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Quantifying Predator Dependence in the Functional Response of 548 Generalist Predators

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Quantifying Predator Dependence in the Functional Response of 548 Generalist Predators 547 Quantifying predator dependence in the functional response of 548 generalist predators. 549 Supporting Information 550 Mark Novak, Christopher Wolf, Kyle Coblentz, Isaac Shepard 551 A. Intuition for the observational approach. 552 B. Derivation of observational approach. 553 C. Functional response parameter estimates. 554 D. Supplementary figures. 555 E. Data tables. 18 556 A Intuition for the observational approach 557 Type II prey dependence 558 Novak & Wootton (2008) introduced an observational method for estimating the prey- 559 specific per capita attack rates of a generalist predator population using estimates of prey 560 abundance (Ni), handling times (hi), and the number of feeding (ni)andnon-feeding 561 (n0) individuals observed during snapshot surveys of a focal predator population. Their th 562 analytical estimator for the attack rate on the i prey (ai) assumes a multispecies Holling 563 type II functional response, aiNi fi(N~ )= , (S1) 1+ k akhkNk 564 and is equivalent to P ni 1 aˆi = (S2) n0 hiNi 565 (Wolf et al.,2015).ThenextsectionoftheSOMprovidesasimplerderivationthanthat 566 used by Novak & Wootton (2008). 567 As described in the main text, intuition for the method may be built by using the 568 estimator to reformulate the type II functional response model in terms of the fraction 569 of predator individuals expected to be observed feeding at any given time. For example, 570 when the predator is a specialist feeding on only one prey species, n aˆ h N 1 = 1 1 1 , (S3) n0 + n1 1+ˆa1h1N1 571 which tends to 1 as a1, h1,orN1 increase. The fraction of individuals observed to be 572 feeding on a particular prey species during a snapshot survey will therefore increase the 573 higher the predator’s attack rate, the longer its handling time, or the more abundant the 574 prey species is (Fig. 1A). Assumptions implicit in applying the approach are discussed in 575 Novak & Wootton (2008). Suffice it to say, the approach has seen independent empirical 576 support (Novak, 2010; Yeakel et al.,2011)andhasbeensuccessfullyappliedincontexts 577 logistically inaccessible to experimentation (Novak, 2013). 578 Ratio dependence 579 A multi-prey extension to the ratio-dependent model of Arditi & Ginzburg (1989) is ↵iNi fi(N,P~ )= , (S4) P + k ↵khkNk 580 where P is the focal predator’s abundance (Abrams,P 1997; Arditi & Michalski, 1995). 581 The attack rate ↵i is often interpreted as the rate at which prey become available to the 582 predator, which di↵ers from the interpretation of ai (Abrams, 2015; Arditi & Ginzburg, 583 2012). In the next section of the SOM we show that the analytical estimator for this 584 parameter is ni P ↵ˆi = . (S5) n0 hiNi 585 When a specialist predator feeds on only one prey species, for example, the fraction of 586 predators that are expected to be feeding at any given time is n ↵ˆ h N 1 = 1 1 1 . (S6) n0 + n1 P + ↵1h1N1 19 587 How quickly the predator population’s feeding rate saturates thus depends on its own 588 abundance. Equivalently, the greater the predator’s abundance, the higher its attack rate 589 must be to maintain a constant feeding rate. 590 Predator dependence 591 The most commonly promoted predator-dependent functional response is the Beddington- 592 DeAngelis model, aiNi fi(N,P~ )= , (S7) 1+ k akhkNk + γP 593 where γ reflects the per capita strength of intraspecificP e↵ects among predators (Bedding- 594 ton, 1975; DeAngelis et al., 1975). Note that ai has the same interpretation as in the type 595 II model and that predators can conceivably have mutualistic e↵ects on their feeding rate 596 (γ<0). This single-predator functional response may be extended to consider multiple th 597 interacting predator species by describing the focal predator j’s feeding rate on the i prey 598 by aijNi fij(N,~ P~ )= , (S8) 1+ k akjhkjNk + p γjpPp 599 where γjp reflects the intra- or interspecificP e↵ect of predatorP species p on the focal predator 600 j’s feeding rate. In the next secton of the SOM we show how the observational framework 601 may be used to derive aˆ n 1 ij = i , (S9) 1+ p γˆjpPp n0 hiNi 602 relating the unknown terms of interest,P a ˆij andγ ˆjp,totheobservednumberoffeedingand 603 non-feeding individuals, the prey and predator abundances, and the handling times. Note 604 that in the absence of mutual predator e↵ects this equation reduces to the estimator fora ˆi 605 of the type II model (eqn. S2). Intuitively, the greater the number or per capita e↵ects of 606 interfering predators, the larger the per capita attack rate must be to maintain the same 607 proportion of feeding individuals. For a specialist predator, for example, the fraction of 608 individuals expected to be feeding at any point in time (Fig. 1B) is n aˆ h N 1 = 1 1 1 . (S10) n0 + n1 1+ˆa1h1N1 + p γˆjpPp 609 The simultaneous estimation ofa ˆij andγ ˆjp terms isP not possible with only one snapshot 610 survey. Rather, this requires surveys replicated in space or in time that di↵er in predator 611 densities. Specifically, we require at least one more survey than the number of predator 612 species. With such information thea ˆij andγ ˆjp terms may be statistically estimated for 613 any number of prey and predator species. 614 In fact, with a sufficient number of surveys, the unknown terms of many any other 615 functional response model may be estimated as well. This is made possible by noting that 616 for a generalist predator the fractions not feeding versus feeding on a particular prey species 617 correspond to the probabilities of a multinomial distribution, just as the fraction feeding 618 corresponds to the probability of a binomial distribution for a specialist predator. Here 619 we thereby also consider a multi-prey extension to the predator-dependent Hassell-Varley 620 style model of Arditi & Ak¸cakaya (1990): ~ ↵iNi fi(N,P)= m . (S11) P + k ↵khkNk 20 P 621 This model exponentiates the predator’s density of the ratio-dependent model (eqn. S4) 622 with parameter m to represent a degree of intraspecific interference among predators when- 623 ever m>0 (Hassell & Varley, 1969). Although m has no specific mechanistic interpre- 624 tation, this model reduces to the type II model when m =0andcorrespondstothe 625 ratio-dependent model when m =1. 626 Function-free feeding rates 627 The next section of the SOM describes the derivation of the last model that we include in 628 our analysis. This model is simpler than the above-described functional response models in 629 that it estimates prey-specific feeding rates across surveys without further describing these 630 by an assumed functional response model. Indeed, this density-independent “function- 631 free” model forms the basis of our statistical framework for estimating the parameters of 632 the density-dependent functional response models. 21 633 B Derivation of observational approach 634 Function-free feeding rates 635 Consider a generalist predator whose diet includes i =1,...,S di↵erent prey species but 636 that can feed on only one prey item at a time. Let i =0denotethestateofnotfeeding. 637 The predator’s feeding status at any point in time will follow a categorical distribution th 638 with probability pi of being in the i state. th 639 Let fi be the feeding rate of the predator on the i prey species and let di be the 640 detection time during which a feeding event is detectable to an observer. Over the course 641 of some time T the predator will, on average, consume fiT individuals of prey i.This 642 will amount to fiTdi total time that the predator will be observable feeding on prey i. 643 The proportion of time that the predator will be observable feeding on prey i will be fidi, 644 while the proportion of time it will be observable not feeding on any prey species will be S 645 1 fkdk. (Note that in principal this framework may be used even when only some − k=1 646 aspectP of the feeding process is detectable.) 647 It follows that if we observe n independent and equivalent predator individuals, the 648 numbers of individuals observed in each state (denoted by subscripts) at any given time 649 will reflect a multinomial distribution, S (n ,n ,...,n ) Mult (1 f d ,f d ,...,f d ). (S1) 0 1 S ⇠ n − k k 1 1 S S Xk=1 S 650 Noting that 0 fkdk 1, this permits us to use snapshot surveys of a predator pop- k=1 651 ulation to estimateP prey-specific feeding rates when detection times are known, regardless 652 of the predator’s true underlying functional response. That is, pˆi ni 1 fˆi = = , (S2) di n di 653 where the notationx ˆ denotes a sample estimate. Equation S2 corresponds to the ‘density- 654 independent (non-functional) model’ to which we refer in the main text. 655 Note that in the main text and henceforth we consider detection times (d)andhandling 656 times (h)tobyequivalentandthusinterchangeable.Forwhelks,forwhichhandlingtimes 657 range from hours to days (Novak, 2013), this is appropriate because the time it takes an 658 individual to chase and capture an encountered prey item is minuscule compared to the time 659 it takes to handle the prey item, and because handling entails both the drilling and digestion 660 of the prey item such that the post-handling digestion time is also minuscule (Novak, 2008). 661 Extensions to predators feeding on multiple prey simultaneously, as needed for the use of 662 gut contents analyses, are also possible with additional assumptions (unpubl.
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