Optimal Foraging in Marine Ecosystem Models: Selectivity, Profitability and Switching

Vol. 473: 91–101, 2013 MARINE ECOLOGY PROGRESS SERIES Published January 21 doi: 10.3354/meps10079 Mar Ecol Prog Ser

Optimal foraging in marine ecosystem models: selectivity, profitability and switching

André W. Visser1,*, Øyvind Fiksen2,3

1Centre for Ocean Life, National Institute for Aquatic Resources, Technical University of Denmark, Kavalergaard 6, 2920 Charlottenlund, Denmark 2Department of Biology, University of Bergen, 5020 Bergen, Norway 3Uni Research, 5020 Bergen, Norway

ABSTRACT: One of the most troubling aspects of ecosystem models is the use of rather arbitrary feeding and preference functions. The predictions of plankton functional type models have been shown to be highly sensitive to the choice of foraging model, particularly in multiple prey situations. Here we propose ecological mechanics and evolutionary logic as a solution to diet selection in ecosystem models. When a predator can consume a range of prey items, it has to choose which foraging mode to use, which prey to ignore and which ones to pursue, and animals are known to be particularly skilled in adapting their diets towards the most profitable prey items. We present a simple algorithm for plankton feeding on a size-spectrum of prey with particular energetic content, handling times and capture probabilities. We show that the optimal diet breadth changes with relative densities, but in a different way to the preference functions commonly used in models today. Indeed, depending on prey class resolution, optimal foraging can yield feeding rates that are considerably different from the ‘switching functions’ often applied in marine ecosystem models. Dietary inclusion is dictated by 2 optimality choices: (1) the diet breadth and (2) the actual feeding mode. The optimality model does not generate ‘safety in low densities’ for prey, as the ‘switching function’ does in ecosystem models, unless predators are shifting feeding mode adaptively. The actual diet, feeding rate and energy flux in ecosystem models can be determined by letting predators maximise energy intake or more properly, some measure of fitness where predation risk and cost are also included. An optimal foraging or fitness-maximising approach will give marine ecosystem models a sound principle to determine trophic interactions.

KEY WORDS: Grazing · Diet composition · Feeding mode · Fitness · Size selection

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INTRODUCTION (Gentleman et al. 2003, Gentleman & Neuheimer 2008, Anderson et al. 2010, Prowe et al. 2012a). The question of how to formulate behavioural or Marine ecosystem models have been strongly adaptive interactions between predators and prey in influenced by Fasham et al. (1990), and the use of ecosystem- or food-web models has been around for a parameterised preference functions has become the long time in the general ecological literature (Abrams standard since then. The preferences can either be a 1984, Schmitz 2010, reviewed in Abrams 2010). More fixed value or a function of relative abundances, recently, the appearance of more ecologically re- often referred to as ‘switching’. Different modelling solved models of marine systems has generated a re- options give different outcomes for the predicted vived interest in the construct and parameterization diversity, structure and stability of the food web and of mixed-diet formulations in these systems as well ecosystem (Anderson et al. 2010, Prowe et al. 2012b).

*Email: [email protected] © Inter-Research 2013 · www.int-res.com 92 Mar Ecol Prog Ser 473: 91–101, 2013

But how do we know which foraging function is the a rescue in low densities if the predators are shifting right one? Clearly, feeding functions that yield lower feeding mode adaptively. The ODB and adaptive food intake for a predator when the abundance or switch among foraging modes could become an inte- availability of other prey increases are flawed (Gen- gral part of current ecosystem models, and replace tleman et al. 2003). Here, we argue that the best diet some of the loosely founded ‘preference functions’. model to use is one that maximises energy intake for Our main goal here is to show how this can be done the predator, based on general optimal foraging in a size-structured ecosystem model using an opti- assumptions. We also point to algorithms where the mality assumption about both diet inclusion and predation risk involved in foraging is included in the foraging mode. choice of foraging mode and diet. Animals have evolved a behavioural rationality in the economics and energetics of how they exploit FUNCTIONAL RESPONSE MODELS their environment. This is the main justification for the optimality principle in behavioural ecology We contend that whatever the specific formulations (Mangel & Clark 1988), and it can be used to predict employed in models, they should be based on mech- what animals do in trade-off situations (e.g. Krebs & anisms that are ecologically meaningful. To this end Kacelnik 1991). Optimality is also a sound principle we assert that Holling type II functional response to apply in ecosystem or food-web models (Smith et (‘Holling’s disk equation’; Holling 1959) should be al. 2011) where organisms (or the modellers) face the primary structure of all trophic interactions. We decisions about which organisms to include in their make this assertion in the light that Holling II is diet and which to ignore. According to classical opti- inherently individual-based, wherein traits of preda- mal foraging theory (Charnov 1976), prey should be tors and prey can be incorporated in a transparent ignored if its energy value relative to the time it takes manner. It is also explicitly mechanistic. It is based on to consume the prey is low compared to alternative measurable subprocesses (encounter rates and han- prey in the environment. By the same reasoning, a dling times) and can be readily tailored to include low probability of capturing a given prey type may changing environmental conditions (e.g. light, turbu- also make it unprofitable if the time could be better lence, predator and prey abundances) or shifts in spent searching for alternative prey. Optimal feeding behaviour (e.g. foraging effort, feeding mode, defen- on a mixture of prey types is often referred to as opti- sive strategies). With regard to the latter, it is partic- mal diet breadth (ODB). This topic has gained much ularly amenable to include adaptive behaviour, attention in simulations of complex systems in ecol- wherein the predator and prey jockey respectively ogy (Petchey et al. 2008, Heckmann et al. 2012), for their best foraging and defence strategies in the where its role in ecosystem stability and biodiversity fitness stakes. has been highlighted. Holling I (with saturation) is a variant of type II and A further facet of optimal foraging in the plankton can be derived from essentially the same principles is the issue of foraging mode (e.g. ambush versus where the predator has the ability to process >1 prey cruise feeding, habitat selection, etc.). Indeed, simi- item at a time (Sjöberg 1980, Wirtz 2012a). The Ivlev lar optimality reasoning may be applied to determine type functional response falls somewhere between a when zooplankton (e.g. a copepod) should switch be- Holling type I and II with a specific processing capac- tween foraging modes when exposed to a mixed diet ity (Wirtz 2012a). In contrast, a Holling type III func- (e.g. phytoplankton and smaller zooplankton) (Visser tional response involves some behavioural response et al. 2009, Mariani & Visser 2010). Specifically, dif- of the predator to the prey field, either through a ferent foraging modes target different sections of the habitat shift (e.g. Murdoch et al. 1975) or learning prey field, usually through the mechanics of the (Real 1977) or from a shift in the predator’s foraging encounter process, and while the optimal diet for a mode (e.g. cruising versus ambush; Kiørboe et al. given feeding mode can be estimated by the ODB 1996, Gismervik & Andersen 1997). That is, a Holling model, the actual choice of feeding mode depends on type III is an emergent functional response dictated the fitness afforded the grazer—a trade-off between by encounter rates, processing time and the adaptive benefit (ingestion), costs and risk. We show that the fitness-seeking behaviour of grazers. It should not be ODB model will not generate ‘safety in low densities’ prescribed a priori in model applications; rather, the for prey (i.e. reduced grazing pressure when alter- underlying mechanisms from which it arises should nate prey is more plentiful) as the ‘switching func- be explicitly modelled to insure some fidelity in the tion’ does in ecosystem models, but that there may be ensuing trophic dynamics. Visser & Fiksen: Optimal foraging in marine ecosystem models 93

Finally, a popular class of functional response but more transparent and mechanistic Holling type formulations follow a Michaelis−Menten or Monod II descriptions. description. While it is true that these descriptions are equivalent to a Holling type II for a single prey MODELS type, their interpretation becomes clouded when multiple prey types are considered. Specifically, Optimal diet breadth and profitability encounter rates and handling times can be readily classified for different prey types, but what is the The concept of prey profitability (Charnov 1976) half saturation concentration (K) for a prey mix- becomes evident when considering the optimal deci- ture? It is not constant but varies with the relative sion a predator should make when confronted by a proportion of various prey types in the diet (Prowe newly encountered prey—whether it should include et al. 2012a). Many (if not most) modellers use the it in its diet or not. In its original form, the argument estimate of K from Hansen et al. (1997)—despite is that if g is the mean ingestion rate, ei is the cell con- large scatter (orders of magnitude) and unclear tent (e.g. in terms of carbon or energy) of the newly meaning—concluding that the half saturation in encountered prey type i, and hi is the time needed for terms of biomass is the same across 8 orders of the predator to handle it, then the criterion for inclu- magnitude of predator mass. To avoid this con- sion is ei/hi > g, while if ei/hi < g then the prey item founding effect, we suggest that Michaelis−Menten should be rejected (see Table 1 for definitions, or Monod descriptions of functional responses for dimensions and units). This is the diet that maximises mixed diets be abandoned in favour of equivalent food intake rate.

Table 1. Glossary of terms

Symbol Description Units

−3 ai Prey abundance for prey class i m β Encounter kernel (maximum clearance rate) m3 s−1 β 3 −1 i Encounter kernel for prey class i m s β 3 −1 i,m Encounter kernel for prey class i and feeding mode m m s −1 cm Cost for feeding mode m gC s −3 CTOT Total biomass concentration gC m ei Prey carbon content for prey class i gC e(r) Prey carbon content as a function of cell size gC ε i,m Escape probability of prey type i when exposed to feeding mode m Dimensionless fm Fitness for feeding mode m gC g Grazing rate gC s−1 ≤ −1 gk Grazing rate for all prey class i k gC s −1 gmax,m Maximum grazing rate for feeding mode m gC s θ φ −1 gΘ0 Grazing rate for non-switching preference function; i = i gC s θ φ −1 gΘ1 Grazing rate for switching preference function; i = i gC s

hi Handling time for prey class i. This can be subdivided into time accruing to all attacks (hia; s pursuit, capture, etc.) and time devoted to processing captured prey (hid, ingestion, digestion, etc.) h(r) Handling time as a function of cell size s ≥ i Profitability index, arranged in descending order so that pi pi+1 Dimensionless −1 μ0 Baseline mortality rate, independent of feeding mode s μ −1 m Additional mortality rate associated with feeding mode m s n(r) Cell size spectrum giving the abundance of cells (m−3) per unit size (m) m−4 −4 NTOT Scaling factor for cell size spectrum m −1 pi Profitability for prey class i gC s φ i Catchability for prey class i Dimensionless φ(r) Catchability as a function of cell size Dimensionless r Size (equivalent spherical radius) of cells m R Size (radius) of grazer m θ i Innate availability for prey class i Dimensionless Θ i Preference function for prey class i Dimensionless −1 um Grazer swimming speed in feeding mode m (e.g. um = 0 for ambush feeding mode) m s −1 vi Swimming speed for prey type i (e.g. vi = 0 for non-motile prey) m s 94 Mar Ecol Prog Ser 473: 91–101, 2013

To this we add the concept of catchability, assum- In this formulation we assume that handling time ing that an attack on a prey will not always lead to a accrues to all attacks (i.e. pursuit and capture). In successful capture. If the probability of an attack order to include handling times related to processing φ being successful is i, then the expected value of captured prey (i.e. ingestion and digestion), one can φ return on an attack is i ei. We define prey profitabil- introduce a slightly different formulation: hi = hia + ity as: φ , where is the time to pursue and capture a e i hid hia p = φ i (1) prey and h is the time it takes to ingest and digest it. iih id i While we do not specifically pursue this here, we The optimal diet of the predator should include all note that the distinction between capture and pro- prey items with profitability above some critical cessing times should be given due attention in prac- value, and exclude less profitable ones. The critical tical applications. profitability value is equal to the grazing rate While the decision to include a prey or not does not attained by including all available prey items of specifically depend on the abundance of that prey superior profitability. This can be shown graphically type, the grazing rate to which profitability is com- by arranging prey according to profitability pi pared and the ensuing inclusion criterion does (Fig. 1). There is a prey profitability where including depend on the abundance of all included prey types. it in the diet will lower the ingestion rate. Specifi- An intuitive result from this model is that large prey cally, writing ai as the abundance of prey within each will be ignored as they become difficult to catch, β class, and i as the encounter kernel (search rate) of while easy-to-catch but small prey will be ignored as the grazer with prey in size class i, and ordering prey their energy content becomes too low to warrant the classes such that pi > pi+1, then the optimal grazing time it takes to catch them, leading to a preferred size rate is: range. Note however that the breadth of this pre- k ferred range is expected to be dynamic, changing its φβiiiiea ∑ i=1 width in accordance with the abundance of prey ggmax ==k (2) k types available to the grazer. This algorithm can be 1+ haiiiβ ∑ i=1 extended to also include a continuous (every time where k is such that gk is a maximum, a condition step) assessment of all alternative foraging modes ensured by gk > pj for j > k. We will provide a more (one ODB for each) and then choose the best one—a concrete example in the following section ‘Size- well-informed predator using simple fitness-seeking based application’. heuristics alternating among foraging modes and diets. ) –1 10–3 Size-based application 10–4 While the general concept of prey selectivity in 10–5 terms of profitability can be applied to any number of cases, we wish to highlight its role in deter- ofitability (g C s ofitability (g –6 10 mining a grazer’s size preference and size class breadth of diet (Hansen et al. 1997, Fuchs & Franks 10–7 2010, Wirtz 2012b). Size after all is the key trait 10–8 governing trophic interactions, particularly in determining the carbon and energy content of a prey 10–9 item. Further, it can be supposed to influence both

Grazing rate and pr 20 40 60 80 100 handling time and catchability and so structure the Profitability index profitability of a predator’s diet. We envisage a Fig. 1. Relationship between prey profitability (solid line) predator (e.g. adult copepod) of size R (equivalent and optimal grazing rate g (dashed line), both of which are spherical radius) feeding on a range of prey of vari- ranked according to descending profitability p. The point ous size classes r. We assume that both handling where the 2 lines cross defines the critical profitability index time and catchability depend on the size of the k (vertical dotted line) and critical profitability p . It is also k prey relative to that of the predator, and the the point where g attains its maximum value gmax (horizontal general size-dependent carbon content, handling dotted line). That is gmax = pk. Including prey with p < pk (i.e. with profitability index > k) will invariably reduce g < gmax times and catchability are given by: Visser & Fiksen: Optimal foraging in marine ecosystem models 95

n er()= c r e 0 (3) We choose an illustrative size spectrum of the form: n ⎛ r ⎞ h () nnud hr=+ h01 h ⎜ ⎟ (4) ⎛ r ⎞ ⎛ r * ⎞ ⎝ R ⎠ nr()= N (6) TOT ⎝⎜ rr* + ⎠⎟ ⎝⎜ rr* + ⎠⎟ n ⎛ r ⎞ p φ()r = 1− (5) defined over the size range r ∈ [r0, r1], and expressed ⎝⎜ R ⎠⎟ in particles number density per length category, m−3 −1 The exponents ne, nh and np control the shape of m . When nu and nd are both positive, this gives a dis- these functions, and illustrative examples together tribution with a peak centred on r*. The total biomass

with the resulting profitability function are shown in concentration in the system CTOT is thus given by: Fig. 2. For particular applications, these functions r1 should reflect the groups of predators and prey in Cernrr= ()⋅⋅ ()d (7) question, and while handling times and catchability TOT ∫ r may be difficult to specify, it is certainly possible to 0

estimate or model them for specific groups of organ- We can thus set CTOT to environmentally relevant isms (Paffenhöfer 1984, Wirtz 2012b). In fact, these values, from which the scaling factor NTOT and thus processes depend on real traits of prey organisms quantified particle size spectra are determined. Fig. 3 (e.g. spines or exoskeletons), the effect of these on shows an example, and is qualitatively similar to handling time (both capture and digestion times), many instances of observed particle size spectra catchability, and even energy content, and provide a found in coastal and shelf seas (Sheldon et al. 1972). mechanistic link to the survival benefits the traits In choosing this example, we are mindful that in a confer on prey organisms in the framework sug- full size-based ecosystem model (e.g. Fuchs & Franks gested here. 2010), the actual size spectrum will be dynamic, being governed by the trophic interac- 103 tions of growth and mortality within a b 2 and between prey size classes. 10–3 10 101 Provided we choose the width of –6 10 100 each size class to be small (i.e. in gen- 10–9 10–1 eral this means a large number of size 10–2 classes), then the integration in Eq. (7) 10–12 10–3 Handling time (s) time Handling can be approximated by the sum: Cell content (g C) content Cell 10–6 10–5 10–4 10–3 10–2 10–6 10–5 10–4 10–3 10–2 i ) max 0.00010 1.0 –1 Cea= ⋅ (8) c d 0.00008 TOT ∑ ii 0.8 imin 0.6 0.00006 where a = n δ is the abundance of 0.00004 i i i 0.4 δ particles in the size class i, and i is the 0.00002 0.2 width of the size class. 0.0 0.00000

Capture probability –6 –5 –4 –3 –2 –6 –5 –4 –3 –2 10 10 10 10 10 10 10 10 10 10 s C (g Profitability Cell radius (m) Cell radius (m) Preference and switching functions in recent marine ecosystem models Fig. 2. (a) Cell content is estimated as being proportional to cell volume (ne = 3) 5 −3 so that c0 = 10 gC m is the cell carbon:volume ratio (Hansen et al. 1997). (b) For handling time, we assume there is a minimum handling time per prey Preference functions generally pro- ≈ item, h0 1 ms, while for larger particles, handling time increases cubically ceed from the concept of innate avail- (nh = 3) with prey:predator ratio to a maximum of h1 = 1000 s at a predator:prey ability θ (Stock et al. 2008) indicating ratio of 1. Maximum handling time is somewhat greater than that associated i with copepods capturing large food items (e.g. order 100 s; Williamson 1980), the proportion of prey in a particular but is consistent with digestion and gut passage times for large prey (800 to class that is available to a predator. 1200 s; Dutz et al. 2008). (c) Following Petchey et al. (2008), catchability re- Dietary preference based on innate flects the general trend that prey much smaller than the predator are easily availability is termed passive selection caught (φ→1 for r/R << 1), while prey of size approaching the predator are φ→ → (Gentleman et al. 2003). The underly- more difficult to catch ( 0 as r/R 1). Parameter setting is np = 3 (Eq. 3). (d) The resulting profitability shows the profitability of prey of a given size com- ing concept here is that the predator bining the effects of energy content, handling time and capture probability can only perceive and/or capture a 96 Mar Ecol Prog Ser 473: 91–101, 2013

limited range of prey due to some physical restric- (Stock et al. 2008). The exponents ns and ms can be tions in its foraging mechanisms (e.g. inability to han- varied to give a stronger or weaker switching mech- dle extremely large or extremely small prey, conflicts anism. While it has the same functionality, this for- between feeding modes and encounter rates on cer- mulation is slightly different from that given in Stock tain prey classes). While this can be seen as being et al. (2008), so that for ns = ms = 1, the preference φ related to the catchability i we have introduced function of Fasham et al. (1990) is returned, while ms above, it is not identical. Preference functions of the = 0 returns a preference function without active passive selection kind are generally prescribed as selection (e.g. Banas 2011). being strongly size-dependent, reflecting general The ensuing grazing rate in existing models is gen- trends of dietary inclusion found in nature (e.g. erally formulated as:

Cohen et al. 1993). For example, Banas (2011) intro- imax Θiiiieaβ duces a preference function that is a log-Gaussian ∑ i=1 (10) gΘ = distribution, centred about one order of magnitude imax 1+ Θiiiihaβ smaller that the predator. ∑ i=1 Marine-food web models often also include a factor Note how the preference function enters this formu- indicating a preference for prey classes that dominate lation; it does so as a pre-multiplier to the en counter the prey community in terms of biomass. This is gen- rate. That is, the predator essentially ignores a pro- erally referred to as switching (Murdoch et al. 1975) portion of the prey encountered in accordance with its or active selection (Gentleman & Neuheimer 2008), in preference. This is in contrast to our catchability for- that it reflects behavioural changes in predator feed- mulation (Eq. 2), which does not automatically predi- ing modes that actively target certain prey classes — cate that prey of low catchability should be ignored. the underlying assumption being that it is best for the predator to target those prey classes that are most abundant. This, together with innate availability leads Optimal diet breadth to a general preference function of the form: Using the size-profitability relationships together m ⎛ ⎞ s with an illustrative size spectrum, we can now calcu- n −1 n n Θ = θθ⎜ ss()ea/ ( θ ea ) s⎟ (9) late ODB for a range of total biomass concentrations. iii⎜ ii ∑ iii ⎟ ⎝ i ⎠ Without going into specific details of the feeding mode employed by the predator (this will be discussed in ‘Results and discussion’ section 0.0035 ab‘Comparing predictions for optimal foraging to prescribed preference and switching func- ) 1012 0.0030 –1 tions’), we choose a constant specific maxi- m mum search rate (i.e. volume searched for –3 0.0025 ) β

–3 prey per body volume per unit time) of /V = 109 106 d−1, where V is the predator’s body vol- 0.0020 ume. This appears to be a relatively robust relationship—albeit with considerable vari- 106 0.0015 ance—across a wide range of marine species (Kiørboe 2011). There is also a considerable Biomass (g C m 0.0010 range in the search rate among predators; 103 they may swim faster if prey are scarce, for

Number spectrum (ind. m Number 0.0005 instance (Follows & Dutkiewicz 2011). This requires more computations, but can readily 0 be included, although we leave that element 10 0.0000 10–6 10–5 10–4 10–3 10–2 10–6 10–5 10–4 10–3 10–2 out here. Similarly, the issue of digestion limitation and how gut constraints influence Prey size (m) Prey size (m) ODB is left out, since this would require a Fig. 3. (a) Number density spectrum n(r) following from Eq. (6), and (b) second state variable in addition to organism the associated biomass density per size class ei ai used in the illustrative example. In the latter, we used 100 size bins of logarithmically increas- size. In general, digestion limitation will in- ing width. Parameter setting: nu = 1, nd = 5, r* = 10 µm. Total biomass crease the benefits of adaptive prey selection ∑ −3 in the system is set at CTOT = ei ai = 0.1 gC m if prey differ in quality. Visser & Fiksen: Optimal foraging in marine ecosystem models 97

RESULTS AND DISCUSSION This change in ODB is moderate; it still spans >2 orders of magnitude in prey size even at extremely Should zooplankton be selective in their diet? high total carbon concentrations and is not particularly sensitive to the shape of the prey size spectrum. For a large adult copepod (R = 10−2 m), we calculate This is consistent with general observations that the ODB using the size-based profitability schedule illus- ODB of copepods span nearly 3 orders of magnitude trated in Fig. 2d. We do this keeping the shape of the (Fuchs & Franks 2010), and are food-limited with half size spectrum constant, but vary the total biomass of saturation constants of the order of 200 mgC m−3 prey. The maximum size of prey included in the diet (Hansen et al. 1997). That is, copepods live in a dilute is slightly less than the size of the predator (Fig. 4a), environment (Kiørboe 2011) and simply cannot reflecting the high profitability of large prey despite afford to be too selective; they should consume most low catchability and long handling times (Fig. 2d). of what they encounter. In the vocabulary of optimal The diet breadth initially spans the entire size spec- foraging theory, they are search-limited and not −5 −3 trum up to about CTOT = 10 gC m , after which an handling-limited. increasing portion of prey items from the small end of the size spectrum (i.e. low profitability) become −2 excluded from the diet. From about CTOT = 10 gC Comparing predictions for optimal foraging to m−3, prey from large size classes also become prescribed preference and switching functions excluded, reflecting increased handling times and decreased catchibility. That is, there is an overall nar- This gives us 3 different estimates of ingestion rowing of ODB with increased total prey biomass. rates: (1) gmax as per Eq. (2); (2) gΘ0 as per Eq. (10) assuming no ‘switching’ (ms = 0, only innate availability); and (3) gΘ1 as per Eq. (10) including ‘switch- 10–2 ing’ that is kept at minimal complexity, with ns = ms = a 1 (i.e. both ‘switching’ and innate availability). To 10–3 make an internally consistent comparison, we use the same function (Eq. 5) to describe both catchabil- φ θ 10–4 ity ( ) and innate availability ( ) in both models. The rationale here is that whatever other factors may be φ θ –5 involved, we can say with some certainty that > ,

Prey size (m) 10 so that setting φ = θ gives the most generous estimates of grazing (gΘ and gΘ ) using the prescribed 10–6 0 1 10–8 10–7 10–6 10–5 10–4 10–3 10–2 10–1 10–0 preference approach. We normalise all grazing rate estimates with respect to β CTOT (Fig. 4b). 1.0 As might be expected, grazing rate gmax determined by prey profitability gives the highest estimated en- 0.8 b ergy intake across all total biomass concentrations. 0.6 The innate availability estimate gΘ0 follows closely at low concentrations but diverges to lower estimates at 0.4 high concentrations. The estimate with switching,

gΘ1, is consistently lower by an order of magnitude. 0.2 The result that gmax > gΘ0 reflects the mechanism that for a prescribed preference function, the preda- Normalized ingestion rate ingestion Normalized 0.0 10–8 10–7 10–6 10–5 10–4 10–3 10–2 10–1 10–0 tors waste valuable time handling small prey with C (g C m–3) low energy content. This is an effect that becomes more pronounced at high total biomass concentra- Fig. 4. (a) Optimal diet breadth as delimited by the minimum tions when profitability optimisation dictates the rmin (solid line) and maximum inclusive size rmax (dashed line), and (b) normalised ingestion rate for a 1 cm predator exclusion of a growing fraction of small prey classes feeding on a prey size spectrum (shape according to Fig. 3) (Fig. 4a). In general, a prescribed preference function −8 −3 for variable total prey biomass ranging from 10 to 1 gC m can at times give similar grazing rates as optimality- (= mgC l−1). In (b), variously determined grazing rates are derived grazing, but will always give suboptimal gmax (solid line) based on profitability, preference with no rates at other times. A broad preference function will switching gΘ0 (dashed line) and preference with switching gΘ1 (dot-dashed line) give suboptimal performance at high prey concen - 98 Mar Ecol Prog Ser 473: 91–101, 2013

trations as time is wasted on handling low-energy- as the averaged sum. These effects can of course be content prey, while a narrow preference function will calibrated out in specific model applications, and most be suboptimal for low prey concentrations where published models use subdivisions that are not arbi- potentially valuable are excluded from the diet. trary. However, there is something very unsettling in

That the switching estimate gΘ1 is less than the opti- the idea that model architecture can have such an ef- mal grazing rate gmax is at first not surprising, but why fect on seemingly fundamental ecological processes. is it so much less? It turns out that this is symptomatic This underscores an important concept for both of a systematic failure of the commonly applied modellers and ecologists. There are demonstrable switching formulation (from Fasham et al. 1990) and density-dependent effects in nature (e.g. Murdoch et variants thereof (e.g. Stock et al. 2008). Specifically, al. 1975, Gismervik & Andersen 1997), but the den- using abundance-weighted preference functions sity of what exactly? Is it the density of conspecifics, leads to a dependence of estimated grazing rates on similar functional types, members of the same or the number of prey classes included in a model. To some alternate targeted prey group? We have no illustrate this, consider the simple thought experiment immediate answer to this, but in what follows, we presented in Fig. 5. We consider 2 models of exactly make a suggestion as to how this may be addressed the same system, one where there are 2 prey classes A for mixed diets in marine ecosystems. and B, and the other where we arbitrarily subdivide prey class A into 2 subclasses A1 and A2. The first con- sequence of this is that the most preferred prey item Switching and optimal foraging mode shifts from A in Model 1 to B in Model 2. More prob- lematic is that the predicted total grazing rate of the We now turn to the question of how switching can predator is consistently lower for Model 2 than for be incorporated in grazing rates according to our Model 1, and all because we chose an arbitrary sub- optimality criterion. Again, we wish to derive this in a division of the prey classes in our model. Exactly the way that is as mechanistic and evolutionarily consis- same problem crops up whether the weighting func- tent as possible. Our premise is that realised switch- tions are constructed in terms of abundance, clearance ing on various prey types is an emergent property of rate or ingestion rates, and it becomes more pro- the predator switching between various feeding nounced as the prey field is subdivided into more and modes, and it is the feeding modes that dictate more classes. It arises essentially from the nonlinearity dietary inclusion. of the functional response—Holling type III for Copepods for instance can in general choose instance—where the sum of averages is not the same between 3 different feeding modes (Kiørboe 2011):

0.8 a A B 0.6 M1

0.4

A1 A2 B 0.2 M2

Relative grazing per grazing class Relative 0.0 α AA(1−α) 1.0 b Fig. 5. Two model constructions, M1 and M2 for the same 0.9 system, the difference being that in M2, the prey class A is subdivided according to a partitioning function α. All other 0.8 parameters are equivalent, and the models are identical when α = 0 or 1. Grazing rate is calculated according to a ‘switching function’ where prey preference is a function of 0.7 prey abundance. In this context, the choice of α has no ecological significance and yet the model predicts different rel- 0.6 ative grazing pressure per prey class (a): B (solid), A1 (dot- dash) and A2 (dash), as well as total grazing rate (b) of model 0.5 M2 compared to M1 as a function of α 0.0 0.2 0.4 0.6 0.8 1.0 Total grazing (Model 2/Model 1) grazing (Model Total α Visser & Fiksen: Optimal foraging in marine ecosystem models 99

suspension (scanning-current) feeding, cruise feed- whether the predator perceives the prey before it ing and ambush predation. Each of these preferen- escapes or not. If it escapes before, then the effect is tially selects for certain prey types predicated prima- on the innate availability of that prey type; the pred- rily on prey motility. For instance, ambush feeding ator simply doesn’t see it so it does not figure in its selects for motile prey but is pretty much useless in profitability schedule. If it escapes after, then it finding immotile prey. On the other hand, a suspen- enters into catchability and profitability, presumably sion feeding current, while good for capturing im - with a lower value due to both lowered expectation motile prey, may alert motile prey to predation risk of successful capture, and possibly also a higher han- and so elicit an escape reaction. Furthermore, differ- dling time to account for pursuit. ent feeding modes also incur greater or lesser ener- Maximum ingestion is not the sole ingredient in getic costs, as well as exposing the predator to differ- determining feeding mode optimality — it includes a ent predation risks of its own. Ambush predation will trade-off between benefits, risks and costs. We make have low energetic costs and low risk, while both will this explicit in the formulation: be higher for suspension feeding and higher still for gcmax, − cruise feeding. Thus optimality of feeding mode is f = mm (13) m μμ+ not simply dictated by the maximisation of ingestion 0 m rate, but should also reflect the inherent trade-offs it where fm is a measure of fitness, cm is cost (in terms of μ poses against costs and risks. carbon loss rate), and m is the marginal increase in Predicated on the same principle of profitability put mortality associated with feeding mode m. The forward in section ‘Optimal diet breadth and profit - assumption here is that cost and risk are functions of ability’, the maximum ingestion rate for each feeding feeding mode, and not dependent on the prey en - mode can be calculated as: counter process. Essentially, Eq. (13) expresses the k instantaneous contribution to the predator’s repro- φβiiiimea , ∑ i=1 ductive value, and in the absence of any other expec- ggmax,mkm== , (11) k tations, should be optimised (Gilliam & Fraser 1987, 1+ haiiβ im, ∑ i=1 Visser 2007). The optimal feeding mode m* is thus where the subscript m indicates an estimate for a that for which fm* is maximised, and the optimal diet specific feeding mode. The feeding mode will have is that dictated by profitability and the optimal feed- no direct effect on prey profitability, although in gen- ing mode. eral, motile prey will have lower catchability and Finally, we offer a fitness-weighted grazing esti- higher handling time than non-motile prey and thus mate: lower profitability in general (food quality notwith- Σmmfgmax, m g* = () (14) standing). Neither will it affect abundance. It will f β Σm m however, affect the encounter kernels i,m. For instance, a simple kinetic encounter kernel formula- where each feeding mode and subsequent grazing is tion may be: proportionally enacted according to their relative fit- 12/ 22 ness. The concept here is that grazers continually βσim,,= im⋅−1 ε im ,⋅ uv m+ i (12) ()() explore their fitness landscape, sacrificing absolute where vi is the speed of the prey, um is the speed of optimality in exchange for information. This provides σ the predator relative to the fluid, i,m is the percep- a simple myopic heuristic that can be implemented in tion cross-sectional area of the predator in mode m ecosystem models to mimic adaptive, risk-sensitive ε on prey type i, and i,m is the probability that the prey foragers. More time is spent in feeding modes will make a successful escape before it can be per- imparting high fitness and less time in lower fitness ceived by the predator in mode m. For copepod feed- feeding modes. A formulation along these lines ing modes, a possible set of interaction rules could be would reintroduce some of the features of ‘switching’ ε ≈ posed as follows. For ambush feeding (um = 0), i,m 0 functions without incurring the prey-type partition- for all prey types, while for scanning and cruise feeding problem illustrated in Fig. 5. The ingestion term ε ≈ ε ≈ ing (um > 0), i,m 0 for non-motile prey and i,m 1 for in Eq. (13) can be elaborated to include a more com- motile prey. These rules would reproduce the plete bioenergetic model, with a gut state variable, motility−feeding mode interactions alluded to in the and where ingestion can more properly be replaced previous paragraph. by growth. If growth satiates at increasing prey den- Note that there is a distinction between the effect sity, this algorithm will lead grazers to shift to less μ of escape and catchability which has to do with efficient foraging modes with lower cm and m, and 100 Mar Ecol Prog Ser 473: 91–101, 2013

an extended benefit of higher food availability rate of the grazer. For the prey field, grazing pressure through reduced mortality from behavioural change shifts not only from non-motile to motile prey, but (see Fiksen & Jørgensen 2011). also changes from small to larger prey sizes. This approach, either through fitness maximisation Although beyond the scope of this present work, or a fitness-weighted estimate (Eq. 14), could result profitability and fitness considerations would also in switching where predators switch from one prey lead to switching among prey types as a result of type (e.g. phytoplankton) to another (e.g. microzoo- changes in the abundances of other actors in the food plankton) as a result of relative abundances. This web. Specifically, the marginal increase in mortality μ switching would bring about something akin to a ( m) would depend on the abundance of top preda- refuge at low numbers, but only insofar as it extends tors present, and thus factor into the grazers’ decision to all prey with a similar susceptibility to a given as to which feeding mode to adopt. Such behav- feeding mode. As a proof of concept, we include iourally (and trait-) mediated indirect interactions are model results for a predator feeding in either ambush apparently active in marine ecosystems (e.g. Kaart - or cruise mode, on a mixture of motile and non-motile vedt et al. 2005), and are thought to be important prey of different size classes (Fig. 6). We also supply shapers of food-web architecture and dynamics the MATLAB code for these calculations as supple- (Schmitz et al. 2008). mentary material (www.int-res.com/articles/suppl/ m473p091_supp/). In this scenario, we assign both an added cost and an increased mortality risk to cruise CONCLUSIONS feeding over ambush feeding. These are set as repre- sentative for an adult copepod (e.g. Acartia sp.) in It is becoming apparent that the cha racteristics of a typical predator field (Visser 2007). There is an the marine pelagic eco system predicted by models abrupt transition from cruise to ambush mode at a are critically dependent on the form of functional critical value of the ratio of motile to non-motile prey response curves used to describe the impact of graz- which is associated with a change in the ingestion ers on a diverse prey community (Mitra & Flynn 2006, Mitra et al. 2007). There is a clear and immediate need to place these descriptor functions on a mechanistically credible and theoretically sound foundation. In this, we follow Smith et al. (2011) in espousing optimality in terms of fitness as a powerful conceptual tool in model development. We propose a general algorithm for esti- mating optimal foraging mode on a mixed prey diet composed of 2 steps. Firstly, for each feeding mode (e.g. cruise, scanning current, cruise), possible ingestion rates are determined by prey profitability (from most profitable to some critical cut-off), Fig. 6. Implication of fitness-regulated feeding mode switching (Eq. 13) the abundance of prey types, and their on (a) grazing pressure (gC s−1) per size class and (b) the net ingestion encounter kernel (a function of feeding rate (gC d−1) for fitness maximisation (dot-dash) and fitness-weighted (Eq. 14; dot-dot-dash) estimates respectively, for varying mixtures of mode and prey motility for instance). Sec- motile and non-motile prey. The dotted and dashed lines to the left of ondly, the actual feeding mode (and thus panel (a) indicate the profitability size cut off for non-motile and motile actual ingestion rate, dietary inclusion prey respectively. In this example, we assume exclusive mode-dependent and grazing pressure per prey class) is feeding (i.e. cruise on non-motile prey only and ambush on motile prey only), although this exclusivity can be readily relaxed (cf. supplement determined by the feeding mode that available at www.int-res.com/articles/suppl/ m473 p091_supp/ ). The size maximises fitness. This can be expressed spectrum for both prey motility types is linear with equal biomass per either as a single optimal mode, or a logarithmic division, and the total prey biomass in the system is kept con- weighted ensemble based on relative fit- stant at 100 mgC l−1. Cruise feeding is given a cost penalty of 10−10 gC s−1 compared to ambush, as well as a 4-fold increase in predation risk. Based ness afforded different feeding modes. on these fitness considerations, there is a transition from cruise to Implementing these algorithms will ambush when the motile to non-motile ratio is about 70% make ecosystem models more consistent Visser & Fiksen: Optimal foraging in marine ecosystem models 101

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Editorial responsibility: Jake Rice, Submitted: June 11, 2012; Accepted: July 27, 2012 Ottawa, Canada Proofs received from author(s): January 7, 2013