Foraging arena theory - Wiley Online Library

F I S H and F I S H E R I E S , 2012, 13, 41–59

Foraging arena theory Robert N M Ahrens1, Carl J Walters2 & Villy Christensen2 1

Fisheries and Aquatic Sciences, IFAS School of Forest Resources and Conservation, University of Florida, 7922 NW 71st

St, Gainesville, FL 32653, USA; 2Fisheries Centre, University of British Columbia, 2202 Main Mall, Vancouver, BC, Canada V6T 1Z4

Abstract There is a critical need for quantitative models that can help evaluate trade-off decisions related to the impacts of harvesting and protection of aquatic ecosystems within an ecosystem context. Ecosystem models used to evaluate such trade-offs need to have the capability of capturing the dynamic stability that can arise when predator-prey interactions are restricted to spatial and temporal arenas. Foraging arenas appear common in aquatic systems and are created by a wide range of mechanisms, ranging from restrictions of predator distributions in response to predation risk caused by their own predators, to risk-sensitive foraging behaviour by their prey. Foraging arenas partition the prey in each predator-prey interaction in a food web into vulnerable and invulnerable states, with exchange between these states potentially limiting overall trophic flow. Inclusion of vulnerability exchange processes in models for recruitment processes and food web responses to disturbances like harvesting leads to very different predictions about dynamic stability, trophic cascades and maintenance of ecological diversity than do models based on large-scale mass action (random mixing) interactions between prey and predators. Although a number of methods to estimate these critical exchange rates are presented, none are considered fully satisfactory. The most important challenge for the practical application of models that incorporate foraging arena theory today is not only developing new or improved methods for measuring exchange rates but also evaluating how such rates vary in responses to major fishery-induced changes in abundances of predators.

Correspondence: Robert N M Ahrens, Fisheries and Aquatic Sciences, IFAS School of Forest Resources and Conservation, University of Florida, 7922 NW 71st St, Gainesville, FL 32653 USA Tel.: +1 (352) 2733630 Fax: +1 (352) 3923672 E-mail: rahrens@ ufl.edu

Received 30 Nov 2010 Accepted 1 Mar 2011

Keywords Compensatory responses, Ecosim, food web dynamics, predator-prey interactions, ratio dependence

Introduction

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Basic models of foraging arena theory

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Mechanisms that cause prey population partitioning and vulnerability

44

exchange processes Arena structure caused by restricted spatial distribution of predators relative to prey Restricted predator distribution in response to predation risk caused by its predators Restricted predator distribution caused by limited predator mobility or habitat requirements Restricted prey distribution and/or activity Time allocation to safe/resting sites

2011 Blackwell Publishing Ltd

45 45 45 47 47

DOI: 10.1111/j.1467-2979.2011.00432.x

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Foraging arena theory R N M Ahrens et al.

Vulnerability exchange associated with dispersal behaviours

47

Vulnerability exchange caused by agonistic behaviours

47

High proportion of individual mass not accessible or digestible

47

Spatial displacement of predators and prey by physical transport processes

48

Foraging arena predictions for a range of scales

48

Predator-prey cycles should be rare in aquatic ecosystems,

50

and no paradox of enrichment (instability at high productivity) should occur along spatial or temporal gradients in primary productivity Trophic cascades (Carpenter and Kitchell 1993) should be common at least in simpler

50

aquatic ecosystems The Gauss ‘competitive exclusion principle’ (Hardin 1960) should fail

50

In harvested systems, surplus production of predators should be created by immediate

50

compensatory responses to increased per-capita food density (availability) in foraging arenas Assessment of vulnerability exchange rates for ecosystem management models

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Direct assessment of exchange rates for spatially simple arena structures

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Empirical relationships between prey mortality rates and predator abundances

51

Fitting ecosystem models to time-series data

51

Using complex individual-based spatial models

52

Discussion

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Box 1: Ecopath with Ecosim software for food web modelling

53

Acknowledgements

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References

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Introduction The appeal to incorporate ecosystem considerations into fisheries management is unmistakable (FAO, 1995, US Commission on Ocean Policy, 2004). To facilitate the selection of appropriate management action, there is a critical need for quantitative models to aid in evaluating trade-off decisions related to impacts of harvesting and protection of aquatic ecosystems within an ecosystem context (Sainsbury et al. 2000; Pikitch et al. 2004). Simple statements of principles about the need and means to protect such systems (e.g. Francis et al. 2007) do not offer guidance to managers about how to rank alternative management options, because such an evaluation requires at least the relative quantification of alternative choices relative to management objectives. A diversity of models has been developed to help evaluate the ecosystem-level effect of fishing, incorporating varying degrees of realism about ecological processes (see reviews in Hollowed et al. 2000; Whipple et al. 2000), and it is important to consider the underlying ecological theory upon which such models were derived and the challenges faced when parameterizing such models. 42

Over the past decade, there has been increasing use of ecosystem models based on the Ecopath with Ecosim software package to provide ecosystem management advice (Christensen and Walters 2011). Ecosim, the dynamic modelling part of that software, is built around what is termed ‘foraging arena theory’, in which it is assumed that trophic interactions are largely limited to spatially restricted foraging arenas (Fig. 1). Ecosim models appear to be capable of at least fitting historical data on responses of multiple fish populations to harvesting and changes in primary production regimes and of explaining why aquatic ecosystems do not exhibit dynamic instability predicted under simpler assumptions about interaction rates. Often, the mass action assumptions inherent in such models, in conjunction with type II foraging response limits on predator feeding rates, predict dynamic instability ‘predator-prey cycles’, particularly in more productive ecosystem (the ‘paradox of enrichment’ Rosenzweig 1971), or unstable community structure (Abrams and Holt 2002). Foraging arena theory emerged through a series of studies during the 1990s (Walters and Juanes 1993; Walters et al. 1997; Walters and Korman 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59


Figure 1 Aquatic organisms have evolved a diversity of behaviours that limit their exposure to predation risk. The use of spatial refuges from predation is likely to restrict foraging to limited volumes (V) nearby and limit predator-prey interaction. In some instances, predator-prey interactions may be severely limited as with juvenile fish moving out from cover to feed on drifting or emerging aquatic insects.

1999). The general predictions of foraging arena theory have been fairly widely used by fisheries scientists, mainly through the application of Ecosim, to explain and model responses of harvested ecosystems (review in Walters and Martell 2004). The potential for the underlying ecological theory upon which foraging arena theory is based to help to understand a broad range of aquatic ecosystem behaviours has apparently not been widely recognized. Here, we describe the basic models of foraging arena theory. We review the various mechanisms that can lead to these models, list the main predictions they imply and discuss the practical difficulties that have been encountered in estimating critical vulnerability exchange rate parameters that appear to limit trophic interaction rates. Basic models of foraging arena theory The basic assertion of foraging arena theory is that spatial and temporal restrictions in predator and prey activity cause partitioning of each prey population into vulnerable and invulnerable population components, such that predation rates are dependent on (and limited by) exchange rates between these prey components. Trophic interactions take place in the restricted ‘foraging arenas’ where vulnerable prey can be found (Fig. 1). That is, if the total prey population is N, and V of these are vulnerable to predation at any moment (i.e. are in the foraging arena for interaction with some predator whose abundance is P), total prey consumption rate Q should be predictable as the mass action product Q ¼ aVP

ð1Þ

where the predator rate of effective search a has units area or volume per time searched by the 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59

predator divided by the area or volume (A) of the foraging arena. Note here that Q is predictable as Q = aNP only when V = N, i.e. when all N prey and predators are randomly distributed or well mixed. This argument remains the same if the predator exhibits type II behaviour, i.e. if a is reduced when search time is lost because of prey handling (Holling 1959a,b). We might represent such effects for example with the multispecies disc equation (May 1973). Two specific models have been proposed for predicting changes in vulnerable prey densities V in foraging arenas (Walters and Christensen 2007). The first or ‘continuous exchange’ model (Walters et al. 1997) proposes that prey exchange between the vulnerable and invulnerable states at instantaneous rates v and v¢, so V gains individuals at rate v(N ) V) and loses them at rates v¢V and aVP. This results in the rate equation: dV=dt ¼ vðN VÞ v0 V aVP

ð2Þ

If the vulnerability exchange and predation rates are high compared to overall rates of change of N and P, V is predicted to remain close to the moving equilibrium (with N and P) given by solving (2) with dV/dt = 0: V¼

vN v þ v0 þ aP

ð3Þ

The second or ‘bout feeding’ model proposes that prey (or predators) regularly (e.g. daily at dawn and/or dusk) enter the foraging arena for short temporal feeding bouts, depleting V exponentially during each bout such that the mean prey density seen by the predator during each bout of duration T is given by 43


V¼

vNð1 eaPT Þ aP

ð4Þ

and initial vulnerable prey abundance vN is some fraction of the total prey population N. Note that both of these models imply two alternative ways to precisely define the phrase ‘limited food supply’, found in ecological arguments (e.g. Abrams and Ginzburg 2000) but generally lacks a formal definition. The supply of food may be defined as a temporal rate vN of food delivery to foraging arenas or alternatively as the limited food density V that results from the balance of supply rate and removal processes. Note that an immediate and crucial prediction of models (3) and (4) is that there can be strong negative effect of predator abundance P on vulnerable prey density V and feeding rate per predator Q/P, whether or not predators have any substantial impact on total prey abundance N (as suggested in Abrams and Ginzburg 2000). Substituting Equation (2) into (1) results in the ‘functional response’ prediction, Q=P ¼

avN v þ v0 þ aP

ð5Þ

That is, the basic foraging arena models predict strong ‘ratio dependence’ in the predation rate Q, with attendant consequences for predator-prey stability (Hassell and Varley 1969; Deangelis et al. 1975; Getz 1984; Arditi and Ginzburg 1989; Akcakaya et al. 1995; Abrams and Walters 1996). Further, these models do not depend on specific assumptions about predator behaviour, such as interference or contest competition. Unlike models based on substituting P/N (prey per predator) ratios into functional response models (e.g. Beddington 1975; Deangelis et al. 1975), they can be derived from fine-scale arguments about behaviour and spatial organization of interactions and so are not subject to Abrams (1994) very valid criticisms about the simplistic ratio formulations. Foraging arena models assert that competition between predators is intensified from the spatial restriction of interactions into arenas; however, there is no one factor that restricts foraging activity and restriction may arise because of factors such as prey and/or predator behaviours, or specific habitat requirements. Another basic prediction is that interaction rates Q can vary between ‘bottom-up’ controlled and ‘top-down’ controlled depending on v and a. This is easiest to see with equation (2): If v is small and a is large, Q approaches the ‘donor controlled’ 44

limiting rate vN as P increases; but as v increases, Q approaches the mass action rate aNP. The predictions from the foraging arena equations extend across a wide range of scales. Before describing these predictions in more detail, we think it important to demonstrate that the fundamental assumption of partitioning of prey into V and N ) V components, with attendant exchange processes that can limit trophic interactions, is very widespread or potentially universal at least in aquatic ecosystems. Partitioning resulting from exchange processes implies a basic reversal of the idea that small proportions of prey may be in safe refuges so as to cause predation rates to have type III functional response form (e.g. Sih 1987b). Under the forging arena assumption, it is far more common for the bulk of prey to be in refuges at any moment, particularly when exchange rates are low. Intense completion for resources within the foraging arena potentially results in increased foraging times by prey (see Walters and Juanes 1993) as prey density increases, resulting in the type III form of the functional response because of changes in prey behaviour rather than predator behaviour. Mechanisms that cause prey population partitioning and vulnerability exchange processes Predation plays an important role in shaping the short-term and long-term behavioural characteristics of organisms. There have been extensive theoretically (see review by Mcnamara and Houston 1992; Abrams 1986) and field (as reviewed in Lima and Dill 1990; Sih 1987a; Kacelnik and Bateson 1996) investigations identifying how and under which circumstances individuals alter their behaviour in response to predation. When the foraging arena equations above were first being developed, we envisioned them narrowly as a way to represent risksensitive foraging behaviours by juvenile fish, which typically avoid predators by hiding in shallow waters, bottom structure or schools and make brief foraging trips to feed on zooplankton in arenas near such relatively safe locations (Walters and Juanes 1993); in such cases, vulnerable prey densities V/A can obviously be depleted without any substantial impact on larger-scale zooplankton abundance. Since then, many other examples of spatial restriction in predator behaviour causing limited access to prey have been documented (e.g. cases reviewed by Heithaus et al. 2008), and in the course of developing ecosystem 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59


models like Ecosim (Walters et al. 1997) a much wider array of behaviours that can lead to the vulnerability exchange structure of Equation (1) have been identified. Indeed, when Ecosim models have been fit to time-series data by estimating vulnerable exchange rates, many predator-prey linkages are predicted to involve a vulnerability exchange structure (estimated low v’s, e.g. Cox et al. 2002; Heymans 2003; Heymans et al. 2009). A critical point about vulnerability exchange structures is that restriction in activity by any one species is likely to induce the exchange structure represented by Equation (1) for at least two trophic linkages, between the species and its prey and between the species and its predator(s). Consider, for example, the simple food chain zooplankton ﬁ small fish ﬁ piscivore. If the small fish ‘chooses’ to restrict its activities so as to forage only near hiding places, most of the small fish become invulnerable at any moment to piscivores. Likewise, then most of its zooplankton prey population becomes invulnerable to it at any moment. This ‘cascade’ of forging arena structures results in spatially limited interactions between predator and prey occurring on time scales of minutes/hours and at the spatial scale of metres (Fig. 2), intensifying competition between predators when exchange processes limit the rate at which prey are replenished. In the following section, a simple classification of behaviours that can lead to vulnerability exchange dynamics is presented, based on experience to date with ecosystem model development. This classification is not complete or exhaustive, but it does cover a wide variety of trophic interactions in aquatic systems and demonstrates the broad applicability of foraging arena theory; a diversity of relevant examples is referenced. Arena structure caused by restricted spatial distribution of predators relative to prey This category includes the original situation mentioned above, where the predator distribution covers only a small proportion of the area or volume occupied by prey organisms. But such restricted overlap can be caused by a variety of factors of which two appear to be particularly common. In all such cases, the vulnerability exchange rates v and v¢ are likely to have values determined mainly by physical transport (advection, diffusion) and random movement processes of the prey and can be 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59

extremely low proportions of the overall prey population in physically large systems. Restricted predator distribution in response to predation risk caused by its predators The behaviour of post-larval juvenile fish is likely dominated by a need to reduce predation risk (see review in Sogard 1997), and this is likely also the case for juveniles of mobile invertebrates (as demonstrated by Stein and Magnuson 1976; Pierce 1988; Main 1987). So far, as we know from many examples, post-larvae move into highly restricted habitats (e.g. structure, schools) and spend relatively little time foraging. For most fish, increase in body size is associated with ontogenetic habitat shifts to use much larger foraging arenas and multiple habitat types (Mittelbach 1986; Love et al. 1991). Many mobile aquatic invertebrates (see reviews in Forward 1988; Haney 1988; Lampert 1989) and fish (Neilson and Perry 1990; Watanabe et al. 1999; Scheuerell and Schindler 2003) exhibit strong vertical migration behaviours, apparently in response to predation risk but perhaps also as a way to manage metabolic costs (Wurtsbaugh and Neverman 1988; Jensen et al. 2011) or gain a horizontal transport advantage (Champalbert and Koutsikopoulos 1995). Such behaviours result in temporally limited periods of overlap with prey, leading to diurnal foraging bouts and possibly localized prey depletion as represented by Equation (4). Restricted predator distribution caused by limited predator mobility or habitat requirements Many ‘predators’ have limited or no mobility, for example, sessile invertebrates that filter-feed the water column above their resting site. Such restriction in vertical access to prey obviously creates a foraging arena exchange structure with algal and detritus ‘prey’ distributed over the whole water column (e.g. Richardson and Bartsch 1997; Dolmer 2000; Sara and Mazzola 2004). It is not necessary for the predator (grazer) in such cases to be entirely sessile, as demonstrated by the elegant experiments of Arditi et al. (1991) and Arditi and Saiah (1992). Their weak swimming species (Simocephalus and Scapholeberis) foraged mainly in a restricted arena at the bottom of their experimental containers, with the vulnerability exchange (v) of dead algal prey into the container bottom arena being caused by sinking of algae, with 45


Flux prediction: Q = aVP

Hours/m

Available prey density (V)

Fine scale

Predator density (P)

Mortality rate

Recruitment (Intraspecific competition)

(trophic interactions)

Adjusts foraging time?

v




Activity depend. mortality

Size depend. mortality

No

Predator cons. rate (Q/P)

Prey mortality rate (Q/N)

Predator activity

Years/km

No

Predator average weight

Yes

Handling time?

Yes

Prey density (N)

Community properties

Recruitment

Abundance

Recruitment relationships

Time Predator density (P)

Equilibrium predator

Decades/Regions

Productivity

High

N Low

Coarse scale

Diversity

V1

P1

V2

P2

V3

P3

Equilibrium prey

Figure 2 Foraging arena predictions across a range of space/time scales. The restriction of predator-prey interaction to ‘foraging arenas’ results in a decreasing hyperbolic relationship between available prey density (V) and predator density (P) at fine space/time scales. Intra-specific competition within these arenas leads to the commonly observed Beverton-Holt recruitment relationship. For inter-specific interactions, the exchange of prey into and out of these arenas limits predation mortality resulting in community stability. 46

2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59


little or no re-suspension (v¢ = 0 in Equation 1). This limited exchange of prey organisms with the bottom is also apparent at lager scales. Muschenheim and Newell (1992) demonstrated that the effective ‘feeding zone’ of blue mussels was restricted to 3–5 cm off the bottom and that prey in this region were depleted quickly. In some cases, apparently mobile predators still concentrate their activities in particular habitats even when not faced with obvious predation risk, perhaps as a way to manage energetic costs (e.g. Inoue et al. 2005) and/or places for ambushing prey (as observed by Savino and Stein 1989; Smale et al. 1995). Many reef and demersal fish like rockfishes tend to hold and forage near bottom structure, even while taking mainly planktonic prey; one reason for this is that the ocean bottom acts as a trap to concentrate vertically migrating prey species like euphausids (Genin et al. 1988; Brodeur 2001). Such behaviour may be optimal under certain conditions (Sih 1984) and establishes an arenatype structure. Restricted prey distribution and/or activity This category represents situations where predators may be widely distributed, but their prey show possibly severe restriction in spatial distribution and activity. Predators that restrict the availability of their prey due to ‘risk-sensitive’ behaviours in turn restrict interactions with the species that prey on them. Time allocation to safe/resting sites This is the interesting case from an evolutionary perspective, where the same behaviours used to acquire resources (movement into foraging arenas to feed) cause some creatures to be the resources of other species (predation risk concentrated in the same arenas). Obviously, such situations create trade-off relationships for which we can expect strong natural selection for optimized time allocation. The optimal allocation of foraging time has been explored extensively (see discussions in Abrams 1991; Ludwig and Rowe 1990; Mangel and Clark 1986; Matsuda and Abrams 2004; Stephens and Krebs 1986; Werner and Anholt 1993) and has been demonstrated to be context dependent. It is difficult to generalize about the amount of time spent individuals under predation threat spend in refuge habitats. Studies have show noticeable variation between individuals (e.g. Fraser 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59

and Gilliam 1987; Gilliam and Fraser 1987; Morgan 1988). There is an indication that for juvenile fish, the optimum appears to typically be a small time allocation to foraging particularly when foraging is restricted to crepuscular periods (Eggers 1978; Clark and Levy 1988; Parrish 1992; Metcalfe et al. 1999; Scheuerell and Schindler 2003). Vulnerability exchange associated with dispersal behaviours The acquisition of resources is not the only behaviour that can expose organisms periodically to predation risk. Dispersal behaviours are also dangerous and can occur for a wide variety of reasons (ontogenetic changes in habitat requirements or opportunities, response to locally high densities of competitors, movement to reproductive sites, etc.). Perhaps the most obvious example in aquatic systems is drift of benthic stream insects (Palmer et al. 1996) that spend most of their time in interstitial microhabitats where they are safe from most fish predation, but occasionally leave such sites to drift downstream. In this case, the V of Equation (1) is literally the concentration of drifting (and emerging) insects, and the drift entry rate v can be limiting to potential abundance of stream predators like trout. Vulnerability exchange caused by agonistic behaviours Many aquatic organisms defend restricted resting or mating sites and exhibit strong aggressive behaviours towards nearby conspecifics. In such cases, there can be strong density-dependent increase in agonistic activity with increasing density of conspecifics, leading to increased predation risk and density-dependent mortality at high densities. Indeed, one of the first attempts to test foraging arena predictions involved a small benthic fish (cunner, Tautogolabrus adspersus), which has exhibited exactly such increase in activity and mortality when its densities were experimentally manipulated or varied among reef sites (Nitschke et al. 2002; Juanes 2005). High proportion of individual mass not accessible or digestible Some predators take only parts of their prey without normally killing the prey (Mouritsen and Poulin 2003; Bowerman et al. 2010). For example, browsing herbivores often select only particular plant parts that are physically accessible (not too high off the ground, not underground) or high in quality 47


(seeds, leaves and active growth tips that are high in protein), leaving most of the prey growth/production system intact. In such cases, the v process represents prey body growth. Such dynamic structures are much more common in terrestrial than aquatic environments, but they do occur with grazers on macrophytes and macroalgae, and even with animal–animal interactions like fishes that nip at the siphons of buried molluscs (e.g. Tomiyama and Omori 2007) and ‘graze’ on parts of corals (Miller and Hay 1998). Spatial displacement of predators and prey by physical transport processes It is common in aquatic ecosystems for production dynamics to be ordered in a physical flow pattern, where nutrient delivery at the head of the flow gives rise to primary production peaks downstream some distance (as primary producers are advected away from the nutrient source as they grow) and to secondary production peaks still further downstream as animals grow in response to the primary production as they are advected. This structure occurs in situations ranging from freshwater springs (Vannote et al. 1980) to estuaries (Winemiller and Leslie 1992; Speckman et al. 2005) to the large upwelling areas off the west coasts of South America and Africa (Pauly et al. 1989; Shannon 1985, Gremillet, 2008). In such flow structures, smaller organisms may be able to partially control their downstream positions through counter-current movements [vertical migration (e.g. Albert 2007; Kamykowski and Zentara 1977)], emergence and flying upstream (e.g. Bird and Hynes 1981; Williams and Williams 1993). If these behaviours are not completely successful at bringing organisms to centres of prey abundance, such counter-current movements can result in organisms being concentrated in areas along the flow such that their food species appear to exhibit largely donor-controlled dynamics, i.e. to have concentration dynamics V with the same dominant terms (exchange in and out, predation loss) as in Equation (1). A similar concentration dynamic is observed when physical flow processes concentrate organism at frontal zones (Olson et al. 1994). These areas of higher food concentrations appear to be are important foraging areas for higher trophic level organism such as sea birds (Hoefer 2000; Xavier et al. 2004), tunas (Royer et al. 2004) or whales (Bluhm et al. 48

2007). In these structures, the concentration of production from a much wider areas establishes a foraging arena as organism exchange into frontal areas either through physical transport or through directed movement. Foraging arena predictions for a range of scales A fundamental assumption of foraging arena theory is that predator-prey interactions occur at the scale of hours and metres through various behavioural and physical mechanisms potentially restricting prey exposure to predation and intensify competition between predators. This foraging arena formulation provides a structure that can be used to predicting observed states across a range of scale from the individual up to the ecosystem level. At the scale of the individual, foraging arena theory can be invoked to explain the failure of at least fishes to consume nearly as much food as we would predict to be possible based on large-scale sampling of prey abundances. Back calculation of food intake rates from observed growth in the field, using laboratory-based bioenergetics models, indicates that fish typically feed at much lower rates than predicted from laboratory estimates of maximum ration (Schindler and Eby 1997; report average intake rates only 26% of maximum, over 186 cases representing 18 fish species). Fish biologists routinely encounter this phenomenon where a high proportion of the fish stomachs examined are empty (Roger 1994; Arrington et al. 2002; Yankova et al. 2008; Ainsworth et al. 2010). Foraging arena theory argues that the phenomenon is a symptom of evolutionary adaptation to predation risk (Lima and Dill 1990) and involves two distinct and possibly interacting causes: spatial restriction in activity that leads to local prey depletion (low V) where foraging does take place, and/or temporal restriction in foraging activity also so as to reduce predation risk (Werner et al. 1983a). Each of these causes can lead to the observation of empty stomachs or apparent reduced food intake. Suboptimal foraging has also been observed in the absence of predation although these observations have been for small individuals that may have restricted opportunity to select which areas to forage in (Mittelbach 1981; Werner et al. 1983b). In addition, individuals commonly stop or reduce feeding during spawning, brood rearing and during migration, or may receive less that optimal ration because 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59


of territorial behaviours or dominance hierarchies (see Huntingford 1993). The theory makes two broad predictions about what we should find when short-term (seasonal, annual) observations are collected across a range of predator densities. First, mean available food density per predator (V) should decrease in an inverse hyperbolic pattern as predator density P increases, with the first increments in predator abundance causing the greatest incremental decreases in available food density, whether or not there is any impact of P on the overall prey population N (Fig. 2). This prediction is dependent on the exchange rates (v) and approaches a linear decrease in V with increasing P at higher exchange. Second, instantaneous prey mortality rate (Q/N) should increase in a hyperbolic pattern towards a maximum rate (v) as P increases, rather than simply being proportional to predator abundance P (Fig. 2). When applied over longer time scales (as discussed later), this second prediction is the basic reason that predator-prey models based on foraging arena equations tend not to show cycles, even when handling time effects (reduction in predator search rate a with increasing N) are included in the predictions provided exchange rates (v) are low (right column of Fig. 2). On time scales of one to a few years, Walters and Korman (1999) argue that the hyperbolic relationship between V and P, along with predator behaviour and predation risk, is likely to lead to the most commonly observed form of stock–recruitment relationship in fish populations, namely the flat-topped curve called the Beverton-Holt relationship (left column of Fig. 2). Hundreds of empirical stock– recruitment relationships have been assembled for fish (Myers et al. 1999), and most of these show net recruitment to harvestable ages (typically 2–4 years) to be largely independent of parental spawning abundance or egg production. Such independence implies strong density dependence in survival rates from egg to recruitment (else recruitment would on average be proportional to egg production, not independent of it). Beverton and Holt (1957) showed that this pattern is expected if juveniles die off over time before recruitment according to a quadratic mortality model of the form dP/dt = )(M0 + M1P)P. Further, Walters and Korman (1999) showed that exactly this linear relationship between instantaneous mortality rate M0 + M1P is expected if (1) food density V available per P decreases as predicted by Equation (2), juvenile fish adjust their foraging times 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59

per day so as to try and achieve a base growth rate needed to complete their ontogeny, and (3) mortality rate is proportional to time spent foraging. Such predictions about individual and population scale patterns may help to interpret some patterns in field data, but the really interesting predictions from foraging arena theory arise when models are developed for predicting impacts of changing trophic interactions in multispecies fisheries and whole aquatic food webs. Using a convenient, static mass-balance model like Ecopath (Christensen and Pauly 1992) to estimate initial abundances (N, P) and trophic flow rates (Q) for a whole food web, changes in these abundances response to disturbances like fishing and changes in nutrient loading can be simulated over time. Many such static models have been constructed (Christensen and Walters 2004), typically representing 20–50 species and aggregated biomass groups. If we predict the changes in Q’s using simple mass action rules (Q = aNP, all species acting as though they were randomly mixed over the ecosystem), it is easy to demonstrate that simulated competition and predation effects quickly result in substantial loss in food web structure (one consumer–one resource, limiting similarity, dominant predator effects). Such model pathologies only become worse when we include more realistic, type II functional response effects representing limitation on predator feeding rates because of handling times and adjustments in foraging times to achieve target food consumption rates; the typical result is to predict at least some predator-prey oscillations, along with ‘paradox of enrichment’ effects (increasing dynamic instability as simulated primary productivity is increased). Such oscillatory effects have been avoided in some simple nutrient-phytoplankton-zooplankton (NPZ) models by assuming quadratic, density-dependent zooplankton mortality (so-called model ‘closure’), but there is little theoretical or empirical support for such assumptions (Franks 2002). When food web models like Ecosim (Box 1) are used to predict effects of dynamic changes in predator-prey interaction rates Q using the foraging arena vulnerability exchange Equations (1–5), there is a dramatic reversal of the difficulties encountered with models based on simple mass action interaction rates. Models with low vulnerability exchange rates (v’s) routinely make four key predictions that are difficult to obtain with simplified mass action models: 49


Predator-prey cycles should be rare in aquatic ecosystems, and no paradox of enrichment (instability at high productivity) should occur along spatial or temporal gradients in primary productivity We know of only one case where long-term data on marine fish showed a cyclic pattern of the amplitude and period expected from simple predator-prey models (between herring and cod off the British Columbia coast, Walters et al. 1986) and that pattern broke down not long after it was published. In an Analysis of cyclical population, Murdoch et al. (2002) suggest that the cyclic dynamics in fish population analysed were not consumer-resource cycles. The paradox of enrichment has not been observed in lake fertilization experiments (as discussed in Abrams and Walters 1996) and is not evident in highly productive marine systems (e.g. upwelling areas). Foraging arena theory is of course not the only explanation for stability in predator-prey relationships but it does contain the prey refuge structure identified as creating stability in predator-prey interactions (e.g. McNair 1986; Sih 1987b; Holt and Hassell 1993). Other mechanisms such as spatial heterogeneity (e.g. McMurtrie 1978; McLaughlin and Roughgarden 1991), prey switching (Murdoch 1969) or age-structure, (Smith and Mead 1974) but also see (Smith and Wollkind 1983), have been shown to cause stability. Trophic cascades (Carpenter and Kitchell 1993) should be common at least in simpler aquatic ecosystems This has been found in comparisons by Shurin et al. (2002) and highly disturbed ocean systems (e.g. Frank et al. 2005; Daskalov et al. 2007) although the influence of cascade effects relative to other factors (e.g. environmental variability, species diversity, behavioural responses, variation in primary productivity, and habitat structural complexity) may not necessarily have an overriding effect on system structure. But even in these cases there should be positive covariation in abundances at all trophic levels when measured along productivity gradients. The Gauss ‘competitive exclusion principle’ (Hardin 1960) should fail Multiple predators can be sustained by a single prey type (or same diet mix of prey types), provided each 50

predator feeds at least partly in its own distinctive foraging arena and none of these predators is capable of depleting the overall prey population. More generally, classical food web theory (May 1973; Cohen et al. 1990) should not work properly for aquatic systems, as suggested by Link (2002). In harvested systems, surplus production of predators should be created by immediate compensatory responses to increased per-capita food density (availability) in foraging arenas Surplus production should occur in the form of improved growth and/or reduced foraging time and associated predation mortality rates, particularly for younger animals, whether or not there are longerterm increases in food availability through numerical responses of prey populations (Aydin 2004). A worrisome corollary of this prediction is that we should not expect to see a species exhibit similar surplus production rates wherever it is found, i.e. compensatory responses may vary widely with the particular spatial foraging arena structures found in different locales; such unpredictability in productivity from taxonomic information is obvious in metaanalyses of stock–recruitment and surplus production data from exploited fish populations (Myers et al. 1999; Dorn 2002; Goodwin et al. 2006; Walters et al. 2008). There are an uncomfortable number of ‘may, maybe, and should’ in these predictions, because they depend on quantitative assumptions about vulnerability exchange rates (v) being low enough to severely limit trophic interaction rates. Such sensitivity to quantitative detail is illustrated by the Arditi et al. (1991) laboratory experiments, where they describe having to take great care in setting volume exchange and mixing rates in their experimental chambers so as to obtain the abundance gradients that they measured across the chambers. In the following section, we address the issue of assessing v’s from field data. Assessment of vulnerability exchange rates for ecosystem management models There is a clear need for quantitative models to evaluate the various trade-offs involved in aquatic ecosystem management, so as to provide advice that can at least rank the relative impact of management options and to expose critical uncertainties that may trigger precautionary or experimental management 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59


policies. We doubt that any natural historian who has looked closely at spatial and temporal organization of aquatic trophic interactions would question the need to represent such interactions as being restricted to at least some degree to what we have called foraging arenas, whether or not such arenas can be precisely defined and measured under field conditions. It is likely that interactions between predators and prey are occurring at the scale of hours and metres. But in practice, there is a huge gulf between knowing that interaction rates are likely to be restricted to some degree by vulnerability exchange rates (v), vs. being able to quantify such rates well enough to say whether they are low enough to require abandonment of simpler mass action predictions of interaction rates, and to make useful predictions about compensatory responses (surplus production) to various disturbance regimes. A variety of approaches have been tried for estimating vulnerability exchange rates from field data. None of these has been fully satisfactory, at least partly because arena structures in the field are spatially and temporally complex; indeed, one reason to call the foraging arena arguments a ‘theory’ is that arena structures are ‘theoretical entities’ that are practically difficult or impossible to directly observe (Maxwell 1962). Three main methods have been used to provide estimates of apparent v’s using field data, and a fourth is under development. Direct assessment of exchange rates for spatially simple arena structures For small fishes foraging on zooplankton in welldefined water volumes adjacent to refuges from predators (weed beds, reefs, etc.), we have attempted to estimate exchange rates of zooplankton because of physical mixing (eddy diffusion, advection) processes. Such calculations give v’s that depend strongly on total pelagic area or volume, implying strong predation linkages in small systems (e.g. ponds) vs. very low exchange rates for fish feeding near small reefs or shoals embedded in large oceanic flow structures. Walters et al. (2000) used data on biomass and drift concentrations of insects in the Colorado River in Grand Canyon to estimate food concentrations available to competing native and non-native fishes, in a multispecies model aimed partly at predicting effects of the exotics on endangered native fishes like the humpback chub (Gila cypha). We estimated v’s 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59

on order 6–8% of the unavailable (buried, under rocks) insect biomass per day, enough to stabilize interactions involving short-lived insects (mainly chironomids) but not enough to prevent depletion of longer lived insect species given high abundances of exotics fishes trout and catfishes. Unfortunately, these relatively ‘clean’ predictions about invertebrate–fish interactions were much less important to overall model predictions about native–exotic interactions than were highly uncertain predictions about predation impacts of the exotics on juvenile native fishes. Unfortunately, simple demonstrations that vulnerable prey densities (V) are depressed in known foraging arenas are not clear evidence of limited exchange rates. For example, we monitored the responses of zooplankton communities in coastal B.C. lakes to fish introductions (Walters et al. 1987) and found around 80% lower densities of limnetic cladocerans and copepods along littoral transects than open-water transects. But we observed the same differences before trout introductions, implying that either the limnetic zooplankton managed to actively avoid littoral areas or encountered high predation rates by an assortment of bottom-associated predatory insects like dragon fly nymphs that were quickly eradicated by trout. Empirical relationships between prey mortality rates and predator abundances For a few large marine ecosystems (e.g. North Sea, Bering Sea), time series of total mortality rates of various fish species have been estimated from survey data on changes in size-age composition, under widely varying combinations of prey and predator abundances (Mackinson and Daskalov 2007). Rearranging Equation (5), the basic foraging arena models predict that predation mortality rates M = Q/N should vary as M = vP/(k + P), where k is the combined parameter k = (v + v¢)/a. So regressions of Q/N vs. P should show a saturating relationship, with maximum value v and ‘Michaelis constant’ k (Fig. 2). Fitting ecosystem models to time-series data Ecosim biomass dynamics and multistanza agestructured models have been fitted to time-series data for multiple fish (and a few invertebrate) species for a wide variety of marine ecosystems (see examples in Walters and Martell 2004), by 51


using non-linear estimation procedures to vary selected v’s. Assuming very low v values for prey of exploited species typically causes Ecosim to predict the exploited species to ‘flatline’ in response to measured historical changes in fishing mortality, i.e. to show much stronger compensatory improvement in growth/mortality rates under fishing than is consistent with historical abundance trends. Likewise, assuming very high v values typically causes exploited species to be more sensitive to fishing than is evident from historical data. In cases where historical predator abundances have varied widely, assuming very high v’s of exploited species to these predators often leads to larger predicted changes in response to these predators than is evident in abundance trend data. Although there are other ways to generate compensatory responses in Ecosim (e.g. foraging time adjustment), typically it is the v’s that are adjusted to fit time-series data. When v’s are tuned to time-series data, fitting procedures have typically indicated relatively low v’s for most trophic linkages, to explain the apparent compensatory responses that most exploited fish species have shown to exploitation (recruitment staying high until stock size is relatively low), and also some apparently weak responses to changes in predator abundances. Estimated v’s are most commonly in the range 1.1– 5.0 times the base mortality (M) estimates from Ecopath, i.e. v’s are estimated to be not much higher than the base empirical Q0/N0 estimates from baseline abundance, feeding rate and diet composition data provided to Ecopath. However, the fitted v estimates typically vary widely among trophic linkages, are sensitive to Ecopath input values and to other Ecosim model assumptions like foraging time adjustments, and are often very large (indicating apparent mass action variation in M). In short, no consistent pattern of ecosystem-scale variation in v’s has yet emerged from Ecosim fitting exercises, rather we see vulnerabilities from fitting that reflect how far the given predators were from their carrying capacity in the given ecosystem (further away equates to higher v’s and less compensatory response). Using complex individual-based spatial models A potential fourth approach for estimating vulnerability exchange rates may be to predict them as emergent properties of multispecies, individualbased models that represent foraging interactions 52

on very short time scales (minutes to hours) over very detailed spatial maps (1 m2 or smaller grid cells). Individual-based models have been used to explore emerging population-level behaviour accounting for individual variability (see a review in Tyler and Rose 1994; Clark and Rose 1997; Shin et al. 2004, p158, 2004; Brodeur 2001; Sable and Rose 2008). K. Rose and Shaye Sable (Louisiana State University) are currently exploring this option with a 6-species model of shrimp and fish species in salt marshes. In that model, individual movements, growth and mortality events are linked to food availability, predation risk as modified by habitat structure (marsh plant cover) and physical factors that drive movement (tidal variation in water depth, temperature). Limited computational experience with that model does indicate that the main predictions of the arena models (decreasing local food availability where local densities of foraging animals are high, upper limits on predation mortality rates) do emerge from the calculations, with hyperbolic functional forms as expected, when model rates are averaged over longer times and to whole-system scale. Discussion The main modelling argument for assuming mass action in predictions of predator-prey and food web interaction effects has never been that such a simplistic assumption is warranted based on field data; rather, the use of such models can be justified mainly because of analytical and computational tractability, i.e. the notion that robust and general predictions cannot be easily derived for more realistic models. The models of foraging arena theory, and associated software like Ecosim for examining dynamic scenarios, largely eliminate such excuses. We assert that the issue now for ecosystem modelling is not whether to bother including vulnerability exchange effects in trophic interaction predictions (it is plainly unwise to ignore them), but rather how to estimate exchange rates and their impacts. The most important challenge for practical application of the theory today is not just estimating foraging arena parameters (vulnerability exchange and predator rates of effective search), but also measuring how such rates change with risk-sensitive behavioural responses to major fishery-induced changes in abundances of larger predators (Heithaus et al. 2008), e.g., smaller fishes and inverte 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59


brates may forage over larger areas (lower a because of larger A) but with higher prey exchange rates (v) into such areas, possibly causing predator-prey instability between the smaller species and their prey. For predicting such changes, it is not really helpful to know that software like Ecosim allows users to assume arbitrary functional forms for such responses (Box 1); we should be trying to develop a sound theoretical and empirical basis for such forms. One approach to this problem is to assume that foraging behaviours vary so as to maximize simple fitness measures and to incorporate optimization calculations directly into the model dynamics (Matsuda and Abrams 2004). Another is to gather field measurements of foraging arena sizes and foraging times along spatial and temporal gradients in predator abundances or with deliberate predator manipulation experiments. Experience with this approach to date suggests that the functional form of responses to varying predation risk may be highly non-linear (Heithaus et al. 2008). For example, Kitchell and Carpenter (U. Wisconsin, personal communication) have ‘titrated’ a small lake by adding large bass a few at a time, then observing behavioural and spatial distribution responses by small minnows. They observed a complex response pattern, where the minnows first shifted from widely distributed in the absence to bass to shoaling behaviour in the presence of a few bass; then adding a few more bass caused the minnows to suddenly abandon the open-water zone in favour of littoral hiding places (where most of them then starved). Power et al. (1985) observed a strong reduction in foraging by grazing minnows to the presence of bass. Similarly, Werner et al. (1983a) demonstrated experimentally how a prey species would minimize predation risk by selecting a habitat with less exposure to predation risk, even if it came at the cost of reduced growth. One potential consequence of risk-sensitive foraging is that predators may choose to ‘manage’ prey behaviour by moving over their home range to avoid frequent exposure to any given part of the prey population (Lima 2002). There are at least two options for avoiding the foraging arena equations in making quantitative predictions about multispecies ecological responses to disturbances like fishing. One option is to move towards simpler modelling frameworks than Ecosim (and similar whole-ecosystem interaction models), for example, by just linking a set of simple age 2011 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 13, 41–59

structured population models through Beverton-Holt recruitment relationships that are modified to include effects of predation risk, habitat area and food availability in a functional structure like that recommended by Walters and Korman (1999). A second option is to move in the opposite direction, to develop very complex spatial models that represent individual-based behavioural responses to mosaics of food, predation risk and habitat structure. We suspect that this approach will be computationally hopeless (the divergence between space-time scales of metres– minutes where behaviour should be explicitly modelled, to scales of ecological response in populations and communities, is just too large), but it is at least worth trying as a way to obtain insights about how to develop better simplified models. A key feature of the foraging arena models is equifinality (a given end state can be reached by many potential means), in the sense that a wide variety of particular hypotheses about the causes of limited access to prey and vulnerability exchange can lead to the same basic equations for variation in vulnerable prey densities V. Such equifinality in models is both blessing and curse. From an applied perspective, it is a blessing in the sense that at least the qualitative form of model rate predictions is robust to lack of information about causes of the exchange process. But from a hypothesis testing perspective, equifinality is a curse as it implies that agreement of data with any particular hypothesis about causes of limited vulnerability provides no support whatsoever for that hypothesis.

Box 1: Ecopath with Ecosim software for food web modelling The computer software package called Ecosim was explicitly developed for exploring the time dynamic consequences of foraging arena vulnerability exchange processes and risk-sensitive foraging (as represented by foraging time and area adjustments), for food webs subject to disturbances from fishing and changes in primary production processes (nutrient loading and production regime shifts). State-rate information needed to initialize Ecosim dynamic simulations is provided by Ecopath mass-balance assessments. The simplest, default Ecosim models represent only biomass dynamics Bi for a set of biomass groups or pools, i = 1…n, where n is

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most often in the range 20–50 groups, using differential equations of the form X X dBi =dt ¼ ei j Qji j Qij FBi M0 Bi ð6Þ where ei is a conversion efficiency of food intake P summed over prey groups ( jQji), predation P losses summed over predator groups ( jQij), F is (time varying) fishing mortality rate summed over fishing gear types and M0 is unaccounted natural mortality. For primary producers, the food intake term is replaced by a biomass- and nutrient concentration-dependent production function. The Qij are time-varying trophic flows, varying with prey biomasses Bi and predator biomasses Bj using the foraging arena rate Equations (1–5). Heithaus et al. (2008) incorrectly assert that ‘each species’ diet is inflexible’ in Ecosim; in fact, simulated diet compositions typically change dramatically over time as the Qij are predicted to vary with prey and predator abundances and with adjustments in predator and prey foraging times. For exploring issues related to population age– size-structure within biomass pools, particularly implications of trophic ontogeny, users may specify that the biomass rate equations for some species be replaced by monthly size–age-structure accounting equations. Users may then specify ranges of ages that represent life history ‘stanzas’, within which individuals are then assumed to have similar diet preferences, predation risks and vulnerability to harvest. Potentially destabilizing type II functional response effects are included as options in Ecosim, through two mechanisms. First, handling time effects can be represented using the multispecies disc equation, by replacing a in Equations (2,3) with the search time to total P time ratio a/(1 + aijhiVi) where the denominator accounts for handling times hi summed over various prey types i, for each predator type j. Second, the model can adjust foraging times dynamically so as to try and maintain base or target food consumption rates per predator (Q/P), and such adjustments tend to make Q/P independent of prey abundance over at least some range of prey abundances. Foraging time adjustments in particular are predicted to have strong destabilizing effects when prey vulnerability exchange rates are assumed to be high, even though they result in lowered predation 54

risk when a species is rare (and hence induce an apparently stabilizing type III functional response form between that species and its predators). Potentially stabilizing effects of riskdependent foraging can be represented by directly linking foraging times to predation risk (to predator abundances). Vulnerability exchange rates (v) are parameterized in Ecosim as multiples of Ecopath base estimates of prey mortality rates. That is, each v is specified as v = KQ0/N0 where model users enter the multipliers K, and Q0/N0 are Ecopath base estimates of instantaneous prey mortality rates. Obviously, K must be >1.0 to predict the base rates, and very high K values imply mass action (Lotka-Volterra) interaction rates. Ecosim has a non-linear estimation routine for trying to estimate subsets of the K values by fitting predicted biomass (and catch and total mortality rates) to time-series data, in the same way that fisheries stock assessment scientists commonly try to estimate surplus production parameters or the steepness of stock–recruitment curves by fitting single-species models to data. Vulnerability exchange and search rates (v, a) could change substantially because of changes in risk-sensitive foraging behaviours following fishery-induced changes in top predators. Such effects can be included in Ecosim scenarios by using arbitrarily shaped ‘mediation functions’ that directly link (as multiplicative factors) the v, a values to simulated abundances of the top predators (and also to other creatures like habitat forming plant and coral species).

Acknowledgements CJW and VC acknowledge funding from the National Science and Engineering Research Council of Canada, and VC also from the Sea Around Us project, a scientific cooperation between the Pew Environment Group and the University of British Columbia. In addition, we appreciate the comments of two anonymous reviewers. References Abrams, P.A. (1986) Adaptive responses of predators to prey and prey to predators – the failure of the arms-race analogy. Evolution 40, 1229–1247.



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