Popul Ecol (2012) 54:557–571 DOI 10.1007/s10144-012-0325-6

ORIGINAL ARTICLE

Biological pest control by investing crops in pests Hiroshi C. Ito • Natsuko I. Kondo

Received: 23 January 2011 / Accepted: 6 May 2012 / Published online: 26 May 2012 Ó The Society of Population Ecology and Springer 2012

Abstract We propose a biological pest control system that invests part of a crop in feeding a pest in a cage. The fed pest maintains a predator that attacks the pest in the target area (i.e., the area for storing or growing crops). The fed pest cannot leave the cage nor the target pest cannot enter the cage. The predator, however, can freely attack both the fed and target pests in the target area. By introducing a refuge in the cage against the predator for the fed pest, the fed pest and predator may be stably sustained. In this study, we analyzed the potential performance of this system by modeling the population dynamics of the target pest, fed pest, and predator as differential equations. First, we show analytically that the target pest can be suppressed at extremely low abundance by adjusting both refuge efficiency and crop investment. Second, we show numerically that crop damage by the pest may be effectively suppressed by investing only small amounts of the crop. Third, we show numerically that the magnitude of required crop investment can be estimated by an index comprising of the predator’s searching cost for prey and the relative growth efficiency of the predator with respect to the pest. Even if the system structure is changed or its population dynamics is modeled based on host–parasitoid interactions, crop damage can be suppressed effectively by small amounts of crop investment. Keywords Apparent competition  Beddington– DeAngelis  Host–parasitoid  Predator–prey  Refuge

H. C. Ito (&)  N. I. Kondo Environmental Biology Division, National Institute for Environmental Studies (NIES), 16-2 Onogawa, Tsukuba, Ibaraki 305-8506, Japan e-mail: [email protected]

Introduction Biological control of pests by their natural enemies (e.g., predators, parasites, and parasitoids) is an important strategy to minimize the burden on the environment caused by pesticides (Bale et al. 2008). Control of pests by releasing their natural enemies is called augmentative biological control (Landis et al. 2000; van Lenteren 2000). While short-term performance achieved by this method can be high after the release of the pest’s enemies, it is not easy to establish the enemies in the target areas (i.e., areas for growing or storing crops), at least in cases of arthropod predators (Pickett and Gilstrap 1986; Minkenberg et al. 1994; Grasswitz and Burts 1996; Norton and Welter 1996; Kehrli and Wyss 2001). One reason for this difficulty is that the pest may not be attractive to the predator as a food source compared with other prey available outside the target area. As a result, the predators emigrate. Even if they do not emigrate and successfully exterminate all pests, the absence of the pest can cause the predator’s extinction by starvation. Then, the predator’s absence makes the crop once again vulnerable to new arrivals of pests. If the pest density is kept extremely low by maintaining the population of the predator, crop damage can be effectively suppressed. However, a strong trade-off seems to exist between stability of their population dynamics and suppression of the pest equilibrium density (Luck 1990; Murdoch 1990). This tendency is known as the ‘‘biological control paradox’’ (Arditi and Berryman 1991). As a potential solution to this difficulty, we propose a biological pest control system that keeps a separate population of the pest to maintain the population of its predator (Fig. 1). In this system, the separate pest population is fed in a cage by investing a part of the crop. The fed pest cannot leave the cage nor the target pest (i.e., the pest in the

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a

b

Target area

P

Target area

Introduction

Predator

P Predator

Cage

NT

Stored crop

N

Fed pest

R

Invested crop

Refuge effect Target pest

Stored crop

C

Predator

PT

Refuge effect Target pest

Cage

Investment

NT

N

Fed pest

R

Invested crop

C Investment

Fig. 1 A BBP system and its variation. a A BBP system (Eqs. 7–11). The stored crop, invested crop, fed pest, target pest, and predator are indicated with circles. Predation or exploitation is indicated with a solid arrow. A box with broken lines indicates a cage, in which a refuge reduces predation pressure on the fed pest. The broken arrow

toward the invested crop indicates investment of the stored crop into the cage to sustain the fed pest. b A separated BBP system (Eqs. 64– 68 in Appendix 6), in which the cage is separated from the target area. A certain fraction of the predator is introduced from the cage to the target area either continuously or at regular intervals

target area) cannot enter the cage. However, the predator can freely attack both fed and target pests. To avoid extinction of the fed pest by predation, which in turn results in the predator’s extinction or emigration, a refuge against the predator is introduced in the cage. High refuge efficiencies may yield stable coexistence of the fed pest and predator. At the same time, high refuge efficiencies restrict the amount of available fed pest for the predator, resulting in small predator population size with low predation pressure on the target pest. By increasing crop investment, predation pressure may then be kept sufficiently high even under high refuge efficiencies. We call this system the bad bug paradise (BBP) system, because in the cage, the pest (i.e., the bad bug) is not only fed crops but also partially protected against predators. The main advantage of the system is that it may suppress the target pest to an extremely low abundance, which is enabled by spatial separation between the fed and target pests and by adjusting refuge efficiency and crop investment. A plausible example would be controlling pests of stored crops, such as seed beetles, in which the seed beetles, parasitoid wasps, and seeds in storage correspond to the pest, predator, and crops in our system (Fujii 1983). If the predator is larger than the pest, it may be difficult to design the cage to stop only the pest. One solution for this may be separation of the cage from the target area (Fig. 1b), which is also briefly explained in this study. In this study, we analyze the potential performance of the BBP system as our first step. To this end, the BBP system is described with differential equations that are analytically tractable. First, we derive conditions under which the target pest can be suppressed to extremely low abundance by adjusting refuge efficiency and crop

investment. Second, we illustrate with simulations how the crop can be protected against the target pest in the BBP system. Third, we investigate numerically how the amount of the required crop investment can be estimated from the interaction strengths of the pest and predator and their life history traits. Fourth, we examine the robustness of the BBP system by either changing its structure (Fig. 1b) or describing it with difference equations based on host–parasitoid interactions. Finally, we discuss the necessary factors for future practical use of the BBP system. We also address the relationship between our method and other biological control methods, including augmentative biological control (Landis et al. 2000; van Lenteren 2000) and conservation biological control (Jonsson et al. 2008; Frank 2010).

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Model For simplicity, we assume that the target area is a crop storage facility in which the crop does not grow (the result does not change even if the amount of crop increases, because the increase does not affect the dynamics of the invested crop, fed pest, and predator, as analyzed below). In the target area, the biomass (cal/g dry weight) of the stored crop, invested crop, target pest, fed pest, and predator are denoted by C, R, NT, N, and P, respectively (Fig. 1). For simplicity, we assume a small migration rate for the predator so that its effect on the population dynamics in the target area can be neglected. We first describe the dynamics in the invested crop area. To sustain the fed pest, part of the stored crop C is invested at a constant

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investment rate h per unit time. By exploiting invested crop R, fed pest N is maintained, which further maintains predator P. We express the changes in these variables with respect to time as differential equations. dC ¼ h; dt dR ¼ N  gN ðN; RÞ þ h; dt dN ¼ kN N  gN ðN; RÞ  P  gP ðP; /NÞ  dN N; dt dP ¼ kP P  gP ðP; /NÞ  dP P; dt

ð1Þ ð2Þ ð3Þ ð4Þ

where kN and kP are the assimilation efficiencies of the pest and predator, respectively. These efficiencies are positive constants less than 1. Similarly, dN and dP are biomass losses of the pest and predator, respectively, as a result of metabolism and natural death. Without loss of generality, dN is set to 1. In this case, the time unit corresponds to one pest generation, if predation and metabolism are both neglected (i.e., dN NDt ¼ N for Dt ¼ 1). The generation time can become shorter or longer than the time unit when predation and metabolism are considered. For simplicity, we treat unit time as one pest generation. The gN and gP are the functional responses of the pest and predator, respectively, which are defined as a Beddington–DeAngelis type (Beddington 1975; DeAngelis et al. 1975) as gN ðN; RÞ ¼

a0 R ; a1 þ a2 N þ R

ð5Þ

gP ðP; NÞ ¼

b0 N ; b1 þ b2 P þ N

ð6Þ

respectively. This function is a general form of the functional response that includes both saturation of consumption rate and interference competition (Abrams and Ginzburg 2000). Here, a0 denotes the maximum consumption rate of N on R, which is attained at high levels of the biomass of the invested crop (i.e., R ! þ1), while b0 denotes the maximum consumption rate of P on N, which is attained at high levels of the biomass of the fed pest (i.e., N ! þ1), and a1 and b1 correspond to searching costs of the pest and predator, respectively. The a2 and b2 can be interpreted as energy loss via interference competition (Beddington 1975; Ito et al. 2009). Equations 5 and 6 become type-II responses (Holling 1959) with a2 = 0 and b2 = 0, respectively. Without loss of generality, the a1 and a2 have been normalized by division with the handling time for unit biomass of the crop, so that the coefficient of R in the denominator of Eq. 5, becomes equal to 1. The b1 and b2 in Eq. 6 have been normalized in the same manner. The constant parameter / (0 B / B 1) denotes the vulnerability of the fed pest, which is defined as the fraction of the

fed pest that is available to the predator. This vulnerability can be adjusted by means of a refuge. The refuge efficiency is calculated as r = 1 - /. When pest individuals arrive in the target area, their population may rapidly increase by exploiting the crop, unless they are attacked by the predator. We assume that the predator does not distinguish between the target and fed pests. In this case, the total predation rate is given by P  gP ðP; NT þ /NÞ; and the fractions of the predated target and fed pests are given by NT/(NT ? /N) and / N/(NT ? /N), where NT denotes the target pest biomass. The entire population dynamics of the system is then written as dC ¼ NT  gN ðNT ; CÞ  h; ð7Þ dt dR ¼ N  gN ðN; RÞ þ h; ð8Þ dt dN /N ¼ kN N  gN ðN; RÞ  P  gP ðP; NT þ /NÞ  dt NT þ /N ð9Þ  dN N; dNT NT ¼ kN NT  gN ðNT ; CÞ  P  gP ðP; NT þ /NÞ  dt NT þ /N  dN NT ; ð10Þ dP ¼ kP NP  gP ðP; NT þ /NÞ  dP NP : dt

ð11Þ

Stored crop C is reduced by the target pest as well as by crop investment. If the amount of crop damage can be effectively suppressed by a small investment rate h, the BBP system may be economically feasible. In this study, we refer to h and r (=1 - /) as control parameters and the other parameters as ecological parameters. All parameters in our model are positive. In this model, a spatially uniform environment in the target area is assumed for simplicity. This assumption may be plausible when every location in the target area can be reached by the predator with a small cost in a short time compared with the pest’s generation time. In this case, the target area may be treated as a single patch for the predator, in terms of the optimal foraging theory (MacArthur and Pianka 1966). For larger target areas, such uniformity may be attained by introducing several evenly spaced BBP systems in these areas.

Basic idea of analysis For effective protection of the stored crop, high predation pressure should be maintained so that arriving pests cannot increase in biomass. To evaluate the efficiency of

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protection, we examined the initial growth rate of the target pest when it is rare. For controllability of predation pressure, it is better to maintain the population dynamics of the BBP system stable, otherwise the system can easily be affected by external disturbances and demographic stochasticity in practice. If the initial growth rate of the target pest is stably kept negative, the arriving pest can be swiftly eradicated and crop damage will be small. If the amount of invested crop is also small, it can be judged that the system has a good potential for pest control.

Analytical results Conditions for stable equilibrium First, we explain the conditions under which the BBP system has a stable equilibrium with positive biomasses. We refer to an equilibrium with R [ 0, N [ 0, and P [ 0 as positive. By adjusting refuge efficiency and investment rate, it is shown in Appendix 1 that the system can always ^ N; ^ PÞ; ^ have a unique locally stable positive equilibrium ðR; if the ecological parameters fulfill fN ¼

lim

1 dN ¼ kN a0  dN [ 0; dt

R!þ1;P!0 N

1 dP ¼ kP b0  dP [ 0: N!þ1 P dt

fP ¼ lim

ð12Þ ð13Þ

Here, fN and fP are the maximum growth rates of the pest and predator, respectively, which are attained under abundant resources and no predation on the pest. We assume that inequalities 12 and 13 hold, otherwise the pest cannot cause any damage and the predator cannot be maintained as a control agent. Conditions for negative initial growth rate of the target pest Here, we assume that the system is in a positive and stable ^ N; ^ PÞ: ^ The initial growth rate of the target equilibrium ðR; pest when rare can be approximately described as  1 dNT  fT ðh; /Þ ¼ lim NT !0 N dt  ^ ^ ^ T

R¼R;N¼N;P¼P

C b0 P^  dN  ¼ kN a0 a1 þ C b1 þ b2 P^ þ /N^ b0 P^ : ’ fN  b1 þ b2 P^ þ /N^

ð14Þ

The above approximation C/(a1 ? C) ^ 1 assumes that searching cost a1, is significantly lower than the crop amount, which seems valid as the target pest just exploits the stored

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crop. It is derived in Appendix 2 that the fT (h, /) \ 0 can be attained by adjusting the refuge efficiency and investment rate, if ecological parameters fulfill fP [ fN : kP b2

ð15Þ

This inequality is also a necessary and sufficient condition for N and P to achieve positive equilibrium under a very high investment rate with no refuge (i.e., h ¼ þ1 and / = 1) (Appendix 3). Although this equilibrium may be unstable, at least a monotonic increase of the fed pest is prevented, resulting in oscillation. Thus, inequality 15 can be examined experimentally in a system with abundant crop and no refuge. Some parasitoids can cause local extinctions of their host populations (McCallum and Dobson 1995; Lei and Hanski 1997), and may thus fall under this scenario. Apparent competition Because the fed and target pests share the same predator, they are apparent competitors of each other. From this viewpoint, the following insight is obtained. When a small amount of the target pest arrives, its growth rate relative to the fed pest under equilibrium ^ N; ^ PÞ ^ is given by ðR;   1 dNT 1 dN lim  NT !0 NT dt N dt R¼R;N¼ ^ ^ N;P¼ P^ ¼

^ kN a0 ða1 þ a2 NÞ a1 þ a2 N^ þ R^ ð1  /Þb0 P^  : b1 þ b2 P^ þ /N^

ð16Þ

Since the growth rate of the fed pest is equal to zero under equilibrium, the sign of Eq. 16 provides the sign of the initial growth rate of the target pest. Here, the first term on the right, which is always positive, is the relative advantage of the target pest due to abundant resources in the target area compared to the limited resource in the cage. The second term is its disadvantage through predation due to the absence of refuge in the target area. If the cage has no refuge (i.e., / = 1), its disadvantage (i.e., the second term) disappears and Eq. 16 cannot be negative. Therefore, the refuge is necessary not only for stability of the system but also for the negative initial growth rate of the target pest.

Numerical results In this section, we first explain how the pest control performance of the BBP system depends on investment rate h and refuge efficiency r by showing the population

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dynamics of the component species (see Appendix 4 for details). Second, we show how the investment rate required to attain a negative initial growth rate of the target pest may be estimated from ecological parameters. Dependency of pest control performance on control parameters Figure 2 shows the initial growth rate of the target pest for various combinations of control parameters (investment rate h and refuge efficiency r = 1 - /) for a given set of ecological parameters. Some combinations of h and r provide positive and stable equilibria with negative initial growth rates of the target pest (dark gray area) or those with positive growth rates (light gray area). We refer to the set of control parameters that minimizes h (indicated with b) as the minimum investment rate, hmin. Figure 3 shows the outcomes for various control parameters, which correspond to the letters in Fig. 2. Without investment, the outbreak of the target pest (dashed-dotted curve in Fig. 3a) immediately results in destruction of the entire crop within three pest generations (gray curve in Fig. 3a). With the minimum investment rate hmin, the outbreak of the target pest is effectively suppressed for 25 generations after its arrival, resulting in Pˆ = 0 1.0

fT (h,φ ) = 0

Refuge efficiency (r)

h

Re(λ0) = 0

e

0.01 % damage to the initial amount of crop (Fig. 3b). Because the amount of investment through the 50 generations (50 hmin) is only 4.9 %, the total loss is only 4.91 %. If the investment rate is smaller than hmin, either the predator cannot be maintained (Fig. 3c) or the outbreak of the target pest cannot be suppressed (f in Fig. 2, not shown in Fig. 3). High investment rates can destabilize the system (Fig. 3d). Low (or high) refuge efficiencies seem to have a similar effect as high (or low) investment rates. For example, dynamics under the two sets of control parameters indicated as g and h in Fig. 2 are qualitatively the same as those under d and f, respectively (not shown). A possible reason for this similarity is that both high investment rates and low refuge efficiencies increase the sustained population size of the fed pest. Since the predator’s functional response to the pest becomes saturated as the pest population becomes large, the mechanism of destabilization seems to be explained by the ‘‘paradox of enrichment’’ (Rosenzweig 1971). In reality, pests may arrive more frequently and in larger biomass. In addition, crop investment into the cage may not be continuous but discrete, e.g., once per pest generation. Both of these possibilities affect pest control performance under hmin. To attain more stable performance, an investment rate somewhat higher than hmin should be chosen, as indicated by e in Fig. 2. The outbreak is then effectively blocked even with the frequent arrival of pests with larger biomasses under discrete investment (Fig. 3e). Even in this case, the damage is suppressed to 0.7 % while the investment rate is 7.5 %, resulting in an 8.2 % total loss. Dependency of the required investment rate on ecological parameters

0.5

c

f

b

d

g 0.0 0.1

1.0

10.0

Investment rate (h) Fig. 2 Expected pest control performance of a BBP system at various combinations of investment rate h and refuge efficiency r. Light and dark gray areas indicate the existence of a locally stable equilibrium with positive biomasses (R^ [ 0; N^ [ 0; P^ [ 0; and Re (k0) \ 0). The Re (k0) is the real part of the leading eigenvalue at the equilibrium. A dark gray color indicates a negative initial growth rate of the target pest (fT (h, /) \ 0). A broken curve indicating fT (h, /) = -0.3 is also plotted to indicate the negative side of the zero-contour fT (h, /) = 0. Model parameters: kN ¼ 0:3; kP ¼ 0:5; dN ¼ 1:0; dP ¼ 2:0; a0 ¼ 20:0; a1 ¼ 1:0; a2 ¼ 2:0; b0 ¼ 10:0; b1 ¼ 0:3; b2 ¼ 0:5; h ¼ fhmin ; 0:4; 3; 15; 12; hmin ; hmin g; r ¼ fr  ; r  ; r  ; r  ; 0:6; 0:85; 0:2g; where hmin = 7.798, and r ¼ 0:444

For practical realization of BBP systems, it is important to find appropriate predators that provide sufficient pest control performance with small investment rates. An important ecological parameter is the searching cost of predators b1, because even a small biomass of the predator can provide high predation pressure if b1 is low (Holt and Lawton 1994; Yano 2006). In this case, the required amount of the fed pest can be small, resulting in a small value of minimum investment rate hmin. On the other hand, a high b1 may cause strong instability of the system, and the attainment of the system’s stability may require high refuge efficiency, resulting in a high hmin. In addition, hmin may also depend on other ecological parameters. We calculated hmin under 50,000 randomly generated sets of ecological parameters that may cover ecologically plausible ranges (see Appendix 4 for details). For these parameter sets we calculated hmin and analyzed how it depends on ecological parameters. In 63.0 % of the total

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Invest: 4.9% Damage: 0.01%

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50,000 sets, BBP systems exhibited positive and stable equilibria with negative initial growth rates of the target pest, achieved by hmin lower than 1,000. Among these sets, hmin depended not only on the searching cost of the predator b1 (correlation coefficient: 0.844) but also weakly on a growth efficiency ratio defined as G ¼ fP =dP  ðfN =dN Þ1 (correlation coefficient: -0.139), as shown in Fig. 4b, c. The predation performance index IP ¼ b1  G shows a strong relationship with hmin (correlation coefficient: -0.877) and the magnitude of hmin can thus be

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Biomass

Biomass (104)

b

Biomass (104)

Fig. 3 Simulated pest control performance of a BBP system at different combinations of investment rate h and refuge efficiency r. In a, dynamics of the stored crop (C) and the target pest (NT) without the BBP system are plotted. The biomass of the target pest is plotted at a magnification of three times. Biomasses lower than ed ¼ 0:002 are removed and treated as extinction. The small semicircle at 25 generations indicates the arrival of target pests with a small biomass (2ed ) at the target area. Extinctions or exterminations are indicated with filled triangles. Invest denotes the percentage of the crop invested with respect to the total crop amount at the initial condition, and Damage denotes the percentage of crop exploited by the target pest with respect to the total crop. In b–e, the dynamics of stored crop (C) and target pest (NT) with the BBP system are plotted on the left side, while that of the invested crop (R), the fed pest (N), and the predator (P) are plotted on the right side. The values for h and r in each of a–e are indicated by triangles in Fig. 2 with the corresponding letters. In e, after 25 generations, the target pest arrives at the target area 30 times with random timings. The biomass at each arrival is randomly chosen from ed to 200ed (see Appendix 4 for details). In e, the crop investment is once per pest generation (h per each); hence, the time average is equal to h. The values of the model parameters are the same as those in Fig. 2

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20

30

40

50

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estimated from IP. We found no clear dependence on other ecological parameters (see Fig. 4a as an example). If data on G are lacking, then b1 itself can serve as a good magnitude estimator. Figure 4e shows the fraction of parameter sets with hmin \ 1,000, for each value of b1. Figure 4f is plotted in the same manner against IP. This fraction drops when b1 is low (Fig. 4e) or IP is high (Fig. 4f), but is not small anyway. Thus, attaining negative initial growth rates does not appear difficult even under high IP or low b1. Therefore, a

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b Required investment rate (hmin)

Required investment rate (hmin)

a 100.0

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4

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10+2

10+4

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f

e 1.0

Frequency

1.0

Frequency

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c Required investment rate (hmin)

Fig. 4 Dependency of the required investment rate on ecological parameters. a–d Distributions of the minimum investment rate, hmin, against the maximum growth rate of the pest, fN ; the growth efficiency ratio, G ¼ ðfP =dP Þ  ðfN =dN Þ1 ; the searching cost, b1, and the predation performance index, IP ¼ G  b1 ; respectively. The scales of both axes are different among the panels. Darker colors indicate higher frequencies (the saturation is negatively proportional to the frequency). Zero frequencies are indicated with white. In each of a–d, the average and the region containing 95 % of plotted data (upper 47.5 % and lower 47.5 % from the average) for each value of the horizontal axis are plotted with solid and broken curves, respectively. e The fraction of parameter sets with hmin \ 1,000, at each value of b1. f is plotted in the same manner against IP. See Appendix 4 for details of the analysis

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good strategy for finding appropriate predators is to select those that have high IP or low b1. Alternative 1: Modeling dynamics based on host–parasitoid interaction It may be easy to design the cage with an appropriate mesh size if the predator is a parasitoid of the pest and is

10-2

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10+4

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smaller than the pest. Difference equations based on host– parasitoid interactions may thus be more directly applicable than our basic model. We tested the behavior of such host–parasitoid system by extending Rogers (1972) (Appendix 5). As shown in Fig. 5a, this host–parasitoid system has a similar feature to the original system (Fig. 2). Without investment, arrival of the pest induces outbreak, which

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Fig. 5 Pest control performance of a BBP system modeled with host–parasitoid interaction. In a, gray and black areas indicate the stable maintenance of the predator. A black area indicates negative initial growth rates of the target pest. b–c are plotted as in Fig. 3a–e. b Dynamics without the BBP system. c Dynamics of the BBP system under control parameters indicated with a triangle in a. Frequent arrivals of the target pest are simulated as in Fig. 3e. Model parameters: a0 = 50; other parameters have the same values as those in Fig. 3, except dN and dP, which are not used in this model

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immediately exploits the entire crop (Fig. 5b). On the other hand, appropriate sets of refuge efficiency and investment rate (inside the black area in Fig. 5) effectively suppress the crop damage with small investments (Fig. 5c).

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Moreover, even if investment of the crop and introduction of the predator are discrete (e.g., twice per pest generation), the outbreak of the pest (Fig. 6b) is effectively suppressed with small crop investments (Fig. 6c).

Alternative 2: A separated BBP system Discussion In contrast to the previous subsection, if the predator is larger than the pest, it may be difficult to design the cage to stop the pest. One possible solution is to separate the cage from the target area (Fig. 1b), although this requires the additional work of introducing the predator from the cage into the target area. Even in this case, the condition regarding the ecological parameters for pest controllability can be obtained as fP  dP [ fN ; kP b 2

ð17Þ

which is similar to that of the original BBP system, i.e., inequality 15 (Appendix 6). The separate BBP system provides similar features of population dynamics in terms of refuge efficiency and investment rate (Fig. 6a).

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It is not easy to suppress pests at extremely low abundance by introducing their natural enemies (Luck 1990; Murdoch 1990). In this study, we have proposed a new system of biological pest control, the BBP system, which maintains both the pest and its predator by investing part of a crop and introducing a refuge for the fed pest. Analytical and numerical results showed that small crop investments can suppress the target pest at extremely low abundance. In this system, the trade-off between stability of the system and suppression of the pest at a low abundance (Arditi and Berryman 1991) is avoided by the existence of a fed pest that stably maintains the predator without damaging the crop. It was also numerically shown that the magnitude of the investment rate required for sufficient pest control

Popul Ecol (2012) 54:557–571

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40

Time (Generation)

performance can be estimated by the predator performance index, which can be examined experimentally. Thus, the BBP system appears to have a good potential, which further merits theoretical and empirical investigation of its control efficacy and economics with a focus on each specific pest species. In this section, we discuss further theoretical analyses, the practical use of the system, and its relationships with other methods of biological control. Further theoretical analyses In this study, we primarily modeled BBP systems with rather simple differential equations without specifying the pest and its predator species. The system appears to maintain its good performance even if it is modeled with difference equations based on host–parasitoid interactions or if the cage is separated from the target area. The next step is to model BBP systems that focus on specific life histories of pests and their predator species and analyze their pest control performance, as in other biological control methods (Hawkins and Cornell 1999; Murdoch et al. 2006; Kidd and Amarasekare 2012). In addition, spatial structure should be incorporated explicitly

R N P

4

Biomass

c Biomass (104)

Fig. 6 Pest control performance of a separated BBP system. In a, expected pest control performance is plotted as in Fig. 2. b and c are plotted as in Fig. 3a–e. b Dynamics without the BBP system. c Dynamics with the BBP system under control parameters indicated with a triangle in a. In c, frequent arrivals of the target pest are simulated as in Fig. 3e. The biomass of the introduced predator in the target area (PT) is magnified by 10 times. In c, the crop investment and introduction of the predator are twice per pest generation, i.e., 0.5h and 0.5k, respectively. Hence, the time averages of the investment and introduction rates are equal to h and k, respectively. Model parameters: k = 1, b1 = 0.05; other parameters have the same values as those in Fig. 3

565

10 5 0 10

20

30

40

50

Time (Generation)

into these models, because limitations on the searching range of predators will provide another restriction on pest control performance, which is not considered in this study. Demographic and environmental stochasticity should also be considered in order to evaluate the robustness of BBP systems. For simplicity, in this study we assumed that the fed pest is fed the crop itself. Different feeds may also be used unless it makes the predator strongly prefer the fed pest to the target pest. Feeds cheaper than the crop can make BBP systems more practicable. Thus, such an extension of the system is also worthwhile. Practical use For practical use of the BBP system, the most important aspect is to find appropriate predators. According to our numerical results, predators with low searching costs b1 are preferable. Predators that can keep the attack rate high even at low pest densities are expected to have a low searching cost. Insect parasitoids are strong candidates with such a property and are already used for pest control (Wajnberg et al. 2008).

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566

Other candidates may be found among predators that can cause local extinction of the target pest. These candidates may be identified by examining the spatial structure of the target pest and its host plant in their abundances. Strong spatial heterogeneity in the pest density without correlation with its host plant distribution may be due to predators’ strong predation pressures (Cronin and Reeve 2005). If indigenous predators are used for a BBP system, the system is free from potential ecological risks that can be serious in the case of foreign species (Howarth 1991; Bigler et al. 2006). Crop quality for the fed pest will be easy to maintain if the crop is a grain in a storage facility. A BBP system to protect a bean crop (e.g., Vigna unguiculata) from a bean beetle (e.g., Callosobruchus maculatus or C. chinensis) may be designed relatively easily by introducing as a refuge other beans that are harder than the crop bean (e.g., Vigna angularis). This is because parasitoid wasps (Anisopteromalus calandrae or Heterospilus prosopidis) may experience difficulty in attacking pest larvae inside the hard beans (Fujii 1983). To our knowledge, there is no report that these wasps prefer beetle larvae in hard beans. From this point of view, hard beans can be used. However, a pest fed with hard beans may have a lower growth rate, and higher refuge efficiency may be required for maintaining both the fed pest and wasp. In this case, the larger amount of biomass of the fed pest may be lost by natural death without contribution to the wasp’s production. Therefore, the required amount of crop investment may become larger. If a crop is a living plant in a farm, the cage requires a special device that keeps the crop alive but prevents the pest from passing through it. In addition, if the predator is larger than the pest, it may be difficult to design the cage to stop only the pest. In these two cases, separation of the cage from the target area (Fig. 1b) may be one solution because this arrangement still seems to result in good pest control performance (Fig. 6c). However, except for pests of stored crops, nutritional as well as ecological requirements of many agricultural pests differ among different developmental stages. Thus, the practical use of our system for these pests may be difficult without further improvement. Other methods of biological pest control Major methods for biological pest control are augmentative biological control (Landis et al. 2000; van Lenteren 2000) and conservation biological control (Jonsson et al. 2008; Frank 2010). One advantage of augmentative biological control is its easy adoption because its usage is similar to that of pesticides. However, like pesticides, natural enemies have to be purchased and released

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repeatedly or in large numbers for effective pest control (Frank 2010). Furthermore, the purchase cost is generally higher than that of pesticides (Collier and VanSteenwyk 2004; VanDriesche and Heinz 2004; Vasquez et al. 2006; Frank 2010). In conservation biological control, natural enemies are not purchased. Instead, they are attracted and maintained by providing them with alternative resources (e.g., nectar and alternative prey) and by reducing negative cultural practices (Landis et al. 2000; Frank 2010). Although conservation biological control often increases natural enemy abundance, further development is necessary for effective pest control (Jonsson et al. 2008). Among the various systems of conservation biological control, banker plant systems are most similar to our BBP system. Banker plant systems typically consist of a noncrop plant that is deliberately infested with a nonpest herbivore (Frank 2010). The nonpest herbivore serves as an alternative diet for the predator of the target crop pest. Some banker plant systems use the pest itself as an alternative diet for the predator (Pickett et al. 2004). Refuges, however, are not introduced in these systems, making them different from BBP systems. While banker plants can suppress the pest population in the long term, the possible target pest is restricted (Frank 2010). This restriction may stem from the complexity of biological communities in banker plants, which can be difficult to control. By restricting component species and adjusting the refuge, our BBP systems might have higher controllability. However, a drawback of BBP systems is that it is difficult to design cages and refuges. In this system, the difficulty of the ‘‘biological control paradox’’ is transformed into the complexity of two different types of ‘‘barriers’’, i.e., the cage and refuge. The cage allows only the predators to pass and the refuge allows only the pests to pass. The cost of producing these barriers may be sufficiently low for some combinations of pests and their predators while it may be too high for others. The separated BBP system may reduce the cost, because such selective barriers are needed only for the pest. Instead, it requires the additional step of introducing the predator to the target area from the cage. The system becomes identical to augmentative biological control, if the cage is maintained not by farmers but by companies that sell the predators to farmers. Thus, the separated BBP system has an advantage when the maintenance cost of the system is lower than the cost of purchasing the predator. Because the BBP system and other biological control methods can be complementary to each other, further studies of BBP systems, in particular those targeting various pest species, would be worthwhile for the development and integration of biological control methods.

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Acknowledgments The authors thank the editor and two anonymous reviewers for their valuable comments on earlier versions of this manuscript. H.C.I. also thanks David Munro for inventing a tool for numerical analysis and visualization with high performance and flexibility, named Yorick, and distributing it for free. All figures in this paper except Fig. 1 were produced with Yorick.

B¼

1 : b1 þ b2 P^ þ N^

ð32Þ

Clearly, a1 ; . . .; a7 are all positive except a4. The eigen equation is given by ðk þ a1 Þ½ðk þ a4 Þðk þ a7 Þ þ a5 a6 Þ þ a2 a3 ðk þ a7 Þ ¼ k3 þ ða1 þ a4 þ a7 Þk2

Appendix 1: Condition for stable equilibrium We derive a condition for Eqs. 2–4 having a stable equilibrium with positive biomasses. The equilibrium that may fulfill all conditions, i.e., R^ [ 0; N^ [ 0 and P^ [ 0; is uniquely given by R^ ¼

hðkN kP a2 b2 h þ a1 fP / þ kP a1 b2 dN þ ðkP b2 fN  fP /Þh þ b1 a0 dP

b 1 a2 dP Þ

;

ð18Þ

k N k P b h þ b1 d P N^ ¼  2 ; f P / þ kP b 2 dN

ð19Þ

kP ðkN fP h/  b1 dN dP Þ P^ ¼ ; ðfP / þ kP b2 dN ÞdP

ð20Þ

where fN and fP are given by Eqs. 12 and 13, which are assumed to be positive. In this case, N^ is always positive. For R^ and P^ to be positive,

þ ða1 a7 þ a1 a4 þ a2 a3 þ a4 a7 þ a5 a6 Þk þ a1 a4 a7 þ a1 a5 a6 þ a2 a3 a7 ¼ k3 þ D1 k2 þ D2 k þ D3 :

ð33Þ

According to Hurwitz’s theorem, all eigenvalues have negative real parts if and only if ð34Þ

D1 [ 0;    D1 1     D3 D2  ¼ D1 D2  D3 [ 0;    D1 1 0    D3 D2 D1  ¼ D3 ðD1 D2  D3 Þ [ 0    0 0 D3 

ð35Þ

ð36Þ

are fulfilled. If a4 is positive, D1 ¼ g þ a4 þ a1 [ 0;

ðkP b2 fN  fP /Þh þ b1 a0 dP [ 0;

ð21Þ

D1 D2  D3 ¼ a4 a27 þ a1 a27 þ a5 a6 a7 þ a24 a7 þ 2a1 a4 a7

kN fP h/  b1 dN dP [ 0

ð22Þ

þ a21 a7 þ a4 a5 a6 þ a1 a24 þ a2 a3 a4 þ a21 a4

are necessary and sufficient conditions, respectively. To check local stability, we express the Jacobian matrix ^ N; ^ PÞ ^ as at equilibrium ðR; 0 oR_ oR_ oR_ 1 0 1 0 a1 a2 oR oN oP B oN_ oN_ oN_ C ¼ @ a3 a4 a5 A; @ oR oN oP A oP_ oP_ oP_ 0 a6 a7 oR

oN

oP

^ ^ R¼R;N¼ N;P¼ P^

ð23Þ where ^ 0 NA ^ 2; a1 ¼ ða1 þ a2 NÞa

ð24Þ

^ 2; ^ 0 RA a2 ¼ ða1 þ RÞa

ð25Þ

^ 0 A2 N; ^ a3 ¼ kN ða1 þ a2 NÞa

ð26Þ

^ ^ 2  b0 /2 PB ^ 2 ÞN; a4 ¼ ða0 a2 RA

ð27Þ

^ ^ 2 N; a5 ¼ b/ðb1 þ NÞB

ð28Þ

^ 2 P; ^ a6 ¼ kP b0 /ðb1 þ b2 PÞB

ð29Þ

^ ^ 2 P; a7 ¼ kP b0 b2 /NB

ð30Þ

1 ; A¼ a1 þ a2 N^ þ R^

ð31Þ

þ a1 a2 a3 [ 0; D 3 ¼ a1 a4 a7 þ a2 a3 a7 þ a1 a5 a6 [ 0 ð37Þ ensure that all eigenvalues have negative real parts. Thus, a4 [ 0 is a sufficient condition for local stability of the equilibrium. The condition a4 [ 0 is expressed as ! 2^ ^ R P k a a b / N 0 2 0 a4 ¼ N^  [ 0: ^ 2 ðb1 þ b2 P^ þ /NÞ ^2 ða1 þ a2 N^ þ RÞ ð38Þ Substitution of Eqs. 18–20 into Eqs. 2 and 4 gives 1 h ¼ ; a1 þ a2 N^ þ R^ a0 N^R^

ð39Þ

/ dP ¼ : b1 þ b2 P^ þ /N^ kP b0 N^

ð40Þ

By exploiting these relationships, inequality 38 is transformed into kN k2P a2 b0 h2 [ 1:  a0 dP2 P^R^

ð41Þ

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Substitution of Eqs. 18 and 20 into this inequality yields ð42Þ

/\K; where K¼

kN kP a2 b0 h fP / þ kP b2 dN  a0 dP kN fP h  b1 dN dP ðkP b2 fN  fP /Þh þ b1 a0 dP /  ; kN kP a2 b2 h þ a1 fP /2 þ ðkP a1 b2 dN þ b1 a2 dP Þ/

where the right side is positive under inequality 15. We suppose a sufficiently large ~h that fulfills both inequalities ~ 44 and 47 under / = 1. With this ~h; a sufficiently small / fulfills inequality 42, keeping inequality 47 fulfilled. Thus, ~ a positive by choosing (h, /) such that h/ ¼ ~h and / ¼ /; and stable equilibrium with negative initial growth rate of the target pest is attained, as long as inequality 15 holds.

ð43Þ Appendix 3: Equilibrium under no refuge

fulfilling lim K ¼ K0 ¼

/!þ0

k2P b0 b2 dN fN h [0 a0 dP ðkN fP h  b1 dN dP Þ

ð44Þ

under inequality 22. Here, h = h/ is assumed to be independent of / by an appropriate adjustment of h, without loss of generality. Since qK/q/ is finite at / = 0, Eq. 43 can be expanded as  oK  K ¼ K0 þ  / þ    ð45Þ o/ /¼0 Thus, if h is sufficiently large such that Eq. 44 is positive and if / is sufficiently small, then inequality 42 is fulfilled. Therefore, this equilibrium is locally stable if / is sufficiently small and h is sufficiently large. On the other hand, there exists another equilibrium that may be locally stable, which is given by ðR; N; PÞ ¼ ~ N; ~ 0Þ with R~ ¼ ða1 dN þ kN a2 hÞ=fN and N~ ¼ kN h=dN : ðR; The initial growth rate of the predator having a very small biomass against this equilibrium is given by  1 dP kN kP b0 h kN fP h  b1 dN dP  d : lim ¼ ¼ P P!0 P dt R¼R;N¼ b1 dN þ kN h b1 dN þ kN h ~ N~

If there is no refuge in the cage (/ = 1), but there is abundant crop there (R ! þ1), the equilibrium of N and P fulfills 1 dN b0 P ¼ kN a0  dN  ¼ 0; N dt b1 þ b2 P þ N

ð48Þ

1 dP kP b0 N ¼  dP ¼ 0; P dt b1 þ b2 P þ N

ð49Þ

which have a unique solution N¼

kP b1 fN ; fP  b2 kP fN

ð50Þ

P¼

b1 dP : fP  b2 kP fN

ð51Þ

Thus, the system has an equilibrium with N [ 0 and P [ 0, if fP [ fN b2 kP

ð52Þ

holds, which is identical to inequality 15. Under this condition, monotonic growth of N is expected to be prevented by P, although oscillation can be produced if this equilibrium given by Eqs. 50 and 51 is not locally stable.

ð46Þ Clearly, the growth rate is positive if inequality 22 holds, in ^ N; ^ PÞ ^ is which case this equilibrium is not stable. Thus, ðR; ^ N; ^ and P^ are all a unique locally stable equilibrium if R; positive.

Appendix 2: Condition for negative initial growth rate of target pest Here we derive the condition that ensures negative initial growth rate of the target pest as well as the positive equilibrium. As for the initial growth rate of the target pest, Eq. 14, fT (h, /) \ 0 is transformed into    b1 dN dP 1 b1 dP fN / fP  þ  \  fN kP b2 ; ð47Þ h h kN kN kP b2

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Appendix 4: Method of numerical analysis Population dynamics in Fig. 3 are calculated with the Runge–Kutta method with fixed time step Dt ¼ 0:0001: Initial values for C, R, N and P are 8,000, 5.0, 0.1 and 0.01, respectively. Biomasses lower than ed ¼ 0:002 are removed and treated as extinction. The target pest is introduced only once at time t = 25 at twice the extinction threshold in Fig. 3a–d, or 30 times randomly after t = 25 in Fig. 3e (and Figs. 5c, 6c), where the biomasses of the arrived pests and their times are randomly assigned within the range ½2ed ; 200ed  and within [25, 50], respectively, according to uniform distributions. To obtain Fig. 4, 50,000 sets of ecological parameters are randomly generated, except dN, which is fixed at 1.0 without loss of generality. (kNa0 - dN)/kNa0, i.e., the

Popul Ecol (2012) 54:557–571

569

amount of biomass production divided by the amount of assimilation, gives the upper limit for net production efficiency of the pest. Because net production efficiencies of animal populations are less than 90 % according to Humphreys (1979), we assign the range for (kNa0 - dN)/kN a0 from 0 to 0.9. This means that the range of (kNa0/dN) spans from 1.0 to 10.0. In the same manner, we assign the range for (kPb0/dP) from 1.0 to 10.0. Assimilation rates kN and kP are within the range from 0.1 to 0.95, according to Humphreys (1979). In addition, according to the relationship between metabolic rates and body sizes (Willmer et al. 2000), dP/dN = (xP/xN)-0.75 approximately holds, where xP and xN are the body lengths or equivalent particle diameters of the predator and the pest, respectively. Here, we assign the range of dP/dN from 0.05 to 200, which provides the range of xP/xN from about 0.001 to 1,000. As for a1, a2, b1 and b2, information about their ranges was not obtained from the literature. Thus, we assign a broad range, from 0.0001 to 100, to these parameters. Values for parameters (kNa0/dN), (kPb0/dP), kN, kP, dP, a1, a2, b1 and b2 are randomly chosen on the basis of uniform distribution within the assigned ranges. Then, a0 and b0 are calculated by a0 ¼ ½kN a0 =dN   dN =kN and b0 ¼ ½kP b0 =dP   dP =kP :

Appendix 5: Modeling BBP system based on host–parasitoid interaction

l ¼ P  gP ðP; NT þ /NÞ  ¼

/N 1  NT þ /N N

/b0 P ; b1 þ b2 P þ ðNT þ /NÞ

lT ¼ P  gP ðP; NT þ /NÞ  ¼

NT 1  NT þ /N NT

b0 P ; b1 þ b2 P þ ðNT þ /NÞ

Ns ¼ Nel ; NTs ¼ NT e

lT

ð53Þ ;

ð54Þ

where l and lT are the average numbers of parasitoid eggs laid on one individual of the fed and target pests, respectively. These values are given by

ð56Þ

where the functional response of the predator, gP, is given by Eq. 6. By attacking the fed and target pests, whose total amount is (N - Ns) ? (NT - NTs), the parasitoid produces the next generation, P0 ; which is given by P0 ¼ kP ½ðN  Ns Þ þ ðNT  NTs Þ:

ð57Þ

Furthermore, the surviving fed and target pests also increase by exploiting the invested or stored crops, respectively, and produce the next generations N 0 and Nw0 given by N 0 ¼ kN G N ;

ð58Þ

NT0

ð59Þ

¼ kN GNT ;

where GN and GNT are the amounts of crop exploited by the fed pest on R and of the target pest on C, respectively. Through these exploitation, the stored and invested crops decrease, given as C 0 ¼ C  GN  h;

By modifying Rogers’ (1972) system into a tritrophic system, we model the BBP system as difference equations based on host–parasitoid interaction. Let N and NT denote the biomasses of the fed and target pests at the age just before they are vulnerable to the parasitoid. The parasitoid’s biomass is denoted by P. The biomasses of the crop and the invested crop at the same time are denoted by C and R, respectively. The reason for the choice of Rogers (1972) among functional responses of host–parasitoid interaction is that all ecological parameters in the original BBP model can be used as they are, except dN and dP, which are not used here. Now we introduce two variables Ns and NTs, which denote the biomasses of the fed and target pests, respectively, that survive the attack by the parasitoid. Only individuals (of the pest) with no egg laid by the parasitoid can survive. We assume that the unit of the biomass is adjusted such that it corresponds to one individual pest in the system. Ns and NTs are given by

ð55Þ

0

R ¼ R  GNT þ h:

ð60Þ ð61Þ

In reality, the amount of exploited crop cannot be larger than the standing crop. For this to hold true, GN and GNT are given by GNT ¼ minðNTs  gN ðNTs ; CÞ; C  hÞ;

ð62Þ

GN ¼ minðNs  gN ðNs ; RÞ; R þ hÞ;

ð63Þ

where the functional response of the pest, gN, is given by Eq. 5.

Appendix 6: A separated BBP system We assume that the cage is separated from the target area. The cage prevents both the pest and predator from entering and leaving it. However, some fraction of the predator is introduced from the cage to the target area, which is represented by the constant k. Here, we denote the biomass of the introduced predator as PT. In this system, the introduced predator attacks only the target pest while the predator in the cage attacks only the fed pest. Then slight modification of Eqs. 7–11 gives the population dynamics of this system,

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dC ¼ NT  gN ðNT ; CÞ  h; dt dR ¼ N  gN ðN; RÞ þ h; dt dN ¼ kN N  gN ðN; RÞ  P  gP ðP; /NÞ  dN N; dt dP ¼ kP P  gP ðP; /NÞ  dP P  kP; dt dPT ¼ kP PT  gP ðP; NT Þ  dP PT þ kP; dt dNT ¼ kN NT  gN ðNT ; CÞ  PT  gP ðPT ; NT Þ  dN NT : dt

ð64Þ ð65Þ ð66Þ ð67Þ

ð68Þ For the stability of the system in terms of R, N and P in the absence of the target pest, the analysis is the same as that of the original BBP system, if dP ? k is denoted by d~P : Clearly, P at its equilibrium P^ also provides equilibrium of ^ In addition, the initial growth rate of PT at P^T ¼ ðk=dP ÞP: the target pest is given by, f~T ðh; /Þ ’ fN 

b0 ðk=dP ÞP^ ; b1 þ b2 ðk=dP ÞP^

which is always smaller than that in the original BBP system, fT (h, /), if k C dp. Then if k = dP, the inequality corresponding to inequality 15 is given by f~P [ fN ; kP b2

ð69Þ

where f~P ¼ kP b0  d~P ¼ fP  dP : Therefore, in the same manner as that of the original BBP system, it is analytically derived that inequality 69 ensures that the negative initial growth rate of the target pest can be stably maintained by adjusting the investment rate and refuge efficiency.

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