The interplay among Allee effects, omnivory and inundative releases in a pest biological control model

Accepted Manuscript The interplay among Allee effects, omnivory and inundative releases in a pest biological control model Lucas dos Anjos, Michel Iskin da S. Costa PII: DOI: Reference:

S1049-9644(16)30178-5 http://dx.doi.org/10.1016/j.biocontrol.2016.09.010 YBCON 3491

To appear in:

Biological Control

Received Date: Revised Date: Accepted Date:

18 March 2016 22 September 2016 24 September 2016

Please cite this article as: dos Anjos, L., Iskin da S. Costa, M., The interplay among Allee effects, omnivory and inundative releases in a pest biological control model, Biological Control (2016), doi: http://dx.doi.org/10.1016/ j.biocontrol.2016.09.010

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The interplay among Allee effects, omnivory and inundative releases in a pest biological control model

Lucas dos Anjos Corresponding author: Lucas dos Anjos Laboratório Nacional de Computação Científica, Av. Getúlio Vargas, 333 Quitandinha, Petrópolis (RJ), 25651-070 - Brazil. [email protected]

Michel Iskin da S. Costa Laboratório Nacional de Computação Científica, Av. Getúlio Vargas, 333 Quitandinha, Petrópolis (RJ), 25651-070 - Brazil. [email protected]

31 pages 14 figures

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Abstract Failures of biological control strategies are attributed, among varied explanations, to the action of Allee effects on the agent and the evidence that these released enemies not only interact with the target pest, but also with other native species of the local ecosystem. By means of a theoretical omnivory dynamical model we show that these explanations are qualitatively coherent with the results of the proposed theoretical omnivory dynamical model and that pest eradication or suppression is strongly related to inundative releases of the pest natural enemies. Keywords: Cooperation, agent, massive introduction, native enemies.

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Introduction Allee effects have played a major role in the understanding of extinction of endangered, rare or declining populations (Courchamp et al., 2008), and their importance has also been acknowledged in the dynamics of invasive species (Taylor and Hastings, 2005; Tobin et al., 2011). A technical term for Allee effects is positive density dependence which can be seen, for instance, as a positive relationship between the overall individual fitness, usually quantified by the per capita population growth rate, and population size or density – the so called demographic Allee effect. Much of what is known about Allee effects comes from mathematical models (Courchamp et al., 2008), and on account of that they have been an important means to assess the potential importance of Allee effects for population and community dynamics (Peng and Zhang (2016), Wang et al. (2014), Morozov et al. (2004), Lewis and Kareiva (1993)). Biological control agents are natural enemies of invasive species (the pest) that are deliberately introduced to control (or eradicate) the pest (Blackwood et al., 2012; Suckling et al., 2012). Cases of success of this pest biocontrol tactic have been reported, but failures have been equally reported (see Hopper and Roush (1993) and references therein). An Allee effect acting on the agent is frequently put forward as a possible cause of these failures (Boukal et al., 2007, Bompard et al., 2013). In addition, it has been put forward that released pest enemies not only interact with the target pest, but also with other native species of the local ecosystem (Wajnberg et al. (2001)). For instance, these native species could interact with the agent as an intraguild (or omnivorous) predator or as a hyperpredator (as shown in figure 2.1 on page 17 in Wajnberg et al. (2001)). Based on experimental evidence, Boukal et al. (2007) also suggested the negative effects of interaction between native species and the introduced agent as factors of pest control failure. However, in their work they analyze a predator– prey model with Allee effect only in the prey and without this additional trophic level of the native species. Given these two factors – Allee effect in the agent and interaction of the agent with native species – that may impair pest biological control, in this work we put forward an omnivory dynamical model where the basal species is the pest and the intermediate consumer is the deliberately introduced biological control agent. The omnivorous predator is assumed to be a native species of the ecosystem which, in the proposed model, portrays a natural enemy of both the pest and the agent. One feature of the model concerns the assumption that the omnivorous predator is generalist in the sense that it does not have a response from the consumption of its exotic prey (pest and agent). Therefore, its population is considered to be constant over time. This is in accordance with the modeled biological setup, once a native predator already possesses its original prey items (to which it certainly responds with variation in its density, but it is not explicitly considered in the proposed model). The introduced agent and the pest, both exotic species, are in this case additional items of the omnivorous predator's diet. 3

Hence, an eventual extinction of the agent and the pest cannot cause the omnivorous predator extinction. Another characteristic of the proposed model is the assumption that the intermediate consumer (the agent) is subject to an Allee effect, which is mathematically translated by a hyperbolic function that modulates its conversion efficiency as a function of its density (Zhou et al., 2005; Verdy, 2010; see also Bompard et al., (2013) for a discrete time host – parasitoid model ). With respect to the pest populations two biological setups will be assumed: (i) pest without an Allee effect; (ii) pest with a strong demographic Allee effect (Boukal et al., 2007). In practical terms, confirming and identifying the pest invasion the manager proceeds to pinpoint a proper agent to control the invasion (for instance, managers introduce exotic agents such as the wasp parasitoid Ooencyrtus kuvanae (Elkinton and Liebhold, 1990), and the carabid predator Calasoma sycophanta (Alalouni et al., 2013) so as to deter the invasive species gypsy moth (Lymantria dispar) in North America). However, to undertake this procedure the manager can be faced at least with two problems: (i) the existence and intensity of an Allee effect in the agent, which will depend on its choice and (ii) the native generalist omnivorous predator population density, which is beyond the manager's control. Therefore, given these characteristics of this biological framework, we propose to assess the efficiency of the released enemy strategy to control pest outbursts (or preferably, to induce pest suppression) by means of the generalist omnivorous predator population size (assumed to be constant over time) and the intensity of the agent’s Allee effect. It is important to remark that the population dynamical models used in this work are of strategic type (May, 2001). Strategic models do not usually describe the dynamics of a specific real community. Instead, they provide a conceptual framework to understand some relevant aspects of the species dynamics of a relatively vast class of communities. Moreover, the proposed models involve a large number of parameters. Hence, this study intends to show some possible outcomes related to the hypothetical chosen sets of parameter values of these strategic models, rather than provide an exhaustive study of conditions required for all possible outcomes generated by these models (Abrams and Roth, 2004). It is almost certain that some parameter values in real systems are quite different from the ones used in the analyzed models and therefore they might not reproduce the dynamical results to be presented in this work. However, it is important to emphasize that the intention of this study is to demonstrate through numerical bifurcations of strategic models the potential of dynamical behaviors generated by these hypothetical parameter values and their possible qualitative interpretations in pest biological control. Nonetheless, we call attention to the fact that the choice of the parameter values was primarily guided by an intention to create, when possible, a high number of coexistence populations of pest and agent in the analyzed models. Such setup would probably describe a complex pest control scenario. On the other hand, future studies of the proposed models containing extended parameter spaces with regard to additional dynamical behaviors would surely be of great interest.

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Methods An omnivory food web model with Allee effect in the consumer population The omnivorous food web to be analyzed is schematically displayed in figure 1.

Figure 1. A generalist omnivorous predator (P) acts upon the consumer (C) and the prey (R) with a multispecies functional response and the consumer (C) preys upon the prey (R) with a functional response type 2. Allee effects act on C. Arrows represent consumption.

Figure 1 consists basically of an omnivory model where both prey (pest, R) and consumer (agent, C) suffer from an Allee effect. The omnivorous predator P is a natural and native enemy of both R and C. A time continuous dynamical model for the trophic scheme of figure 1with multispecies functional responses of P on R and C and functional response type 2 of C upon R can be given by: dR R  aCR R a PR R  =rR1 − C− P − dt K  1 + aCR ThCR R 1 + aPR ThPR R + aPC ThPC C 

(1)  C   dC aCR R a PC C   C − =eRC  P − mC C dt 1 + aPRThPR R + a PC ThPC C  θ C+ C   1 + aCR ThCR R 

The densities of prey and consumer will be denoted by R and C, respectively (the terms density and size will be used interchangeably throughout this work), while P represents the constant density of an omnivorous predator; r is the maximum per capita rate of prey growth and K is its carrying capacity; a CR represents the attack coefficient of the consumer C upon the prey R; ThCR is the manipulation time of R by C, while eRC C is the predator food–to–offspring conversion coefficient; describes the Allee θC + C effect in the response of C where θC denotes its intensity (Zhou et al., 2005; Verdy,

5

2010);

a PC C is a multispecies functional response (Case, 2000) of the 1 + aPCThPC C + aPRThPR R

omnivorous predator P upon the consumer C, where a PC is the attack coefficient, ThPC its manipulation time. Since P also preys upon R, aPR is the attack coefficient of P on R and ThPR is the manipulation time of R by P; mC is the density independent per capita mortality rate of C. Model (1) will be supposed to represent a pest biological control scenario. The predation of the agent C by a native omnivorous predator P is corroborated by the experimental evidence that natural native enemies of agents are very common in natural systems (Boukal et al. (2007) and references therein). Since C represents an agent, it is supposed to prey upon the pest with a functional response type 2 because this response exerts a strong predation pressure when pest population is low (that is, when the pest population is low, the risk of an individual pest being killed by the agent is very high (see Case (2000))) – an appropriate feature of an agent to carry out suppression or pest eradication. The assumption of the generalist character of P (i.e., a predator with multiple prey items and conveyed in this work by its constant density over time) can be borne out by the fact that it is a native species, therefore its diet contains other items besides the biological control agent and the pest itself, which are both exotic species. Hence, model (1) precludes that the extinction of an exotic species (C) causes the extinction of a native species (P). The analysis of the influence of the density of the omnivorous predator and the intensity of the agent Allee effect on the pest dynamics in model (1) will consist of one– parameter bifurcation diagrams of the proposed model. These diagrams are drawn by means of the software XPPAUT (Ermentrout, 2002). The respective phase planes to present the dynamical results of the bifurcation diagrams will be drawn by the software MATLAB.

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Results Model (1) – Bifurcation along P Given that in our assumed biological framework P is a natural native enemy of both pest and agent, we intend first to assess the efficiency of a pest biological control strategy – pest suppression or economically tolerable pest levels in the environment (Boukal and Berec, 2009; Suckling et al., 2012) – as a function of the omnivorous native enemy density. To this end, a bifurcation diagram of model (1) as a function of P is drawn in figure 2.

(a)

(b) Figure 2. Bifurcation diagram of R (a) and C (b) in model (1) as a function of P. Solid lines: stable equilibrium points; dashed lines: unstable equilibrium points. Parameter values: r =3; K = 1; aCR=4.9823; ThCR=1.2444; efRC=1; mC=0.4; θ C = 0.21; aPR=0.032; ThPR=6.25;aPC=0.72; ThPC=2.5.

Figure 2 shows that along the region 0< P < 0.435 the sole outcome is the extinction of the agent and the persistence of the pest. For 0.435 < P < 0.496 sustained oscillations set in. At P= 0.496 there occurs a Hopf bifurcation and the system stabilizes. Note that for 0 = P < 0.496 extinction of the agent and persistence of the pest is still possible for some initial populations – a result not appropriate in pest biological control. In the region 0.496 < P < 0.588 there occurs multistability formed by a positive equilibrium point (R ,C>0) and a boundary equilibrium point (R>0,C=0). For P > Pmax=0.588 all initial conditions lead to the extinction of the agent and the stabilization 7

of the pest near its carrying capacity – again, a result not appropriate in pest biological control. Recall that when C=0 (agent extinction) model (1) reduces to dR R  aPR R  =rR1 − P. − dt K 1 + aPRThPR R  

If the parameter values are such that a positive stable equilibrium R* exists, then R* < K (Turchin, 2003). Figure 3 shows one phase plane of model (1) for P = 0.4 (0< P < 0.435).

Figure 3. One phase plane of model (1) for P = 0.4 (0< P < 0.435), where agent extinction occurs and the pest stabilizes near its carrying capacity. Other parameter values are the same as in figure 2. ’ ’ – initial conditions.

Agent extinction (an inappropriate result in pest biological control) occurs for all presented initial conditions which include inoculative releases (small amounts) as well as inundative (large amounts) releases of agent individuals. Probably this is due to the fact that the density of the omnivorous predator is such that it cannot lessen the magnitude of the oscillations inherent to the model (1) without omnivorous predator (i.e., with P=0).

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Figure 4 shows one phase plane of model (1) for P = 0.45 (0.435< P < 0.496).

Figure 4. One phase plane of model (1) for P = 0.45 (0.435< P < 0.496), where sustained oscillations of pest and agent or agent extinction occurs dependent on initial conditions. Other parameter values are the same as in figure 2. ’ ’ – initial conditions.

Agent extinction occurs for inoculative releases as well as inundative releases. In between these two sets there occur sustainable oscillations of pest and agent. The magnitude of these oscillations will dictate whether they are appropriate or not to pest biological control since extremely high peaks can describe pest outbursts.

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Figure 5 shows one phase plane of model (1) for P = 0.55 (0.4965< P < 0.5882).

Figure 5. One phase plane of model (1) for P = 0.55 (0.496< P < 0.588), where stable coexistence or agent extinction occurs dependent on initial conditions. Other parameter values are the same as in figure 2. ’ ’ – initial conditions.

Likewise as in figure 4, note that agent extinction (an inappropriate result in pest biological control) occurs for inoculative as well as inundative releases. In between these two sets there occurs stable coexistence of pest and agent. Depending on the established pest economic thresholds (Boukal and Berec, 2009), the pest levels of this stable coexistence will dictate whether they are appropriate or not to pest biological control.

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Figure 6 shows one phase plane of model (1) for P = 0.6 (P >Pmax=0.588).

Figure 6. One phase plane of model (1) for P = 0.6 (P>Pmax=0.588), where only agent extinction occurs. Other parameter values are the same as in figure 2. ’ ’ – initial conditions.

The density of the omnivorous predator is so high that the predation pressure upon the agent brings about its extinction for all initial conditions. Unlike figures 3, 4 and 5, note that R and C isoclines in figure 6 do not intersect in this case, and therefore there are no coexistence equilibrium populations (i.e., there are no positive equilibrium points R, C > 0). As a consequence, the system tends to the sole stable equilibrium point – agent extinction and stabilization of the pest.

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Insofar as species extinction is concerned, the results of this section bring out the possibility of one sole common dynamical outcome along the whole analyzed range density of P: agent extinction and stabilization of the pest near its carrying capacity dependent on initial conditions. Agent inoculation (level of the initial condition of C) plays a determinant role in this result: inoculative and inundative agent releases can cause agent extinction while intermediate releases can promote pest and agent coexistence (see the trajectories generated by increasing initial conditions of C with the same initial condition of R in figures 4 and 5). This result is not in accordance with the view that agent inundative release can be related to pest control success. In addition, elevation of P density is detrimental from the pest control viewpoint since it causes an increase in pest density (R) in the stable equilibrium points region (see bifurcation diagram as a function of P in figure 2(a)). Since the extinction of R and C is of interest in our biocontrol setup, it is important to study the local stability of point (0,0) in model (1). The trace of the Jacobian of model (1) at the origin (J(0,0)) is given by: Trace( J (0, 0)) = rR − aPR P − a PC P − mC , and the respective determinant is given by Determinant ( J (0, 0)) = (rR − aPR P)( − aPC P − mC ), which shows that the origin is stable for densities of P such that P>

rR . aPR

This means that when the density of the omnivorous predator is above

rR it a PR

can cause the extinction of both pest and agent. The agent extinction (C=0) and stabilization of the pest near its carrying capacity is due, in part, to the Allee effect experienced by the agent. Therefore, it is also important to investigate how the intensity of the Allee effect on the agent ( θ C ) influences the overall dynamics of model (1) (Taylor and Hastings, 2005). This task will be tackled in the next section.

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Influence of the intensity of the agent’s Allee effect ( θ C ) in the dynamics of model (1) – bifurcation along θ C

By the same token as in section the previous section, the current investigation will be undertaken by a bifurcation analysis of model (1) as a function of the intensity of the agent’s Allee effect θ C . Figure 7 shows a bifurcation diagram of model (1) as a function of θ C .

(a)

(b) Figure 7. Bifurcation diagram of R (a) and C (b) in model (1) as a function of

θ C . Solid lines:

stable equilibrium points; dashed lines: unstable equilibrium points. Parameter values are the same as those of figure 2; P = 0.16.

The qualitative similarity between figure 2 and figure 7 shows that the abundance of predator and the intensity of the Allee effect in the agent play a qualitatively similar role in the dynamics. Another point worth mentioning is that the agent’s coefficient of attack (aCR) plays the same role of P and θ C in figures 2 and 7, but in a reverse order, i.e., increasing the magnitude of a CR can destabilize the system so that the agent itself goes extinct (bifurcation diagrams as a function of aCR, not shown). Figure 7 also depicts that along the region of locally stable nontrivial equilibrium points (i.e., R, C>0), R increases while C decreases with augmentation of θ C , which is not an appropriate result in pest biological control. Accordingly, beyond a maximum value 13

of θ C ( θ C max=0.513), there occurs only agent extinction and pest stabilization near its carrying capacity. Again, this result is not appropriate in terms of pest biological control. These features point to the fact that the choice of an agent with an intense Allee effect in its response may cause the failure of pest biological control (corroborating the hypothesis raised in Boukal et al., 2007). Moreover, given the qualitative similarity of figures 7 and 2, inundative releases (massive releases) of agent may be deleterious to pest biological control as was shown in the previous section by means of phase planes of model (1).

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Strong demographic Allee effect in the pest population Bifurcation along P Model (1) with strong demographic Allee effect in the prey can be given by: dR R  aCR R aPR R  R − RC  =rR C− P 1 − − dt K  K  1 + aCR ThCR R 1 + aPRThPR R + aPCThPC C 

(2)

 C   aCR R a PC C dC   C − =eRC  P − mC C dt 1 + aPRThPR R + a PC ThPC C  θ C + C   1 + aCR ThCR R  The threshold level of the strong demographic Allee effect in the pest population is conveyed by the term RC (RC=0.1). When pest density is below this level (R < RC), the pest population decreases (dR/dt < 0). Figure 8 shows a bifurcation diagram of model (2) as a function of P.

(a)

(b) Figure 8. Bifurcation diagram of R (a) and C (b) in model (2) as a function of P. Solid lines: stable equilibrium points; dashed lines: unstable equilibrium points. Parameter values are the same of those of figure 2 except for RC=0.1.

For the chosen set of parameter values the qualitative behavior of figure 8 is similar to figure 2, the only difference being the locally stable characteristic of the point (0,0), indicating thus the possibility of overall species extinction. In quantitative terms, the range of P for sustained oscillations and stable steady states is much shorter in 15

figure 8 than in figure 2. Moreover, the amplitude of sustained oscillations is also much shorter in figure 8 than in figure 2. Figure 9 shows one phase plane of model (2) for P = 0.1 (0< P<0.234), where agent extinction or both agent and pest extinction occurs depending on initial conditions.

Figure 9. One phase plane of model (2) for P = 0.1 (0< P<0.234). Parameter values are the same as those of figure 8. ’ ’ – initial conditions.

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Figure 10 shows one phase plane of model (2) for P = 0.2345 (0.234< 0< P<0.235), where (i) unstable coexistence; (ii) agent extinction; (iii) both agent and pest extinction occurs depending on initial conditions.

Figure 10. One phase plane of model (2) for P = 0.2345 (0.234< 0< P<0.235). Parameter values are the same as those of figure 8. ’ ’ – initial conditions.

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Figure 11 shows one phase plane of model (2) for P = 0.236 (0.235< P<0.238), where (i) stable coexistence; (ii) agent extinction; (iii) both agent and pest extinction occurs depending on initial conditions.

Figure 11. One phase plane of model (2) for P = 0.236 (0.235< P<0.238). Parameter values are the same as those of figure 8; ’ ’ – initial conditions.

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Figure 12 shows one phase plane of model (2) for P = 0.3 (P>0.238), where (i) agent extinction; (ii) both agent and pest extinction occurs depending on initial conditions.

Figure 12 One phase plane of model (2) for P = 0.3 (P>0.238). Parameter values are the same as those of figure 8; ’ ’ – initial conditions.

Figures 8 –12 show that the dependence on the initial population of agent is of crucial importance to the ultimate dynamics of the analyzed model. Hence, for the case of figure 11 (stable coexistence of pest and agent) we plot a bifurcation diagram showing the equilibrium pest and agent population in model (2) as a function of the initial agent population (i.e., C(0)) as depicted in figure 13.

Figure 13. Bifurcation diagram of the equilibrium populations of R and C in model (2) as a function of C(0) in the stable case of figure 11. Parameter values are the same of those of figure 9.

The bifurcation diagram of figure 13 consists of simulation runs of model (2) for a fixed value of R(0) with a duration of 2000 time units in order to eliminate any transient phase. After each simulation, the 200 last values of variables R and C (considered as their respective steady states) are plotted against the respective C(0) value (to this end we used the software MATLAB). Figure 13 displays four behaviors as C(0) increases: C extinction, stable coexistence of C and R, C extinction again, and finally C and R extinctions (note that to visually enhance these four behaviors we resorted to a shorter interval of C(0) [0.53–0.57] than that used in figure 11). In terms of pest control this sequence of results points to the fact that failure of pest biocontrol can 19

be interspersed with pest and agent coexistence (which can portray a possible success of pest control depending on the stable pest level) as more agents are introduced in the system. This non–monotonic behavior is at variance with the view that agent inundative releases are proper to pest eradication or control. According to model (2) only extremely high levels of agent inundative releases can guarantee pest eradication (and in this case, concomitant agent extinction). Corroborating the bifurcation diagram of figure 8, figures 3, 4, 5 and 6 differ from their counterparts 9, 10, 11 and 12 in the instance of both pest and agent extinction for some initial conditions. This is the impact of the strong Allee effect on the pest: once the pest level falls below RC in model (2), it tends inexorably toward extinction and since the agent is a specialist, it tends also to extinction. Note that this scenario is also possible when there are coexistence equilibrium points (i.e., R, C > 0), On the other hand, in model (1) overall species extinction occurred only for densities of r omnivorous predator such that P> R . aPR Since the extinction of R and C is of interest in our biocontrol setup, it is important to study the local stability of point (0,0) in model (2). The trace of the Jacobian of model (2) at the origin (J(0,0)) is given by: R Trace( J (0,0)) = − rR c − aPR P − aPC P − mC , KR and the respective determinant is given by Determinant ( J (0, 0)) = ( − rR

Rc − aPR P )( − aPC P − mC ), KR

which shows that the origin is locally stable for any set of parameter values of model (2). This is in stark contrast with the local stability of the origin point in model (1) where there is no is strong Allee effect on the pest. The local stability of the origin point in model (2) states that pest eradication is possible when driving the pest level below RC by means of an agent. Moreover, once this scenario is attained (i.e., the pest population lies below RC), the pest would go extinct even in the absence of the agent (conveyed in model (2) by C=0).

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Bifurcation along θ C in model (2)

Likewise as in model (1), the dynamical results of the previous section are also due, in part, to the Allee effect experienced by the agent. Therefore, it is also important to investigate how the intensity of the Allee effect on the agent ( θ C ) influences the overall dynamics of model (2). Figure 14 shows a bifurcation diagram of model (2) as a function of θ C .

(a)

(b) Figure 14 Bifurcation diagram of R (a) and C (b) in model (2) as a function of θ C . Solid lines: stable equilibrium points; dashed lines: unstable equilibrium points. Parameter values are the same as those of figure 8; P = 0.001.

Figure 14 is qualitatively similar to figure 7. Hence, for the chosen set of parameter values, the Allee effect in the agent creates similar dynamics in the analyzed omnivorous setup regardless of the Allee effect in the pest. The sole difference is the possibility of overall species extinction in model (2) for any set of parameter values. Beyond a maximum value ofθ C ( θ C max=0.255), there occurs agent extinction and pest stabilization near its carrying capacity or both agent and pest extinction depending on initial conditions. Again, this result is not appropriate in terms of pest biological control. These features point to the fact that the choice of an agent with an intense Allee effect in its response may contribute to pest control failure. Moreover, given the qualitative similarity of figures 14 and 8, the phase planes corresponding to figure 14 will also show that inundative releases of agent may be deleterious to pest biological control.

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Discussion

In the context of pest management strategies, release of pest natural enemies is usually undertaken in order to drive pest populations to extinction or to economically tolerable levels. However, biocontrol managers repeatedly report that the possibility of Allee effects in the agent may be a relevant cause for failure of inoculative enemy releases (Hopper and Roush, 1993). In addition, it seems that the enemies released interact not only with the target pest but with native species of the local ecosystem as well, for instance, by being a prey to native intraguild predators (Boukal et al., 2007). In this paper, we analyzed an omnivory dynamical model with the following features: (i) the absence or the presence of a strong demographic Allee effect can be a characteristic of the pest population; (ii) the agent possesses an Allee effect in its response; (iii) and both are victims of a native omnivorous top predator with constant density over time. One of the points of the analysis centered upon how the constant number (or density) over time of individuals of the generalist omnivorous predator influenced the pest dynamics. We think it relevant in pest biological control, since this is a variable beyond the manager's control. A result that stood out was that the density of the omnivorous predator changes qualitatively the possible dynamics of the system, which consisted of: (i) pest and agent extinction; (ii) stable and unstable coexistence of pest and agent; (iii) agent extinction and stabilization of the pest near its carrying capacity. It is important, however, to recall that in some instances the occurrence of (i), (ii) and (iii) depended also on the initial population levels. These ecological terms are associated with multistability (extinction/coexistence) and basins of attraction of equilibrium points (dependence on initial conditions). Given that the analyses of model (1) and (2) draw on the parameters θ C and P, it may be argued how the results of (i), (ii) and (iii) would be affected by uncertainties in the value of these parameters. However, if the lower and upper bounds of these uncertainties are relatively close, the dynamics of model (1) and (2) will remain qualitatively unaltered because the uncertainties will not be strong enough to change the characteristic of the equilibrium points. The same is valid for the pest initial population and the density of the natural enemy if these uncertainties are not strong enough to cause a change of basin of attraction of the initial condition. These features point to a certain degree of resilience of such biological control strategies. Costa and Anjos (2015) performed a similar analysis for a tritrophic food chain where the agent suffered predation from a native hyperpredator with constant density over time. Interestingly, dynamical outcomes (i), (ii) and (iii) were also found in that biological context, indicating thus a robustness of the results concerning Allee effects in the pest and agent in two different biological setups. As to stable coexistence of pest and its agent, a trophic cascade occurred with and without Allee effect in the pest as the density of the omnivorous native enemy P increased. Consequently, the pest population increased while the agent's decreased, evidencing thus an outcome that is not appropriate for pest biocontrol objectives. In fact, in these cases the effects of the omnivorous predator and the agent control on the 22

pest are said to be not additive. Such qualitative behavior was found experimentally in an intraguild system composed by a carabid species (primarily Pterostichus melanarius, the generalist omnivorous predator), a wasp species (Aphidius ervi, a natural enemy of the pest) and an aphid species (Acyrthosiphon pisum, the pest) (Snyder and Ives, 2001). In that experiment, the carabid not only consumed the aphid but also the pupae of the wasp. Since the pupae are immobile they become an easy target for the omnivorous carabid. In turn, this consumption impairs the aphid control performed by the wasp. Moreover the aphid control by the carabid is partially hindered by the anti–predatory behavior of the aphid. Consequently, these three factors contribute to higher growth rates of aphid population in the presence of the carabid than in its absence. This result is in qualitative accordance with our model that displays an increase in the pest population when the population of the omnivorous predator is also increased. With regards to unstable coexistence of pest and its agent, the sustained oscillations occurred only in a very limited interval of the intraguild predator density P for the chosen set of parameter values and the magnitude of the oscillations was rather small (see a qualitative similar result in a pest biological control context within a tritrophic food chain (Costa and dos Anjos, 2015); see also van de Koppel et al., (1996) for a specific ditrophic food chain model within a plant–herbivore context). This result seems to be in qualitative agreement with some field observations that indicate that cyclical parasitoid populations are not commonly observed (Hassell, 2000). Moreover, our proposed model attests that successful biological control does not only depend on the traits of the introduced pest natural enemies but on the quantities of these released enemies as well. The phase planes of all analyzed cases with strong demographic Allee effect in the pest showed that pest eradication and agent extinction – an appropriate goal in pest management – can be achieved when the introduced agent is released in relatively high densities. This dependence on the initial population of C theoretically supports pest biocontrol failures (e.g., agent extinction and pest persistence) when agents are released in small numbers (Boukal et al., 2007). Therefore, in the case the pest suffers a strong demographic Allee effect, our model suggests that the introduction of large numbers of agent individuals in a given area will lead to success of pest biocontrol. Certainly, this measure is feasible only when mass rearing is efficient to enable the introduction of agent in large densities. On the other hand, in some instances mass rearing of the agent may not be so efficient or economically feasible. In this case, according to our models, the inoculative releases of agent are not always successful when a strong demographic Allee affects the pest population, since they may bring about either biocontrol failure (i.e., agent extinction and stabilization of pest near its carrying capacity) or stabilization of pest and agent. In terms of pest control the effectiveness of this stabilization of pest and agent would depend on pest tolerable economic thresholds (Boukal and Berec, 2009). Gauging pest biocontrol success by simultaneous pest eradication and agent extinction, our model also points out that the number of individuals released can affect the success of an agent if an Allee effect is 23

experienced by both the agent and the host species and not necessarily experienced only by the agent as suggested in Taylor and Hastings (2005). Running counter to the above considerations concerning massive releases of the natural enemy of the pests, this study showed that such protocol can be detrimental to pest control if the pest itself does not possess an Allee effect. If the density of the r natural enemy (P) of the agent is below the level R , inundative releases of agent in a PR the analyzed cases without strong demographic Allee effect in the pest proved to be generally deleterious to pest biological control objectives since they mostly led to agent extinction and stabilization of pest near its carrying capacity. At this juncture, it is important to recall that the topic of agent release has centered upon punctual inundative and punctual inoculative release. On the other hand, a specified number of agent individuals of a punctual inundative release could correspond to this same amount being released as a function of time (or a periodic introduction of any deliberately number of agent individuals as propounded by augmentative biological control (van Lenteren, 2000)). This feature would change the C dynamical equation by adding, say, an input function u(t) of agent individuals explicitly time dependent, turning then models (1) and (2) into non–autonomous systems of differential equations. With these new models in hand, comparisons of the results of the two kinds of release could be performed. Nonetheless, it should be kept in mind the possibility of complex dynamical behavior in the latter model on account of two–dimensional non–autonomous systems of nonlinear differential equations and their implications to pest control strategies. This work is amenable to extensions. Because species across taxa suffer the influence of different mechanisms that may generate different Allee effects (Courchamp, 2008; Tobin et al., 2011), other mathematical expressions describing Allee effects both in the invasive and in the agent species could be employed. By means of these analyses it would be possible to investigate whether these proposed mathematical expressions of Allee effects qualitatively alter or corroborate the results presented in this work. The multispecies functional response (Case, 2000) used here embeds a fixed preference of the omnivory (P) predator upon its prey (R and C). However, for instance, zoophytophagy, the habit of consuming plant and animal tissue, is common in agroecosystems. Usually, plants are assumed to be resources of low quality, which are exploited largely when animal prey (e.g., insects) is at low abundance (Hunter, 2009). In this context, preference of the predator seems to vary according to the densities of its prey items. This consumption setup would probably be more properly described by a switching preference functional response (Faria and Costa, 2009). Finally, with respect to the theory of control of invasive species, the performed analysis presented some features that show the influence of Allee effects in the efficacy of pest biological control. More precisely, we think this study provided some theoretical 24

results that contribute to a better understanding of the dynamics of pest biocontrol when Allee effects are present both in the target pest species and in the agent.

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We devised an omnivory dynamical model to describe biological pest control. Pest and agent have Allee effects and are consumed by an omnivorous predator. Pest extinction is related to inundative releases of agent. Allee effect in the agent and omnivorous predator density generate similar dynamics

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