A theoretical study on effects of cultivation management on biological pest control: A spatially explicit model

Biological Control 93 (2016) 37–48

Contents lists available at ScienceDirect

Biological Control journal homepage: www.elsevier.com/locate/ybcon

A theoretical study on effects of cultivation management on biological pest control: A spatially explicit model Yusuke Ikegawa a,⇑, Kotaro Mori b, Makiko Ohasa b, Ippei Fujita c, Takeo Watanabe d, Hideo Ezoe a, Toshiyuki Namba a a

Graduate School of Science, Osaka Prefecture University, Sakai, Osaka, Japan Central Research Institute, Ishihara Sangyo Kaisha, Ltd., Kusatsu, Shiga, Japan Graduate School of Agriculture, Kagawa University, Miki, Kagawa, Japan d Kagawa Agricultural Experimental Station, Ayauta, Kagawa, Japan b c

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


Cultivation management such as

thinning may interfere with biological control.
A metapopulation model is built for control of herbaceous thrips by predatory thrips.
Too intensive thinning of juvenile strawberry fruits disrupts biological control.
Spatial structure and high mobility of enemies can offset the negative effect.
Dispersed introduction of sufficiently abundant enemies is optimal for pest control.

a r t i c l e

i n f o

Article history: Received 14 May 2015 Revised 2 November 2015 Accepted 19 November 2015

Keywords: Alternative diet Fruit thinning Harvesting Metapopulation model Natural enemy Optimal release strategy

a b s t r a c t Cultivation management to improve quality or yield of crops causes periodic disturbances in agricultural fields and increases mortality of arthropods. Thus, it may interfere with biological control of pests. An herbivorous pest thrips (Frankliniella intonsa (Trybom)) is a significant pest for strawberry (Fragaria x ananassa) in a greenhouse and a polyphagous predatory thrips (Haplothrips brevitubus (Karny)) is its natural enemy. Because strawberry flowers bloom sequentially, thinning of juvenile fruits and harvesting of mature fruits are also sequentially implemented. Such management practices work as periodic disturbances for both the pest and natural enemy. We constructed a spatially explicit metapopulation model to examine effects of periodic disturbances on efficiency of biological control. The natural enemy was more susceptible to periodic disturbances than the pest, because the former lost diets whereas predation pressure on the latter was weakened. However, the high mobility and alternative diet of the natural enemy could compensate for the negative effect of periodic disturbances. If economic value of individual harvested fruits was diminished by a burden of excess fruits, total yield could increase with the intensity of fruit thinning. The more efficient release strategy of the natural enemy was either the clumped or dispersed depending on the mobility and initial number of the natural enemy and the intensity of periodic disturbances. Thus, for consistently practicing cultivation management and biological control, it is important to apply appropriate intensity of fruit thinning and to release a sufficient number of natural enemies in proper arrangement depending on the intensity of periodic disturbances. Ó 2015 Elsevier Inc. All rights reserved.

⇑ Corresponding author. E-mail address: [email protected] (Y. Ikegawa). http://dx.doi.org/10.1016/j.biocontrol.2015.11.008 1049-9644/Ó 2015 Elsevier Inc. All rights reserved.

38

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

1. Introduction Studies on biological control have a tendency to focus on ecological properties of pests and natural enemies (Murdoch and Briggs, 1996; Janssen et al., 1998). However, it has recently been suggested that the nature of crop plants (e.g., either resistant or not to pests) or cultivation management by farmers (e.g., thinning, harvesting, weeding, and tillage) can affect efficiency of pest control (Verkerk et al., 1998; Ferron and Deguine, 2005). Such cultivation management, as well as spraying chemical pesticides against arthropods (Hardin et al., 1995; Morse, 1998; Cohen, 2006), are often recognized to work as periodic disturbances toward arthropods in agricultural fields and increase their mortality (Stinner and House, 1990; Thomas and Jepson, 1997; Thorbek and Bilde, 2004). Thus, quantitative examination of effects of the periodic disturbances on population dynamics of the pests and natural enemies is crucial for establishing efficient biological control practices. Some theoretical studies have investigated the effect of periodic disturbances on population dynamics of prey and predators (or hosts and parasitoids) (Matsuoka and Seno, 2008; Sabatier et al., 2013). Matsuoka and Seno (2008) found that instantaneous reductions in host (pest) numbers due to periodic disturbances caused paradoxical increases (resurgence) in the host at equilibria even if the disturbances were not effective against parasitoids (natural enemies). However, in their studies, spatial scale and heterogeneity of agricultural fields are not considered. Other theoretical studies have dealt with spatially explicit metapopulation models, in which organisms were spatially localized and allowed to move in space (Sherratt and Jepson, 1993; Ives and Settle, 1997; Childs et al., 2004; Strevens and Bonsall, 2011). Some of them could explain the pest resurgence after the implementation of periodic disturbances (Sherratt and Jepson, 1993) and show differential effects of their intensity and frequency on suppression of pests at equilibrium (Ives and Settle, 1997; Childs et al., 2004; Strevens and Bonsall, 2011). Short-term transient-phase population dynamics is crucial rather than long-term one in the case of annual crops because the crops are harvested before the population dynamics achieves equilibrium. Some empirical studies showed that the transient population dynamics strongly depended on initial numbers and distributions of pests and natural enemies (Yasuda and Ishikawa, 1999; Alatawi et al., 2011). However, the short-term population dynamics have rarely been examined in theoretical studies (but see Bommarco et al., 2007). In this study, we investigate effects of periodic disturbances on pest suppression during a season in a greenhouse by numerically analyzing a mathematical model. An herbivorous thrips (Frankliniella intonsa (Trybom)) is a significant pest of the strawberry (Fragaria x ananassa) and a polyphagous predatory thrips (Haplothrips brevitubus (Karny)) is its natural enemy. Clusters of strawberry flowers emerge sequentially, and thinning of juvenile fruits and harvesting of mature fruits are successively implemented. Pests and natural enemies on the thinned or harvested fruits are removed by the sequential fruit thinning and harvesting because they typically inhabit and reproduce on the flowers and fruits in strawberry fields (Ohasa unpublished data). Thus, the fruit thinning and harvesting cause periodic disturbances against the pest and natural enemy. In order to examine effects of intensity of the fruit thinning and harvesting on efficiency of the biological control, we construct a spatially explicit metapopulation model and numerically simulate it. For numerical calculations, we parameterize life histories and interactions of both species, following some previous empirical studies. We first examine effects of the intensity of periodic disturbances and mobility of the pest and natural enemy. Second, we examine optimal release strategies of the

natural enemy which decrease cumulative damage or increase total yield in order to effectively control the pest by assuming a few different initial numbers and distributions of the natural enemy. Third, we investigate how the negative feedback from damaged plants on fecundity of the pest influences the cumulative damage and total yield. Finally, we will discuss implications of our results for the efficient biological control in agricultural fields under some cultivation management. 2. Materials and methods 2.1. Ecological properties of thrips and strawberry An herbivorous thrips, F. intonsa, is a widespread agricultural pest for a broad spectrum of crops including 41 families and 108 species in Japan (Murai, 1988). For example, strawberry fruits sapped by larvae of F. intonsa get dull-hued color and lose commercial values. A polyphagous predatory thrips, H. brevitubus, is widely observed in the temperate zone in Japan and both the adults and larvae can prey on a broad range of thrips including F. intonsa (Kakimoto et al., 2006; Baba et al., 2008; Fukuda et al., 2008). It can survive and develop on pollen of some crops including strawberries even if its animal prey are scarce (Morita et al., 2008). Both the pest and natural enemy have relatively short developmental time from egg to adult emergence (10:3  0:6 days in F. intonsa and 18:9  0:3 days in H. brevitubus) and high survival from egg to adult emergence with a sufficient amount of diet (85.0% in F. intonsa and 94.7% in H. brevitubus) (Murai, 1988; Kakimoto et al., 2006). Thinning of juvenile fruits done in order to enhance growth of the residual (non-thinned) fruits is a common management practice for many crops, and individuals of the pest and natural enemy on thinned fruits are removed together with the thinned fruits. Because clusters of flowers of the strawberry sequentially emerge and grow, farmers carry out the fruit thinning and harvesting cluster by cluster and the inhabiting pest and natural enemy are exposed to instantaneous reductions in number. Thus, the cultivation management can cause periodic disturbances on population dynamics of the pest and natural enemy. 2.2. Model description 2.2.1. Temporal population dynamics of pests and natural enemies We consider a two-dimensional square lattice space with the size L by L, and assume that each lattice point is occupied by a plant. We assume three developmental stages of the strawberry, i.e., flower, juvenile fruit and mature fruit (Fig. 1). At the start of a simulation (t = 0), there emerges the first cluster bearing K flowers and no fruit. Then, all the flowers are pollinated and grow simultaneously to juvenile fruits at t = s (=7 days). Then, K0 juvenile fruits are thinned (K P K 0 ; K0 = 0 means no fruit thinning). The remaining K  K0 juvenile fruits grow into mature fruits after further s time steps (t = 2s). At the same time when the old cluster enters the mature-fruit stage, a new cluster bearing K flowers emerges (there exist K  K0 mature fruits in the old cluster and K flowers in the new cluster simultaneously). After further s time steps (t = 3s), all the mature fruits in the old cluster are harvested and the flowers in the new cluster develop to juvenile fruits (all procedures are schematically shown in Fig. 1). Hereafter, we call an individual flower, juvenile fruit, or mature fruit a ‘site’ as a unit habitat of the pest and natural enemy. Intensity of periodic disturbances is defined as the proportion of juvenile fruits that are removed by the fruit thinning, K0 /K. We assume that individuals of the pest and natural enemy on the thinned K0 sites are simultaneously

39

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

harvesting K - K’ fruits from 1st cluster

thinning K’ fruits

1st flower

cluster

juvenile fruit

mature fruit

2nd

emerging K flowers

flower

harvesting K - K’ fruits from 2nd cluster

thinning K’ fruits

cluster

juvenile fruit mature fruit

thinning K’ fruits

3rd

emerging K flowers

flower

cluster

harvesting K - K’ fruits from 3rd cluster

juvenile fruit mature fruit

emerging K flowers

flower

………

emerging K flowers

0

2

3

6

5

4

Time (t) Fig. 1. Schematic representation of developmental stages of the strawberry along time course (t). s indicates a duration of each stage of the strawberry.

removed when the fruit thinning is carried out. Likewise, individuals on the harvested K  K0 sites are removed when the harvesting is carried out. Population dynamics of the pest and natural enemy at each site is described by a discrete-time-stage-structured prey-predator (pest-enemy) model by expanding discrete-time prey-predator (Neubert and Kot, 1992) and stage-structured host-parasitoid models (Murdoch et al., 1992; Tuda and Shimada, 2005). We assume that both the pest and natural enemy have four life stages, i.e., egg, larva, pupa, and adult, and population densities of the pest at these stages at time step t are described as C te , C tl , C tp , and C ta , respectively. Likewise, population densities of the natural enemy at egg, larva, pupa, and adult stage at time step t are described as N te , N tl , N tp , and N ta , respectively. Here, we count only female individuals of the pest and natural enemy. We assume that population densities of the pest and natural enemy vary during a developmental stage of the strawberry. Population density of the pest at stage i at the next time step (C tþ1 ; i 2 fe; l; p; ag) is expressed as follows, i except for time steps when periodic disturbances are applied:

C etþ1 ¼ C te þ 0:5ra C ta  ðre þ de ÞC te ; C ltþ1 ¼ C tl þ r e C te  ðr l þ dl þ

X

ð1-aÞ

f li Nti ÞC tl

ð1-bÞ

C ptþ1 ¼ C tp þ r l C tl  ðr p þ dp þ f pa Nta ÞC tp ;

ð1-cÞ

C atþ1 ¼ C ta þ r p C tp  ðda þ f aa Nta ÞC ta ;

ð1-dÞ

i¼l;a

where ra is the oviposition rate of the adult pest; ri is the development rate of the pest at stage i (i 2 fe; l; pg); di is the densityindependent mortality of the pest at stage i (i 2 fe; l; p; ag); fij is the predation rate of the natural enemy at stage j on the pest at stage i (i 2 fl; p; ag; j 2 fl; ag). We assume that female-to-male sex ratio is 1:1. Because the adult pest oviposits eggs under the epidermal tissue of crops, we assume that no egg can be preyed upon. The larval natural enemy can only prey on the larval pest because of a constraint on size while the adult natural enemy can prey on all stages of the pest, except for its eggs. We assume no density dependent regulation in population growth.

Population density of the natural enemy at each stage at the next time step is expressed as follows, except for time steps when periodic disturbances are applied: 0

Netþ1 ¼ Nte þ 0:5r 0a ðC tl ; C tp ; C ta ÞNta  ðr 0e þ de ÞN te

ð2-aÞ

0

ð2-bÞ

0

ð2-cÞ

Nltþ1 ¼ Ntl þ r 0e N te  fr 0l þ dl ðC tl ÞgNtl ; Nptþ1 ¼ Ntp þ r 0l Ntl  ðr0p þ dp ÞNtp ; 0

Natþ1 ¼ Nta þ r 0p Ntp  da Nta ;

ð2-dÞ

where ri0 is the development rate of the natural enemy at stage i (i 2 fe; l; pg); di0 is the density-independent mortality of the natural enemy at stage i (i 2 fe; l; p; ag). We assume that female-to-male sex ratio is 1:1. We also assume that mortality of the larval natural enemy dl0 increases as the pest density decreases, but it is moderated by the constant supplements from flowers (i.e. pollen): 0

dl ðC tl Þ ¼ dmax  ðdmax  dmin Þ

sl þ f ll C tl ; 1 þ sl þ f ll C tl

ð3Þ

where sl is the amount of supplements (pollen) from flowers to the larval natural enemy (sl = 0 in periods without flowers); dmax and dmin are mortalities of the larval natural enemy in the absence of all diets and in the presence of sufficiently abundant diets, respectively. Likewise, we assume that the oviposition rate of the adult natural enemy ra0 depends on the supplements and population densities of the pest:

sa þ r0a ðC tl ; C tp ; C ta Þ

¼ r min þ ðr max  r min Þ

X

f ia C ti

i¼l;p;a

1 þ sa þ

X

f ia C ti

ð4Þ

i¼l;p;a

where sa is the amount of supplements from flowers to adults of the natural enemy (sa = 0 in periods without flowers); rmax and rmin are the oviposition rates of the adult natural enemy in the presence of sufficiently abundant diets and in the absence of all diets, respectively.

40

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

2.2.2. Dispersal of pests and natural enemies Larvae and adults of the pest and natural enemy are assumed to be mobile and they leave a site at per capita departure rates gi and gi0 at stage i, respectively (i 2 fl; ag). An individual that has left a site reestablishes itself at a site on the same plant or on one of the four neighboring plants. Since larval mobility is low, we assume that larvae of both species can never transfer to the adjacent plants and the departure rates are low; gl = gl0 = 0.05 (Table 2). The ratio of between- to within-plant reestablishment of adults is assumed to be ha to 1  ha for the pest and ha0 to 1  ha0 for the nat0 ural enemy (0 6 ha ; ha 6 1). Because it was observed that both the pest and natural enemy prefer flowers to juvenile and mature fruits (Fujita unpublished data), we assume that individuals of the pest and natural enemy leaving flower sites do not get reestablished at mature fruit sites. For simplicity, we assume that within the same plant, individuals can colonize any site except for the departing one at an equal probability under the constraints explained above. Likewise, we assume that individuals that move to the adjacent plants can choose any of the four neighboring plants with the same probability and then colonize any site on the chosen plant with an equal probability under the constraints explained above. Finally, we assume that boundaries of the field are reflecting for both the pest and natural enemy because a greenhouse is a relatively closed system, in comparison with an open field. 2.3. Parameterization and analysis

Table 2 Dispersal parameters of the pest and natural enemy. Value

Emigration rates: gl 0.05 ga Variable gl 0 0.05 ga 0 Variable

Description Departure Departure Departure Departure

rate rate rate rate

of of of of

Symbol

Value

Developmental rates re 0.3453 rl 0.1417 rp 0.1417 9.600 ra de 0 dl 0 dp 0 da 0.01923

the the the the

larval pest adult pest larval natural enemy adult natural enemy

Intra-plant dispersal rates: ha Variable Between-plant reestablishment rate of the adult pest ha 0 Variable Between-plant reestablishment rate of the adult natural enemy

Description of the pest: Hatching rate of the pest eggs Pupation rate of the larval pest Eclosion rate of the pupal pest Oviposition rate of the adult pest Mortality of the pest eggs Mortality of the larval pest Mortality of the pupal pest Mortality of the adult pest

Developmental rates of the natural enemy: 0.2222 Hatching rate of the natural enemy re0 eggs 0.1041 Pupation rate of the larval natural rl0 enemy 0.2083 Eclosion rate of the pupal natural rp0 enemy rmax 3.600 Maximum oviposition rate of the adult natural enemy rmin 0 Minimum oviposition rate of the adult natural enemy de0 0 Mortality of the natural enemy eggs dmax

0.1

dmin

0

dp0 da0

0 0.02865

Predation rates: fll 0.1350

We need demographic parameter values of the pest and natural enemy to numerically analyze the model. Developmental rates and density-independent mortalities (per day) of the pest and natural enemy are derived from Murai (1988) and Kakimoto et al. (2006), respectively (Table 1). Since animals are usually provided with sufficient diet in experiments for examining the performance of the natural enemy, we have no data for the maximum mortality (caused by starvation) of the larval natural enemy dmax. Therefore, we assume a hypothetical value shown in Table 1. Likewise, because the minimum oviposition rate of the adult natural enemy rmin is not known, we assume that the adult cannot oviposit in the absence of any diets at each time step (rmin = 0). Because H. brevitubus can be reared and bred only by feeding on pollen in experimental conditions (Morita et al., 2008), we assume that the amount of constant supplements (pollen) to the larval and adult natural enemies is enough for them to survive and reproduce. Because mortalities of the egg, larva, and pupa of the pest and egg and pupa of the natural enemy are very low in experimental conditions, we fixed them to zero (shown in Table 1). Predation rates of the natural enemy on the pest per day are derived from Ohasa (unpublished data). Because there is no data on the predation rate of the adult enemy on the pupal pest, we assume an intermediate value between those on the adult pest and on the larval pest. All the above values are fixed over all simulations, if not specified

Symbol

Table 1 Demographic parameters of the pest and natural enemy.

fla

0.5625

fpa

0.2913

faa

0.0200

Maximum mortality of the larval natural enemy Minimum mortality of the larval natural enemy Mortality of the pupal natural enemy Mortality of the adult natural enemy Predation rate from the larval natural enemy on the larval pest Predation rate from the adult natural enemy on the larval pest Predation rate from the adult natural enemy on the pupal pest Predation rate from the adult natural enemy on the adult pest

Supplements to the natural enemy: sl 5.000 Supplements to the larval natural enemy sa 5.000 Supplements to the adult natural enemy

Reference Murai (1988)

Kakimoto et al. (2006)

Arbitrarily assigned Kakimoto et al. (2006) Arbitrarily assigned Kakimoto et al. (2006)

Mori (unpublished data) Arbitrarily assigned Mori (unpublished data) Arbitrarily assigned Arbitrarily assigned

otherwise (shown in Table 1), and we examine effects of changing these values later. Parameters describing movement of the pest and natural enemy are difficult to estimate even in a greenhouse. Thus, we consider ga, ga0 , ha, and ha0 as variable parameters and examine effects of their magnitude on success of biological control. We assume that adult pests and natural enemies are distributed in specific patterns described below at the start of simulations, and the numbers of individuals over the whole lattice space are set at 10, if not specified otherwise. Because the initial number and distribution of the pest and natural enemy may significantly affect outcomes of biological control as shown in previous empirical studies (Yasuda and Ishikawa, 1999; Alatawi et al., 2011), we distinguish two patterns of initial distributions of the natural enemy; the first is the ‘clumped distribution’ where all enemies are located at sites on the central plant (Fig. 2a), and the second is the ‘dispersed distribution’ where enemies are equally allocated to sites on nine equally spaced plants (Fig. 2b). For simplicity, we assume that the initial distribution of the pest is clumped (identical with Fig. 2a) in all simulations, and all individuals are located at the central plant. We numerically calculate the mean population densities (total numbers of individuals over the whole space divided by the number of sites) of the larval pest and adult natural enemy,

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

(a)

(b)

Clumped

Dispersed

Fig. 2. Two typical patterns of initial distributions of natural enemies. Each lattice indicates an individual plant and natural enemies are initially added to the plants colored with dark or light gray. (a) In the clumped distribution, natural enemies are placed at sites on the central plant. This is also the initial distribution of adult pests in all simulations. (b) In the dispersed distribution, natural enemies are placed at sites on nine equally spaced plants.

and examine effects of mobility of the adult pest and natural enemy, intensity of fruit thinning, and initial number and distribution of the natural enemy on the mean densities. In addition, we trace the sum of larval pest numbers experienced by harvested fruits on a plant from flower emergence to fruit harvesting, and use it as an index of fruit damage per plant (the damage is assumed to be mainly caused by the larval pest not by the adult one). We express the mean fruit damage (MFD) per harvesting as follows:

MFD ¼

ðsum of fruit damage over all plantsÞ L  L  ðK  K 0 Þ

ð5Þ

We fix the lattice size at 121 (L = 11) in all simulations. In order to more realistically evaluate performance of management practices, we use total yield which incorporates both the harvest and commercial value of each harvested fruits, in addition to cumulative MFD. As the number of harvested fruits decreases by fruit thinning, individual fruits become bigger and potentially more valuable. However, the bigger a fruit is, the slighter damage impairs its commercial value. Thus, we assume a threshold damage for the harvested fruits to be accepted by the market, ðK  K 0 Þa=K, in proportion to the harvested fraction. a is the acceptable damage when all the fruits are harvested (no fruit thinning; K0 = 0). Even if fruit damage is below the threshold, commercial value of an individual fruit is assumed to decrease as the number of harvested fruit b

increases as 1=ðK  K 0 Þ , where b is the exponent of compensation of the commercial value for reduction in harvested fruits due to fruit thinning. Thus, multiplying the value with the number of harvested fruits, the total yield over 20 times harvest becomes 1b

20L2 ðK  K 0 Þ . If b > 1, the total yield increases as the number of harvested fruits decreases, which means that fruit thinning can compensate for reduction in the number of harvested fruits. The total yield does not depend on and decreases with the intensity of fruit thinning respectively if b = 1 and b < 1.

41

showed unimodal patterns irrespective of the departure rate of the adult enemy (Fig. 3a, c, d, and f). The maxima of the larval pest density and MFD became higher and appeared later as the fruit thinning became more intensive. Intensive fruit thinning interrupted population growth of adult natural enemies because of the reductions in their diet (Fig. 3b, and e), and released the pests from control by their enemies. Furthermore, if the departure rate of the adult natural enemy was low (ga0 = 0.2; lower panels in Fig. 3), intensive fruit thinning nearly extirpated the natural enemy (dotted-grey line in Fig. 3e) and caused an outbreak of the pest (dotted-grey line in Fig. 3d) and monotonic increases in MFD (dotted-grey line in Fig. 3f). When the departure rate of the pest was higher than that of the natural enemy and the fruit thinning was intensive, the pests could move to enemy-free sites and escape from biological control. When thinning was not so intensive and the departure rate was high, the natural enemy could grow even if the pest became nearly extinct (upper panels in Fig. 3). This was because the natural enemy could feed on alternative diet (pollen). An index of fruit damage, the cumulative MFD (sum of MFD over 20 harvest times), was much higher when pollen was not available (Fig. 4c, and d) than when it was available (Fig. 4 a, and b) and thus the alternative diet could contribute successful biological control. Sufficiently high departure rate (departure rate was the lowest for lines with cross symbols in Fig. 4) and proportion of between-plant reestablishment (Fig. 4b, and d) of the natural enemy were also necessary to suppress fruit damage. However, too high departure rate of the natural enemy made its number too low at the center of pest infection, and rather weakened the pest control (circles slightly higher than boxes in Fig. 4). In general, the higher proportion of between-plant reestablishment of the natural enemy than that of the pest was much more effective for biological control, but the lower departure rate (ga0 = 0.2, cross symbols in Fig. 4) of the natural enemy than that of the pest and intensive fruit thinning weakened the effect. In contrast, the total yield of harvested fruits did not always decrease with the intensity of fruit thinning (Fig. 5). The total yield first increased and then decreased (Fig. 5b), or monotonically increased (Fig. 5c) as fruit thinning became intensive, if the thinning compensated for the decrease in harvest of mature fruits by increasing commercial value of each fruit (b > 1). However, too intensive thinning reduced the total yield because it decreased the threshold damage and made the harvested fruits commercially valueless. On the other hand, the total yield monotonically decreased with the intensity of fruit thinning (Fig. 5a) if the thinning does not promote increases in commercial value of individual fruits to compensate for the decreases in harvest (b = 1). These suggest that sufficiently high compensation of commercial values for reduction in harvested fruits due to fruit thinning is necessary to offset the negative effect of fruit thinning on fruit damage. The outcome did not qualitatively depend on the acceptable damage a when no thinning was made (figure is not shown). 3.2. Optimal release strategies of the natural enemy

3. Results 3.1. Effects of periodic disturbances and mobility of the natural enemy We found that efficiency of biological control of the herbivorous thrips, F. intonsa, by the polyphagous predatory thrips, H. brevitubus, decreased as the periodic disturbances (fruit thinning) became intensive, when the initial distribution of the natural enemy was clumped and moving individuals were equally reestablished at the original and neighboring plants (ga = 0.5, ha = ha0 = 0.5; Fig. 3). When the fruit thinning was not extensive, the larval pest density and MFD initially increased and then decreased, and

Which initial distribution, clumped or dispersed (Fig. 2), was more effective for biological control depended on the initial number, mobility of the adult natural enemy, and intensity of fruit thinning. Here, we define the optimal release strategy as the initial distribution resulting in a smaller cumulative MFD (Fig. 6a, and b), or in a larger total yield (Fig. 6c, and d). If fruit thinning was absent, the clumped and dispersed distributions were the optimum based on the cumulative MFD respectively in the high-departure rate and low-initial-number region and in the low-departure-rate and high-initial-number region (Fig. 6a). When the fruit thinning became intensive, the optimal release strategy

42

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

Larval pest density log10(1 + Cl)

Time (t)

Time (t)

Harvest times

(e)

(f)

Time (t)

log10(1 + MFD)

Larval pest density log10(1 + Cl)

Adult enemy density log10(1 + Na)

(d)

(c)

log10(1 + MFD)

(b) Adult enemy density log10(1 + Na)

(a)

Time (t)

Harvest times

Fig. 3. (a), (d) Temporal variation in population density of the larval pest, (b), (e) that of the adult natural enemy, and (c), (f) temporal variation in MFD (mean fruit damage). Solid-black, dotted-black, solid-grey, and dotted-grey lines indicate no (K0 = 0), mild (K0 = 3), intermediate (K0 = 5), and intensive fruit thinning (K0 = 7), respectively. The departure rate of the adult natural enemy was high (ga0 = 0.5) in (a) to (c), and low (ga0 = 0.2) in (d) to (f). The other parameter values are as follows: K = 10, ga = 0.5, ha = 0.5, and ha0 = 0.5.

was mainly determined by the initial number of the natural enemy (Fig. 6b). However, if the departure rate of the natural enemy was too low (white region in Fig. 6b), the pest was released from control and an outbreak occurred. These trends did not qualitatively vary depending on the between-plant reestablishment rate of the natural enemy ha0 (figure is not shown). The total yield was also maximized respectively in the highdeparture rate and low-initial-number region for the clumped distribution and in the low-departure-rate and high-initial-number region for the dispersed distribution if fruit thinning was absent (Fig. 6c). However, borders between different strategies were irregular in comparison with those in the case that cumulative MFD was used as the index, probably because value of each harvested fruit became discontinuously zero at the threshold as a function of fruit damage. When fruit thinning became intensive, the clumped distribution was always the optimum even if the departure rate and initial number of the natural enemy were high (Fig. 6d). Even if the cumulative MFD was low, adult fruits harvested at the center of infection became valueless when the initial distribution was dispersed, unless the threshold damage was sufficiently high. If the threshold damage became extremely high (e.g. a = 10000), the dispersed distribution became the optimum when the departure rate and initial number of the natural enemy were high, but the region was narrow (figure is not shown). These trends did not qualitatively differ depending on the exponent of compensation of the commercial value for reduction in harvested fruits due to fruit thinning (figure is not shown). 3.3. Effects of plant damage on pest fecundity Hitherto, the pest and crop dynamics were only indirectly linked through fruit thinning and harvesting, since we assumed that the numbers of flowers and fruits were not affected by the number of pests at the cluster, and that the fecundity of the adult

pest was independent of plant quality (damage suffered from the pests at the site). Here, we examine effects of plant damage at an individual site on the oviposition rate of the adult pest (ra) assuming that

ra ¼ r max

C

 ðrmax

C

 r min C Þ

ðdamageÞ ; 1 þ ðdamageÞ

ð6Þ

where rmax_C and rmin_C are the maximum and minimum oviposition rates of the adult pest. Fixing rmax_C at 9.6, the value when ra was constant, we examined effects of plant damage on the population dynamics and total yield by varying rmin_C. As a result, it was found that intensive fruit thinning did not always deteriorate the efficiency of pest control and reduce the total yield (Fig. 7) when the effect of plant damage on the pest oviposition rate was severe (small rmin_C), even if the increase in commercial value could not balance the reduction in harvest (b = 1). The intensive negative bottom-up feedback, fruit thinning, and low departure rate of the adult pest interrupted population growth of the pest, and the pest density and MFD were reduced to less than one order of magnitude on log scale (Fig. 7a, c), even if the natural enemy was scarce (Fig. 7b). Therefore, the total yield increased since the pest density and MFD decreased, when the intensity of fruit thinning increased from the intermediate (K0 = 5) to severe (K0 = 7) (Fig. 7d). 3.4. Sensitivity analyses We analyzed the sensitivity of our results to changes in the arbitrarily assigned values of some parameters. First, we increased the minimum oviposition rate of the adult natural enemy to the maximum value (rmax = rmin = 3.6). Then, the oviposition rate became high independent of the pest density, and the natural enemy was affected by the pest density only through the mortality. Then, the dispersed and clumped distributions extended their

43

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

sl = sa = 5, ha’ = 0.2

(a)

sl = sa = 5, ha’ = 0.9

log10(1 + cumulative MFD)

log10(1 + cumulative MFD)

(b)

Intensity of fruit thinning (K’)

Intensity of fruit thinning (K’)

sl = sa = 0, ha’ = 0.2

sl = sa = 0, ha’ = 0.9

(d)

log10(1 + cumulative MFD)

log10(1 + cumulative MFD)

(c)

Intensity of fruit thinning (K’)

Intensity of fruit thinning (K’)

Fig. 4. Dependence of cumulative MFD on the intensity of fruit thinning (K0 ), presence or absence of alternative diet (sl and sa are 0 or 5), departure rate, and proportion of between-plant reestablishment. Horizontal axes indicate the intensity of fruit thinning (K0 ) and vertical axes indicate the cumulative MFD. Alternative diet for the natural enemy is present (sl = sa = 5) in (a) and (b), and absent (sl = sa = 0) in (c) and (d). Proportion of between-plant reestablishment of the natural enemy is low (ha0 = 0.2) in (a) and (b), and high (ha0 = 0.9) in (c) and (d). Cross, diamond, box, and circle symbols indicate different values of the departure rate of the natural enemy, ga0 = 0.2, 0.4, 0.6, and 0.8, respectively. The other parameter values are identical with those in Fig. 3.

=2

(b)

log(1 + Total yield) Intensity of fruit thinnig (K’)

(c)

=5

log(1 + Total yield)

=1

log(1 + Total yield)

(a)

Intensity of fruit thinnig (K’)

Intensity of fruit thinnig (K’)

Fig. 5. Dependence of total yield on the intensity of fruit thinning (K0 ) and exponent of compensation of the commercial value for reduction in harvested fruits due to fruit thinning (b). Horizontal axes indicate intensity of fruit thinning and vertical ones indicate total yield. We set the threshold damage (a) to 1 0 0 for all panels, and b to 1 for (a), 2 for (b), and 5 for (c), respectively. Symbols and other parameter values are identical with those in Fig. 4b and d.

distributions respectively in the low and high initial-number regions when the cumulative MFD was the objective, if fruit thinning was absent (Fig. 8a). The qualitative outcomes did not differ even when the total yield was used as the index. However, the

two strategies were equally effective if initial-number and departure rate were high (light grey region in Fig. 8c). If fruit thinning became intensive, similar tendencies appeared and the clumped distribution tended to be optimum in the high departure-rate

44

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

(a)

No fruit thinning (K’ = 0)

(b)

Intensive fruit thinning (K’ = 7)

Dispersed

Departure rate of adult enemy (ga’)

Departure rate of adult enemy (ga’)

Cumulative MFD

Clumped Dispersed

(Uncontrollable) Initial number of the natural enemy No fruit thinning (K’ = 0)

Clumped

Dispersed

Initial number of the natural enemy

Initial number of the natural enemy

(d)

Intensive fruit thinning (K’ = 7)

Clumped

Departure rate of adult enemy (ga’)

Departure rate of adult enemy (ga’)

(c)

Total yield

Clumped

Dispersed (Uncontrollable) Initial number of the natural enemy

Fig. 6. Dependence of optimal release strategies (initial distributions) of the natural enemy on the initial number and departure rate of the natural enemy. Horizontal axes indicate the initial number of the natural enemy and vertical axes indicate the departure rate (ga0 ). Fruit thinning is absent (K0 = 0) in (a) and (c), and intensive (K0 = 7) in (b) and (d). Optimal release strategies are defined as the initial distribution resulting in a lower cumulative MFD in (a) and (b), or in higher total yield in (c) and (d). The optimum is the clumped distribution in black regions, the dispersed one in dark gray regions, and either one in light gray regions. In white regions, the MFD monotonically increases and no biological control is effective. The threshold damage a and exponent of compensation of the commercial value for reduction in harvested fruits due to fruit thinning b is identical with those in Fig. 5a. The other parameter values are the same as those in Fig. 3.

region when evaluated by the cumulative MFD and the dispersed distribution became optimum in a small region when evaluated by the total yield (Fig. 8b, and d). When we decreased the maximum mortality of the larval enemy to the minimum value (Fig. 8a) the mortality became low independent of the pest density, but the qualitative outcomes did not change (figure is not shown). When we increased the maximum mortality dmax from 0.1 to 0.9 and ran a sufficiently longterm simulation (e.g., 100 times harvesting), damped oscillations of the population densities and MFD appeared and the pest was never eradicated even if no fruit thinning was applied and the natural enemy was supplemented with pollen (figure is not shown). However, the dynamics, effects of fruit thinning, and optimum strategy in the early phase did not differ qualitatively. Predation rate of the adult natural enemy on the pupal pest fpa, and departure rate of the adult pest ga did also not influence the qualitative outcomes even if they were changed ±50% of the original values. 4. Discussion Our results showed that fruit thinning could have significant negative effects on efficiency of biological control, and that the

optimal release strategy (initial distribution of the natural enemy) strongly depended on the initial number and mobility of the natural enemy and the intensity of fruit thinning.

4.1. Effects of fruit thinning Our model predicted that the pest density and MFD per harvesting, at least transiently, increased regardless of the intensity of fruit thinning, because the pest could avoid predation from the natural enemy by escaping to enemy-free space (Jeffries and Lawton, 1984). When the fruit thinning was absent, the natural enemy spread over the whole space due to the advance of the colonization front after a lapse of sufficient time if the departure rate and proportion of between-plant reestablishment were sufficiently high. Then, the pest lost enemy-free space and finally went extinct (Fig. 9). However, the population density of the pest and MFD increased with the intensity of fruit thinning unless the negative feedback from plant damage to pest fecundity was incorporated. In general, natural enemies tend to be more vulnerable to fruit thinning than pests, since they lose their diets concurrently with their conspecifics whereas pests can enjoy the benefits of reduced

45

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

(b)

Larval pest density log10(1 + Cl)

Adult enemy density log10(1 + Na)

(a)

Time (t)

(d)

log10(1 + MFD)

log(1 + Total yield)

(c)

Time (t)

Harvest times

Intensity of fruit thinning (K’)

Fig. 7. (a) Temporal variation in population density of the larval pest, (b) that of the adult natural enemy, (c) temporal variation in MFD, and (d) total yield depending on intensity of fruit thinning (K0 ) and the departure rate of adult pest. Minimum oviposition rate of the adult pest rmin_C was set to 0 (the pest hardly oviposits if its resident site suffers sufficiently high damage). Cross, box, and circle symbols indicate different values of the departure rate of the adult pest, ga = 0.05, 0.2, and 0.5, respectively. Lines in (a), (b), and (c) are the same as those in Fig. 3. The other parameter values are as follows: K = 10, ga = 0.05, ga0 = 0.5, ha = 0.5, ha0 = 0.5, a = 10, and b = 1.

predation pressure. This idea could be tracked back to the Vito Volterra’s law of the disturbance of the averages; If an attempt is made to destroy the individuals of the two species uniformly and in proportion to their number, the average of the number of individuals of the species that is eaten increases and that of the individuals of the species feeding upon the other diminishes (Volterra, 1928). On the contrary, the total yield did not monotonically decrease with the intensity of fruit thinning. If the increases in price could compensate for the reduction in harvest (b > 1) or the negative feedback of fruit damage on pest fecundity was severe, unimodal or monotonic increases in the total yield with the intensity of fruit thinning were observed. Therefore, intensive fruit thinning should be implemented only if the compensatory increase in value is sufficient or it enhances the negative feedback of fruit damage on pest fecundity. The negative effect of periodic disturbances on biological control is also known as the paradox of pesticides (Jansen and Sabelis, 1992; Sherratt and Jepson, 1993; Sabelis et al., 1999; Li and Yang, 2013). Sherratt and Jepson (1993) assumed periodic disturbances against both the pest and natural enemy by spraying pesticides, and showed that the regular application of pesticides could cause prey metapopulations to fluctuate at densities higher than they would otherwise do. However, in a set of simulations, they obtained a qualitatively different result from ours; low dispersal rates of the prey and high maximum dispersal rates of the

predator resulted in the greatest mean prey densities. The difference between their assumption that only 50% of fields were sprayed and ours that all the clusters experienced fruit thinning might influence the results. Some theoretical works suggested that effects of periodic disturbances could be different depending on whether the disturbances were synchronous or asynchronous in space (Ives and Settle, 1997; Childs et al., 2004). Childs et al. (2004) considered a host-parasitoid metapopulation model and examined spatially synchronous and asynchronous periodic disturbances. They demonstrated that the asynchronous disturbances resulted in more efficient suppression of the host than the synchronous ones when only the parasitoid dispersed. This is because parasitoids dispersed into young patches had a large impact on the subsequent development of host populations, thereby reducing their mean densities. However, when both species dispersed, the situation was more complex, and the relative level of suppression depended upon the amount of parasitoid aggregation. Incorporating spatial asynchrony of periodic disturbances into our model might be a future problem. Some empirical researchers revealed that some cultivation managements (e.g., thinning, harvesting, tillage, and weeding, etc.) caused increases in mortality of arthropods (including both pests and natural enemies) inhabiting the fields (Stinner and House, 1990; Thomas and Jepson, 1997; Hossain et al., 2002; Thorbek and Bilde, 2004). However, they rarely quantified the

46

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

No fruit thinning (K’ = 0)

(b)

Departure rate of adult enemy (ga’)

Clumped Departure rate of adult enemy (ga’)

Cumulative MFD

(a)

Dispersed

Intensive fruit thinning (K’ = 7)

Clumped

Dispersed (Uncontrollable)

Initial number of the natural enemy

(c)

No fruit thinning (K’ = 0)

Initial number of the natural enemy

(d)

Intensive fruit thinning (K’ = 7)

Either Dispersed

Departure rate of adult enemy (ga’)

Departure rate of adult enemy (ga’)

Total yield

Clumped Clumped Dispersed (Uncontrollable) Initial number of the natural enemy

Initial number of the natural enemy

Fig. 8. Dependence of optimal release strategies (initial distributions) of the natural enemy on the initial number and departure rate of the natural enemy when the oviposition rate of the natural enemy does not depend on pest densities (rmax = rmin = 3.6). Light grey regions indicate that the two strategies were equally effective. Axes, the other regions and parameter values are identical with those in Fig. 6.

negative effect of such management on consequences of biological control. Empirical studies to explore effects of periodic disturbances on biological control are urgently needed.

and natural enemy on efficiency of biological control should be studied more extensively (but see van Maanen et al., 2012). 4.3. Optimal release strategies of the natural enemy

4.2. Mobility and alternative diet of the natural enemy The negative effect of periodic disturbances on biological control could be offset by high mobility of the natural enemy. We showed that not only the sufficiently high departure rate but also the significant proportion of between-plant reestablishment was necessary for successful biological control. This result demonstrated the importance of spatial or metapopulation structure for biological control and the effect was reinforced by the presence of alternative diet for the natural enemy. This is because the natural enemy frequently dispersing toward neighboring plants without pests can establish itself even without the pest due to its alternative diet (supplements from plants). Similar outcomes were already reported by previous metapopulation models. Namba et al. (1999) considered a prey-predator system comprised of two sedentary prey and one mobile shared predator, and demonstrated that one of the two prey regionally went extinct due to apparent competition mediated by the shared predator as the mobility of the predator increased. They also showed that the result also depended on spatial arrangements of patches the prey and predator inhabit. As shown in our model, the exclusion of the pest by the highly mobile polyphagous natural enemy can also occur even if the pest is also mobile. Effects of mobility and spatial arrangements of the pest

When fruit thinning was less intensive, the clumped initial distribution was more effective than the dispersed one if the initial number of the natural enemy was low and the departure rate was high, since the clumped natural enemy at the center of infection could feed on most pests there and efficiently trace the dispersing pest. If the natural enemy was abundant, the dispersed distribution could intercept the pest spreading from the center of infection, and became the optimum. If the fruit thinning was intensive, the expected optimum strategies were different depending on response variables (cumulative MFD and total yield) especially when the initial number and departure rate of the natural enemy were high. When the initial number of the natural enemy was low or high, the clumped or dispersed distribution respectively lowered the cumulative MFD. This is because fruit thinning not only decreased the numbers of the pest and natural enemy at each site but also played a role to reduce the numbers of individuals that successfully arrived at another cluster. Thus, the dispersed release of the sufficiently abundant natural enemy can more successfully suppress the damage equally over the space, while the clumped release targeted toward a plant infested by the pest can more successfully suppress the damage if the natural enemy was less abundant. On the other

47

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

t = 40

(b)

t = 60

(c)

t = 80

Larval pest density log10(1 + Cl)

(a)

(e)

(f)

Adult enemy density log10(1 + Na)

(d)

Fig. 9. Snapshots of a typical development of spatial distributions of (a)–(c) the larval pest and (d)–(f) adult natural enemy. Bars indicate population densities of the pest and natural enemy on individual strawberry plants. Fruit thinning is absent (K0 = 0) and other parameter values are identical with those in Fig. 3a–c.

hand, the clumped distribution generally brought the higher total yield than the dispersed one, regardless of the initial number and departure rate of the natural enemy. This might be because the clumped natural enemy can suppress damage of harvested fruits on the center plant below the economic threshold, while the dispersed one cannot sufficiently protect any fruits on any plants. If we assumed that the oviposition rate of the adult natural enemy was high independent of the amount of diets and fruit thinning was absent, the dispersed strategy tended to be the optimum for the higher departure rate in the low initial-number region, because the high oviposition rate supplemented the low initial number if the cumulative MFD was used as the criterion. In contrast, the clumped strategy tended to be optimum in the high initial-number and high departure-rate region since the advantage of the dispersed strategy was lessened because of the high oviposition rate. The fruit thinning weakened the advantage of the dispersed strategy to decrease the cumulative MFD in the low initial-number region. The dispersed strategy could bring the higher total yield only in a limited region under the intensive fruit thinning because the effect of high oviposition rate was weakened. Bommarco et al. (2007) examined effects of the degree of aggregation or heterogeneity of the initial distribution of the pest assuming a random distribution of the natural enemy. They showed that the evenly distributed pest was always suppressed more efficiently than the aggregated one, because the former was more prone to be detected by the natural enemy than the latter. In our setting, the pest was distributed on the same plant with the clumped natural enemy or surrounded by the dispersed natural enemy, and thus there appeared no detection problem. To incorpo-

rate the uncertainty of the pest distribution into our model may be a future problem. In an empirical study of pest control, Alatawi et al. (2011) considered two patterns of initial distributions (clumped and even) of the pest and natural enemy and examined their effects on the mean number of pests and mean plant damage. They showed that the pest number and crop damage were efficiently suppressed when the patterns of the initial distributions of the pest and natural enemy were matched (i.e., even-even, or clumped-clumped combination), and that the evenly distributed natural enemy against the clumped pest performed the poorest control efficiency. In contrast, our spatial model showed that the dispersed (evenly distributed) natural enemy could more efficiently suppress the clumped pest in some cases. In Alatawi et al. (2011), the overall predator number was sufficiently high and the pests were undetectable in all treatments at 18 days after releasing predators. Since the peripheral plants were adjacent to the central plants and easily accessible, the initial overlap between the pest and abundant natural enemy might be more important.

5. Conclusions In this study, we constructed a metapopulation model of the herbivorous pest (F. intonsa) and its polyphagous predatory natural enemy (H. brevitubus) in a greenhouse strawberry field, and examined effects of periodic disturbances (sequential fruit thinning and harvesting) on biological pest control. Although the periodic disturbances increase fruit damage, high departure rate and proportion of

48

Y. Ikegawa et al. / Biological Control 93 (2016) 37–48

between-plant reestablishment and an alternative resource for the natural enemy improved efficiency of the control. In addition, sufficient compensation of commercial value of each harvested fruits for the reduction in harvest increased the total yield even under severe periodic disturbances. Finally, we showed that the optimal release pattern of the natural enemy was the dispersed distribution for a large number of less mobile natural enemies and the clumped one for a small number of highly mobile ones. However, it can qualitatively differ depending on the intensity of fruit thinning and response variables. Thus, for consistently applying cultivation management and biological control, it is important to carry out adequate intensity of fruit thinning and to release natural enemies in proper number and spatial pattern depending on their mobility and the intensity of periodic disturbances. Currently, K. M. and M. O. are experimentally examining effects of fruit thinning on efficiency of pest control in actual strawberry fields, and applicability of the results to other crops is a next problem to be examined. Collaboration of experimental and theoretical studies would help us to understand proper implementation of cultivation management and biological control. Acknowledgment This research was financially supported by Grants-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science (JSPS) KAKENHI 23570034 to HE and 26440247 to TN. References Alatawi, F., Nechols, J.R., Margolies, D.C., 2011. Spatial distribution of predators and prey affect biological control of twospotted spider mites by Phytoseiulus persimilis in greenhouses. Biol. Control 56, 36–42. Baba, H., Sakamaki, Y., Tsuda, K., Kusigemati, K., Kakimoto, K., 2008. The repertoire of potential prey of Haplothrips brevitubus (Karny) and its seasonal abundance in Kagoshima (in Japanee with English summary). Bull. Exp. Farm Faculty Agric. Kagoshima Univ. 30, 1–6. Bommarco, R., Firle, S.O., Ekbom, B., 2007. Outbreak suppression by predators depends on spatial distribution of prey. Ecol. Model. 201, 163–170. Childs, D.Z., Bonsall, M.B., Rees, M., 2004. Periodic disturbance in host-parasitoid metapopulations: host suppression and parasitoid persistence. J. Theor. Biol. 227, 13–23. Cohen, E., 2006. Pesticide-mediated homeostatic modulation in arthropods. Pestic. Biochem. Physiol. 85, 21–27. Ferron, P., Deguine, J.P., 2005. Crop protection, biological control, habitat management and integrated farming. A review. Agron. Sustain. Dev. 25, 17–24. Fukuda, T., Inoue, H., Kakimoto, K., Kashio, T., 2008. Predatory ability of Haplothrips brevitubus (Karny) (Thysanoptera: Phlaeothripidae) on three species of greenhouse thrips (in Japanese with English summary). Kyushu Plant Protect. Res. 54, 74–77. Hardin, M.R., Benrey, B., Coll, M., Lamp, W.O., Roderick, G.K., Barbosa, P., 1995. Arthropod pest resurgence: an overview of potential mechanism. Crop Protect. 14, 3–18. Hossain, Z., Gurr, G.M., Wratten, S.D., Raman, A., 2002. Habitat manipulation in lucerne Medicago sativa: arthropod population dynamics in harvested and ‘refuge’ crop strips. J. Appl. Ecol. 39, 445–454. Ives, A.R., Settle, W.H., 1997. Metapopulation dynamics and pest control in agricultural systems. Am. Nat. 149, 220–246. Jansen, V.A.A., Sabelis, M.W., 1992. Prey dispersal and predator persistence. Exp. Appl. Acarol. 14, 215–231.

Janssen, A., Pallini, A., Venzon, M., Sabelis, M.W., 1998. Behavior and indirect interactions in food webs of plant-inhabiting arthropods. Exp. Appl. Acarol. 22, 497–521. Jeffries, M.J., Lawton, J.H., 1984. Enemy free space and the structure of ecological communities. Biol. J. Linn. Soc. 23, 269–286. Kakimoto, K., Inoue, H., Hinomoto, N., Noda, T., Hirano, K., Kashio, T., Kusigemati, K., Okajima, S., 2006. Potential of Haplothrips brevitubus (Karny) (Thysanoptera: Phlaeothripidae) as a predator of mulberry thrips Pseudodendrothrips mori (Niwa) (Thysanoptera: Thripidae). Biol. Control 37, 314–319. Li, Y.C., Yang, Y. 2013. On the paradox of pesticides. arXiv:1303.2681v1 (q-bio.PE). Matsuoka, T., Seno, H., 2008. Ecological balance in the native population dynamics may cause the paradox of pest control with harvesting. J. Theor. Biol. 252, 87– 97. Morita, S., Kashio, T., Inoue, H., Kakimoto, K., 2008. Development and fecundity of Haplothrips brevitubus (Karny) (Thysanoptera: Phlaeothripinae) on the pollen of some fruit vegetables (in Japanese with English summary). Kyushu Plant Protect. Res. 54, 78–84. Morse, J.G., 1998. Agricultural implications of pesticide-induced hormesis of insects and mites. Hum. Exp. Toxicol. 17, 266–269. Murai, T., 1988. Studies on the ecology and control of flower thrips, Frankliniella intonsa (in Japanese with English summary). Bull. Shimane Agric. Exp. Station 23, 1–73. Murdoch, W.W., Briggs, C.J., 1996. Theory for biological control: recent developments. Ecology 77, 2001–2013. Murdoch, W.W., Nisbet, R.M., Luck, R.F., Godfray, H.C.J., Gurney, W.S.C., 1992. Sizeselective sex-allocation and host feeding in a parasitoid-host model. J. Anim. Ecol. 61, 533–541. Namba, T., Umemoto, A., Minami, E., 1999. The effects of habitat fragmentation on persistence of source-sink metapopulations in systems with predators and prey or apparent competitors. Theor. Popul. Biol. 56, 123–137. Neubert, M.G., Kot, M., 1992. The subcritical collapse of predator populations in discrete-time predator-prey models. Math. Biosci. 110, 45–66. Sabatier, R., Meyer, K., Wiegand, K., Clough, Y., 2013. Non-linear effects of pesticide application on biodiversity-driven ecosystem services and disservices in a cacao agroecosystem: a modeling study. Basic Appl. Ecol. 14, 115–125. Sabelis, M.W., van Baalen, M., Bruin, J., Egas, M., Jansen, V.A.A., Janssen, A., Pels, B. 1999. The evolution of overexploitation and mutualism in plant-herbivorepredator interactions and its impact on population dynamics. In: Hawkins, B.A., Cornell, H.V. (eds.) Theoretical Approaches to Biological Control. Cambridge University Press, Cambridge, pp. 259–282. Sherratt, T.N., Jepson, P.C., 1993. A metapopulation approach to modelling the longterm impact of pesticides on invertebrates. J. Appl. Ecol. 30, 696–705. Stinner, B.R., House, G.J., 1990. Arthropods and other invertebrates in conservationtillage agriculture. Annu. Rev. Entomol. 35, 299–318. Strevens, C.M.J., Bonsall, M.B., 2011. The impact of alternative harvesting strategies in a resource-consumer metapopulation. J. Appl. Ecol. 48, 102–111. Thomas, C.F.G., Jepson, P.C., 1997. Field-scale effects of farming practices on linyphiid spider population in grass and cereals. Entomol. Exp. Appl. 84, 59–69. Thorbek, P., Bilde, T., 2004. Reduced numbers of generalist arthropod predators after crop management. J. Appl. Ecol. 41, 526–538. Tuda, M., Shimada, M., 2005. Complexity, evolution, and persistence in hostparasitoid experimental systems with Callosobruchus beetles as the host. Adv. Ecol. Res. 37, 37–45. van Maanen, R., Messelink, G.J., van Holtstein-Saj, R., Sabelis, M.W., Janssen, A., 2012. Prey temporarily escape from predation in the presence of a second prey species. Ecol. Entomol. 37, 529–535. Verkerk, R.H.J., Leather, S.R., Wright, D.J., 1998. The potential for manipulating croppest-natural enemy interactions for improved insect pest management. Bull. Entomol. Res. 88, 493–501. Volterra, V. 1928. Variations and fluctuations of the number of individuals in animal species living together. Journal du Counseil international pour l’exploration de la mer, III, vol. 1. Translated by Mary Evelyn. Republished as an appendix to Animal Ecology, (ed.) Chapman, R.N. 1931. McGraw-Hill, New York, pp. 409– 448. Yasuda, H., Ishikawa, H., 1999. Effects of prey density and spatial distribution on prey consumption of the adult predatory ladybird beetle. J. Appl. Entomol. 123, 585–589.