Journal of Plankton Research Vol.22 no.10 pp.1871 1900, 2000 Modelling the attack success of planktonic predators: patterns and mechanisms of prey size selectivity Philippe Caparroy, Uffe Høgsbro Thygesen and André W.Visser 1 Department of Marine Ecology and Aquaculture, Danish Institute for Fisheries Research, Kavalergaarden, 6, DK 2920 Charlottenlund, Denmark 1 To whom correspondence should be addressed Abstract. A mathematical model of the attack success of planktonic predators (fish larvae and carnivorous copepods) is proposed. Based on a geometric representation of attack events, the model considers how the escape reaction characteristics (speed and direction) of copepod prey affect their probability of being captured. By combining the attack success model with previously published hydrodynamic models of predator and prey perception, we examine how predator foraging behaviour and prey perceptive ability affect the size spectra of encountered and captured copepod prey. We examine food size spectra of (i) a rheotactic cruising predator, (ii) a suspension-feeding hovering copepod and (iii) a larval fish. For rheotactic predators such as carnivorous copepods, a central assumption of the model is that attack is triggered by prey escape reaction, which in turn depends on the deformation rate of the fluid created by the predator. The model demonstrates that within a species of copepod prey, the ability of larger stages to react at a greater distance from the predator results in increased strike distance and, hence, lower capture probability. For hovering copepods, the vorticity field associated with the feeding current also acts in modifying the prey escape direction. The model demonstrates that the reorientation of the prey escape path towards the centre of the feeding current s flow field results in increased attack success of the predator. Finally, the model examines how variability in the kinetics of approach affects the strike distance of larval fish. In cases where observational data are available, model predictions closely fit observations. Introduction Predation is a major force in controlling the dynamics and structure of planktonic communities (Steele and Frost, 1977). In an effort to understand the structuring role of predation, ecologists have long aimed at predicting the risk of a particular prey to specific predators. Selective feeding is considered an important mechanism whereby planktonic communities are structured (Greene, 1983). Therefore, understanding the mechanism of selective feeding is important for interpreting the dynamics of predator prey populations. In a review of the mechanisms and patterns of prey selection in carnivorous calanoid copepods, Greene (Greene, 1988) proposed that the vulnerability of a prey to a predator should be considered as the product of its encounter rate with the predator and its capture probability. Therefore, Holling s (Holling, 1966) predation cycle model appears to be an appropriate conceptual approach to advance understanding of the mechanisms of prey selectivity (Price, 1988), as it dissociates predatory events into component processes of encounter, attack, pursuit and capture. Encounter processes have frequently been modelled (Gerritsen and Strickler, 1977), whereas only a few theoretical studies have concerned the subsequent steps of pursuit (MacKenzie et al., 1994) and attack/capture (Beyer, 1980; Heath, 1993). These models are limited in that they consider only visual predators and Oxford University Press 2000 1871
P.Caparroy, U.H.Thygesen and A.W.Visser do not take either the prey perceptive abilities or escape reaction characteristics into account, parameters critical for the outcome of an attack event. There is increasing evidence in the literature (Kirk and Gilbert, 1988; Yen and Fields, 1992; Fields and Yen, 1997b; Kiørboe et al., 1999) that rheotactic planktonic prey (e.g. copepods and microzooplankton) react to components of the hydrodynamic signal generated by moving predators in the form of an escape reaction. In a recent experimental study (Kiørboe et al., 1999) isolating the different components of the velocity gradient created by a planktonic predator, it was demonstrated that the copepod Acartia tonsa reacts specifically to the deformation rate of the flow field. These authors further demonstrated that the threshold deformation rate eliciting escape reaction scaled inversely with the size of the copepod. This implies that larger developmental stages of A.tonsa react to lower deformation rates and, hence, escape at greater distances from an approaching predator. It is therefore critical to describe how prey size and perceptive ability of the predator control the distance between predator and prey at the time of prey escape/predator strike, as this distance contributes to the success of the predator s approach or attack. In carnivorous copepods that create a feeding current, vorticity and shear of the flow field can influence the escape direction of prey (Fields and Yen, 1997a), orientating the prey s body axis towards the centre of the feeding current. In this case, the escape direction of the prey is the result of the interaction of its body shape, its (possible) eccentric centre of mass and the flow field. That is, the feeding current potentially controls the capture success of the suspension-feeding predator. The aim of this study is to develop a model of the attack process, taking into account the characteristics of the planktonic predator s flow field, its perceptive ability and characteristics of the prey s escape reaction. In the following, we first present a model of attack success. We then combine it with the model of predator flow field of Kiørboe and Visser (Kiørboe and Visser, 1999) in order to examine how predator and prey perception of hydromechanical signals contribute to patterns of prey size selectivity in carnivorous copepods and fish larvae feeding on copepod nauplii and copepodites. Modelling attack success The model considers two basic elements: the predator attack and the prey escape. In the simplest case, prey is attacked when it is within the strike zone of the predator, whereupon it perceives a hydromechanical signal alerting it to a probable attack and eliciting an escape jump. Prey is captured when it stops within, or passes through, the predator s capture zone. The outcome of the attack escape event can be determined geometrically (Figure 1), and depends on the predator s strike distance r s, its capture distance r c, the strike duration t and the attack speed v a, as well as the prey s escape speed w, and jump angle,, relative to the attack direction. This latter may also involve a random element, so that the actual jump falls within a three-dimensional cone 1872
Modelling prey size selectivity centred on mean and with spread ±. A mathematical description can be found in the Appendix, and parameter definitions are listed in Table I. The capture success for various parameter values can be evaluated through numerical simulation of attack escape events with randomly distributed jump angles. General behaviour of the model For a predator faced with the whole size spectrum of A.tonsa developmental stages, escape velocities are within the range 0 100 mm s 1 (Buskey, 1994). The capture radius equals the length of the maxillipeds and is r c = 0.66 mm (Kiørboe et al., 1999). From the detailed description of the attack process given by Yen (Yen, 1988), we further assume that Euchaeta rimana lunge at prey from a distance r s = 2 mm, with an attack speed v a = 37.5 mm s 1 and a strike duration t = 0.05 s. Fig. 1. The geometry of the attack/escape. The frame of reference follows the predator, and the x- axis is defined by the predator s attack direction. The strike distance is r s, r c is the predator s capture radius and r d is the relative displacement between predator and prey during the attack. The prey escape speed is w, the predator s attack speed is v a and the strike duration is t. The escape direction of the prey is given by the angle between the directions of attack and escape, as well as by the angle from the x-y plane spun by the attack and escape velocity vectors. The figure is drawn for = 0; non-zero corresponds to a rotation around the x axis. lies in the cone centred on mean with spread. 1873
P.Caparroy, U.H.Thygesen and A.W.Visser Table I. Glossary Symbol Meaning (dimensions) a Prey estimated spherical radius (mm) c Radius of sphere, length scale of predator (mm) C Prey concentration (prey ml 1 ) d Offset between centre of buoyancy and geometric centre of the prey (mm) F Clearance rate of copepod predator (ml copepodite 1 day 1 ) g Gravitational acceleration 980 cm/s 1 k Cruising predator deceleration rate (mm s 2 ) l Prey bodylength (mm) L Predator body length (mm) n Number of simulations of the approach process performed n sa Number of successful approaches simulated n eda Number of escapes during approach P capt Expected capture probability/attack success P eda Probability of prey escape during the approach P sa Probability of successful approach R Visual predator s perception distance (mm) r ap Predator prey relative distance during approach (mm) r c Predator s capture sphere radius (mm) r s Predator s strike distance (mm) S * Threshold deformation rate signal strength, velocity difference (mm s 1 ) S Searching activity of cruising copepod t a Approach time (s) t Strike or attack duration (s) T h Handling time of a prey item (s) U Cruising copepod searching velocity/maximal feeding current speed u Copepod prey swimming speed (mm s 1 ) v a Predator s attack speed (mm s 1 ) v ap Predator s approach speed (mm s 1 ) <v s > Cruising predator mean searching speed (mm s 1 ) w Prey escape speed (mm s 1 ) s Search rate volume of copepod predator; encounter kernel (ml s 1 ) Deformation rate of the fluid at prey position (s 1 ) * Threshold deformation rate eliciting prey escape reaction (s 1 ) Length scale of rheotactic predator s perceptive field (mm) Dynamic viscosity of sea water (1 to 2 10 2 cm 2 s 1 ) Scatter in the escape directions vs Standard deviation in searching velocity Time ellapsed since the start of approach (s), Polar and azimuthal escape angles of the prey Horizontal component of the feeding current vorticity (s 1 ) Angular velocity of the prey (s 1 ) Orientation of nauplius with respect to the vertical Results Independent of the range of escape directions available for the prey ( i ), there is a critical escape speed below which capture is systematically successful (Figure 2a). Above this critical speed, capture probability decreases with increasing escape velocities, and increases with the random spread of escape direction. For completely random escapes ( i = 180 ), the capture probability tends asymptotically towards a minimal value defined by the fraction of prey escape paths that are intercepted by the predator s capture sphere. 1874
Modelling prey size selectivity Fig. 2. Capture probability versus the prey escape velocity (w) for various parameters in the models of escape and capture. (a) Effect of the scatter in escape directions ( ). The mean escape direction is mean = 0. (b) Effect of variations in the mean escape angle: mean ( = 45 ). (c) Effect of variations in the predator s attack velocity. (d) Effect of variations in the predator s capture sphere radius (r c ) or strike distance(r s ). mean = 0, = 180. Figure 2b presents simulations of the capture probability for prey jumping in different mean directions ( mean ) with a constant scatter in escape ( = 45º). Increasing the mean escape angle, i.e. jumping towards the predator (0º = directly away, 180 = directly towards), results in increasing capture probability. For the simpler situation where the predator encounters and attacks prey orientated at random ( = 180 ), the capture probability increases with increasing attack speed (Figure 2c) as a result of increased volume swept clear by the capture sphere during the attack. Similarly, an increase in capture sphere radius or decrease in predator strike distance (Figure 2d) results in increased capture probability, as the model is equally sensitive to both parameters. However, this effect is more important for high prey escape velocities where both parameters define the range of escape directions that will result in crossing the predator s capture sphere. 1875
P.Caparroy, U.H.Thygesen and A.W.Visser Application of the model to more complex situations The results of the simple capture probability model are all consistent with intuition. However, in order to examine mechanisms and patterns of prey size selectivity, it is necessary to examine how the different parameters of the model depend on the size of the prey. Although there is no precise description of onthogenic changes in prey escape abilities, it appears reasonable to assume that the escape speed increases with prey size, as does swimming speed (Buskey et al., 1993; Buskey, 1994). The relative distance between predator and prey at the start of an attack, r s, is not a fixed parameter but depends on predator and prey perceptive abilities. If the predator perceives the prey first, this distance is identical to the predator s perception distance of the prey. For a rheotactic predator, it depends on the size and swimming speed of the prey (Jonsson and Tiselius, 1990; Kiørboe and Visser, 1999). Conversely, if the prey perceives the predator first, and the prey escape triggers an attack by the approaching predator (Yen, 1988), then the strike distance becomes dependent on the distance at which the prey perceives the predator. Although this latter scenario appears more convoluted, there are arguments in the literature to support its validity (Landry, 1978; Yen and Strickler, 1996). Finally, the flow field generated by the approaching predator may affect the escape direction of the prey (Fields and Yen, 1997a), and the magnitude of the effect depends on the size of the prey. In the following, we examine these more complex predatory interactions and combine the capture probability model with a model of the flow field created by a moving predator (Kiørboe and Visser, 1999). Specifically, we consider a cruising copepod, a hovering copepod with a feeding current, and a larval fish. In each case, we apply the model to specific predator prey interactions, and compare model output with observations of prey size selection reported in the literature. Cruising copepod There is evidence in the literature that cruising copepods attack escaping prey and do not react to uniformly moving prey until they perform an escape jump. Yen and Strickler, describing the attack behaviour of E.rimana, indicated that direct contact with the prey Acartia fossae did not result in a strike, and that the attack was only triggered by a real escape reaction which resulted in shedding a wake against the setae of the predator (Yen and Strickler, 1996). A similar description of the attack process has been given for Paraeuchaeta norvegica (Tiselius et al., 1997) and Labidocera trispinosa (Landry, 1978). In this scenario, it is the prey s perception (and hence reaction) distance to the approaching predator that defines the strike distance, and prey reacting closer to the predator are therefore attacked from a shorter distance. In order to explore the implications of such a process for the capture success of a cruising copepod, we use the hydrodynamic model of Kiørboe and Visser (Kiørboe and Visser, 1999). We assume that the cephalic anterior extremity of a swimming copepod can be modelled as a translating hemisphere (Tiselius and Jonsson, 1990), and that the motion of the predator in an otherwise still fluid will 1876
Modelling prey size selectivity generate velocity gradients in the ambient fluid. Details and equations of the model are given in Kiørboe and Visser (Kiørboe and Visser, 1999) and will not be developed again here. In a recent experimental study, Kiørboe et al. (Kiørboe et al., 1999) demonstrated that of the different components of a velocity gradient (acceleration, vorticity and deformation), the copepod A.tonsa reacts to the deformation rate only. This can therefore be considered as the hydrodynamic signal that elicits prey escape reaction in copepods (Haury et al., 1980; Yen and Fields, 1992). Assuming that a copepod can sense the velocity difference created over its body by the deformation of the fluid (= signal strength), these authors also demonstrated that the threshold signal strength required to elicit an escape reaction was invariant with developmental stages of A.tonsa. Using the mathematical formalism proposed by Kiørboe and Visser (Kiørboe and Visser, 1999), we therefore consider that within a particular species of copepod prey with fixed threshold signal strength, S *, the critical deformation rate * required to elicit a response is given by: * = S * /a (1) where a is the equivalent spherical radius of the prey. Thus, * increases with decreasing prey size. The surface surrounding the predator where the deformation rate equals the critical deformation rate * can therefore be seen as the boundary of an effective perceptive field of the predator (Figure 3). This surface is axially symmetric, as is the flow field model of Kiørboe and Visser (Kiørboe and Visser, 1999), and the surface is therefore described by the predator s perception distance of the prey, R( ), as a function of angle. This function can be obtained by solving (R( ), ) = * (2) numerically for R( ). The radius of the cross-sectional perceptive area of the predator, (Figure 3), is then given by: = R * sin * = max{r sin : (R, ) = * } (3) and the volume swept clear by the perceptive field of the cruising predator per unit time (search rate volume) is computed as: s = 2 (u 2 + U 2 ) 1/2 (4) where u and U are, respectively, the prey and predator s swimming velocities. We assume the prey velocity u to be negligible and use u = 0 in the following. Because prey are assumed to be encountered at random, prey escape angles are randomly distributed, = 180. Finally, to account for changes in the prey s reaction distance/predator s strike distance with variations in the approach angle, the expected capture probability is computed as: 1877
P.Caparroy, U.H.Thygesen and A.W.Visser Fig. 3. The perceptive field of a cruising copepod. The cephalic extremity of the cruising predator is modelled as a translating hemisphere of radius c, which creates a velocity gradient (Kiørboe and Visser, 1999). Perception of rheotactic prey is assumed to be a result of prey escape reaction, which is elicited by the critical deformation rate: * (s 1 ). The outermost limit of the predator s perceptive field is the isoline (R, ) = *, and the volume swept clear by the perceptive field of the predator is a cylinder of radius [see equations (2) and (3) for details]. P capt = 0 * P ( R ( ), ) sin R( ) d * 2 ( ) sin d The integrals in equation (5) are evaluated numerically. # capt 0 # 2 R (5) Results We consider the case of the cruising predator Euchaeta elongata feeding on the developmental stages of Calanus pacificus (Greene and Landry, 1985). Values of the different parameters of the model are given in Table II and are estimated from the literature. The size-dependent prey reaction distance (Kiørboe et al., 1999) is illustrated in Figure 4a and shows how such a process results in drastic changes of the predator s perceptive field as a function of prey size. The predator s perception distance of escaping prey increases from naupliar to adult stages of C.pacificus and varies from 0.7 to 3.5 mm, which is in the range of values observed by Yen (Yen, 1988) for the congeneric species E.rimana. The effect of this perceptual bias is illustrated in Figure 4b, which shows that the volume explored by the searching predator increases by a factor of 16 when the prey changes from naupliar to adult stages. However, the increased ability to perceive and encounter larger prey does not necessarily translate into an increased capture rate (Figure 4c and d). The simulated changes in capture probability over the whole size spectrum of C.pacificus suggest that E.elongata captures all developmental stages of C.pacificus smaller than 0.6 mm. For bigger prey size classes, the combined effect of increased strike distance and escape velocities results in a continuous decrease in capture probability, down to a minimal value of 20% for the adult stages. The capture rate volume, computed as the product of the search rate volume and the capture 1878
Modelling prey size selectivity Table II. Model parameter values and symbols for the cruising predator E.elongata feeding on C.pacificus prey Symbol Parameter Value Dimensions c Radius of the cephalic extremity of the cruising predator 0.5 a mm l Prey body length 0.22 3 a mm L Predator body length 7.4 a mm r c Capture sphere radius. Length of the extended maxillipeds 3 b mm S * Threshold deformation rate signal strength 0.8 c mm s 1 t E.elongata strike duration 0.1 d s U E.elongata swimming speed L s 1 mm s 1 v a E.elongata attack speed 17 L s 1d mm s 1 w Prey escape speed 100 l s 1c mm s 1 a Greene and Landry (Greene and Landry, 1985); b Landry and Fagerness (Landry and Fagerness, 1988); c Haury et al. (Haury et al., 1980); d Yen (Yen, 1988). probability, exhibits a dome-shaped pattern (Figure 4d), with a peak value of 3500 ml copepodite 1 day 1 for stage I copepodites of C.pacificus, and is similar to the maximal clearance rate of E.elongata feeding on Pseudocalanus sp. measured by Yen (Yen, 1983). The pattern of prey selectivity simulated is therefore qualitatively consistent with the conceptual scheme proposed by Greene (Greene, 1988): (i) the search rate volume increases with prey size; (ii) conversely, the capture probability, or susceptibility of prey to a predator, decreases with prey size; (iii) the resulting product, the instantaneous capture rate volume, exhibits a dome-shaped pattern with the maximum value indicating the most preferred prey. In order to compare our quantitative results with the observations of Greene and Landry (Greene and Landry, 1985), we need to consider that different handling times for specific prey items can contribute to net ingestion rates. This process appears potentially important in the case of E.elongata, where handling time is known to vary by three orders of magnitude for the range of prey sizes considered here (Yen, 1983). According to Holling s conceptual scheme of predation, the clearance rate, F (ml copepodite 1 day 1 ), of E.elongata can be expressed as: F = S s = s /(1 + s T h C) (6) where T h (day prey 1 ) is the handling time of an individual prey, C (prey ml 1 ) is the prey density and S (%) is the proportion of total time spent searching for prey. From the data in Table I of Yen (Yen, 1983), we fitted the power function: T h = a (l/l) b (7) where l and L are prey and predator body length, respectively. Finally, by combining the fitted model of handling time (a = 45.79 days; b = 3.62; R 2 = 0.99; P < 0.05) with equation (6), we simulate the decrease in searching activity with increasing prey sizes (Figure 4e), which, in turn, allows a reasonable fit to the measured daily clearance rates (Greene and Landry, 1985) (Figure 4f). 1879
P.Caparroy, U.H.Thygesen and A.W.Visser Fig. 4. (a) Perceptive field of the cruising predator E.elongata feeding on the different developmental stages of C.pacificus. The dashed lines are critical deformation rate isolines for different size classes of this prey [Table 2 in (Greene and Landry, 1985)]. Effect of prey length on (b) search rate volume, (c) capture probability, (d) capture rate volume, (e) searching activity and (f) daily clearance rate. Prey density is assumed to be 6.25 and 12.5 prey l 1, respectively, for the naupliar and copepodite stages (Greene and Landry, 1985). 1880
Modelling prey size selectivity Hovering copepods: prey rotation Another foraging strategy exhibited by carnivorous calanoid copepods is the hovering mode. Typically, hovering copepods hang vertically in the water while creating a feeding current, the thrust from which counterbalances gravity. Although this behaviour has been described in herbivorous copepods (Strickler, 1985), it also characterizes some carnivorous copepods such as Neocalanus cristatus (Greene and Landry, 1988) and E.rimana (Fields and Yen, 1997a). Similar to the cruising strategy previously described, prey entrained within the flow field of a hovering predator should detect its presence by the rate of fluid deformation. However, in order to discuss the consequences of this particular predatory behaviour on the outcome of the capture event, an additional process has to be taken into account. Because the feeding currents of copepods are sheared and therefore contain local vorticity components within the flow (Fields and Yen, 1993; Kiørboe and Visser, 1999), prey are both advected and rotated. This process, which has been demonstrated in the case of naupliar stages of Acartia hudsonica entrained within the feeding currents of E.rimana (Fields and Yen, 1997a), affects the prey orientation at the time of escape and has potential implications for the capture success of the predator. In the following, we therefore examine how a detailed representation of the feeding current flow field affects the capture success and pattern of prey size selectivity for carnivorous hovering copepods. As an example, we consider a stationary hovering copepod, E.rimana, orientated vertically and creating a feeding current entraining prey. The prey are naupliar and copepodite stages of A.tonsa. We assume that they are vertical when entrained within the feeding current, doing so during the sink phase of their hop and sink motion (Fields and Yen, 1997a). The passive trajectory of prey (Figure 5) is defined by the stream function of the feeding current, and, their orientation with respect to the vertical, is determined by the balance of gravitational torque T g and frictional torque T v. The equations describing the trajectory and rotation are listed in the Appendix. At each prey position, the deformation rate of the flow field is computed according to Kiørboe and Visser (Kiørboe and Visser, 1999), noticing that the translating sphere model of an approaching predator gives rise to the same deformation field as the spherical pump model of a feeding current. When the critical value * (equations 1 and 2) is reached, the prey jumps at a speed w, during a time interval t e (escape time). The direction of the jump is defined by the orientation of the prey relative to the vertical at the time that the critical deformation rate is encountered. During the escape reaction, we assume that the speed of the prey is too high to allow any additional rotation due to vorticity. The trajectory of the escape jump can therefore be calculated (cf Appendix) and the capture success estimated for various parameter values. Model parameter values based on information in the literature are given in Table III. Results Because the vorticity varies spatially, each entrainment path is characterized by a unique vorticity pattern. This is illustrated in Figure 6a c, which shows the effect 1881
P.Caparroy, U.H.Thygesen and A.W.Visser Fig. 5. The trajectory and rotation of an eccentric particle (e.g. nauplius) in an idealized copepod feeding current. The pumping volume where the motion of the cephallic appendages creates the feeding current is modelled as a sphere of radius c, generating a flow field with deformation (Kiørboe and Visser, 1999). The particle is neutrally buoyant and follows a streamline. Its orientation relative to the vertical,, is determined by a balance between gravitational torque, T g, acting on its nonuniform distribution of mass, and frictional torque T v, arising from its relative rotation with respect to the fluid. An escape reaction is elicited when the particle crosses the critical deformation rate ( * ) isoline. This in turn elicits an attack response from the copepod, i.e. the * isoline also defines the predator s perception limit. The outermost limit of the predator s perceptive field is defined by the streamline * intersecting the critical deformation rate isoline (R, ) = *. This outermost streamline defines a virtual pumping sphere of radius * which encompasses all prey trajectories within the feeding current resulting in perception by the predator. Table III. Model parameter values and symbols for the hovering predator E. rimana feeding on A.tonsa prey Symbol Parameter Value Dimension c and r c Radius of the pumping/capture sphere volume 0.6 a mm d Offset between the centre of buoyancy and geometric centre of the prey 2 10 3 l b mm g Gravitational acceleration 9.81 m s 2 l Prey body length 0.5 1 mm Dynamic viscosity of sea water (15 C, 20 p.s.u.) 1.19 10 2 g cm 1 s 1 c Density of the prey 1.05 10 3 g cm 3 S * Threshold deformation rate signal strength 0.15 c mm s 1 t esc Prey escape (jump) duration 0.1 s U Feeding current velocity scale 9 b mm s 1 w Prey escape speed 100 l s 1 mm s 1 a Kiørboe and Visser (Kiørboe and Visser, 1999); b Fields and Yen (Fields and Yen, 1997a); c Kiørboe et al. (Kiørboe et al., 1999). of the feeding current flow field on the orientation of a 100 m A.tonsa nauplius. Prey approaching the predator from ahead where the vorticity is zero [cf equation (a.13)] remain vertically orientated, whereas prey entrained within the lateral (left) part of the feeding current flow field experience high vorticity and are rotated clockwise. For the range of entrainment paths which result in a prey escape reaction (streamlines crossing the critical deformation rate isoline), the 1882
Modelling prey size selectivity outermost trajectory results in a clockwise rotation of 40. Similar results were obtained by Visser and Jonsson (Visser and Jonsson, 2000). Therefore, in agreement with the observations of Fields and Yen (Fields and Yen, 1997a), the vorticity field acts in re-orientating prey towards the centre of the feeding current at the time of escape. In our simulations, this effect decreases with increasing prey size (Figure 6c), due both to longer reaction distances (Figure 6d) and longer Fig. 6. Rotation of a spherical nauplius (100 m) entrained within the feeding currents of E.rimana with pumping sphere radius c = 0.6 mm. (a) and (b) The dotted line represents the isotach where feeding current velocity is 10% of its maximal value (0.9 mm s 1 ). The solid line is the critical deformation rate isoline (2 s 1 ) for this prey size class. The dashed line is the trajectory of the nauplius following a feeding current streamline. The circles mark positions of the prey during the approach of the predator, the arrow indicating its body axis orientation. (a) A general trajectory. (b) The limiting trajectory where the nauplius will experiences a critical deformation rate and initiate a jump. (c) and (d) Effects of varying angle of penetration within the feeding currents for a 0.1 mm (diamonds) and 0.2 mm (triangles) nauplii. (c) Effect on, prey body axis angle relative to the vertical; (d) effect on reaction distance. 1883
P.Caparroy, U.H.Thygesen and A.W.Visser separation distances between the geometric and buoyancy centres [equation (a.15)]. The effects of the vorticity-induced rotation of the prey on the pattern of escape reaction are illustrated in Figure 7, where the positions of a 100 m nauplius before and after an escape reaction are shown for different trajectories in the left side of the feeding current. It is clear that the vorticity-induced asymmetry in prey swimming directions results in focusing the prey in the central region of the predator s flow field. Furthermore, prey which arrive in the lateral part of the feeding current are less efficient in increasing the radial distance from the predator at the end of an individual escape reaction than those arriving front on. For the particular simulation presented here, a nauplius following the outermost streamline even results in being closer to the predator s capture sphere after the escape reaction. If the escape reaction of the prey creates a hydrodynamic signal that triggers an attack by the predator, the predation risk experienced by a prey is potentially dependent on its path within the feeding current. According to the previous results, prey arriving in the lateral part of the feeding current should, for instance, always be attacked from shorter strike distances than those arriving front on. In order to test this hypothesis, we have coupled this model of prey trajectories with the capture probability model. In this, we consider multiple escape jumps where the predator attacks, jumping towards the prey at the initial position of its second escape jump, with an attack speed of 60 body lengths s 1 (Yen, 1988). Fig. 7. Initial (filled circles) and final (open circles) positions of a 0.1 mm naupliar prey, during an escape reaction in the feeding current of E.rimana (the hemisphere represents the pumping volume of the predator). From left to right, initial positions of the prey correspond to decreasing angles of penetration within the feeding current and are situated on the critical deformation rate isoline for this particular size class of prey. Escape trajectories are represented for the outermost (dotted line) and innermost (continuous line) prey positions. 1884
Modelling prey size selectivity For different size classes of prey (developmental stages of A.tonsa), the trajectory within the feeding current s flow field, and a first escape reaction, are simulated using equations (a.10) (a.21). The attack process is simulated using the capture probability model, simply assuming that prey will escape a second time in the direction defined by its body axis orientation at the end of the first escape (which is still the one defined by the vorticity-induced rotation). The mean escape angle to vertical is therefore given by: mean = (8) where is the polar angle (spherical co-ordinate) of prey position at the start of attack. Fields and Yen observed that an A.tonsa nauplius did not simply leap forward, but rather leapt at an average angle within ±10 from its pre-escape orientation, suggesting a scatter angle of = 10 (Fields and Yen, 1997a). Figure 8a presents the simulated variations of capture probability with the angle of penetration within the feeding currents for the younger developmental stages of A.tonsa. Size classes smaller then 250 m (NIV) are systematically captured (not shown on the graph), independently of the angle of penetration. However, for bigger prey sizes, the model predicts that the capture risk is spatially varying and is highest in the external part of the flow field. This spatial variability in capture probability is the result of the vorticity and its systematic rotation of prey prior to escape. This is shifted by up to 50 towards the feeding current axis (Figure 8b). Simultaneously, the decreased ability of the prey to counteract the radial component of the flow field during their first jump results in shorter strike distances for those prey which travel within the lateral part of the flow field (Figure 8c). For the range of prey sizes considered here, the vorticity-induced reorientation of escape direction is slightly lowered for larger prey (Figure 8b). At the same time, the increase in escape speed with prey size results in longer distances travelled during the first escape reaction and, hence, in longer strike distances (Figure 8c). Hovering copepods: prey length In order to examine how the size dependence of prey capture probability affects prey selectivity, we consider how the mechanism of prey perception affects the predator s perceptive field and encounter rate. As with the cruising predator, we consider that prey perception leads to a prey escape reaction. Therefore, the perception distance of the prey by the predator, R, equals the distance where the deformation rate within the feeding current equates to the threshold deformation rate ( * ) of the prey, (R, ) = *. Because prey passively follow streamlines before escape, the outermost streamline allowing prey perception (cf Figure 5) is defined by the stream function as * = (R *, * ), with R * sin * = max{r sin : (R, ) = * } as in equation (3). For a particular prey size with a threshold deformation rate eliciting escape reaction ( * ), the flux of fluid passing through the perceptive field of the predator can be expressed as: 1885
P.Caparroy, U.H.Thygesen and A.W.Visser Fig. 8. Changes in (a) capture probability, (b) mean escape angle ( ) relative to predator position at the start of attack and (c) strike distance with varying angle of penetration within the feeding current for three size classes of A.tonsa developmental stages. 1886
which gives: R V * S W 2 2 = S c U+ 2 # rui (, r ) drw 2 S c W T X * 2 2 3 - c s = cu * 2 Modelling prey size selectivity (9) (10) where * (Figure 5) represents the radius of a sphere which encompasses all prey trajectories (feeding current streamlines) resulting in escape reaction/perception by the predator, i.e. ( *, /2) = *. There are no analytical solutions to these equations so that values of R *, * and * are computed numerically. Results The simulated search rate volume and capture probability of E.rimana suspension feeding on A.tonsa developmental stages are presented in Figure 9. For the range of parameter values chosen in these simulations, the model suggests that E.rimana should not be able to capture copepodite stages of A.tonsa (size class >0.3 mm) using its feeding currents in the hovering mode. Conversely, the increase in prey reaction distance with prey size leads to a subsequent increase in the predator s search rate volume (Figure 9b). Therefore, the concurrent effects of prey size on the capture probability and the kinetic component of the encounter rate result again in a dome-shaped pattern of the capture rate volume (Figure 9c). Unfortunately, there are no observational data available for comparison. Modelling the approach and attack success of a fish larva In a recent experimental study, Viitasalo et al. (Viitasalo et al., 1998) showed that the ability of juvenile fish to successfully approach, attack and capture copepod prey was mainly dependent on the fish s final approach speed. This is qualitatively in accordance with the theoretical predictions of the hydrodynamic model of Kiørboe and Visser (Kiørboe and Visser, 1999). In the following, we examine how this hydrodynamic model can be used to predict the prey size spectrum from a predator s approach strategy and the perceptive ability of the prey. Dynamic description of fish approach We consider a cruising predator, such as a herring larva (Kiørboe and Mackenzie, 1995), searching for prey at a speed v s and locating prey at a distance R, which represents the radius of a spherical visual field of the predator. Upon locating prey, fish larvae typically approach head on, decelerating so as to be nearly motionless at the strike distance (r s ), where they bend their body into an S-shape, primed to propel them during the final strike. 1887
P.Caparroy, U.H.Thygesen and A.W.Visser Fig. 9. Changes in (a) expected capture probability over the whole feeding current flow field, (b) search rate volume and (c) capture rate volume, with prey size, for E.rimana suspension feeding on the developmental stages of A.tonsa. 1888
Modelling prey size selectivity We will assume that the failure of an approach is due to the prey escaping before the predator is motionless and ready to strike. According to the results of Viitasalo et al. (Viitasalo et al., 1998) and Kiørboe et al. (Kiørboe et al., 1999), prey escape during approach arises when the predator s approach speed is too high and generates a deformation rate at the prey s position exceeding the critical rate for detection. To represent the dynamics of the approach phase, we will simply assume that as soon as the prey is perceived by the cruising predator, this latter decelerates at a constant rate, k. The instantaneous approach speed, v, of the predator is therefore given by: v( ) = v s k (11) where (s) is the time elapsed since the start of approach. While this linear model is consistent with the observed approach behaviour of the three-spined stickleback (Viitasalo et al., 1998), no direct values of k are available in the literature. We therefore estimate k from observed approach times, t a (MacKenzie et al., 1994; Kiørboe and Mackenzie, 1995), which is the time elapsed between the perception of the prey and the formation of the S-shaped position. By considering that the predator is motionless at the end of the approach, equation (11) gives: k = v s /t a (12) To describe the position of the prey relative to the predator during the approach, we use the model of Kiørboe and Visser (Kiørboe and Visser, 1999), which considers that the prey is subject to the action of the flow field created by the head of the approaching fish larva, modelled as a translating sphere. The time rate of change of the relative predator prey separation distance, r ap, is given by: dr d ap R S 3c =- v S1 - + 2rap S T X where c is the radius of the fish larva s head and 0 r ap R. Equation (13) assumes that during the approach, the prey is situated directly in front of the head of the predator, so that there is no tangential component to the flow field experienced by the prey. Finally, the time rate of change of the instantaneous deformation rate at the prey s position during the approach is given by: 2 c r 2 ap 3 V W W W R d 3 c S ( c r )( r c )( r c) () 2 2 2 + ap - ap ap - = S-o d 2 8 S rap T 2 2 2 2 2 2 r - c ap - k 4 r ap V W W W X (13) (14) 1889
P.Caparroy, U.H.Thygesen and A.W.Visser Probability of successful approach To compute the probability of a successful approach, we include some variability around the deterministic pattern of approach described above. As suggested by Viitasalo et al. (Viitasalo et al., 1998), we assume that the success of the approach depends on the predator s approach speed. Thus, we introduce a random component in the fish approach speed v s, being normally distributed with a mean value, <v s >, and a standard deviation, vs. The approach process is simulated n times using equations (13) and (14), with varying initial speed at the start of approach (searching speed). The probability of a successful approach, P sa (dimensionless), is given by the ratio P sa = n sa /n where n sa is the number of successful approaches simulated. The condition for a successful approach is that ( ) does not exceed *. Probability of escape during the approach To describe the whole set of events which can be realized during the approach, we must consider the probability that the prey escapes during the approach (unsuccessful approach), P eda (dimensionless), evaluated by P eda = n eda /n where n eda is the number of escapes occurring during the approach. The condition for escape is that ( ) exceeds or equals * at some time during the approach. Probability of escape before the start of approach The model also allows the computation of the probability that a prey will escape before the start of an approach, or before the predator perceives the prey. This will occur if the prey s sensitivity is high relative to the visual capability of the predator. Escape of the prey before the start of the approach occurs if the instantaneous deformation rate created by the searching predator at a distance equivalent to its own reaction distance (R) is higher than the threshold deformation rate of the prey ( * ). The condition for escape before approach is therefore given by: (R) = 3 v s c ((R + c) 2 c 2 )/(2 (R + c) 4 ) * (15) Probability of successful attack For successful approaches, the model of capture probability is used in its initial form to compute the probability of a successful attack, given an approach. The strike distance is defined, for a particular size class of prey, as the mean distance at the end of successful approaches (output of the approach simulations). This results in the following expression for the attack duration: t s = r s /v a (16) Prey are considered encountered while swimming in random directions, so that 1890
Modelling prey size selectivity Table IV. Model parameters and variables for the cruising herring larva feeding on A.tonsa Symbol Parameter Value Dimensions c Fish larva head radius 0.1 L mm Deformation rate at prey position [Equations (14) and (15)] s 1 k Cruising predator deceleration rate [Equation (12)] mm s 2 L Fish larva body length 5 or 10 mm n Number of simulations of the approach process 100 R Predator s reaction distance 0.5 L mm r ap Predator prey relative distance during approach [Equation (13)] mm t a Approach time 1 s Time elapsed since the start of approach 0 t a s t s Strike duration (attack duration) [Equation (16)] s v a Predator s attack speed 20 L s 1 mm s 1 v Predator s approach speed [Equation (11)] mm s 1 vs Standard deviation in searching speed 0.1 L s 1 mm s 1 the direction of escape upon perception of the predator by the prey can be characterized by mean = 0 and =. A list of parameter values and symbols is presented in Table IV. Results Figure 10 presents the approach of a 5 mm body length herring larva, having perceived a copepodite (0.5 mm length) of the copepod A.tonsa. The deformation rate at the position of the prey (Figure 10a) shows a maximum of 0.58 s 1. Thus, prey with a critical deformation rate lower than this maximal value cannot be approached successfully. However, with this simple representation, the probability of a successful approach is still either 0 (maximum deformation rate during approach bigger than prey critical deformation rate) or 1 (maximum deformation rate during approach smaller than prey critical deformation rate). Figure 10b presents results of simulations made for various values of the variability in searching speed at the start of approach ( vs ). The probability of a successful approach is now smaller, and decreases with higher variability in approach velocities. Above a certain variability (~20%), escape of the prey before the start of approach occurs simply because some approach velocities are high enough to simulate escape reactions of the prey before it comes into the predator s visual field. Since increased prey reaction distance potentially leads to a decrease in the probability of successful approach, and increased predator s strike distance potentially contributes to a decrease in the probability of successful attack, these result suggest that increased variability in the approach pattern results in a decrease in the final capture probability for a particular type of prey. Variability in the approach speed may, for instance, be related to the developmental stage of the fish larva. In order to see how this representation of the approach process affects the size spectra of prey available for attack and/or capture, we now run the model for 1891
P.Caparroy, U.H.Thygesen and A.W.Visser Fig. 10. Dynamic representation of the approach of a fish larva (cruising strategy) towards a prey. Parameter values given in Table IV. (a) Instantaneous deformation rate at prey position (s 1 ). (b) Variations in the probabilities of successful and unsuccessful approaches, with variability in the initial approach speed of the predator. various size classes of the copepod A.tonsa. We make simulations for the whole size range of A.tonsa (0.1 1 mm) and for two sizes of predators (5 and 10 mm body length). For each prey size, we realize 100 simulations (n = 100) and introduce a theoretical variability in the initial searching speed of 10%. The probability of a successful approach decreases with increasing prey size, approaching zero for prey exceeding 0.8 mm (Figure 11a). This suggests that for prey sizes smaller then 0.6 mm, all unsuccessful approaches result from escapes during the approach (Figure 11b), whereas bigger prey are sensitive enough to escape before perception by the predator. Therefore, adults of A.tonsa should never be seen by our theoretical herring larva. The model also predicts a shift of sucessfully attacked prey towards bigger prey with increasing predator size 1892
Modelling prey size selectivity (Figure 11c). Figure 11d shows simulated changes in the attack success of a 13.5 mm herring larva as a function of the prey size/predator size ratio compared with the observations of Munk (Munk, 1992). The similarity between the observations and the model results suggests that when considering approach and attack success alone, the pattern of prey size selectivity should be systematically shifted towards the lower end of the prey size spectrum. When considering capture rate, however, Munk (Munk, 1992) observed a dome-shaped relationship to prey size. As with cruising and hovering copepods, such a pattern is probably due to an increase in the predator s perception distance and, hence, encounter rate, with prey size. For visual predators such as fish, perception distance is known to be positively correlated with prey size (Werner and Hall, 1974). Therefore, the use of visual-based encounter rate models (Aksnes and Utne, 1997) is probably necessary to reproduce the final shape of the larval fish prey size spectrum. Discussion The goal of our model was both to take into account the detailed mechanisms which determine the success of attack and capture events, and to examine how onthogenic changes in prey characteristics (escape speed and perceptive ability) may account for observed patterns of vulnerability to a given predator. The general features of the size selectivity conceptual model proposed by Greene (Greene, 1988) are qualitatively reproduced, thus confirming the possibility of representing the scale dependency between the processes of encounter and capture occurring at the time scale of a second, and the classical experimental results obtained over 24 h using a black-box approach. Perceptive ability and predator prey orientation A central assumption of the model is introduced by the representation of a constant threshold signal strength required to elicit prey escape reaction. By combining the hydrodynamic model of a predator s flow field proposed by Kiørboe and Visser (Kiørboe and Visser, 1999) with our representation of the attack success, the ability of larger prey to react at greater distances from the approaching predator affects the encounter and capture processes in counteracting ways. This results in a dome-shaped prey size spectrum. The assumption that prey escape reactions elicit perception and attack by the predator provides a simple mechanism to explain the apparent increase in encounter rate with prey size observed in most studies of predatory copepods, feeding rates (Landry, 1980; Greene, 1983; Greene and Landry, 1985). This conceptual scheme of the attack process was described by Yen (Yen, 1988) for the cruising copepod E.rimana feeding on copepod prey. However, the readiness of the lunge upon prey perception led this author to suggest that the predator had prior knowledge of the presence of prey in close proximity. An alternative mechanism of prey perception would therefore be that rheotactic predators perceive the absolute velocity signal created by the approaching prey (Kiørboe and Visser, 1999). Because the magnitude of this signal scales with prey size and 1893