The functional response of a predator provides important information on mechanisms underlying predator–prey dynamics. In unstructured models, sigmoid (type III) functional responses have the potential to stabilize predator–prey dynamics due to the density-dependent predation mortality. In contrast, a type II functional response destabilizes the dynamics because the predators cause an inverse density-dependent mortality of the prey (e.g. Murdoch & Oaten 1975; Hassell 1978). The shape of the functional response curve is dependent on several characteristics such as encounter rate, capture efficiency and handling time of the predator (Holling 1965; Hassell 1978). These characteristics may differ within a species as a consequence of predator or prey size, interference among predators or presence of alternative prey (Spitze 1985; Chesson 1989; Safina 1990; Tripet & Perrin 1994; Elliott 2003)

Detailed information on size-dependence in the functional response is scarce, as most of the studies performed have incorporated a small variation in predator and prey sizes (Thompson 1975; Moran 1985; Hewett 1988; Eggleston 1990). A small prey size range may exclude information concerning presence of maxima or minima in the responses. In systems where the prey population has a large size range, the attack rate as a function of size should have a humped-shaped form (Persson *et al*. 1998). Prey may be too small to be efficiently encountered, or too large to be captured (Hyatt 1979). Persson *et al*. (1998) argued that handling time per prey should have a minimum. For very small prey, handling time decreases with prey size due to difficulties in handling very small prey. Above the minimum, handling time increases as the prey size approaches the predator gape size. In addition, the digestive capacity contributes to an increasing handling time per prey as prey becomes bigger.

The size-dependence in the attack rate may have implications for predator–prey dynamics. In a size-structured consumer-one resource model by Persson *et al*. (1998) the attack rate increased slowly with predator size, and this resulted in unstable recruiter driven predator–prey dynamics. When the attack rate was reduced for the small predators the newborn predators became less dominant, which stabilized the dynamics. Thus, the size scaling of the attack rate and its implication for the functional response provides valuable information for the understanding of dynamics in size-structured populations. In order to perform the corresponding analysis of both predator and prey size on population dynamics we need to know the form of a size-dependent functional response. Kooijman (1993) derived a functional response that accounted explicitly for predator size, but no functional response function has yet been derived to incorporate both predator and prey size.

The aim of this study was to derive a functional response function that included both predator and prey size. In addition, we investigated experimentally the size-dependent functional responses of the benthic isopod *Saduria entomon* feeding on *Monoporeia affinis* amphipods. Both isopods and amphipods had a large variation in body size. The idea of deducing a size-dependent functional response from basic principles is that if it fits well to experimental data, the parameter values give us valuable biological information about the predator and prey behaviour. For modelling purposes it is important that the parameter values are reliable for extrapolation to the real world. For that purpose we performed a reliability analysis of our achieved functional response that allowed rescaling to match survival and growth data from the field.

#### derivation of a size-dependent attack rate

To derive a mathematical function for a size-dependent attack rate we divided the function into two components; encounter rate and capture efficiency. The sequence of prey capture could be described briefly by the following. A predator encounters and detects a prey, there is then some probability of attacking it, and when the predator attacks the prey there will be a capture efficiency, a probability of capturing the prey. The encounter rate arises from encounters within the predator's search area due to movements of predators and (or) prey. The velocity, ν, of aquatic animals generally scales with body mass to the power 1/3 (Peters 1983). Following Skellam (1958), Werner & Anholt (1993) and Crowley & Hopper (1994) we approximated the encounter rate per prey by:

- (eqn 1)

where *g*(*W*_{N}, *W*_{P}) is the search area of a predator as a function of prey and predator size and the proportion of time they are active, ν is the velocity of the prey and the predator, indexed by *N*, *P*, respectively, *W*_{N} and *W*_{P} and is the body mass of the prey and the predator, respectively, and ϑ_{N} and ϑ_{P} are constants. The search area is given by the reactive distance of the predator and this distance often changes with prey size (Pastorok 1981; Crowl 1989; Parker 1993; Streams 1994). The size scaling of the search area is likely to change with the predator's foraging mode and with variations in prey size availability. Thus, the reactive distance for visually hunting predators is expected to be proportional to the prey area (silhouette), i.e. it scales to . This would imply the search area to be proportional to when the search area is proportional to the squared reactive distance. In general, the search area should be proportional to *W*^{2s}, when the allometric scaling of the stimulus is given by *s*. For tactile hunters, reacting to vibrations of the prey, *s*= 1. In contrast, for filter feeders prey size may be less important, *s*= 0, and search area should rather scale with predator size as as indicated in the functional response given by Kooijman (1993). Because we expect the size scaling of the search area to vary between species, we made the mathematical function general and included a prey size-scaling constant (γ) in the function for the search area. To complete the function *g*(*W*_{N},*W*_{P}), information on the size-dependence of the predator and prey activity patterns was required. It was difficult to define a general expression for such a size-dependence. Here we included the prey size-dependence into the constant γ, while for the predator size-dependence we entered a new size-scaling constant *.* Thus, the search area was defined as: *g*(*W*_{N},*W*_{P}) = , where *d* is a combined constant for the two allometric relationships.

For the attack probability and capture efficiency, we assumed that it could be expressed as a hump-shaped function of the prey/predator mass ratio (cf. Munk 1992). The decrease towards a low ratio, implying small prey, is due to reduced prey detectability, catchability. In contrast, the decrease towards a large ratio should be due to difficulties to handle large prey, and for some predator species an upper ratio may be set by gape limitation. A mathematical function that satisfies these conditions is:

where *c* is a scaling constant, *e* is the natural base for the exponential function, *k* is the rate at which the combined attack probability and capture efficiency declines with increasing size ratio, .

The complete attack rate function, eqn 1 × eqn 2 yields a hump-shaped response, both in relation to prey and predator size, producing a ridge in a three-dimensional space. This ridge, i.e. the maximum attack rate, becomes a linear function in the prey–predator mass plane when the species are completely isometric. To allow for possible nonlinearity in this maximum attack rate, we let the ratio *x* be defined by , where β is a size-scaling constant. Then the maximum attack rate scales with predator mass as , where *z* is a constant, when either the predator or prey velocity is zero. The complete function for the attack rate becomes:

- (eqn 3)

This attack rate is density independent. For the density-dependent case, as in the sigmoid functional response, the attack rate in eqn 3 needs to be multiplied by the prey density.

#### derivation of a size-dependent handling time

The functional response accounting for predator size given by Kooijman (1993) had a size scaling of the handling time expressed as . Thus, handling time for a fixed prey size should decrease with predator size. The handling time results from physical handling time which involves capture and eating, and the time to digest the prey (Jeschke, Kopp & Tollrian 2002). A complicating fact is that these two components may overlap. That is, while physically handling one prey, the predator may digest another prey. A reasonable assumption for the physical handling is that it is proportional to the ratio of prey size to predator size. An extra term is necessary to allow the handling time to increase for very small prey in relation to predator size. Multiplying each of the two terms with separate constants adjusts the levels and allows the result to be expressed in correct units. The digestion time should be proportional to prey mass when the predator consumes the prey in pieces, as in *S. entomon*. However, it is difficult to deduce a complete handling time function based on the current knowledge. We therefore prefer to keep the function short and flexible to capture most of the variation in the observed data. The digestion and the physical handling time will therefore be confounded in our handling time function. The function we chose to describe the size-dependence in the handling time is therefore partly phenomenological:

- (eqn 4)

where *h*_{h}, and *h*_{d} are constants, η_{N} and η_{P} are the size-scaling constants with indices *N* and *P* corresponding to the prey and the predator, respectively.

#### models

In contrast to the experiments, the predators in the model had access to all three age-classes of *M. affinis* simultaneously, and we used the median mass for each of these. The total number of consumed prey of one age-class, *j*, per day of all predators belonging to size-class *k* was formulated as:

- (eqn 8)

Here *a* is attack rate, *h* is handling time, *N* is prey density, *P* is predator density, and the indices denote size-class of predator and prey, respectively. The subscripts *m*, *j* denote that prey mortality of size-class *j* is given by the function. A constant *q* was introduced as a correction factor to adjust for potential bias in the attack rate estimates. The total daily prey consumption in terms of prey biomass per predator was estimated from:

- (eqn 9)

Here the subscripts *g*, *k* denote that the function is used to calculate growth of a predator belonging to size-class *k*, *W*_{i} is the individual prey mass of age class *i*.

The prey consumption of the predator was modelled as a continuous process, using daily rates. For *M. affinis* with a fecundity of about 30 eggs and a 3-year semelparous life cycle there is little room for mortality. We therefore assumed that the mortality of *M. affinis* due to other factors than predation by *S. entomon* was negligible. Our model handled the change in the actual size-structure of the prey throughout the year, accounting for spawning with male death (mid-November), female death after release of her offspring (1 March) and growth (March–September). Growth of *M. affinis* is concentrated to the spring–early summer period following the spring bloom (Elmgren 1978; Sarvala 1986).

The field data available indicate that the spawning of the predator is distributed all over the year. Moreover, the size-distributions from 1995 and 1996 in our field data indicated that growth compensated for mortality, leading to a stable size-distribution. For this reason we chose to keep the predator size-distribution constant in the model. Collecting information on prey biomass consumption by each predator size-class in the model allowed us to construct growth curves for the predators. These growth curves were one part of the check for realism in our results.

A modelling schedule with more detailed description of the events throughout the year is as follows. The consumption simulation starts with the initial prey and predator size distributions in October and covers 340 days. At day 40, 2+ males (40% of the age-class, estimated from samples collected in 1996) mate and die. At day 160, the offspring are released and the spent females die. Until now winter prevails and no growth has taken place. The remaining 180 days of the simulation covered the growing season. The growth rate of the prey was defined using a log-normal distribution (*LND*) to mimic the seasonal growth and reach the observed sizes in September. The growth rate was given by; *G*_{0} + *z*_{i} * *exp* (*x*) * *pdf* (*LND*, *x* = ln(day number), µ, σ), where *G*_{0} also provides growth of 0+ during late summer, *z*_{i} is a scaling constant for each age-class (*i*) to reach the correct final size, *x* is day number, pdf is short for probability density function, µ and s are parameters in the lognormal distribution. The parameters for the growth function are given in Table 2 together with the other parameters used in the model. We performed several simulations by varying the correction factor, *q*, to obtain an estimate of the sensitivity of the predicted results in relation to the experimentally derived attack rate estimate. After each simulation, we compared the predicted number of surviving prey of each age-class with that observed from the field samples.

Table 2. Description of parameters and their numerical values used in the consumption model, and in the predator growth model Symbol | Value | Unit | Description |
---|

Prey parameters |

*x* | 1–340 | | Day number, starting mid-October |

µ | 210 | | Day number when maximum growth occurs |

σ | 0·3 | | Parameter in the lognormal distribution |

*G*_{0} | 0·002 | | Basic growth for 0+. *G*_{0} is zero for 1+ and 2+ |

*z*_{0} | 0·0125 | | Constant, scaled for 0+ to reach correct final size |

*z*_{1} | 0·00486 | | Constant, scaled for 1+ to reach correct final size |

*z*_{2} | 0·00322 | | Constant, scaled for 2+ to reach correct final size |

*f* | 30/20 | | Fecundity, eggs per female. Two scenarios |

*m* | 0·01 | mg | Newborn dry weight |

Predator parameters |

*ĉ* | 0·7/*a* | dim. less | Conversion coefficient, where a = 180 converts prey mass from mg dry weight to g wet weight |

*m*_{1} | 0·00005 | | Size-independent metabolic cost |

*m*_{2} | 0·85 | | Allometric scaling of metabolic cost |

*W*_{P} | | g | Predator mass, wet weight |

The second part of the model concerned the growth rate of the predator, which needed to be realistic to rely on the correction for possible bias in the experimentally derived attack rate. The growth model for the predator followed:

- ( eqn 10)

where ĉ is a conversion coefficient, *m*_{1} is the size-independent metabolic cost and *m*_{2} is the allometric scaling of metabolic cost. These parameters were estimated from Sparrevik & Leonardsson (1999); see Table 2. We obtained the predator growth curves by integrating eqn 10 over time, using prey densities and prey sizes from the consumption model (extended to cover 365 days). The smallest predators, i.e. less than 10 mm, are probably not efficient consumers of *M. affinis* due to these predators’ soft exoskeleton. For this reason extrapolations for the first year of growth would be too uncertain. Also the size at 1 year of age is not clear from the literature or from our data. Because of lack of knowledge of how large 1-year-old *S. entomon* are at these deep bottoms we chose to use two different 1-year-old predator sizes (9 mm and 15 mm) as starting points for the growth curves. Using these sizes at year 1 we predicted the size at year 1 by integrating eqn 10, and then repeatedly predicted the size at next year using *W*_{t} of previous year as a new starting point.