How Predator Functional Responses and Allee Effects in Prey Affect the Paradox of Enrichment and Population Collapses

ARTICLE IN PRESS

Theoretical Population Biology 72 (2007) 136–147 www.elsevier.com/locate/tpb

How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses

David S. Boukala,b,Ã, Maurice W. Sabelisc, Ludeˇk Bereca

aDepartment of Theoretical Ecology, Institute of Entomology, Biology Centre, Academy of Sciences of the Czech Republic, Branisˇovska´ 31, 37005 Cˇeske´ Budeˇjovice, Czech Republic bInstitute of Marine Research, Postboks 1870 Nordnes, 5817 Bergen, Norway cPopulation Biology Section, Institute of Biodiversity and Ecosystem Dynamics, University of Amsterdam, NL-1090 GB Amsterdam, the Netherlands

Received 4 October 2006 Available online 19 December 2006

Abstract

In Rosenzweig–MacArthur models of predator–prey dynamics, Allee effects in prey usually destabilize interior equilibria and can suppress or enhance limit cycles typical of the paradox of enrichment. We re-evaluate these conclusions through a complete classification of a wide range of Allee effects in prey and predator’s functional response shapes. We show that abrupt and deterministic system collapses not preceded by fluctuating predator–prey dynamics occur for sufficiently steep type III functional responses and strong Allee effects (with unstable lower equilibrium in prey dynamics). This phenomenon arises as type III functional responses greatly reduce cyclic dynamics and strong Allee effects promote deterministic collapses. These collapses occur with decreasing predator mortality and/or increasing susceptibility of the prey to fall below the threshold Allee density (e.g. due to increased carrying capacity or the Allee threshold itself). On the other hand, weak Allee effects (without unstable equilibrium in prey dynamics) enlarge the range of carrying capacities for which the cycles occur if predators exhibit decelerating functional responses. We discuss the results in the light of conservation strategies, eradication of alien species, and successful introduction of biocontrol agents. r 2007 Elsevier Inc. All rights reserved.

Keywords: Allee effect; Functional response; Predator–prey model; Stability; Bifurcation analysis; Population cycles; Conservation; Pest management

1. Introduction passed theoretical and empirical scrutiny: sigmoidal or other similar non-linear functional responses of the Predator–prey models of the Rosenzweig–MacArthur predators (Murdoch, 1969; Oaten and Murdoch, 1975; type (i.e. including logistic growth of the prey and a type II Hassell and Comins, 1978; Nunney, 1980; Abrams and functional and numerical response of the predator) predict Roth, 1994; Sugie et al., 1996; Collings, 1997; Oksanen destabilization of interior equilibria and the emergence of et al., 2001; Gross et al., 2004), mutual interference stable limit cycles with increasing carrying capacity of the between predators (Ruxton et al., 1992), hiding of prey environment (Rosenzweig, 1971; Gilpin, 1972). This (Ruxton, 1995), induction of defenses (Vos et al., 2004; phenomenon, known as the paradox of enrichment, did Verschoor et al., 2004), limiting nutrients and more not show up in empirical tests with aquatic predator–prey generally the prey quality (Sommer, 1992; Loladze et al., systems (Murdoch et al., 1998; McCauley et al., 1999). 2000; Andersen et al., 2004), heterogeneity in the prey Several explanations of this apparent contradiction have population with respect to edibility (Abrams and Walters, 1996; Genkai-Kato and Yamamura, 1999; Bohannan and ÃCorresponding author. Department of Theoretical Ecology, Institute Lenski, 1999; Persson et al., 2001), and patterned distribu- of Entomology, Biology Centre, Academy of Sciences of the Czech tion in space (Scheffer and De Boer, 1995; Nisbet et al., Republic, Branisˇovska´31, 37005 Cˇeske´Budeˇjovice, Czech Republic. Fax: +420 385310354. 1998; Holyoak, 2000; Jansen and de Roos, 2000). E-mail addresses: [email protected] (D.S. Boukal), Another mechanism that can prevent predator–prey [email protected] (M.W. Sabelis), [email protected] (L. Berec). systems from exhibiting sustained cycles is a (demographic)

Allee effect, i.e. positive density dependence in prey 2. Model population growth at low prey densities (Stephens et al., 1999). Strong Allee effects—with negative population We base the analysis on a simple model of a predator– growth at low densities—lead to extinction if the popula- prey interaction of the type tion falls below a threshold size or density. If troughs in dx otherwise plausible prey cycles extend below the Allee ¼ gðxÞFðxÞy, dt threshold density, both prey and predators go extinct. This mechanism has already been shown to lead to a reduced dy ¼my þ eFðxÞy, (1) potential for cycles and an increased propensity for system dt collapse (Courchamp et al., 2000; Kent et al., 2003; Webb, in which x and y are the prey and predator densities, gðxÞ is 2003; Zhou et al., 2005). the density-dependent prey growth rate, m is the predator The Allee effect in prey may be caused by predation or per-capita mortality rate, e is the predator’s food-to- by processes inherent to the prey life history (Dennis, 1989; offspring conversion efficiency, and FðxÞ is a laissez-faire Sinclair et al., 1998; Stephens et al., 1999; Boukal and type of predator functional response, assuming no inter- Berec, 2002; Liermann and Hilborn, 2001; Gascoigne and ference among individual predators. We use a general Lipcius, 2004). Empirical evidence for Allee effects in representation of the functional response single-species animal and plant populations is widespread (Lamont et al., 1993; Hopper and Roush, 1993; Groom, lxn FðxÞ¼ (2) 1998; Kuussaari et al., 1998; Hackney and McGraw, 2001; 1 þ lhxn Morris, 2002; Liebhold and Bascompte, 2003). Some of introduced by Real (1977). Parameter h can be interpreted these observations (e.g. Hackney and McGraw, 2001)in as the handling time, and l and n as scalings of the fact point to a weak rather than a strong Allee effect, such predator–prey encounter rate. The exponent n describes the that the prey per-capita population growth rate is reduced shape of the functional response, yielding decelerating at low densities but remains positive. responses for 0onp1, including the Holling type II Considerable attention has been given to the role of response for n ¼ 1, and Holling type III (sigmoidal) human exploitation in promoting collapse of animal responses for n41. For n ! 0, Eq. (2) approaches a populations subject to strong Allee effects (e.g. Rowe et al., density-independent functional response regarded as a 2004; Hutchings and Reynolds, 2004) as well as the role of good approximation for certain predatory mites (van predators (Sinclair et al., 1998; Courchamp and Macdo- Baalen and Sabelis, 1995). A plausible mechanistic nald, 2001; Gascoigne and Lipcius, 2004; Mooring et al., explanation for integer values of n involves the Holling 2004; Sarnelle and Knapp, 2004). However, theoretical disk equation with the prey encounter rate equal to lxn,in investigations of predator–prey dynamics have only dealt which n can be related to the number of encounters a with a limited set of predator functional responses and predator must have with the prey before becoming strong Allee effects (references above). Given that preda- maximally efficient (Real, 1977). For 0onp1, per-capita tion is a general ecological mechanism, a more thorough predation rates of the prey are higher at low prey densities, analysis is needed, especially because insights in the role of i.e. predation alone leads to a component Allee effect in the predation and Allee effects for ecosystem persistence may sense of Stephens et al. (1999) if the feedback through well be important for population management and species changing predator densities is disregarded. conservation (Sinclair et al., 1998; Courchamp et al., 1999; We further assume that the prey exhibits a demographic Gascoigne and Lipcius, 2004). Allee effect at low population densities due to reasons In this article, we examine how system collapse and limit other than predation by the focal predator. The equation cycles, characteristic of the paradox of enrichment, are describing the prey dynamics can be interpreted, for influenced by Allee effects inherent to prey life history on example, as a heuristic approximation of a two-sex the one hand and the (constant, decelerating or sigmoidal) population dynamics where the differences between the shape of the predator’s functional response on the other. In sexes are not explicitly modelled (Dennis, 1989), or as an particular, we ask to what extent these mechanisms reduce impact of another, generalist predator with a Holling type the propensity to exhibit sustained cycles with increasing II functional response (Gascoigne and Lipcius, 2004). prey carrying capacity (or decreased predator mortality) in Alternatively, it can be seen as an approximation of group the framework of a Rosenzweig–MacArthur predator–prey dynamics under obligate cooperation (Courchamp et al., model. This is done by bifurcation analysis of this model, 2000). extended to include a continuum of functional response Given these assumptions, Eq. (1) becomes shapes and a wider variety of Allee effects than usually dx x A þ c lxn considered in other model exercises. This type of analysis ¼ rx 1 1 y, has not been carried out before. As we will show, it is dt K x þ c 1 þ lhxn critically important to distinguish between weak and strong dy lxn Allee effects and different types of predator functional ¼my þ y, (3) responses. dt 1 þ lhxn ARTICLE IN PRESS 138 D.S. Boukal et al. / Theoretical Population Biology 72 (2007) 136–147 where r scales the prey growth rate, K is its intrinsic 3. Results carrying capacity, A is the Allee threshold, and c is an auxiliary parameter (c40 and AX c, Boukal and Berec, The standard Rosenzweig–MacArthur model exhibits 2002). The auxiliary parameter c affects the overall shape three types of behavior (Krˇivan, 1996): predator extinction of the per-capita growth curve of the prey. As c increases, for Kpm=lð1 hmÞ, which also follows from (7) with n ¼ the curve becomes increasingly ‘‘flatter’’ and reaches lower 1 (see Appendix B); stationary coexistence at E% for maximum values (Fig. 1). Predator’s conversion efficiency e m=lð1 hmÞoKpð1 þ hmÞ=lhð1 hmÞ; and a stable limit has been scaled away from Eq. (3) by a simple linear cycle surrounding an unstable equilibrium E% for K4 transformation (t~ ¼ et, y~ ¼ y=e, r~ ¼ r=e, m~ ¼ m=e, tildes ð1 þ hmÞ=lhð1 hmÞ. dropped). Only a part of the functional response eventually plays a role in the dynamics given that viable prey 3.1. Local stability analysis populations become constrained between A and K; the shape of the functional response in that range is discussed We first extend this result by deriving general stability in more detail in Appendix A. conditions for the interior equilibrium E%. Table 1 lists all The Allee effect is strong if A40: the prey exhibits two possible configurations of the predator and prey isoclines non-zero equilibria in the absence of predators and goes and corresponding dynamics. Given the laissez-faire type extinct below the Allee threshold (Fig. 1). For Ao0, the of the functional response and the lack of any other Allee effect is weak: the lower unstable equilibrium density-dependent processes in the predator population, disappears and the origin becomes unstable. For A4 the predator isocline in Eq. (3) is a vertical line at the cK=ðc þ KÞ the prey population exhibits a demographic equilibrium prey density x%. The prey isocline is a hump- Allee effect in the sense of Stephens et al. (1999), i.e. the shaped function of the prey density, positive only for x 2 prey per-capita growth rate is positively density dependent ðA; KÞ (for x 2ð0; KÞ if Ao0). It has one or two (local) at low densities. For smaller A, the prey per-capita growth maxima and zero or one local minimum in that interval. rate becomes negatively density dependent for all popula- The determinant of the Jacobian matrix associated with (3) tion densities. Eq. (3) is the standard Rosenzweig–Ma- is always positive in E% (Appendix C), and E% is therefore cArthur model for the limiting case A ¼c and n ¼ 1. stable if the prey isocline has a negative slope at E % and Eq. (3) has three boundary equilibria: E0 ¼ð0; 0Þ, Eu ¼ unstable if the slope is positive. % ðA; 0Þ and Es ¼ðK; 0Þ, of which E0 and Es are locally stable If the prey isocline has a single maximum, E can change and Eu locally unstable in the prey-only subspace. For a stability only once from stable to unstable as K increases or finite range of the predator mortality rate m (Appendix B), m decreases. If the isocline has two maxima and one Eq. (3) has a unique interior equilibrium minimum, E% can change stability three times (stable ! unstable ! stable ! unstable as K increases or m decreases). 1 % % % 1 gðF ðmÞÞ The prey isocline may also have one maximum and one E ¼ðx ; y Þ¼ F ðmÞ; (4) % m minimum if the Allee effect is weak or absent, such that E changes from stable to unstable to stable. Moreover, the in which the prey equilibrium density equals maximum (or both maxima) of the prey isocline shifts to higher prey densities as the strength of the Allee effect given m 1=n by A increases while the vertical predator isocline remains in x% ¼ F 1ðmÞ¼ . (5) lð1 hmÞ the same position, thereby enlarging the interval of K in which the interior equilibrium E% can become unstable. E% thus requires hmo1 to exist. Inequalities (11)–(14) derived in Appendix C provide general analytic conditions for the local stability of E %. For a prey with logistic growth (A ¼c), the paradox of enrichment can occur if and only if noð1 hmÞ1 and thus becomes less feasible (e.g. the range of m leading to the ) ) x 0 x 0 ( strong Allee effect (A> ) ( weak Allee effect (A< ) paradox of enrichment decreases) when the functional g g

1 1 response becomes more sigmoidal (exponent n increases). 0.75 0.75 Moreover, the interior equilibrium E% remains stable for 0.5 0.5 0.25 0.25 any K and strong Allee effects with small but positive A if c

2 4 6 8 10 12 14 2 4 6 8 10 12 14 and m are sufﬁciently small and n is sufﬁciently large − 0.25 − 0.25 − 0.5 − 0.5 (compare the predator with Holling type III functional prey population density, x prey population density, x response in the right panel of Fig. 3B). prey per-capita growth rate, per-capita growth prey prey per-capita growth rate, per-capita growth prey

Fig. 1. Per-capita growth rate of the prey in (3) in the absence of 3.2. Bifurcation analysis predators. Parameter values: r ¼ 1, K ¼ 10. Left panel: A ¼ 0:5, c ¼ 0:1 (thick curve); A ¼ 0:5, c ¼ 5 (thick dashed curve). Right panel: A ¼0:05, c ¼ 0:1 (thick curve); A ¼1, c ¼ 5 (thick dashed curve). To obtain a complete classiﬁcation of the qualitative Thin line in both panels indicates the limiting case of logistic growth. behavior of model (3), we analyze the bifurcation pattern in ARTICLE IN PRESS D.S. Boukal et al. / Theoretical Population Biology 72 (2007) 136–147 139

Table 1 Five major types of isocline conﬁguration and corresponding dynamics of model (3)

Isoclines Dynamics (areas in Fig. 3)

Case 1 Stable coexistence/cycles (Stable Equilibrium, Locally Stable Equilibrium, Paradox of Enrichment Cycles, Stable Equilibrium/PoE Cycles)

Case 2 Prey-only; predator cannot establish (Prey Only)

Case 3 Collapse (Extinction 2)

Case 4 Collapse or stable coexistence/cycles (Extinction 1, Stable Equilibrium, Locally Stable Equilibrium, Paradox of Enrichment Cycles, Stable Equilibrium/PoE Cycles)

Case 5 Collapse or prey-only; predator cannot establish (Prey Only)

Dotted curve ¼ prey isocline (diagrammatic; the isocline can have one or two peaks; equilibrium E% can be to the left or right of either isocline peak); solid line ¼ predator isocline. Equilibria (see text for notation): black ¼ always stable; transparent ¼ always unstable; gray ¼ stable or unstable.

the XPPAUT package (Ermentrout, 2002), and illustrate bifurcation points tracking a heteroclinic orbit leading the results with two values of c. The classiﬁcation requires from the prey-only stable equilibrium Es to the prey-only up to ﬁve codimension-one bifurcations: branching points unstable equilibrium Eu. There are up to six bifurcation corresponding to predator invasion, ‘‘simple extinction’’ branches for any given two-parametric projection: one branching points corresponding to the (dis)appearance of heteroclinic branch, two branches of Hopf bifurcation the interior equilibrium E% and unconditional collapse of points given by (12), one branch of predator invasion the predator–prey system, Hopf bifurcation points in points given by (7), one branch of limit points of cycles, which E % exchanges stability, limit points of cycles in and one branch of simple extinction points given by (8), which a stable and an unstable cycle collide, and all with inequalities replaced by equalities (Appendix B). ARTICLE IN PRESS 140 D.S. Boukal et al. / Theoretical Population Biology 72 (2007) 136–147

The simple extinction branch is absent if Ap0, the small quantity into the prey-only equilibrium Es; Cases 2 heteroclinic branch is absent if A40, and the limit points and 5). In the domain of Locally Stable Equilibrium, initial of cycles do not occur if K, c or n are sufficiently low. conditions inside an unstable limit cycle lead to the locally Four of the six bifurcation branches are rooted in a stable equilibrium E% while those outside the cycle result in single point mr ¼ l=ð1 þ lhÞ as n ! 0 in the projection to ever-increasing oscillations and, if A40, in a deterministic the ðm; nÞ parameter plane (Figs. 2 and 3A), and in the collapse (Cases 1 and 4). A small inoculum of predators point Ar ¼ Kr with mr (Fig. 3B) or Kr (Fig. 3C) given by introduced into the prey-only equilibrium Es thus drives (7) in projections with n40 (see Appendix D for more the system to collapse if A40. In domains denoted as details). One of the Hopf branches (as well as the branch of Paradox of Enrichment Cycles (PoE Cycles) and Stable limit points of cycles) does not reach the common point. It Equilibrium, both populations coexist after introducing a is the branch approaching m ¼ 0asn !1if both K and c small inoculum of predators (Cases 1 and 4). In Stable are large enough (left panel in Fig. 3A), and the branch Equilibrium domain, population densities reach the equili- approaching m ¼ 1=h as n !1 if K or c is small (right brium E% while populations in the PoE Cycles domain panel in Fig. 3A). reach the well-known cycles of the paradox of enrichment The branches delimit seven major parameter domains in (Rosenzweig, 1971; Gilpin, 1972). Equilibrium E% can be which the interior equilibrium exists or does not exist, is stable and surrounded by an unstable and a stable cycle if stable or unstable, and is or is not surrounded by a stable K, c and n are all sufficiently large (Stable Equilibrium/PoE or an unstable limit cycle or by both. We show all these Cycles domain, Cases 1 and 4). domains in Fig. 3A, which focuses on the predator’s life Overall, predators with type III functional responses and history—its mortality rate m and the shape of the sufficiently low mortality rates induce cycles for a much functional response embodied in the exponent n. The other smaller range of Allee effects and prey carrying capacities four panels in Fig. 3 target the prey life history (K and A, as compared to the Holling type II response (dashed vs. Fig. 3B) and an interplay of the Allee effect in prey and solid lines in Fig. 3B and C). Appendix D further shows predator mortality (A and m, Fig. 3C). and Fig. 3A illustrates that the Hopf point branch (thick In Extinction 1 and 2 domains, the system collapses for black curve) rooted in the common point (mr; 0) in the any initial conditions that involve predators. There is no ðm; nÞ plane ‘‘moves to the left’’ with decreasing c. interior equilibrium in Extinction 2 (Case 3 in Table 1), However, the heteroclinic branch (thick gray curve) while an unstable source exists in Extinction 1 (Case 4). In changes less, thereby leading to a much smaller PoE Prey-Only domain, predators cannot establish (i.e. cannot domain as c decreases. For c ¼ 0:1 (Fig. 3A, right panel), initially increase in density after being introduced in a very the PoE domain for no2 almost disappears; recall that c

A B 10

8 15

6 Prey Only n n 10 Internal Equilibrium 4

5 2 Internal Equilibrium Prey Only Extinction 2 0 0 Extinction 2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.1 0.2 m m C

Extinction 2

n 10

5 Internal Equilibrium

Prey Only 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 m

Fig. 2. Three generic cases showing the region of existence of the interior equilibrium E% in relation to predator mortality (m) and shape of the functional response (n). Solid and dotted curves given by inequalities (7) and (8) in Appendix B, respectively. ARTICLE IN PRESS D.S. Boukal et al. / Theoretical Population Biology 72 (2007) 136–147 141

A c = 5 c = 0.1

Stable Equilibrium 8 Stable Equilibrium

6 n Paradox of Enrichment Locally Stable PoE Cycles Cycles Stable EquilibriumParadox / of Enrichment 4 Equilibrium Cycles Locally Stable Equilibrium Extinction 2 2 Extinction 1 Stable Eq. Extinction 2 Prey Only Extinction 1 Prey Only 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 m

B c = 5 c = 0.1

Locally Stable Extinction 1+2 Equilibrium (n=2)

10 Paradox of Enrichment Cycles collapse K

collapse no dAE 5 Stable Equilibrium infeasible areas Prey Only 0 -5 -4 -3 -2 -1 0 1 2 -4 -3 -2 -1 0 1 2 A

C c = 5 c = 0.1 10

8 Extinction 1+2 6

4 Locally Stable Equilibrium (n=2) A 2 collapse

0 PoE Cycles Prey Only -2 Stable Equilibrium infeasible area -4 no dAE

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 m

Fig. 3. Bifurcation diagrams of model (3) for r ¼ 1, l ¼ 0:25, h ¼ 0:6. Left column: c ¼ 5, right column: c ¼ 0:1. Thick arrows illustrate possible routes to a deterministic collapse of the system. Thick gray solid/dashed curves ¼ branches of heteroclinic orbits; thick black solid/dashed curves ¼ branches of Hopf points given by (12); thin black solid/dashed curves ¼ predator invasion curves given by (7); dot-dashed curve ¼ branch of limit points of cycles; dotted curve ¼ simple extinction curve given by (8), omitted in (B) and (C). (A) K ¼ 10, A ¼ 0:5. (B) m ¼ 0:365 (left panel) and m ¼ 0:331 (right panel). Dark gray areas ¼ infeasible areas (AXK or Apc); light gray areas ¼ no demographic Allee effect (no dAE). Solid curves ¼ Holling type II functional response (n ¼ 1), dashed curves ¼ Holling type III functional response (n ¼ 2). (C) K ¼ 10. Lines and areas as in (B); area of no demographic Allee effect negligible for c ¼ 0:1. characterizes the overall shape of the prey per-capita The analysis highlights two major routes by which growth rate (Fig. 1). In more extreme cases, illustrated by paradox of enrichment and population collapses can arise. c ¼ 0:1 and n ¼ 2, the position of the Hopf point branch Both routes require a decline in the predator mortality and the heteroclinic branch reverse for a wide range of rate or increase in the carrying capacity of the prey other parameter values such that sustained cycles become (Rosenzweig, 1971; Gilpin, 1972) or increase in the severity impossible (dashed lines in right panels of Fig. 3B and C). of the Allee effect (arrows in Fig. 3). We illustrate them by ARTICLE IN PRESS 142 D.S. Boukal et al. / Theoretical Population Biology 72 (2007) 136–147

stability but becomes surrounded by an unstable limit cycle, such that trajectories starting close to the Prey-Only % 4 equilibrium Es no longer approach E but the origin. At sufﬁciently low predator mortality rates, the unstable route to collapse limit cycle disappears in a subcritical Hopf bifurcation 3 before the system collapses (Fig. 4B). These differences arise from changes in the mutual position of the Hopf and prey 2 heteroclinic bifurcation curves (Fig. 3A).

4. Discussion 1 predator and prey density Our analysis re-emphasizes the original idea of Rosenz- predator weig (1971) that enrichment of the environment can lead to 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 a collapse of the entire predator–prey system via increasing m oscillations. Rosenzweig considered the extinction in a stochastic sense. More recent studies showed that the extinction becomes more acute, i.e. deterministic and abrupt, if the prey exhibits a strong Allee effect (Courch- 4 amp et al., 2000; Kent et al., 2003; Webb, 2003; Zhou et al., 2005). When conditions are met for the predator–prey route to collapse system to cycle, the prey population in the trough of the 3 cycle is not only subject to stochasticity but also to a strong Allee effect in growth, and this makes both populations prone to extinction. Most likely, any other mechanism 2 prey causing the populations to cycle would lead to the same outcome. 1 In this paper, we analyzed a wide range of Allee effects predator and prey density and predator functional responses to see how they affect predator the propensity of predator–prey systems to oscillate and/or 0 collapse. Our results thus provide a full classification of 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 m possible dynamics and in turn provide several important messages for applications. Fig. 4. Bifurcation diagrams showing two different routes to collapse Overall, we found that although the exact shape of under strong Allee effects. Dashed lines ¼ prey densities; solid lines ¼ functional response is of limited importance to the global ¼ predator densities; thick and thin lines stable and unstable equilibria/ bifurcation patterns—in agreement with Collings (1997) cycles, respectively; horizontal lines ¼ prey-only equilibria; limit cycles shown as maximum and minimum values; equilibrium E0 (locally or who analyzed a predator–prey system of the Leslie–Gower globally stable for all m) not shown. (A) First route via supercritical type—, its sigmoidal property does matter. For strong bifurcation of the interior equilibrium and subsequent collapse of the Allee effects (A40), increasingly more sigmoidal func- stable limit cycle. Model (3) with r ¼ 1, K ¼ 10, A ¼ 0:5, c ¼ 5, l ¼ 0:25, tional response can enlarge the area in which the interior ¼ ¼ h 0:6, and n 2. (B) Second route via subcritical bifurcation, i.e. equilibrium exists (Fig. 2A), compromise the predator’s disappearance of the unstable limit cycle surrounding the interior equilibrium. Same parameter values except n ¼ 3. ability to establish (Fig. 2B), or strengthen its ability to drive the system to a collapse (Fig. 2C). The three different qualitative impacts are related to which part of the a decreasing predator mortality rate (Fig. 4). The behavior functional response (accelerating vs. decelerating) is high- of the system is initially the same as in the absence of the lighted by the ‘‘focal’’ range of prey densities (A; K). (strong) Allee effect: predators can establish once their mortality rate decreases below a threshold (Rosenzweig, 4.1. System collapses: the importance of functional response 1971; Gilpin, 1972). However, the system eventually collapses at low but positive predator mortality rates if The ability of predators to locate prey at low prey the Allee effect is strong. In the first route for low densities is crucial for system collapses and the paradox of exponents n, equilibrium E% loses stability via a super- enrichment in Rosenzweig–MacArthur models. Type III critical Hopf bifurcation and, before the collapse via a functional responses can create a temporary refuge for the heteroclinic bifurcation, gives rise to stable limit cycles as prey when at low densities and stabilize the interior in the ‘‘standard’’ case of the paradox of enrichment predator–prey equilibrium (e.g. Oaten and Murdoch, (Fig. 4A). This route was included by Webb (2003) among 1975). Other mechanisms, such as the existence of small possible scenarios for evolutionary suicide. In the other spatial refuges for the prey, can have the same effect. The route for sufficiently high n, equilibrium E% retains its severity of the (strong) Allee effect and predation pressure ARTICLE IN PRESS D.S. Boukal et al. / Theoretical Population Biology 72 (2007) 136–147 143 then determine whether the system collapses (Fig. 3C). Hutchings and Reynolds, 2004), the African wild dog Even for potentially high predation pressure, e.g. due to (Courchamp and Macdonald, 2001; Courchamp et al., very low predator mortality, the prey may avoid determi- 2002), and the desert bighorn sheep (Mooring et al., 2004). nistic extinction if the refuge range extends above the Allee One possible counter-measure to save small groups of an threshold (high n in Fig. 3A). endangered species is to cull its predators (Sinclair et al., As a consequence of this stabilization property, the 1998; Courchamp et al., 2002; Mooring et al., 2004). This is collapse of the predator–prey system comes unannounced where our model can have an important message: this for a broad range of type III functional responses. The activity should take into account predators’ efficiencies to impending extinction is only signalled by decreasing prey locate and/or attack prey at densities close to the Allee equilibrium densities and can occur well above the Allee threshold. Culling predators with negligible predation threshold (Fig. 4B). This type of dynamics thus does not pressure at such low densities, e.g. due to a type III yield any qualitative early warning signals—such as the functional response or the existence of prey refuges, may paradox of enrichment—that could be used e.g. in (initially) have little effect on the prey growth (Fig. 5A). On conservation strategies. This phenomenon has not been the other hand, culling predators with type II functional mentioned in previous models focused on the role of response but otherwise identical predator and prey life the Allee effect in predator–prey interactions (Courchamp histories might save the prey from extinction (Fig. 5B). Our et al., 2000; Kent et al., 2003; Webb, 2003; Zhou et al., results emphasize the need for this kind of detailed 2005). information on predator functional responses (Sinclair et al., 1998). Obviously, other direct and indirect food web 4.2. Paradox of enrichment: the importance of weak Allee interactions, such as multiple predators sharing the prey, effects should be carefully examined in studies aiming at specific predator–prey systems. Counterintuitively, we have also found that model (3) Second, invasions of alien species (exotics or genetically can actually exhibit a higher propensity towards the manipulated organisms) represent a risk of considerable paradox of enrichment under the Allee effect. This is economic concern. Appropriate measures can reduce the another new result not covered by previous theoretical risk of such invasions but cannot fully exclude them. Thus, studies on the subject. The populations can cycle for a we need to develop strategies to combat successful larger range of prey carrying capacities and predator invaders. Liebhold and Bascompte (2003) make the mortality rates when the Allee effect is weak and predators important point that contrary to the commonly held belief, have type II or weakly sigmoidal functional responses eradication can be achieved by less than 100% removal of (Figs. 3B and C). The paradox of enrichment is promoted the alien species if it suffers from a strong Allee effect. by flat-shaped prey per-capita growth rates (e.g. by Using a historical data set of the invasion dynamics of increasing parameter c, Fig. 1). We are not aware of any small isolated populations of the gypsy moth, they showed experimental or observational data that would allow that 80% removal can be sufficient in small populations. assessing the impact of the Allee effect on the overall Such mortality levels are still high but may be achieved shape of the population growth rate. Our analyses thereby environmentally safe pesticides or by the collective fore highlight the need to consider such phenomena. action of introduced or natural enemies (Dwyer et al., 2004; Ferguson et al., 1994; Elkinton et al., 2004). In the 4.3. Perspectives for species conservation, pest management, example in Fig. 4B, predators with mortality rate below and biological control m0:18 reduce the equilibrium prey density by 80% and the Allee effect subsequently secures full eradication of The key assumptions and components of model (3) have the pest. received some empirical support and it is therefore worth To drive pest populations subject to a strong Allee effect to point out potential consequences of our theoretical to extinction, natural enemies should be released when pest results for population processes in more detail. Below, we population density is still sufficiently low or they should be consider three areas of applied population biology to efficient and/or abundant enough to bring the pest illustrate what role the insights gleaned from our analysis population below the Allee threshold. If the pest popula- may play in population management. tion is well above the threshold and the enemies are rare, a First, commercial exploitation or habitat loss may bring pest outbreak may be unavoidable before the pest drops populations at and beyond the brink of extinction. Many below the threshold density and collapses. Analysis of proximate mechanisms might be involved in the extinction model (3) stresses that in any case the natural enemies of small populations (inbreeding, demographic stochasti- should be efficient at low pest densities. This excludes some city, etc.) but Allee effects are at least suspected to rank predators and parasitoids with steep type III functional high on the list. Examples of formerly abundant popula- responses, i.e. high n, which cannot reach the pest in some tion, most likely suffering from a strong Allee effect at low refuges. On the other hand, any pest life history (i.e. r, K, densities, include the Atlantic cod and top predatory fish in A, and c) is associated with an ‘‘optimal’’ shape of the general (de Roos and Persson, 2002; Rowe et al., 2004; functional response (exponent n between 1and2 in our ARTICLE IN PRESS 144 D.S. Boukal et al. / Theoretical Population Biology 72 (2007) 136–147

A 7 B 10 6 predators culled 8 5

4 predators not culled 6 predators culled

3 4 prey density prey density 2 2 1 predators not culled 0 0 0 1020304050 0204060 time time

C D 12 10 no AE in natural enemy 10 weak AE in natural enemy 8 weak AE in pest no AE in pest 8 6 6

pest density 4 4 natural enemy density 2 2

0 0 0 20406080 0 20406080 time time

Fig. 5. Population dynamics of the prey in model (3). (A) Recovery after a sudden population crash from equilibrium (t ¼ 10) when their only predator with type III functional response is culled at t ¼ 10 to 33% of its population size (thick line) or not (thin line). Allee threshold indicated by a dashed line. (B) Same as (A) but with type II functional response of the predator. (C) The effect of introducing a small number of predators when the pest ( ¼ prey) exhibits no Allee effect (thin line) or a weak Allee effect (thick line). (D) The effect of top predators on a released natural enemy ( ¼ prey) with no Allee effect (thin line) and with a weak Allee effect (thick line). Parameter values: r ¼ 1, K ¼ 10, l ¼ 0:25, h ¼ 0:6 ((A)–(D)); A ¼ 0:5, c ¼ 0:1, m ¼ 0:8, n ¼ 5 (A); A ¼ 0:5, c ¼ 0:1, m ¼ 0:8, n ¼ 1 (B); A ¼5 (thin line) and 0:5 (thick line), c ¼ 5, m ¼ 0:7, n ¼ 2, xð0Þ¼10, yð0Þ¼0:01 (C); A ¼1 (thin line) and 0:1 (thick line), c ¼ 1, m ¼ 0:8, n ¼ 2, xð0Þ¼0:05, yð0Þ¼0:01 (D). analyses) for which the range of predator mortalities they should have a strong functional and numerical leading to pest eradication is largest (Fig. 3A). response to increasing pest densities. Such inoculative The analysis further shows that pest populations can releases are definitely not always successful. Among the exhibit higher propensity for sustained cycles under a weak many and varied explanations, Hopper and Roush (1993) Allee effect when exposed to enemies with an appropriate proposed—after analyzing a large data set on biocontrol functional response and mortality rate (Fig. 5C). In the attempts—that failures may be due to poor mate finding troughs of these cycles, the pest population grows at chances among the enemies released. Although in need of substantially reduced rates but the peaks of the cycles may further investigation, it is a fact that biocontrol workers actually represent repeated pest outbreaks. Thus, inducing repeatedly rank the possibility of the mate-finding Allee weak Allee effects, e.g. through partial disruption of effect high on their list of possible causes for failure of mating opportunities, along with the introduction of inoculative enemy releases (Hopper and Roush, 1993; natural enemies may give rise to a desirable reduction in M.W. Sabelis, unpublished data). An awareness is also pest abundance in the short run but not prevent outbreaks growing that the enemies released do not only interact with in the long run. the target pest but become an integral part of the local Third, successful biological control does not only depend ecosystem including intraguild predators and hyperpreda- on the traits of the natural enemies but also on the tors. quantities released. In some cases, mass rearing is so Thus, we take here our results one step further and efficient that large numbers can be introduced in a given consider that the released predator becomes a potential area (inundative releases) and for their short-term impact, prey of another top predator, and drop the dynamics of the it is the functional (not the numerical) response that basal prey. For example, failures to control waterlettuce by matters. However, mass rearing is usually not so efficient caterpillars of the moth Spodoptera pectinicornis in Florida or is not (economically) feasible. In that case, the enemies (USA) were attributed to boat-tailed grackles, fire ants, are released in small numbers (inoculative releases) and and spiders (Dray et al., 2001). The latter may predate ARTICLE IN PRESS D.S. Boukal et al. / Theoretical Population Biology 72 (2007) 136–147 145 upon the enemy inoculum much like we have supposed in to accelerating as exponent n increases. However, reversals model (3). The released enemies may be driven to in that sequence may occur if lho1 (details not shown). extinction by their enemies, e.g. due to strong Allee effects, or remain rare and little effective initially, yet become more Appendix B. Necessary and sufficient conditions for the effective later in time. In view of the latter phenomenon, existence of the interior predator–prey equilibrium E% biocontrol workers often decide to look for alternative enemies or suppose that the natural enemy needs to We start with the particularly simple case of n ¼ 0, such gradually adapt to its new environment (M.W. Sabelis, that predator’s functional response becomes independent unpublished data). However, analysis of model (3) hints at of the prey density. Predators can establish if mol= an alternative explanation: the change in effectiveness can ð1 þ lhÞ, and they subsequently drive the entire system to be due to a weak Allee effect in the growth rate of the collapse (i.e. both predators and prey die out determinis- released enemy (Fig. 5D). A tri-trophic model is needed to tically). If the opposite inequality holds, predators go investigate fully this alternative explanation but is beyond extinct and the prey population saturates in the boundary the scope of this paper. Such a model could also equilibrium Es unless A40 and the predators are intro- incorporate Allee-like phenomena which arise through duced in such large numbers that they also drive the prey nutritional constraints for herbivores in autotroph–herbi- below the Allee threshold before both go extinct. vore–carnivore systems (Loladze et al., 2000; Andersen More generally, predators cannot establish if and only if et al., 2004); we have not explicitly considered such lKn mechanisms here. m4m ¼ . (7) max 1 þ lhKn As a final take-home message, we would like to emphasize that such alternative explanations are the prime asset On the other hand, the entire system certainly collapses if of our and similar model exercises. For this reason, feed- predators can establish and there is no interior equilibrium, back from theoretical models provides useful guidelines i.e. if and only if A40 and for practitioners in biological control and other fields of lAn mommin ¼ . (8) applied population biology. 1 þ lhAn Inequalities (7) and (8) provide necessary and sufficient Acknowledgments conditions for the interior predator–prey equilibrium to exist, i.e. we require m 2ðmmin; mmaxÞ if A40. There is no We thank V. Krivan, J. Gascoigne and two anonymous such lower threshold in m if Ap0 and therefore mmin ¼ 0. referees for useful comments on the manuscript. The research of D.S.B. and L.B. was supported by grants from Appendix C. Local stability of the interior equilibrium the Grant Agency of the Czech Republic (201/03/0091) and Institute of Entomology, Biology Centre AS CR The derivation of inequalities (11)–(14) follows the same (Z50070508). lines as outlined e.g. in Oaten and Murdoch (1975) and van Baalen et al. (2001). The determinant and trace of the Appendix A. Properties of the functional response Jacobian matrix Jðx; yÞ associated with Eq. (1) equal Det Jðx; yÞ¼myF 0ðxÞmg0ðxÞþFðxÞg0ðxÞ, The functional response (2) is decelerating for all prey densities for 0onp1 and becomes increasingly sigmoidal if Tr Jðx; yÞ¼m þ FðxÞyF 0ðxÞþg0ðxÞ. (9) n41. The inflection point x ðnÞ of the sigmoidal functional % % % i Substituting the values of E ¼ðx ; y Þ from (4) into (9) response can be found by differentiating (2) twice with and simplifying, we get respect to x and setting the resulting expression equal to 0, % % % 0 % which yields Det Jðx ; y Þ¼gðx ÞF ðx Þ40, 1=n % n 1 % % gðx Þ % % x ðnÞ¼ . (6) Tr Jðx ; y Þ¼ F 0ðx Þþg0ðx Þ. (10) i lhðn þ 1Þ m % A straightforward analysis of (6) shows that for The equilibrium E of (1) is thus locally asymptotically stable if and only if Tr Jðx%; y%Þo0, i.e. if and only if n 2ð1; þ1Þ, xiðnÞ increases monotonically from 0 to 1 if lhX1 but becomes hump-shaped if lho1, increasing g0ðx%Þ F 0ðx%Þ o . (11) initially from 0 above 1 and subsequently decreasing gðx%Þ m asymptotically to 1. The hump is steeper and higher as Using the functions g and F from Eq. (3) and the value of lh decreases. The shape of the functional response in the x% from (5) we obtain that E % is locally asymptotically focal range of prey densities—ðA; KÞ,orð0; KÞ if A 0—is o stable, provided that it exists, if and only if: given by the position of the inflection point xiðnÞ with respect to that range. In general, the shape in that range x% x% x% onð1 hmÞ1. (12) changes from decelerating to sigmoidal and then possibly x% A x% þ c K x% ARTICLE IN PRESS 146 D.S. Boukal et al. / Theoretical Population Biology 72 (2007) 136–147

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