decline of the
Prey equilibrium yield (kt)
A predator–prey model incorporating alternative prey
621
200
(a)
150
100
50
0
0.15
0.1
F2
0.05
0.2
Predator equilibrium yield (kt)
0 0
0.6
0.4
1
0.8
F1
2.5
(b)
2
1.5
1
0.5
0
0
0.02
0.4
F1
0.6
0.15
0.1
0.8
0.05
1 0
F2
Figure 5. Equilibrium yield curves for a system at the asterisk in Figure 4. Two equilibrium yield curves exist for values of F1 and
F2 within the TVR, only the higher of which is presented here. (a) Equilibrium prey yield. At low levels of F2 the TVR will exist
and gradual increases in F1 will eventually ‘‘flip’’ the system to the low prey equilibrium. At high levels of F2 only a single
equilibrium exists and the equilibrium yield curve converges to a parabola when F2 =ec. (b) Equilibrium predator yield. At low
levels of F1, predator equilibrium yield is independent of F1 due to the horizontal predator isocline at high prey abundances. Note
the difference in scale and orientation between (a) and (b).
haddock soon followed in the late 1960s (Fig. 6a). Total
effort remained high after the enactment of the
Magnuson Fishery Conservation and Management Act
in 1977 and the principal groundfish and flounders
(Atlantic cod (Gadus morhua), haddock, redfish
(Sebastes spp.), silver hake (Merluccius bilinearis), red
hake (Urophycis chuss), pollock (Pollachius virens), and
several species of flatfish) have not recovered and are at
historically low levels. Coincident with this decline, the
elasmobranchs (spiny dogfish and seven species of skate)
also increased from low levels in the early 1960s (25% by
weight in trawl surveys) to high levels (75% by weight) in
recent years (NEFSC, 1993) (Fig. 6a). Thus, the Georges
Bank ecosystem today is characterized by low levels
of commercially important groundfish and high levels of
lesser valued species such as the elasmobranchs.
The parameter estimates chosen for the two-species
system represent reasonable first-order estimates appropriate for the spiny dogfish–Georges Bank haddock
interaction. An initial estimate of the intrinsic growth
rate of haddock (r), obtained from a surplus production
model (Collie and Spencer, 1993), was 0.66; this was
increased to 0.9 to adjust for explicitly separating the
mortality due to spiny dogfish predation from other
sources of mortality. Bowman et al. (1984) estimated
an annual consumption rate (c) of spiny dogfish of
622
P. D. Spencer and J. S. Collie
(a)
Biomass (kt)
1000
800
600
400
200
0
1930
1940
1950
1970
1960
Year
1980
(b)
13
Temperature (C)
1990
12
11
10
9
Variance (C2 per year)
1950
1960
1970
Year
1980
1990
(c)
1.0
0.5
0.05
0.10
Frequency (per year)
0.50
Figure 6. (a) Georges Bank haddock age 2+ biomass (——) (from Clark et al., 1982; NEFSC, 1992) and minimum estimates of
spiny dogfish biomass in the northwest Atlantic (– – –) (Gulf of Maine–mid-Atlantic stock) (from Rago, NEFSC, pers. comm.).
(b) Time series of annual mean sea surface temperature (——) from 1)#1) quadrant in Georges Bank having 67)W 42)N as
its northeast corner. A first order autoregressive relationship (– – –) is the optimal fit to the time series. (c) Spectrum of SST
data in (b).
2.7 yr "1; we rounded this value to 3 for this study. The
intrinsic growth rate of spiny dogfish was estimated
from the increasing trend in the minimum biomass
estimates (Fig. 6a) and was approximately 0.18 yr "1.
This growth rate is the product e#c and thus e was
determined to be 0.06. The fishing mortality of haddock
(F1) was set to the mean value observed from 1931–1990
(0.40 yr "1), whereas the fishing mortality for spiny
dogfish (F2) was set to the value observed in 1992 of
0.017 yr "1 (P. Rago, Northeast Fisheries Science Center
(NEFSC), pers. comm.). The degree of coupling (B) is
unknown and is taken here as the length of time spiny
dogfish are residents on Georges Bank, or 5/12 of the
year (0.42). The remaining parameters, tuned to obtain
realistic results in the simulations discussed below,
were K=850 kt, A=48 kt, and D=0.0035 yr "1. This
set of parameter values corresponds to the asterisk in
Figure 4.
Environmental variability with a pattern similar to
that observed in the Georges Bank area was incorporated into the model (Fig. 6b,c). Annual mean sea
surface temperature (SST) from 1946–1990 in a 1)#1)
quadrant on Georges Bank, obtained from the National
Climatic Data Center (NCDC), shows that an autoregressive pattern and a first-order relationship provided
the optimal fit with the autocorrelation coefficient (ñ)
estimated as 0.47. In the frequency domain, this time
series has a red-shifted spectrum with variance decreasing with frequency. This type of variability was used to
force D to vary between high and low values. With the
parameter values above, increasing D to 0.006 results in
a single, high prey equilibrium whereas decreasing D to
0.001 results in a single, low prey equilibrium; the intermediate case of D=0.0035 results in the TVR (Fig. 7a).
The simulations below were produced by varying D
with the first-order autoregressive relationship such that:
A predator–prey model incorporating alternative prey
623
Figure 7. (a) The intermediate configuration of predator (dotted lines) and prey (solid line) isoclines in the TVR corresponds to a
system at the asterisk in Figure 4 with F1 =0.4, F2 =0.017; increases or decreases in D lead to high or low prey equilibrium
abundances, respectively. (b) Simulated predator (dotted line) and prey (solid line) abundances over 300 time steps for the system
in (a) with D0 =0.0035, ñ=0.8, and ì=0.0011. (c) Time series of simulated abundances in the predator–prey phase space; each point
is one year of the 300-yr simulation.
Dt =ñDt"1 +(1"ñ)D0 +ìåt
(7)
where D0 is the mean value of D, ñ is the autocorrelation
coefficient, å is a standardized normal random variable
and ì is a scalar. Simulations were run over 1000 time
steps with system (3) solved by fourth-order RungeKutta numerical integration. With the parameter values
resulting in the triple-value region identified above and
setting ñ=0.80, and ì=0.0011, the system fluctuated
between the high and low prey equilibria; Figure 7b
shows a 300-yr slice of the 1000-yr simulation. The prey
population fluctuated more dramatically than the predator population due to the relatively large difference
between the high and low stable prey equilibrium abundances. A period of favorable environmental conditions
(low values of D) led to predator outbreaks, reducing
the prey to the lower equilibrium. The presence of
alternate prey allowed the predator population to grow
despite low modeled prey. The effect of multiple attractors can be seen clearly when the simulated abundances
are plotted in the predator–prey phase space (Fig. 7c).
Note that the variability present in the system does not
allow the system to equilibrate despite the presence of
stable attractors.
The fluctuations for the prey species obtained from
this simulation are on the same decadal time scales
observed in many marine fisheries (Caddy and Gulland,
1983; Lluch-Belda et al., 1989), and are consistent with
the patterns observed in Georges Bank haddock. The
peak of the simulated prey reaches approximately 360 kt
whereas the lower level is approximately 50 kt, corresponding to roughly the same magnitudes observed in
624
P. D. Spencer and J. S. Collie
major periods; increasing ñ to 0.5 increases the movement between equilibria (Figure 9). At very high levels
of ñ the system approximates a random walk and the
movement between equilibria is decreased, as seen with
ñ=0.99. All of the simulations in Figure 9 were produced with identical parameter values and sequence of
random errors å; the scalar ì was adjusted to maintain
constant variance.
Discussion
Figure 8. Time series of simulated predator (dotted lines) and
prey (solid lines) at two levels of predator fishing mortality F2.
Remaining parameters are as in Figure 7.
the haddock stock. The pattern of predator abundance,
increasing despite the modeled prey species being low,
is consistent with the observed haddock–spiny dogfish
patterns (Fig. 6a).
The simulated abundance of the prey population
was sensitive to the level of fishing mortality on the
predator. With the same parameter values as in Figure 7,
reducing F2 to 0.00 resulted in increasing the proportion
of time the system existed at the low prey equilibrium
(Figure 8). Conversely, increasing F2 to 0.05 decreased
the proportion of time the system existed at the low
equilibrium. These results can also be inferred from
Figure 4, where it is shown that increases (or decreases)
in F2 generally move the system to the high (or low)
equilibrium.
Note that the pattern of variability used to force the
model had an autocorrelation coefficient greater than
that observed in the SST data. Sea surface temperature
was used here to illustrate the pattern of variability in a
readily measured environmental variable. The habitat
relevant to spiny dogfish biology is likely to be subsurface, where environmental variables would be expected
to have a higher autocorrelation coefficient than SST.
The predator–prey dynamics essentially serve as a filter
by which environmental variability is translated to
variability in biological processes, but the exact nature
by which this occurs is unknown. Various assumptions
regarding the nature of variability lead to remarkably
different model outputs. With ñ=0, or white noise, the
prey population exists at the high equilibrium for two
The model developed in this study is useful for describing predators that are partially coupled to a prey population and, specifically, can be applied to marine fish
populations. The dynamics of the prey species are not
changed substantially from the completely coupled
model presented in Collie and Spencer (1994), but the
dynamics of the predator population are considerably
more realistic. For example, the increases in spiny
dogfish abundance in the 1980s occurred when Georges
Bank haddock and other groundfish were at low abundances, a pattern difficult to explain in a completely
coupled system. The pelagic fish Atlantic herring
(Clupea harengus) and Atlantic mackerel (Scomber
scombrus) have also increased in abundance since the
early 1980s (NEFSC, 1993), and spiny dogfish prey
upon these stocks (Overholtz et al., 1991). It is likely that
increases in dogfish abundance depend on predation of
pelagic fish, although there remains some predation on
groundfish stocks. Thus, realistic attempts to model
marine piscivores such as spiny dogfish will require
consideration of alternate prey.
Spiny dogfish have a diverse diet and feed on other
prey besides haddock when on Georges Bank; it is
questionable whether any one species serves as the
‘‘primary’’ prey. Haddock was selected as a representative groundfish species for the purpose of illustrating
the model, but the results seen here can most realistically
be viewed as applying to the aggregate community of
similar groundfish species, including Atlantic cod.
Groundfish are a logical choice for the modeled prey
because of the commercial importance of their dynamics, although one could view pelagic fish as the modeled
prey species. In either case, the effect of alternative prey
must be considered and future work will include aggregating similar species into discrete groups (e.g. groundfish, pelagics) and examining the effect of simultaneous
predation on these groups with the multispecies functional responses discussed by Murdoch and Oaten
(1975).
Critical to the dynamics is the quadratic, or densitydependent, closure term of the predator and the type III
functional response. This formulation follows from the
plankton models of Steele and Henderson (1981, 1992),
in which movement between alternate stable equilibria
in a variable environment was viewed as more consistent
A predator–prey model incorporating alternative prey
350
625
(a)
ρ = 0.0
300
250
200
150
100
Population abundance (kt)
50
(b)
ρ = 0.5
300
200
100
400
(c)
ρ = 0.99
300
200
100
0
0
200
400
600
800
1000
Time
Figure 9. Time series of simulated predator (dotted lines) and prey (solid lines) at various levels of the autocorrelation coefficient
ñ; the scalar ì was adjusted to maintain constant variance. Remaining parameters are as in Figure 7.
with observations than limit cycles. A linear predator
closure term and either a type II or type III functional
response yields unstable equilibria and the associated
limit cycles for a portion of the parameter space (the
former combination corresponds to Rosenzweig’s (1971)
‘‘paradox of enrichment’’); allowing a type III function response and a quadratic predator closure term
effectively eliminates the existence of unstable solutions.
The density-dependent closure term has been explained
in the plankton models as representing an unidentified
predator whose density varies in proportion to the
modeled predator (Steele and Henderson, 1981), and it is
reasonable to assume that density-dependent controls
also exist in marine fish. In particular, it has been
suggested that dogfish recruitment is controlled by com-
pensatory density-dependent fecundity (Holden, 1973).
Compensation could occur through density-dependent
growth and earlier maturation at low population abundance, but there is no direct evidence due to the difficulty
of ageing dogfish. There is some evidence that exploited
dogfish and skate populations have higher fecundity
than unexploited stocks (Holden, 1973). The ability of
dogfish to compensate for fishing mortality is thought to
be weak because of their low maximum fecundity.
There is theoretical and experimental work to suggest
that a type III, rather than a type II, functional response
is most appropriate for marine fish. One, switching
between alternative prey is typically invoked as causing
a type III response and has been observed in laboratory fish (Murdoch and Oaten, 1975). Two, a patchily
626
P. D. Spencer and J. S. Collie
distributed prey combined with aggregation of predators
can produce a type III response (Murdoch and Oaten,
1975), and patchiness is a characteristic feature of
marine systems. Finally, should a predator have a decelerating type II response, it may involve a threshold
below which feeding does not occur; if this threshold is
normally distributed among individual predators a type
III form will result for the population (Peterman, 1977).
Unfortunately, empirical field data for marine fish
populations are usually inadequate to confirm a specific
functional response.
Ecological systems can be classified with respect to the
causal mechanisms leading to variations in abundance,
including stochasticity and overcompensatory biotic
interactions (DeAngelis and Waterhouse, 1987). In the
model presented here, exogenous forcing constantly
moves the system around and between two stable attractors. Exogenous forcing alone can produce a remarkable
array of complex patterns; each of the simulations in
Figure 9 was produced from the same simple model with
identical parameters values, and only the level of autocorrelation in the variability was changed. Allowing
fishing mortality to vary could also result in the endogenous dynamics demonstrated in this study. The limit
cycle in Figure 3 could not occur in this model without
predator fishing, which essentially lowers the effective
growth rate of the predator and limits its ability to
control the prey. Although cyclic patterns in abundance
are suggested for some species, such as the California
Dungeness crab, to our knowledge a pattern of
predator–prey cycles has not been observed in marine
fisheries. The relative importance of endogenous and
exogenous mechanisms has been viewed differently
among ecologists; theoretical and terrestrial ecologists
have concentrated on the former (Turchin and Taylor,
1992; Abrams and Roth, 1994), whereas the latter has
been discussed among marine ecologists (Steele, 1985).
Our emphasis here on exogenous forcing is in recognition of the importance of variability in marine ecosystems, and the range of endogenous behavior is limited
by the two-dimensional differential equation format.
More complex nonlinear models that explicitly consider
three (or more) species may reveal substantially more
complex endogenous dynamics.
The examination of the model with stochastic changes
in the density-dependent term, D, is consistent with
the earlier marine plankton models of Steele and
Henderson, 1981, 1992), but in practice one would
expect environmental variability to affect several processes. Varying the intrinsic rate of growth, r, the
consumption rate, c, or the degree of coupling, B, leads
to similar flips between high and low prey abundances
but not the dramatic increases in predator equilibrium
abundances obtained by varying D. That variations in D
should have this effect is easily seen; D is analogous to
the carrying capacity K in the traditional logistic model.
Environmental variability can affect population growth
rates directly via density-independent rate processes
(Beddington and May, 1977) or via density-dependent
processes and equilibrium abundances (Roughgarden,
1975; Shepherd and Horwood, 1979). The latter case is
consistent with variability at long time scales (Pimm,
1984) and is used here. The highly simplified nature of
the model and uncertainty of marine ecosystems does
not allow specific consideration of how variability affects
populations. Several mechanisms may be important in
the north-west Atlantic, including fish distributions.
Spiny dogfish and their pelagic prey, Atlantic mackerel,
are generally distributed at more northern latitudes
during warm years (Mountain and Murawski, 1992),
and the dramatic increases in spiny dogfish and mackerel
abundances in the 1980s occurred when water temperatures, on average, were higher than in the late 1960s
(Holzwarth and Mountain, 1992).
The model presented here can be used to address the
problem of finding the optimal combination of fishing
mortality rates for the two species. A unique feature of
this model is that nonlinear predation rates can cause
multiple stable equilibria, differing from models with
linear predation rates (May et al., 1979). Increased
harvesting of a predator species generally results in
increased prey yield, but the presence of alternate stable
equilibria makes these increases non-uniform. The presence of fishing increases the potential causes for shifts
between equilibria relative to the earlier plankton
models, as a shift to the low prey equilibrium can now be
caused by either increased fishing on the prey, decreased
fishing on the predator, environmental variability, or
some combination of these factors. For example, the
relatively high fishing mortality rates on spiny dogfish
from 1971–1976 (P. Rago, NEFSC, pers. comm.) may
have kept spiny dogfish biomass low, permitting a
partial haddock recovery in the late 1970s (Fig. 6).
The concept of multispecies management has particular
relevance for Georges Bank fisheries, as the task of
rebuilding the important groundfish stocks will require
consideration of how exploitation of predators such as
spiny dogfish affects recovery rates.
The generality of the results above is limited because
of simplifications made with respect to age structure and
time delays. In particular, spiny dogfish generally do not
consume fish until they are approximately five years old
(Overholtz et al., 1991). This may be stabilizing, as
Maynard-Smith and Slatkin (1973) found in a theoretical model with age-dependent predator hunting abilities.
Predation on predominately young fish, such that prey
age serves as a refuge, would also be expected to be
stabilizing (May, 1974). Conversely, time delays in regulatory processes are generally thought to be destabilizing. Steele and Henderson (1981) found that introducing
a time delay in the predator closure term DH2 increased
the variability of predator–prey responses.
A predator–prey model incorporating alternative prey
In summary, an unsettling fact of any predator–prey
model is that it is impossible to incorporate all the
relevant details into a simplified framework. Like system
(2), system (3) is a grossly simplified depiction of reality,
with the additional concept of alternative prey. From an
ecological viewpoint, it is desirable to have a more
detailed understanding of species interactions but, for
fisheries management, it is necessary to have models that
can be used with the data typically available for stock
assessment. Obtaining unequivocal parameter estimates
is difficult even in the simplest single-species models, and
adding model complexity further exacerbates the problem. The lack of any precise fit of the model to the data
should not be surprising, as it is likely that several
variables vary stochastically, resulting in the unique
realizations seen in nature. With regard to spiny dogfish
abundance, part of the lack of fit can be explained
simply by defining population size; the observed data
correspond to the entire spiny dogfish population in the
northwest Atlantic, whereas the model pertains to only
that portion of the stock that interacts with Georges
Bank haddock. Thus, the modeled spiny dogfish
abundance (Fig. 7) is lower than the total estimated
dogfish abundance (Fig. 6a). The model presented in this
study is a link between the completely coupled case of
Collie and Spencer (1994) and more complex threespecies models. More detailed models will be pursued in
future research, while recognizing that optimal model
complexity is a compromise between desired realism and
analytical tractability.
Acknowledgements
We thank Glenn Strout and the staff at the Fisheries
Climatology Investigation, NMFS, Narragansett,
Rhode Island for providing the SST data. Paul Rago
provided spiny dogfish biomass and fishing mortality
estimates. We are grateful to John Steele and Henrik
Gislason for their comments on an earlier draft. This
publication is the result of research sponsored by
Rhode Island Sea Grant College Program with funds
from the National Oceanic and Atmospheric Administration, Office of Sea Grant, Department of
Commerce, under grant no. NA36RG0503 (project no.
R/F-931).
References
Andersen, K. P. and Ursin, E. 1977. A multispecies extension to
the Beverton and Holt theory, with accounts of phosphorous
circulation and primary production. Meddr. Danm. Fisk.-og
Havunders. N.S., 7: 319–435.
Abrams, P. A. and Roth, J. D. 1994. The effects of enrichment
of three-species food chains with nonlinear effects of predation. Ecology, 75: 1118–1130.
Beddington, J. G. and May, R. M. 1977. Harvesting natural
populations in a randomly fluctuating environment. Science,
197: 463–465.
627
Blinov, V. V. 1991. A robust approach to the analysis of feeding
data and an attempt to link multispecies virtual population
analysis and Volterra predator–prey system analysis. ICES
Marine Science Symposium, 193: 80–85.
Bowman, R. E., Eppi, R., and Grosslein, M. D. 1984. Diet and
consumption of spiny dogfish in the northwest Atlantic.
ICES C.M. 1984/G: 27.
Caddy, J. F. and Gulland, J. A. 1983. Historical patterns of fish
stocks. Marine Policy, 7: 267–278.
Clark, S. H., Overholtz, W. J., and Hennemuth, R. C. 1982.
Review and assessment of the Georges Bank and Gulf of
Maine haddock fishery. Journal of Northwest Atlantic
Fishery Science, 3: 1–27.
Collie, J. S. and Spencer, P. D. 1993. Management strategies for
fish populations subject to long-term environmental variability and depensatory predation. In Proceedings of the international symposium on management strategies of exploited
fish populations, pp. 629–650. Ed. by G. H. Kruse, D. M.
Eggers, R. J. Marasco, C. Pautzke, and T. J. Quinn. Alaska
Sea Grant College Program Report No. 93–02. 825 pp.
Collie, J. S. and Spencer, P. D. 1994. Modeling predator–prey
dynamics in a fluctuating environment. Canadian Journal of
Fisheries and Aquatic Sciences, 51: 2665–2672.
Conroy, M. J., Cohen, Y., James, F. C., Matsinos, Y. G., and
Maurer, B. A. 1995. Parameter estimation, reliability, and
model improvement for spatially explicit models of animal
populations. Ecological Applications, 5: 17–19.
DeAngelis, D. L. and Waterhouse, J. C. 1987. Equilibrium and
nonequilibrium concepts in ecological models. Ecological
Monographs, 57: 1–21.
Dunning, J. B. Jr, Stewart, D. J., Danielson, B. J., Noon, B. R.,
Root, T. L., Lamberson, R. H., and Stevens, E. E. 1995.
Spatially explicit population models: current forms and
future uses. Ecological Applications, 5: 3–11.
Getz, W. M. 1984. Population dynamics: a per capita approach.
Journal of Theoretical Biology, 108: 623–643.
Gislason, H. 1991. The influence of variations in recruitment on
multispecies yield predictions in the North Sea. ICES Marine
Science Symposium, 193: 50–59.
Holden, M. J. 1973. Are long-term sustainable fisheries for
elasmobranchs possible? Rapports et Procés-Verbaux des
reunions du Conseil International pour l’Exploration de la
Mer, 164: 360–367.
Holling, C. S. 1965. The functional response of predators to
prey density and its role in mimicry and population regulation. Memoirs of the Entomological Society of Canada, 45:
1–60.
Holzwarth, T. and Mountain, D. 1992. Surface and bottom
temperature distributions from the Northeast Fisheries
Center spring and fall bottom trawl survey program, 1963–
1987, with addendum for 1988–1990. NEFSC Reference
Document 90–03. 62 pp.
Kareiva, P. 1989. Renewing the dialogue between theory and
experiments in population ecology. In Perspectives in ecological theory, pp. 68–88. Ed. by J. Roughgarden, R. M. May,
and S. A. Levin. Princeton University Press, Princeton, NJ.
394 pp.
Lluch-Belda, D., Crawford, R. J. M., Kawasaki, T., MacCall,
A. D., Parrish, R. H., Schwartzlose, R. A., and Smith, P. E.
1989. World-wide fluctuations of sardine and anchovy
stocks: the regime problem. South African Journal of Marine
Science, 8: 195–205.
Ludwig, D., Jones, D. D., and Holling, C. S. 1978. Qualitative
analysis of insect outbreak systems: the spruce budworm and
the forest. Journal of Animal Ecology, 47: 315–332.
May, R. M. 1974. Stability and complexity in model ecosystems, 2nd ed. Princeton University Press, Princeton, NJ.
265 pp.
628
P. D. Spencer and J. S. Collie
May, R. M. (Ed.) 1976. Theoretical ecology: principles and
applications. Blackwell Scientific Publications, Oxford,
England. 317 pp.
May, R. M. 1977. Thresholds and breakpoints in ecosystems
with a multiplicity of stable states. Nature, 269: 471–477.
May, R. M., Beddington, J. R., Clark, C. W., Holt, S. J., and
Laws, R. M. 1979. Management of multispecies fisheries.
Science, 205: 267–277.
Maynard-Smith, J. and Slatkin, M. 1973. The stability of
predator–prey systems. Ecology, 54: 384–391.
Mountain, D. G. and Murawski, S. A. 1992. Variation in the
distribution of fish stocks on the northeast continental shelf
in relation to their environment, 1980–1989. ICES Marine
Science Symposium, 195: 424–432.
Murdoch, W. W. and Oaten, A. 1975. Predation and population stability. Advances in Ecological Research, 9: 1–132.
NEFSC (Northeast Fisheries Science Center) 1992. Report of
the thirteenth regional stock assessment workshop (13th
SAW). NEFSC. Ref. Doc. 92-02. 183 pp.
NEFSC (Northeast Fisheries Science Center) 1993. Status of
the fisheries resources off the northeastern United States for
1993. NOAA Tech. Mem. NMFS-F/NEC-86. 140 pp.
Noy-Meir, I. 1975. Stability of grazing systems: an application
of predator–prey graphs. Journal of Ecology, 63: 459–481.
Overholtz, W. J., Murawski, S. A., and Foster, K. L. 1991.
Impact of predatory fish, marine mammals, and seabirds on
the pelagic fish ecosystem of the northeastern USA. ICES
Marine Science Symposium, 193: 198–208.
Peterman, R. M. 1977. A simple mechanism that causes collapsing stability regions in expoited salmon populations.
Journal of the Fisheries Research Board of Canada, 34:
1130–1142.
Pimm, S. L. 1984. Food webs. Chapman and Hall, London.
219 pp.
Rosenzweig, M. L. 1971. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science,
171: 385–387.
Roughgarden, J. 1975. A simple model for population dynamics in stochastic environments. American Naturalist,
109: 713–736.
Shaefer, M. B. 1954. Some aspects of the dynamics of populations important to the management of commercial marine
fisheries. Bulletin of the Inter-American Tropical Tuna
Commission, 1: 27–56.
Shepherd, J. G. and Horwood, J. W. 1979. The sensitivity of
exploited populations to environmental ‘‘noise’’, and the
implications for management. Journal du Conseil International de la Mer, 38: 318–323.
Sissenwine, M. P. 1986. Perturbation of a predator-controlled
continental shelf ecosystem. In Variability and management
of large marine ecosystems, pp. 55–85. Ed. by K. Sherman,
K., and L. M. Alexander. Westview Press, Boulder, CO.
319 pp.
Soutar, A. and Isaacs, J. D. 1974. Abundance of pelagic fish
during the 19th and 20th centuries as recorded in anaerobic
sediment off the Californias. Fishery Bulletin, 72(2): 257–
273.
Sparre, P. 1991. Introduction to multispecies virtual population
analysis. ICES Marine Science Symposium, 193: 12–21.
Steele, J. H. 1985. A comparison of terrestrial and marine
ecological systems. Nature, 313: 355–358.
Steele, J. H. and Henderson, E. W. 1981. A simple plankton
model. American Naturalist, 117: 676–691.
Steele, J. H. and Henderson, E. W. 1984. Modeling long-term
fluctuations in fish stocks. Science, 224: 985–987.
Steele, J. H. and Henderson, E. W. 1992. The role of predation
in plankton models. Journal of Plankton Research, 14:
157–172.
Turchin, P. and Taylor, A. D. 1992. Complex dynamics in
ecological time series. Ecology, 73: 289–305.
Vance, P. R. 1978. Predation and resource partitioning in one
predator–two prey model communities. American Naturalist,
112: 797–813.