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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. 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