A tritrophic interaction model for an olive tree pest, the olive moth — Prays oleae

: Encyrtidae) and this, in Trás-os-Montes region, is commonly followed by the facultative hyperparasitoid Elasmus flabellatus (Fonscolombe) (Hymenoptera: Eulophidae). Spiders represent a relevant group of generalist predators in olive agroecosystems and encompass an important predatory action in agroecosystems as well as an ability to reduce the populations of various insect pests. In this context, a mathematical model, considering the population of the olive moth, the two parasitoids populations and the spider population as the variables in our system, was constructed. The ecosystem steady states for feasibility and stability were assessed. The possible pesticide effects, that represent essentially extra mortality rates for each one of the insect populations, and potential abundance variations on their populations under a climate change scenario were included. Results indicate that the most important natural control agent is A. fuscicollis but in certain conditions E. flabellatus or spiders may be relevant contributors for the pest reduction. This approach may provide a useful tool to assist the field researchers on this pest system and its management.


Introduction
The olive tree (Olea europaea L.), one of the oldest and most widespread crops, has characterized economically, socially and culturally the populations of the Mediterranean basin.Nowadays, it is cultivated in all regions with climatic conditions that allow its establishment (Bartolini and Petruccelli, 2002).Portugal is one important olive producer country and Trás-os-Montes region, located in northeastern Portugal, in 2019 was responsible for the production of 914 504 tons of olives in 359 949 ha of groves (INE, 2021).
However, the olive tree is attacked by several pests that result in severe economical losses.The olive moth, Prays oleae (Bernard) (Lepidoptera: Praydidae) is the most damaging pest in Trás-os-Montes.
The insect has three generations a year and their larval stages attack different organs of the olive tree.Eggs of the anthophagous generation are laid on flower buds and, after hatching, larvae feed on flowers.Its adult flight period occurs at the end of the spring, laying the eggs of the carpophagous generation on the olive calyx.The carpophagous generation larvae bore into the olive stone and feed on the seed.At the end of the summer and beginning of the autumn, adults emerge and lay the phyllophagous generation eggs on the olive leaves.The phyllophagous larvae feed on the leaves and dig tunnels into them, where they overwinter until the following spring (Arambourg and Pralavorio, 1986).https://doi.org/10.1016/j.ecolmodel.2021.109776Received 3 August 2021; Received in revised form 21 September 2021; Accepted 3 October 2021 S. Pappalardo et al.P. oleae is naturally controlled by several organisms which include several generalist and specialist parasitoids as well as generalist predators.The most abundant is the parasitoid Ageniaspis fuscicollis (Dalman) (Hymenoptera: Encyrtidae) (Bento et al., 1998;Herz et al., 2005;Villa et al., 2016a).This specialist parasitoid of the olive moth is well synchronized phenologically with the pest (Campos and Ramos, 1982).In Trás-os-Montes region, in some years the second most abundant parasitoid was Elasmus flabellatus (Fonscolombe) (Hymenoptera: Eulophidae) (Villa et al., 2016a).It behaves as a facultative hyperparasitoid, parasitizing some larvae of hymenopteran and larvae and pupae of lepidopteran species (Yefremova and Strakhova, 2010) and references therein.Adult parasitoids need energy for maintenance, locomotion and reproduction that is provided by non-host foods such as flowers and insect honeydews (Jervis et al., 2008(Jervis et al., , 1993)).In Trásos-Montes olive groves, honeydews produced by some secondary pests, such as the black scale, Saissetia oleae (Olivier) (Hemiptera: Coccidae) and the olive psyllid Euphyllura olivina (Costa) (Hemiptera: Psyllidae) and flowers from many plant species within and around the olive groves probably nourish the olive moth parasitoids.
The olive moth is also attacked by generalist predators.Spiders are generalist predators with important predatory action in agroecosystems and ability to reduce the populations of various insect pests (Marc et al., 1999;Nyffeler and Sunderland, 2003;Riechert and Lockley, 1984).In this context, some spiders exhibit a high degree of superfluous killing (individuals attack more prey that they actually consume) at high levels of prey density (Riechert and Maupin, 1997).Spiders constitute one of the most abundant group of predators in the olive agroecosystem and have ubiquitous feeding habits (Pascual et al., 2010;Benhadi-Marín et al., 2016).During the spring and coinciding with the adult flight of the anthophagous generation, several abundant spiders in the olive canopy potentially prey on the olive moth (Pascual et al., 2010).In addition, it is of paramount importance to understand the behavior of this type of tritrophic relations under a climate change context because shifts in plant and insect phenologies, distribution or voltinism might result in modification of the trophic interactions timing and distribution (Castex et al., 2018).Our goal is to develop a mathematical model to generate population behavioral predictions under different abundance scenarios, resulting from the potential effects of climate change, for the various agents involved in the trophic system under consideration (the pest-P.oleae, its main parasitoids, -A.fuscicollis and E. flabellatus, and spiders -as model for an abundant generalist predator in the olive tree canopy).

Materials and methods
We consider the olive moth  population, the E. flabellatus population , the A. fuscicollis population  and the spiders population  as the variables in our system.
The spiders feed on the adult populations of the two parasitoids and on the olive moth.Their hunting rate on  is , the one on  is  and the one on  is , with respective conversion coefficients   ,  ∈ {, , }.But since they are generalist predators, they have also other resources modeled in a logistic fashion with carrying capacity  (i.e. the theoretical amount of individuals of the population that the environmental conditions can sustain) and net reproduction rate  (i.e. the difference between reproduction rate and mortality rate), described by the last term.Their dynamics is expressed in the first equation of the system.
The second equation describes the development of the parasitoid E. flabellatus, .They feed by parasitizing either the other parasitoid , at rate , with conversion coefficient ,or , the moth, at rate  with conversion coefficient ℎ.Consequently they have the resources to reproduce.The first two terms in the second equation describe jointly the parasitizing and reproduction processes, exerted on both prey.In addition they thrive also by feeding on sugary liquids (nectar or honeydews) or parasitizing other insects, that are not explicitly modeled.
This gives the additional third logistic term with net reproduction rate  and intraspecific competition rate  −1 , where  represents the carrying capacity of these extra resources.The last two terms represent mortalities induced by spiders hunting or use of pesticides.
The third equation contains the dynamics of the adult individuals of the parasitoid A. fuscicollis, .They reproduce at rate  by feeding on the moth , with conversion coefficient  and they feed also on sugary liquids in the environment (honeydews or nectar) that, as we said before, we do not include in the model.Sugary liquids would improve the reproduction but A. fuscicollis can survive without them while they cannot survive without the moth.This is because reproduction depends strongly on the moth (being specialist parasitoids, they do not parasitize other insects).Thus, if the moths vanish, even if there are sugary liquids in the environment, A. fuscicollis cannot reproduce, but if the sugary liquids are absent, the moths can still reproduce.This is modeled by the parameter , which can be thought to be split among a baseline value  0 and a boost term  1 provided by the possible presence of sugary liquids.Thus:  =  0 +  1 .In addition, we introduce  , the rate of intraspecific competition of , the natural mortality   and hunting by both  and  at respective rates  and  (as seen above).
The olive moth , last equation, is attacked by both parasitoids larvae  and , as well as , at respective rates ,  and .The moth logistic exploitation of the resources is modeled by the fourth term, where  denotes the carrying capacity represented by the olive crop and  is net reproduction rate.
In all equations, we include also the possible pesticide effects, that represent extra mortality rates for each populations,   ,  ∈ {, , , }.They appear in the system as the last terms of each equation.
The resulting system, where all the parameters are nonnegative and whose interpretation is given in Table Table 1

Analysis of the equilibrium points
In this section we analyze all the possible equilibrium points of the system (2.1), for both feasibility and stability, deferring to Appendix A the more complicated mathematical details.
Then  3 = ( 3 ,  3 , 0, 0), where This equilibrium is feasible if and only if the following conditions hold Thus for   >  the point is unfeasible, while in the absence of spraying, the feasibility condition reduces to  < . (3.5) Next we find  9 = ( 9 , 0, 0,  9 ), with and again for   >  the equilibrium  9 is unfeasible.When no insecticide is used, feasibility simplifies as follows  < . (3.7) For  10 = (0,  10 , 0,  10 ) we have  (3.11)For the points at which just one population vanishes, analytic expressions for the population values can be obtained, and are contained in Appendix A. These equilibria are  11 = ( 11 ,  11 , 0,  11 ),  13 = ( 13 , 0,  13 ,  13 ) and  14 = (0,  14 ,  14 ,  14 ).The coexistence equilibrium instead cannot be assessed and its analysis is obtained through numerical simulations.As seen, not all the possible combinations between the presence and the absence of a population arise.Ecologically, this can be explained by the fact that A. fuscicollis, being a specialist predator, cannot thrive in the absence of its prey P. oleae.

Stability
The Jacobian of (2.1) is where: Let us also introduce the following notation.The square matrix  [;] of order  is obtained by retaining the   -th and   -th rows and columns,  = 1, … , .
Here we report the results for the easier equilibria, where the stability conditions are obtained explicitly and postpone to Appendix A the analysis of the most complicated ones.In any case, all possible equilibria are found to be conditionally stable.Note that the particular case of no insecticide spraying affects somewhat the stability.However, with the exception of the cases that are explicitly mentioned below, the modifications in the conditions is trivial, amounting to setting the extra killing rate to zero, but does not provide particular additional insights in them.Thus, these cases are not explicitly listed.(3.15) Equilibrium  3 The Jacobian's explicit eigenvalues for  3 are − 3 −   −   −  3 and − 3 +  −  3 −   .From the latter the submatrix  [1,2;1,2] of order two is left for which we find that both Routh-Hurwitz conditions hold: The stability conditions are therefore   +  3 +   +  3 > 0 which is trivially satisfied and For  8 the eigenvalues of the Jacobian are    8 +  −   , ℎ 8 +  −   ,  8 − (  +   ),   −  < 0, the latter inequality stemming from the feasibility condition (3.3).The stability conditions are thus (3.17) Again, here we discover that no spraying entails the instability of this equilibrium.The same result is achieved if spiders are not affected by the insecticide.

Verification and model behavior
We address here the issue about the factual existence of the equilibria.As some of the stability conditions are quite involved, and in some cases the feasibility conditions too, it is legitimate to ask the question whether these conditions are indeed verified and do not define the empty set.Now, the equilibria with just one population are all easily attainable as both feasibility and stability conditions simultaneously hold for suitable parameter choices, that are not here reported.Those with two nonvanishing populations are also attained.We report for each one of them a parameter set that satisfies them.Note that for the parameters that can be assessed, we use the values that can be obtained from field data, reported below in Table 3,   The remaining equilibria are seen to arise for suitable hypothetical parameter choices, and the results are reported in Figs.1-2 (3.27)

Bifurcations
The model allows transitions from an equilibrium to another one, when suitable changes in the parameters occur.For instance, from Table 2, it is immediately seen that  0 is incompatible with each one of the points  1 ,  2 and  8 , because its stability conditions are the opposite ones of the feasibility conditions for the latter set of points.This is an indication that transcritical bifurcations indeed occur.In this section we summarize the findings on this issue.
In fact, note that the model (2.1) shows several other transcritical bifurcations.They have been fully analyzed by means of Sotomayor's Theorem (Perko, 2011), and the details are deferred to Appendix B. In addition a few of them have been found by numerical simulations, and are reported in Figs.3-5.It is interesting to note that from  10 , the Elasmus-moth equilibrium, the spiders can invade if their reproduction rate grows past the value 34.6, but if it grows further past 35, Elasmus is wiped out and only spiders and moth persist, Fig. 3 bottom.
In addition to transcritical bifurcations, also persistent oscillations can be determined, originated by suitable Hopf bifurcations.In particular, analytical conditions for the existence of such bifurcations are given in Appendix B, for the equilibria  13 and  14 .For coexistence instead, the oscillatory behavior of the model has been discovered numerically.Indeed, the occurrence of a Hopf bifurcation at coexistence is shown in Fig. 5   Finally, in Fig. 6 we provide a full picture illustrating the general relationships among the equilibria through transcritical bifurcations.

Simulation of realistic scenarios
We now turn to investigate the possible behavior of the system for the ecological application.The first step is the assessment of the model parameters from the available field data.

Parameters assessment
For the spider population, only their density at the beginning and at the end of the spring is known, respectively 58.500 and 476.500 individuals per hectare (Benhadi-Marín et al., 2020).By fitting on these data the simple logistic equation ) we obtain the estimates for their reproduction rate and carrying capacity, that give (92) = 476.4991,which is acceptable.We thus take the values  = 0.029368,  = 980. (3.28) For the other insects, the available field data are reported in Table 3.These parameters include the probability that each insect species has of emerging from a pupa, the mean number of emerged insects from a pupa, the number of eggs laid under an optimal and underfed diet, and the longevity and mortality rates.The probability with which each insect species emerges from a pupa and the number of insects emerging from a pupa were retrieved from previous experimental research conducted by Villa et al. (2016b) that collected pupae of P. oleae in olive orchards and kept them in controlled conditions until emergence.The probability of one of the species emerging from a pupa was estimated considering the emerged organism (i.e.P. oleae or the parasitoids, E. flabellatus or A. fuscicollis), the total number of pupae, and the nonemerging pupae.The number of insects emerging from a pupa was retrieved from the same work.In the case of P. oleae, one individual emerges from each pupa.In the case of the parasitoids, up to 2.31 and 11.83 individuals of E. flabellatus and A. fuscicollis can emerge on average, respectively.Longevity and mortality data were retrieved from Villa et al. (2016a) in the case of P. oleae, from Villa et al. (2017a) in the case of E. flabellatus, and from Villa et al. (2017b) in the case of A. fuscicollis.In these studies, several biological parameters were derived under laboratory conditions.In the case of P. oleae, the number of laid eggs was also recorded.Unfortunately, this parameter is not available for the parasitoids.Now, this information is used to obtain values for some other parameters as follows.
Fig. 6.The graph of the transcritical bifurcations structure: in red the ones found numerically.The nodes represent the various system equilibria, while the arcs denote the existing transcritical bifurcations linking pairs of equilibria, obtained either analytically or numerically.In general it is possible to move from one equilibrium, with some nonvanishing populations, to an ''adjacent'' one, where one of the system populations present in the old equilibrium either vanishes or an altogether new one appears from a former zero value.
For the olive moth, we only consider the case of an optimal diet.The net birth rate  is the difference between the birth rate and the mortality rate.The birth rate ŝ is the product of the number of eggs laid by a moth in its lifetime with the probability that the emerging larvae become indeed adults, namely Elasmus reproduces by parasitizing  and  but the number of eggs laid is unknown, thus an estimate of the net reproduction rate  is not possible.Also for Ageniaspis the number of eggs laid is not available.In a similar way we treat the corresponding coefficient  for Ageniaspis, so that we have Based on Bento (1999), we finally take the following value for the moths carrying capacity: and on previous records of olive moth eggs in the region.

Possible climatic changes
We discuss now how the ecosystem could be affected by possible climatic changes.To this end, we assume that the system parameters will change if the temperature raises.Specifically, for P. oleae we assume that the reproduction rate becomes lower with increasing humidity while instead it increases with raising temperature, at any given degree of humidity.This is based on the data reported in Table 4 (Bueno, 1981).
It is difficult to specifically quantify this information within the model, because the same change in reproduction could arise under different combinations of humidity and temperature.We therefore assume various degrees of P. oleae reproduction rate reduction, namely starting from , we consider in sequence ∕2, ∕4, ∕6, ∕8 and ∕10.At the same time we assume that climatic-induced changes may not or may occur in the parameters of the other species.The results for the various For the Ageniaspis-Prays-only equilibrium,  12 , in Fig. 7 we observe that the levels of both species steadily drop with diminishing moth reproduction rates.The moth population in this situation gets reduced tenfold in the worst situation while Ageniaspis are essentially eradicated, in case only the moth growth rate is affected.When instead all insects suffer the climatic changes, Ageniaspis behaves in the same way, but its reduction is only a third of the standard value, while the moth drops up to being almost eradicated.However, it is interesting to note Fig. 8. Effects of climatic changes for the spider-free equilibrium  14 .Top , center , bottom .Note that in this column the vertical scale for Elasmus starts from 90 and not from 0, The red trajectories represent the current system behavior, with parameters given by (3.26), (3.20), (3.28), (3.29), (3.30), (3.31).The other colors indicated in the legend represent the system trajectories under climate changes that supposedly induce a reduction in one or all insects growth rates, as specified below.Upper frame: only the moth growth rate  is affected; Lower frame: the growth rates of all species are affected in the same way as for moths.Note that in this column the vertical scale for Elasmus starts from 90 and not from 0, to better show the differences in the graphs.that the behavior is more complex, because for a half of the standard parameter values, Ageniaspis is slightly reduced, while the Prays population raises of about 50%.For a fifth of the standard parameter values, both populations essentially disappear.Finally when the parameters attain a tenth of their standard reference value, Ageniaspis rebounds to a third of the starting value, as said above, while Prays attains values slightly different from zero.
For the spider-free equilibrium  14 , Fig. 8 top, Elasmus drops steadily up to about 13% of the standard reference value, with decreasing moth reproduction rates, as does the moth itself, finally attaining vanishing values.For the Ageniaspis instead at first we observe a decrease, but then there is a rather large rebound for ∕8, raising the population about 5 times from the standard reference level and then another drop for ∕10 to almost vanishing values.When all insects growth rates are affected by climatic changes, Fig. 8 bottom, Elasmus is scantly affected, Ageniaspis shows a similar behavior that it has when only  changes, but the rebound for ∕8 attains only the population level corresponding to the original value of .For Prays instead a steady increase is observed The other colors indicated in the legend represent the system trajectories under climate changes that supposedly induce a reduction in one or all insects growth rates, as specified below.Upper frame: only the moth growth rate  is affected.Note that in this column the vertical scale for Elasmus starts from 90 and not from 0. Lower frame: the growth rates of all species are affected in the same way as for moths.In this column the vertical scale for Elasmus starts instead from 80, to better show the differences in the graphs.with a decrease in the parameters, leading to a final value that doubles the standard reference level.
For the Elasmus-Prays-only equilibrium  10 , Fig. 9, when only the moth growth rate is affected, Elasmus drops about 3% while Prays instead initially drops about a fourth of the standard reference for ∕2, and then for further reductions of this parameter, it is essentially eradicated.A similar behavior is observed also in case when all insects suffer climatic influence, but Elasmus drops instead about 17%.
In case of the Elasmus-free point  13 , Fig. 10, spiders are not affected by the moth growth rate changes, Ageniaspis and moths, after a steady decrease, are essentially eradicated when the maximum reduction in the parameter  occurs.When all insects growth rates are affected by temperature and humidity changes, with the reduction of the reproduction parameters, the spiders population seems to rise slower, at least initially, Ageniaspis are scantly affected, the moth population at equilibrium instead experiences a steady increase.
For the Ageniaspis-free point  11 , Fig. 11, by substantial changes only in the moth growth rate, spiders and moths are essentially wiped out, while Elasmus gets reduced about 10%.When all insects feel the effect of climatic variability, spiders again after a steady decrease disappear, Elasmus instead increases steadily about 25%, and also Prays rebound gradually to double their equilibrium values, The spider-moth-only point shows a steady decrease of both populations with decreasing , the former by about 50% the latter by 44%, Fig. 12.If all insects are subject to climate influence, both populations equilibrium values do not change, only the speed at which these values are attained do appear to be affected, being slowed down by the lower parameter values.
For coexistence, by changes only in the moth growth rate, spiders are not affected, Elasmus, Ageniaspis and moths are reduced and finally essentially wiped out.When all insects feel the effect of climatic variability, spiders are significantly reduced, although an increasing trend is observed in the figure, perhaps meaning that only the speed at which equilibrium is reached is slowed down.The remaining populations again show a decreasing trend that pushes them to very low values, Fig. 13.

Discussion
The olive tree damage caused by the olive moth is associated with the feeding behavior of its different generations.The anthophagous generation destroys a variable amount of flowers during the olive tree blooming, reducing the amount of set fruits.The carpophagous generation is responsible for the fruit drop in two periods, the first after the fruit setting in June/July and the second one starting at the beginning of September.The phylophagous generation usually does not originate important fruit losses, although severe attacks may negatively affect the tree development (Bento, 1999).
In view of these remarks, we now investigate the system equilibria to draw inferences on how to fight this olive pest.
The system can attain any one of the 12 equilibria listed in Table 2.But in what follows we should remember that the points  0 ,  8 ,  12 and  14 are achievable only with a substantial use of insecticides.We discuss at first the ones where the olive moth disappears.

The pest-free cases
At  0 = (0, 0, 0, 0) we observe the extinction of the four insect species, including P. oleae.Thus, the olive grove is free from the pest.This equilibrium is always admissible and turns out to be stable if and only if each extra mortality rate   with  = , , , , caused by the pesticides, is larger than the net reproduction rate of the corresponding population.
From the biological point of view, this equilibrium is achieved if the olive groves are subjected to curative treatments based on insecticides.).The other colors indicated in the legend represent the system trajectories under climate changes that supposedly induce a reduction in one or all insects growth rates, as specified below.Upper frame: only the moth growth rate  is affected; Lower frame: the growth rates of all species are affected in the same way as for moths.
The authorized pesticides in Portugal for the olive moth control are based on the neonicotinoid acetamiprid; the pyrethroids cypermethrin, deltamethrin and lambda-cyhalothrin; the organophosphate phosmet; and spinetoram, a multi-component tetracyclic macrolide in the class of spinosyn insecticides (DGAV, 2021).However, pesticides may be responsible for serious toxic effects on human health and the environment.They remain in the ecosystem and hamper the sensitive environmental equilibrium through bio-accumulation, reaching nontarget organisms such as humans and pests' natural enemies (Sharma et al., 2020).For example, Pitzer et al. ( 2021) describes acute and chronic effect of deltamethrin on human brain and behavior.Alternatively, the use of sexual pheromones were investigated but the results indicate that they would be useful and effective only with low populations levels.On the contrary, Bacillus thuringiensis can be used under organic production systems against the anthophagous generation.It was indeed found to reduce the pest up to 80 to 90%, Equilibrium  1 = ( 1 , 0, 0, 0) is better than the former, because the spiders survive.Spiders are generalist predators that feed mainly on insects.They are important natural enemies of relevant pests in agroecosystems because of their ubiquity and abundance (Benhadi-Marín et al., 2016).Usual agricultural management often relies on the use of pesticides, tillage, fertilization or landscape simplification.However, these practices may alter the diversity of spiders and their effectiveness as pest control agents (Benhadi-Marín et al., 2016).At first, for the equilibrium  1 to be feasible, the average reproduction rate must exceed the extra mortality rate due to pesticides  >   .Stability of  1 is achieved by satisfying the conditions (3.14), which explicitly can be rewritten as 0 < (  +  − ) <   and 0 < (  +  − ) <   .Thus, the reproduction rates of the populations  and  must respectively be lower than the extra corresponding mortality rate due to the pesticide plus a certain positive amount that depends on the spider hunting rate,  <   +,  <   +.Further, the mortalities in these right hand sides must be bounded above by the same net reproduction augmented by an extra factor, namely   + <    −1 +,   + <    −1 + .
At  2 = (0,  2 , 0, 0) only the E. flabellatus thrives thanks to the alternative nutritional resources in the environment.As for  1 , here the equilibrium is feasible if the net reproduction rate is larger than the extra pesticide-induced mortality rate  >   .The increase in the number of E. flabellatus leads to an increased parasitism of the olive moth larvae.This entails a further decrease of the moth's adult population, leading to a reduction in fruit damage.Equilibrium  2 is stable if and only if 0 < ( +    −   ) <     and 0 < (  +  − ) <   .The second stability condition requires in particular that the moths net reproduction rate must be lower than the combined mortality rates due to pesticide use and E. flabellatus parasitism, the latter indicating the importance for moth eradication of this parasitoid.Generally, the effect of E. flabellatus on the olive moth has been understood as unwanted because the highest parasitism rates were found in the phyllophagous generation (attacking 10 to 11% of the olive moth) and because its hyperparasitic behavior which may reduce also the populations of A. fuscicollis.Villa et al. (2016a), found that in olive orchards with herbicide application where the emergence of A. fuscicollis was significantly lower than in olive orchards with spontaneous vegetation, E. flabellatus was responsible for almost half of the parasitism.This supports our results and highlights the nonnegligible effect of non-specific natural control agents (which can rely on other resources for survival), which have a higher impact on the S. Pappalardo et al. particular situation of unfavorable conditions for the specific ones (such as low number of the pest or low food resources for adults).
At  3 = ( 3 ,  3 , 0, 0) the spiders and E. flabellatus populations coexist and biologically the situation for the olive trees is excellent with a population of parasitoids and predators persisting ready to fight possible pest infiltration.This equilibrium point undergoes a transcritical bifurcation with  1 ,  2 and  11 when respectively the parameters ,  and  vary as reported in Fig. 6.This equilibrium is unfeasible again if the pesticide extra mortality rate is larger than the E. flabellatus net reproduction rate.The free-pest scenarios highlights the ecological importance of generalist natural control agents such as the spiders or E. flabellatus.They may prove to be effective in the case of pest invasions of free-pest territories such as the Americas or Australia (CABI, 2021), where the olive growing is in expansion, or in the hypothetical case that the pest would become extinct in a territory where it is already present.However, in regions where the olive moth is well established the free-pest scenarios would be very unlikely.Therefore the scenarios described in the next subsection would apply.

The moth persistence scenarios
We now turn to the situations in which the olive pest is endemic.We try to discuss the conditions for which these equilibria would not be attainable, but one should keep in mind that if such an equilibrium becomes unfeasible or unstable, in general it is not possible to establish a priori where the system would settle moving away from it; thus it is very well possible that if one such pest-affected point is prevented to arise, the system would stabilize at another point where the pest is still present, so that eradication is not attained.The bifurcation diagram of Fig. 6 however could constitute a useful guide to the applied ecologist in the choice of the policy to be undertaken case by case.4.2.1. 8 = (0, 0, 0,  8 ) At the equilibrium  8 only the population of the P. oleae thrives.This is possible because it has essentially unlimited food resources within the olive agroecosytem.For its feasibility the insect net reproduction rate must exceed the extra mortality rate due to the pesticide, i.e.  >   .This is trivially satisfied in the absence of insecticide spraying.It is well known that in the absence of natural control agents, the size of the pest population grows rapidly starting in the spring until at the end of the season it reaches a value higher than 300 in real scenarios (Villa et al., 2021).For this equilibrium to be unachievable, we need to ensure that the feasibility or the stability conditions (3.17) must not hold.The former is implied by a spraying rate larger than the moth reproduction rate.Instead, instability is ensured by either one of the conditions  +    8 ≥   ,  + ℎ 8 ≥   or  8 ≥   +   .Thus  8 cannot be attained if, in the third case, the A. fuscicollis feeding rate on moth is larger than its combined natural and pesticide-induced mortalities.The first case amounts to state that the combined spiders' growth rate due to their alternative resources and predation on the moth exceeds their death rate due to insecticides; the second possibility is the corresponding situation for E. flabellatus.Note that in the absence of spraying, both these conditions are satisfied.This result is somewhat counterintuitive, implying that the moth-only point is not naturally achievable.Thus the pest must certainly coexist with one of its natural enemies.
Note that from this point the only moth-free equilibrium achievable is the origin, by suitably acting on the moth reproduction rate , compare Fig. 6.Modifications of either ,  or , i.e. the reproduction rates of spiders, A. fuscicollis and its parasitizing rate would move the system from  8 to other equilibria where however the moth still thrives.4.2.2. 9 = ( 9 , 0, 0,  9 ) Coexistence of spiders and P. oleae populations occurs at  9 .A sufficient condition for it to be unfeasible is given by a pesticideinduced extra moth mortality rate larger than its net reproduction rate.In the opposite case, for  −   being positive, the inequalities in (3.6) could still be violated, by using the corresponding difference for the spiders.The occurrence of equilibrium  9 can be prevented also by destabilizing it, violating the inequalities in (3.18).Thus the total E. flabellatus reproduction rate due to moth parasitism and alternative resources must exceed their total mortality due to spiders predation and pesticide action.Alternatively, a corresponding situation for A. fuscicollis should be ensured, so that its reproduction rate exceeds the natural as well as the combined mortality rates due to pesticides and spiders hunting.It is interesting to note that these conditions involve both species that do not appear in the equilibrium.Although somewhat counterintuitive, they can be explained by the fact that either one of these two species must invade this point, and this can occur only if the model parameters allow their thriving.
Direct moth eradication from this point can only be obtained by action on the parameter , its reproduction rate, to attain  1 , the spideronly equilibrium, see Fig. 6.Changes in , ,  and  would all keep the moth in the ecosystem, although other populations would appear; changing  has the same effect, but with the removal of spiders.Fig. 3 top, shows the bifurcation between  11 and  9 as  increases, while in the bottom frame the same transition occurs in terms of .Fig. 4 in the top frame depicts instead the transcritical bifurcation between  9 and  13 when the parameter  changes.

𝑃
Violating the feasibility conditions for  10 , the point where E. flabellatus and moth thrive, can be obtained exactly as in the corresponding discussion for equilibrium  9 , by substituting the difference between the E. flabellatus reproduction rate and the spraying-induced mortality in place of the spiders' one.More easily, also the equilibrium is unfeasible again if the pesticide extra mortality rate is larger than the moth net reproduction rate.Also the discussion for instability parallels the one of the previous equilibrium.The spiders combined reproduction rates based on feeding on moth and E. flabellatus as well as alternative resources must exceed their spraying-induced mortality, or else the A. fuscicollis natural mortality and the ones due to insecticide action and E. flabellatus predation must be lower than their hunting rate on moths.
Moth eradication can be ensured only by reducing its reproduction rate , and the system would move to the point  2 , where only E. flabellatus thrives, see Fig. 6.Acting on ,  and  would maintain the moth presence, together with some other species.A reduction of  would lead to the extinction of E. flabellatus, but the moths would still persist.In Fig. 3 the bifurcations involving  10 have already been described above when considering  9 .In addition, the bottom frame of Fig. 4 contains the transition from this point to  14 when  decreases.

𝑃
For the A. fuscicollis-free point  11 the interpretation of both feasibility and stability conditions is rather hard.However, the equilibrium is certainly unstable if condition (A.9) is violated.This can be achieved by requiring that the A. fuscicollis growth rate by moth feeding exceeds its combined natural plus spray-induced plus spiders' and E. flabellatus predation related mortalities.In such situation, however, it is likely that A. fuscicollis invades the system and thus coexistence is attained.Therefore pest eradication would not be achieved.Hence, despite the presence of two natural control agents, the pest thrives, this being probably due to the fact that the two natural control agents have alternative nutrients on which they can feed.From Fig. 6 it is seen that three more transcritical bifurcations may occur at this point, on of which in particular leads to the moth-free point  3 .Therefore, destabilizing  11 by violating one of the remaining stability conditions (A.6) or (A.8) will eventually lead to the pest eradication.In view of their rather involved nature, however, it is hard to assess which condition to use, and also on which parameters to act to modify it.
From this point the moth can be eradicated by acting on  or , reducing the moth birth rate or increasing the spiders' one, thereby bringing the system to equilibrium  3 .Note that a large reduction of  instead induces the disappearance of the spiders; the system would then attain  10 .Modifying instead  and  has the effect of removing one of the useful moth natural enemies, keeping the pest in the system.Acting on  would still keep the moth alive, but depending on the change, either spiders disappear, at  10 or A. fuscicollis invades the system and coexistence  15 is attained, Fig. 6.
From Fig. 6 it is evident that no change in any parameter would lead to direct moth eradication.Fig. 3 shows the two bifurcations involving  11 , taking the system to  10 as  decreases, and to  9 when  increases.already discussed been described above when considering  9 .4.2.5. 12 = (0, 0,  12 ,  12 ) To guarantee unfeasibility of the spiders and E. flabellatus-free point  12 , if insecticide is used, the moth spraying-induced mortality being larger than their reproduction rate is sufficient.Further, suitable multiples of the difference between the moth reproduction rate and the corresponding insecticide killing rate establish a range, outside of which the sum of the natural and insecticide-caused mortalities of A. fuscicollis should lie in order to ensure unfeasibility.To violate stability of this equilibrium, the insecticide killing action should be smaller than the combined insects reproduction rates, coming by hunting their prey parasitizing their hosts or feeding on alternative resources, the insects being either spiders or E. flabellatus.In the absence of spraying, this equilibrium is unconditionally unstable, and therefore not achievable.It is likely that acting on  the spiders birth rate due to other resources introduces in the system natural control agent populations, the spiders themselves or E. flabellatus, without however eradicating the pest.From Fig. 6 it is evident that no change in any parameter would lead to direct moth eradication.4.2.6. 13 = ( 13 , 0,  13 ,  13 ) Feasibility of  13 cannot be easily interpreted in ecological terms.In this case, to have instability, the spiders' conversion rate on moths should exceed the product of their conversion rate on A. fuscicollis and conversion rate of the latter on moths, but this is not sufficient.A sure way of unsettling this equilibrium is by ensuring that the total E. flabellatus growth rate by using alternative resources as well as by feeding on moths and A. fuscicollis is larger than their combined mortality due to spiders predation and insecticide action.In this second situation the likely outcome of increasing  is the coexistence equilibrium, with the consequent non resolution of the pest problem, Fig. 6.In this case too, moth eradication cannot be obtained without first moving toward other ''neighboring'' equilibria, see Figs. 5 and 6. 4.2.7. 14 = (0,  14 ,  14 ,  14 ) The feasibility of  14 is again hard to interpret.Instability can be ensured by a low enough spider's killing rate due to the use of insecticides, which should be overcome by their combined reproduction rates on all their available resources, whether explicitly modeled or being alternative food supplies.In addition, one could exploit the situation in which the E. flabellatus conversion rate by feeding on moths could be larger than the product of their conversion rate on A. fuscicollis and conversion rate of the latter on moths, although by itself this condition does not guarantee instability.
From Fig. 6 it is not possible to directly remove the pest.Reducing  as stated above has the effect of removing the E. flabellatus, thereby attaining the point  12 , while increasing it brings the system to coexistence.Further transitions appear in Figs. 4 and 5, respectively with  10 and  12 .

Final remarks
The olive moth is subjected to the attack by several natural control agents in the olive agroecosystem.In the model presented here three of the most important organisms naturally feeding on the olive moth are represented.This model intends to elucidate how the abundance of P. oleae would change depending on corresponding variations in the population sizes of its natural enemies present in the ecosystem and accounted for in the equations.In the model a specific parasitoid, a generalist parasitoid and a generalist predator are included with the aim of gaining information about their effectivity for reducing the pest numbers.Their populations behaviors under different climate change scenarios are also considered.Thus, we compared the population levels attained in the various system equilibria.Starting from the coexistence, when one of the top two generalist natural enemies is removed and A. fuscicollis thrives, we observe that the pest population slightly changes (equilibria  13 and  14 ), although these results are highly affected by the effects of climate change.If instead A. fuscicollis is removed (equilibrium  11 , Fig. 11) the pest population suddenly explodes, reaching values about ten times higher than those attained at coexistence.The same phenomenon is observed when only two populations thrive, one of them being the olive moth.If the pest coexists with just A. fuscicollis ( 12 , Fig. 7), their equilibrium level is scantly affected.However, if they thrive either only with spiders or only with E. flabellatus (respectively  9 , Fig. 12;  10 , Fig. 9), they are heavily affected and their number is highly increased.
In all cases the action of the climate change can be observed.Thus, at  12 , (the moth and A. fuscicollis both thriving) (Fig. 7) and  14 (the spider-free point) (Fig. 8) the moths maintain always low levels.At  10 (the E. flabellatus-moth equilibrium) (Fig. 9) a reduction of the moth to almost vanishing values is observed when the moth reproduction rate reduces between four-and tenfolds.These results indicate the contribution to the pest reduction of both parasitoids under a climate change scenario.Accordingly, at the E. flabellatus-free equilibrium  13 (Fig. 10), the moths increase from almost extremely low values near 0 (with the normal moth reproduction rate) to 100 (with a tenfold reduction of its reproduction rate) and at the A. fuscicollis-free equilibrium  11 (Fig. 11) the moths increase from 50 (with a normal moth reproduction rate) to 100 (after a 10 fold reduction of the moth reproduction rate), suggesting the relevance of the combined action of both parasitoids in the regulation process of the pest in the climate change context.
From these findings, it is apparent that the role of a specialist parasitoid in keeping the pest numbers in check is more important than the one of generalist predators, even if they act jointly in combination in a non-climate change scenario, but both seem to be important when the action of the climate change takes place.Additionally, the  2 equilibrium indicates that under of unfavorable conditions for A. fuscicollis, such as very low number of the pest -which in certain years with unusual climatic conditions occurs, or low food resources for adults originated for example after the herbicide application (Villa et al., 2016b), or even invasions of P. oleae to free-pest territories (and therefore A. fuscicollis-free), the role of E. flabellatus (or other facultative parasitoids) could encompass a higher importance.This is in the line of Dainese et al. (2017) results, who found complementary effects among different guilds of natural enemies and an improvement of the biological control efficacy when increasing the non-crop areas (which may involve extra resources not explicitly considered in our model).
The equilibria  0 -where all arthropods are absent,  8 -where only the moth is present,  12 -where A. fuscicollis and the moth are present, and  14 -where E. flabellatus, A. fuscicollis and the moth are present, are not achievable without spraying.Additionally, in general the equilibria are not feasible if the pesticide extra mortality rate is larger than the net reproduction rate for all or some of the organisms involved.However, the available information about how pesticides affect to the mortality rates of the studied organisms is very limited or absent, pointing at the urgency of evaluating the effect of pesticides in natural control agents.The same occurs in the case of the effects of climate change.
Here, we assumed the natural enemies would show a similar behavior to the olive moth, or that they would not be affected by climate change.However, a most probable real scenario would attain different results.For example, Villa et al. (2017a) indicate that A. fuscicollis hazard mortality rate highly increase with small variations of temperature.Therefore, both pesticides and climate change effects on the olive moth trophic web should be urgently investigated.

Model limitations
We present now a brief discussion of the possible criticism the proposed model could be subject to.
First of all, in the modeling process, using Occam's razor, we attempted to obtain a balance between the actual situation and a mathematically feasible compromise.Indeed, we completely disregarded the fact that insects undergo several development stages.Namely, we ignored the larval and pupa stages and we focused just on the adult individuals.This entails also that possible delays due for instance to hatching of eggs have been disregarded.Also, when developing dynamical models, we opted for having coefficients, i.e. model parameters, that are independent of time, i.e. in particular independent of the season.Using constant values for these coefficients means essentially that we focus just one season with suitable thriving conditions.This makes the system autonomous and renders the analysis possible.In other situations time-dependent parameters can be considered, see e.g.Rossini et al. (2021).
Furthermore, from the mathematical viewpoint, (2.1) suffers from the fact that the mutual pairwise interactions of the species are modeled as bilinear terms.This entails that there is no upper limit in how much a predator can eat of a captured prey.In reality, feeding is subject to a satiation phenomenon, for which, even in the presence of an unlimited amount of food, the actual intake rate drops and the amount eaten attains an upper value.This behavior is captured by a concave response function, called Holling type II (HTII), represented by a hyperbola which raises up from the origin to approach a horizontal asymptote.To introduce such expressions into (2.1) would make the model closer to reality, at the expense of complicating very much the mathematics, so that most likely the analysis would be altogether impossible, and only simulations would provide some information.Moreover, in the HTII formulation, the speed at which the plateau is approached depends on the attack rate.Indeed note that for the hyperbola () = (1 + ) −1 the derivative is  ′ () = (1 + ) −2 and its evaluation at the origin  = 0 gives  ′ (0) = .Thus  here represents the attack rate, and the higher it is, the higher the slope and the faster the asymptote is approached.Moreover, the attack rate in turn largely relies on the encounter rate.The encounter rate in the field may not be large enough to imply a fast approaching of the asymptote.The hyperbola and its linear approximation in this case are therefore about the same.For these reasons, we opted for the current less complicated formulation (2.1).
The simulation of these models requires a deep knowledge about the behavior of the involved organisms in the field.In particular, the specific values of at least some of the parameters appearing in the model would be required.As it is apparent from Section 3.2, there is still an important gap in the information available so far.More dramatic is the lack of knowledge regarding the behavior of these organisms under a range of climate variables values changes, see Section 3.2.2.Additionally, not only the organisms itself, but also their different life cycle stages or different generations along the year may respond differently to changes in climate variables, similarly to the suggested by Pollard et al. (2020) for Phratora vulgatissima L. Therefore, increasing the laboratory and field research on this pest and its associated trophic web becomes an urgent matter.Furthermore, this model did not consider the effect that sex ratios may have on the organism populations, see e.g.Rossini et al. (2021).Finally, in field conditions the system is certainly much more complex.Here, we consider two important parasitoids and spiders as generalist predators.But it is known that other parasitoids or predators [e.g., the Chrysopideae Chrysoperla carnea (Stephens) or the Anthocoridae Anthocoris nemoralis (Fabricius)] may or do have important roles in the trophic web of the olive moth (Bento et al., 2007;Paredes et al., 2014).

Conclusion
The goal of this work is mainly to study the interaction between the natural enemies in the biological control of pest-infested olive trees, assessing the ecosystem steady states for feasibility and stability of a tritrophic system composed by the olive tree (the crop), the olive moth (the pest), A. fuscicollis (the specialist parasitoid), E. flabellatus (the facultative parasitoid) and spiders (generalist predators).
Thus we tried to analyze in detail, from a mathematical point of view, the possibilities to eradicate the pest to avoid possible damage to the olive tree crop.
Our results indicate that in normal circumstances the natural enemy which will perform the highest pressure over the olive moth is the specific parasitoid A. fuscicollis, and under particular conditions, E. flabellatus or spiders might have significant roles in controlling the pest.
In sum, our results indicate the high importance of specific parasitoids.The role of hyperparasitoids or generalist predators under particular circumstances must not be overlooked or disregarded.Additionally, we pointed out the huge gap of biological knowledge that still exists on the trophic web of the olive moth.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

A.2. Stability
The last condition is Extensively, it can be written as  [3,4;3,4] are unconditionally satisfied, tr (𝐽 [3,4;3,4]  In the particular case of no insecticide use, these conditions cannot be satisfied, and this point becomes unconditionally unstable.

Appendix B. Bifurcations
Let us rewrite the model (2.1) in synthetic notation as  ′ () = (), where   = (, , , ) and   = ( 1 ,  2 ,  3 ,  4 ) denotes the right hand side of the system.The analysis of the bifurcations is carried out using Sotomayor's Theorem, Perko (2011) and the notation therein, using subscripts to denote partial derivatives.In particular we will need the following quantities: and where the second derivatives for all the remaining combinations of the variables are zero.Also, all the third order derivatives vanish, i.e.     = 0,  = 1, … , 4 and ,  ,  ∈ {, , , }.This last condition implies that no pitchfork bifurcation is possible.In what follows, the eigenvalues that are negative will not give rise to bifurcations for feasible values of the model parameters and therefore will not be considered.

B.1.1. Eigenvalue 𝜆 1
Take as bifurcation parameter  and let ẑ ∶=   .The right  and left  eigenvectors of the Jacobian are  =  = (1, 0, 0, 0)  .Upon suitable differentiation, in this case we find showing a transcritical bifurcation at the equilibrium point  0 for the critical parameter value  = ẑ, between  0 and  1 for  = ẑ.

B.5.3. Eigenvalue 𝜆 3
Taking as bifurcation parameter   and letting m ∶=  8 −   we find that this threshold is feasible for  8 >   .But the threshold for the parameter values used, is negative, thus no feasible bifurcation can occur in terms of   .

B.6.2. Eigenvalue 𝜆 2
Taking as bifurcation parameter   and letting m ∶=  9 −  9 −   , for the parameter values used the threshold is negative,  9 <  9 +   , thus no feasible bifurcation can occur in terms of   .

Fig. 7 .
Fig. 7. Effects of climatic changes for the Ageniaspis-Prays-only equilibrium,  12 .Top , bottom .The red trajectories represent the current system behavior, with parameters given by (3.23), (3.20), (3.28), (3.29), (3.30), (3.31).The other colors indicated in the legend represent the system trajectories under climate changes that supposedly induce a reduction in one or all insects growth rates, as specified below.Upper frame: only the moth growth rate  is affected; Lower frame: the growth rates of all species are affected in the same way as for moths.

Fig. 9 .
Fig. 9. Effects of climatic changes for the Elasmus-Prays-only equilibrium  10 .Top , bottom .The red trajectories represent the current system behavior, with parameters given by (3.22), (3.20), (3.28), (3.29), (3.30), (3.31).The other colors indicated in the legend represent the system trajectories under climate changes that supposedly induce a reduction in one or all insects growth rates, as specified below.Upper frame: only the moth growth rate  is affected.Note that in this column the vertical scale for Elasmus starts from 90 and not from 0. Lower frame: the growth rates of all species are affected in the same way as for moths.In this column the vertical scale for Elasmus starts instead from 80, to better show the differences in the graphs.

Fig. 10 .
Fig. 10.Effects of climatic changes for the Elasmus-free equilibrium  13 .Top , center , bottom .The red trajectories represent the current system behavior, with parameters given by (3.26), (3.20), (3.28), (3.29), (3.30), (3.31).The other colors indicated in the legend represent the system trajectories under climate changes that supposedly induce a reduction in one or all insects growth rates, as specified below.Upper frame: only the moth growth rate  is affected; Lower frame: the growth rates of all species are affected in the same way as for moths.

Fig. 11 .
Fig. 11.Effects of climatic changes for the Ageniaspis-free equilibrium  11 .Top , center , bottom .The red trajectories represent the current system behavior, with parameters given by (3.25), (3.20), (3.28), (3.29), (3.30), (3.31).The other colors indicated in the legend represent the system trajectories under climate changes that supposedly induce a reduction in one or all insects growth rates, as specified below.Upper frame: only the moth growth rate  is affected; Lower frame: the growth rates of all species are affected in the same way as for moths.

Fig. 12 .
Fig. 12. Effects of climatic changes for the Elasmus-Ageniaspis-free equilibrium.Top , bottom .The red trajectories represent the current system behavior, with parameters given by (3.21), (3.20), (3.28), (3.29), (3.30), (3.31).The other colors indicated in the legend represent the system trajectories under climate changes that supposedly induce a reduction in one or all insects growth rates, as specified below.Upper frame: only the moth growth rate  is affected; Lower frame: the growth rates of all species are affected in the same way as for moths.

Fig. 13 .
Fig. 13.Effects of climatic changes for the coexistence equilibrium.The red trajectories represent the current system behavior, with parameters given by (3.27), (3.20), (3.28), (3.29), (3.30), (3.31).The other colors indicated in the legend represent the system trajectories under climate changes that supposedly induce a reduction in one or all insects growth rates, as specified below.Upper frame: only the moth growth rate  is affected; Lower frame: the growth rates of all species are affected in the same way as for moths.

Table 1
The parameters and their interpretation.

Table 2
Summary of feasibility and stability conditions of the equilibrium points of the model.

Table 3
Available data on the other insects.The first and second row data are fromVilla et al.

Table 4
Percentage of P. oleae egg mortality under various temperature and relative humidity conditions.