Ongoing climate change is projected to raise global temperatures by 2 to 8 oC over the next century, with atmospheric CO2concentrations expected to reach 800 ppm1. This climate warming is leading to changes in precipitation patterns, which can directly impact soil temperature and moisture levels2. Temperature can impact the biology, population dynamics3,4, abundance, species diversity, and richness of soil mite communities5. It also influences the metabolic rate, feeding, locomotor, and searching activity of soil mites6,7. It is closely linked to ecosystem functions such as trophic interactions through the consumption and metabolism of the predator and prey6,8,9. It can affect predatory soil mites’ functional response which is one of the important aspects of quantifying trophic interactions10,11.

The functional response is a critical component of the interaction between predator or parasitoid and prey or host, playing a key role in the dynamics of animal populations and ecological communities11–18. It illustrates how the number of prey captured by a predator or host parasitized by a parasitoid changes with the density of prey or host available in the environment. There are three basic types of functional response, Type I, II, and III, described by Holling11. Over the years, researchers have suggested various modifications to these basic types16,17,18. Type I shows a linear increase in consumption rate until it reaches a plateau; Type II demonstrates a hyperbolic approach to the maximum consumption rate as prey density rises; Type III involves an initial rise in consumption rate, followed by a decrease after reaching a turning point on a sigmoid curve. In insect and mite predators, Type II and III responses are most frequently reported13,14,19. Additionally, there is a Type IV functional response, the domed type, which indicates a decrease in predation efficiency at specific prey densities; this has also been noted in predatory mites20,21.

The effectiveness of a predator can be measured by looking at the functional response parameters, which include the predator’s attack rate, the handling time14, and the predator’s potential for prey mortality22. Predators with high attack rates and short handling times are expected to be the most effective for biological control23,24,25. Alternatively, if a predator is very efficient and causes significant prey mortality, it is likely to have a high potential for mortality to its prey. On the other hand, if a predator is not as efficient and does not cause much mortality, its potential for prey mortality will be lower22. The ecological impact of the predator can be assessed by the functional response ratio (FRR) which is the attack rate or potential of prey mortality divided by the handling time25,26. This parameter is especially useful when handling time and attack rate give the opposite predictions. The higher the value of FRR, the higher the impact of the predator on the ecosystem and vice versa26. In invertebrate predators, the type and parameters of functional response can vary depending on host plant27, temperature28,29, humidity25,30, age of predator31, type of predator and prey14,32,33, and exposure to insecticides34.

Previous studies show that temperature has the potential to alter the type of functional response in insect and mite predators. For instance, rising temperature shifted the type of functional response from Type II to Type III in the pentatomid bugs Podisus maculiventris Say and P. nigrispinus Dallas (Hemiptera: Pentatomidae) preying on larvae of the beet armyworm Spodoptera exigua Hübner (Lepidoptera: Noctuidae)29. On the contrary, the rising temperature changed the functional response type from Type III to Type II in the phytoseiid mite Amblyseius swirskii Athias-Henriot (Acari: Phytoseiidae) foraging on eggs of the two-spotted spider mite Tetranychus urticae Koch (Acari: Tetranychidae)28 and the macrochelid mite Macrocheles muscaedomesticae Scopoli (Acari: Macrochelidae) feeding on eggs of the house fly Musca domestica L. (Diptera: Muscidae)10. On the other hand, the temperature change did not affect the type of functional response in the phytoseiid mite Neoseiulus californicus McGregor (Acari: Phytoseiidae) feeding on eggs, larvae, nymphs, or adults of T. urticae35. This suggests that temperature has varying impacts on the predator-prey system, probably due to species-specific differences in the sensitivity of predator and prey to temperature and foraging behaviour36,37. As temperature may destabilize interactions between predator and prey by either increasing predator activity or boosting prey mortality rates38, valuing the effect of temperature on species-specific responses of the predator-prey system can enable a better understanding of the impact of temperature on food webs.

Predatory mites belong to the family Blattisociidae (Acari: Mesostigmata) which inhabit a diverse array of habitats, including soil, mosses, grasses, and dead organic matter. They can also be found in association with fungi and various plant structures such as flowers, leaves, and tree bark, as well as within rodent and bird nests4,39. These mites are frequently linked to insects facilitating their movement to fragmented habitats39,40,41. Among the Blattisociid mites, the genus Blattisocius, including species such as Blattisocius dentriticus Berlese, B. tarsalis Berlese, B. everti Britto, Lopes and Moraes, B. keegani Fox, and B. mali Oudemans, is particularly well-studied. Although they commonly inhabit edaphic environments, outside of soil, litter, or rotten plant material, they are often reported in storage facilities, where they prey on coleopteran and lepidopteran pests as well as acarid mite pests associated with stored products39.

Blattisocius dentriticus, B tarsalis, B. everti, and B. keegani have been reported to have the potential to control mould mites42,43,44,45,46,47,48. Additionally, B. mali has been reported as a potential biocontrol agent of insects, nematodes, and mites49,50,51,52,53. Notably, the life table parameters of B. mali were much higher than those of B. dentriticus, B. keegani, or Gaeolaelaps aculeifer Raumilben (Acari: Laelapidae) while feeding on the mould mite Tyrophagus putrescentiae Schrank (Acari: Acaridae), which makes this predator an especially promising biological control agent against T. putrescentiae54. The acarid mites can cause serious problems in stored products, mushroom farms, and horticultural crops55,56,57,58. The T. putrescentiaeis an omnivorous acarid mite and common in-house dust, soil with rotting plant material, and vertebrate nests. It is a pest of various stored food products and crop plants such as cucumber, gerbera, or bulbs of many ornamental plants56,59. It can develop in a wide range of temperatures, from 10 °C to 34 °C, and at an optimal temperature of 22 °C and humidity of 85%, it can make one generation in only 4.41 days60,61.

This study aimed to examine the effect of varying temperature levels on the functional response of B. mali preying on T. putrescentiae. This might not only enrich our knowledge about the possible negative effects of extreme temperatures on the stabilization of systems between soil predatory mites and acarid prey but also show at what temperatures this predator is most effective in the biological control of acarid mite pests. In the previous paper25, we demonstrated that a decrease in humidity level not only led to the decrease in B. mali predation rate on the T. putrescentiae eggs but also shifted its functional response from Type III to Type II on this prey. As the temperature has a considerable impact on blattisociid mite activity and development4,62, we hypothesized that this factor, similar to humidity, might significantly affect the interactions between B. mali and its prey, and the functional response of this predator. In our current research, we tested B. mali over a wide range of six temperature levels including extremes at 10 °C and 35 °C where this mite could still develop63. We also used two prey stages, eggs or adult males of T. putrescentiae to examine whether, and to which extent, the functional response of B. mali might change in the presence of smaller immobile eggs as prey or much bigger and movable males as prey, at varying temperature levels. Furthermore, we have compared different models based on the fitness of our data to provide a better understanding and interpretation of the dynamics within the predator-prey system.

Results

The statistical analysis indicated a significant effect of both temperature (χ2 = 148.16; df = 5; P < 0.0001) and the density of T. putrescentiae eggs (χ2 = 925.60; df = 6; P < 0.0001) offered on the mean number of eggs eaten by B. mali. Moreover, there was a significant influence of both temperature (χ2 = 164.18; df = 5; P < 0.0001) and the density of T. putrescentiae males (χ2 = 103.45; df = 6; P < 0.0001) offered on the mean number of males eaten by B. mali. Furthermore, the interaction between temperature and density of prey eaten was found to be significant for both eggs (χ2 = 19.01; df = 30; P < 0.0001) and males (χ2 = 10.34; df = 30; P < 0.0001) as prey, indicating that the mean number of preys eaten by the predator depended not only on the temperature but also on the density of the prey offered. When T. putrescentiae eggs were offered as prey, the mean number of prey eaten by B. mali increased significantly with rising temperature across all tested prey densities except for 10 or 20 eggs (Fig. 1a). On the contrary, the mean number of T. putrescentiae males eaten by B. mali significantly decreased from 10 oC to 15 oC and then rose to 35 oC in most prey densities (Fig. 1b).

Fig. 1
figure 1

The effect of six temperatures and seven densities of Tyrophagus putrescentiae eggs or males on the mean number (±95% CI) of the T. putrescentiae eggs (a) or males (b) eaten by Blattisocius mali over 24 h period. Different lowercase or uppercase letters indicate significant differences between means (P < 0.05; Tukey test) for various prey densities within each temperature or among different temperatures, respectively.

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The estimates of the parameters of the Real64 model showed that the value of the scaling component ‘q’ and handling time ‘Th’ were greater than zero for T. putrescentiae eggs as prey, indicating a Type III functional response at all tested temperatures (Table 1). On the other hand, the value of ‘q’ for T. putrescentiae males as prey was not significantly different from zero, and ‘Th’ was greater than zero across all tested temperatures, indicating a Type II response (Table 2).

Table 1 Estimates of various parameters of the Real64 model, a (attack rate), q (scaling component), and Th (handling time), for the proportion of Tyrophagus putrescentiae eggs eaten by Blattisocius mali relative to the initial number of eggs provided at six temperatures over a 24 h period.
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Table 2 Estimates of various parameters of the Real64 model, a (attack rate), q (scaling component), and Th (handling time), for the proportion of Tyrophagus putrescentiae males eaten by Blattisocius mali relative to the initial number of males provided at six temperatures over a 24 h period.
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The functional response curves were drawn and compared based on the models proposed by Hassell14 and Cabello et al.22 across all tested temperatures when T. putrescentiae eggs were used as prey. The comparison revealed parallel outcomes, indicating that the number of eggs eaten increased with increasing egg densities following a nearly sigmoidal shape (Fig. 2). On the other hand, when T. putrescentiae males were the prey, the curves were drawn based on the Roger65 model for all tested temperatures, indicating that the number of males eaten increased with increasing male densities following a hyperbolic fashion (Fig. 3).

Fig. 2
figure 2

Type III functional responses of Blattisocius mali to the Tyrophagus putrescentiae eggs at six temperatures and seven prey densities predicted from the models. Blue and red lines were drawn based on the models proposed by Hassell14 and Cabello et al.22, respectively.

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Fig. 3
figure 3

Type II functional responses of Blattisocius mali to the Tyrophagus putrescentiae males at six temperatures and seven prey densities predicted from the model proposed by Roger65.

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Based on the Hassell14 model of Type III functional response, B. mali exhibited longer handling times at lower temperatures compared to higher temperatures when preying on T. putrescentiae eggs. However, at 10 °C, the handling time was significantly shorter compared to 15 °C while there were no significant differences in handling times at 25 °C, 30 °C, and 35 °C (Fig. 4). The maximum predation rate of B. mali was significantly influenced by warmer temperatures (χ2 = 43.16; df = 5; P < 0.0001). As the temperature rose, the maximum predation rate increased, peaking at 30 °C and 35 °C, where they remained statistically similar (P > 0.05) (Fig. 4). The parameters estimated from the Cabello et al.22 model showed that B. mali exhibited higher potential for prey mortality values and shorter handling times at higher temperatures when preying on T. putrescentiae eggs as compared to lower temperatures (Fig. 5). The values of potential for prey mortality were low and did not differ significantly at 10 °C and 15 °C (P > 0.05) while these values peaked but still did not show significant difference at 30 °C and 35 °C (P > 0.05). By contrast, the handling time was the longest at 10 °C, decreased with increasing temperature up to 25 °C, and then stabilized without further change up to 35 °C (P > 0.05) (Fig. 5).

Fig. 4
figure 4

The handling time (Th) and maximum predation rate (T/Th) of Blattisocius mali preying on Tyrophagus putrescentiae eggs at six temperatures and seven egg densities, resulting from the Hassell14 model of Type III functional response. Different letters near the bars indicate significant differences between temperatures (P < 0.0001) based on 95% CI.

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Fig. 5
figure 5

The potential of mortality (α) and handling time (Th) of Blattisocius mali preying on Tyrophagus putrescentiae eggs at six temperatures and seven egg densities, resulting from Cabello et al.22 model for Type III functional response. Different letters near the bars indicate significant differences between temperatures (P < 0.05) based on 95% CI.

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Warming had a significant effect on both the FRRs (χ2 = 51.91; df = 5; P < 0.0001) and the maximum predation rates (χ2 = 33.28; df = 5; P < 0.0001) of the predator when exposed to T. putrescentiae eggs. The FRR was the lowest at 10 °C which did not vary significantly from that at 15 °C. However, these values were significantly increased at higher temperatures and achieved the highest at 35 °C (Fig. 6). Also, the maximum predation rates were lower at 10 °C and 15 °C, showing no significant difference (P > 0.05). In contrast, the maximum predation rates were higher at 30 °C and 35 °C, which also did not differ significantly from each other (P > 0.05) (Fig. 6).

Fig. 6
figure 6

The functional response ratio (α/Th) and the maximum predation rate (T/Th) of Blattisocius mali preying on Tyrophagus putrescentiae eggs at six temperatures and seven egg densities, resulting from Cabello et al.22 model for Type III functional response. Different letters above the columns indicate significant differences between temperatures (P < 0.0001) based on the Dunn test with Bonferroni corrections.

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Based on the Roger65 model of Type II functional response, the attack rate of B. mali was significantly higher at 35 °C compared to other temperatures (P < 0.05) when preying on T. putrescentiae males. The attack rates fluctuated between 10 °C and 35 °C; it was significantly lower at 15 °C than at 10 °C. However, it increased again at 20 °C, slightly but significantly decreased at 25 °C, and then increased once more at 30 °C (P < 0.05) (Fig. 7). The handling times were longer at 10 °C and 15 °C, with no significant difference between them. It decreased from 15 °C to 25 °C and then remained consistent up to 35 °C (P > 0.05) (Fig. 7).

Fig. 7
figure 7

The attack rate and handling time of Blattisocius mali preying onTyrophagus putrescentiae males at six temperatures and seven male densities, resulting from the Roger65 model of Type II functional response. Different letters near the bars indicate significant differences between temperatures (P < 0.05) based on 95% CI.

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Temperature significantly affected both the FRRs (χ2 = 51.91; df = 5; P < 0.0001) and the maximum predation rates of B. mali when preying on T. putrescentiae males (χ2 = 33.28; df = 5; P < 0.0001) (Fig. 8). The FRRs were lower at 10 °C and 15 °C, with no significant difference between them (P > 0.05). However, the FRR showed an upward trend at elevated temperatures, reaching its peak at 35 °C. Similarly, the maximum predation rates were lower at 10 °C and 15 °C, where these values did not differ significantly (P > 0.05). On the other hand, as temperatures rose, the maximum predation rate increased, peaking at 25 °C; however, it declined at both 30 °C and 35 °C, with no significant variation between them (P > 0.05) (Fig. 8).

Fig. 8
figure 8

The functional response ratio (a/Th) and the maximum predation rate (T/Th) of Blattisocius mali preying on Tyrophagus putrescentiae males at six temperatures and seven male densities, resulting from Roger65 model of Type II functional response. Different letters above the columns indicate significant differences between temperatures (P < 0.0001) based on the Dunn test with Bonferroni corrections.

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Discussion

The present study demonstrated that the functional response of B. mali did not change with changing thermal conditions ranging between 10 °C and 35 °C but varied with changing prey stages, from egg to adult male of T. putrescentiae. The temperature ranges we tested are relevant across temperate or sub-tropical regions. Across all tested temperatures and prey densities, the predatory females exhibited Type III functional responses when T. putrescentiae eggs were used as prey and Type II responses when T. putrescentiae males were the prey. In addition, the handling times were shorter at 25 °C, 30 °C, and 35 °C compared to lower temperatures, regardless of whether the prey was either eggs or males. The potential for prey mortality and the maximum predation rate, estimated for eggs as prey, were the lowest at 10 °C and 15 °C but peaked at 30 °C and 35 °C. By contrast, the attack rate of the predator exposed to T. putrescentiae males showed fluctuation from 10 °C to 25 °C, with the highest rate occurring at 35 °C. The maximum predation rate was the lowest at 10 °C and 15 °C, peaked at 25 °C, then slightly decreased at 30 °C and 35 °C. For both prey stages, the FRRs increased with rising temperatures, recorded as the lowest at 10 °C and 15 °C and the highest at 35 °C.

In predatory insects and mites, the developmental stage of prey can influence the type of functional response33,35,66. Such a phenomenon has been also observed in this study. The females of B. mali exhibited Type III functional response when preying on T. putrescentiae eggs while Type II response when T. putrescentiae males were offered as prey. Also, when tested across varying humidity levels25, B. mali females initially followed a Type III response when preying on T. putrescentiae eggs; however, when the humidity dropped to a critical level of 33%, they transitioned to a Type II response. This raises an important question about the underlying mechanisms driving shifts in functional response types. In Type III functional response, the proportion of prey eaten initially increases, and generally, this type of functional response is expected when resources or environmental conditions are suboptimal18,67,68. As suggested by Hassell14, at low prey densities, there may be insufficient ‘reward rate” for a predator to continue the constant prey-searching activity. Factors like the necessity of learning to capture prey, the small size of the prey, effective defence mechanisms, or the availability of inaccessible refuges can all hinder predation efforts18,67,68,69,70. According to a study on life table parameters54, the eggs of the T. putrescentiae were less profitable prey for B. mali than larvae. It suggests that, unlike other prey stages, the eggs of T. putrescentiae may be a suboptimal prey stage for B. mali females, leading to a Type III functional response. The eggs are too small to satisfy hunger immediately, and are immobile, making them difficult for predators to detect, especially at low densities. However, the situation may change under worsening environmental conditions such as a drop in humidity. Low humidity may lead to substantial water loss in mites, including predatory soil mites. At 33% humidity, B. mali females significantly decreased predation rate, most presumably to conserve energy25. Nonetheless, they also shifted to Type II functional response, indicating that their efforts in searching for prey remained low regardless of whether the egg densities were high or low.

Similar to humidity, temperature also affects the functional response of insect and mite predators10,28,29. At suboptimal temperatures, the cost associated with searching for food may exceed foraging rewards due to longer handling times. Additionally, rising temperatures might lead to the shift from Type II to Type III functional response or vice versa18,71,72. Contrary to our initial hypotheses, B. mali females did not change the functional responses when preying on either T. putrescentiae eggs or males across the tested temperatures, including extremes of 10 °C and 35 °C. However, this does not mean that a shift in functional response will not occur in this predator if only the range of tested temperatures is widened even further from the optimal values. It should be emphasized that B. mali was tested under conditions of optimal humidity of 85 ± 5%. In another study involving a soil mite, M. muscadomesticae that was preying on eggs of M. domesticae, the shift from Type III to Type II functional response was noted at 33 °C10. However, the mite was deprived of food before the experiment and tested at a much lower humidity level of 65.5%. This combination of high temperature and low humidity might have affected both the searching rate and functional response of this predator.

Studying functional responses not only enhances our understanding of how predator-prey interactions can fluctuate at the population level but also sheds light on the factors that may disrupt the stability of these systems18,67,71,72,73. Type II and Type III functional responses show distinct differences in terms of the stability of the predator-prey system. Type II response is characterized by a gradual decrease in the proportion of prey killed, indicating inverse density dependence. By contrast, Type III responses exhibit positive density dependence up to a certain threshold prey density, which may help in stabilizing the system when the average prey densities fall below this threshold18,67,74. A recent study by Daugaard et al.72 on the effect of warming on the functional response of the ciliate predator, Spathidium sp. and its prey Dexiostoma campylum (Stokes) Jankowski (Hymenostomatida: Tetrahymenidae), have confirmed that shifts from Type III to Type II responses may destabilize the predator-prey system. Simulation studies on population dynamics indicated that shifting to a Type II response resulted in increased prey consumption at low densities, ultimately leading to extinction in nearly all scenarios. Our findings suggest that T. putrescentiae eggs which constitute nearly 50% of a prey population75, may play an important role in stabilizing the B. mali-acarid mite system. However, to verify this hypothesis, younger and smaller developmental stages of prey such as acarid mite eggs should be used in the tests on the functional response of B. mali.

The phenomenon of warming has been shown to accelerate the metabolic rate, feeding rate, and energy gain requirements6,76, which the predators may meet by consuming more prey, possibly explaining our results of increased predation observed under warming. Tyrophagus putrescentiae performed well within a wide range of temperatures from 20 °C to 32.5 °C77, promoting prey population growth and increasing prey availability which coincides with the higher predation by the predator. The increased predation under warming has been observed for the predatory mites M. muscaedomesticae preying on the immatures of M. domestica78; A. swirskii preying on eggs of T. urticae28; Neoseiulus barkeri Hughes (Acari: Phytoseiidae) preying on nymphal stages of T. urticae79; Amblyseius longispinosus Evans (Acari: Phytoseiidae) preying on active life stages of the bamboo spider mite Aponychus corpuzae Rimando (Acari: Tetranychidae)80, indicating the widespread nature of this phenomenon.

The magnitude of functional response can be described by the predator’s attack rate, handling time, and maximum predation rate14. In this study, we also used the potential of prey mortality (α), a parameter of the expression for the Hassell14 Type III functional response model developed by Cabello et al.22. In alignment with our previous study25, this model fitted well with our data on the functional response of B. mali when preying on T. putrescentiae eggs. Also, α, which corresponds to the potential of prey mortality in a Type III response turned out to be a useful parameter in the interpretation of the effectiveness of B. mali exhibiting Type III functional response. In our study, handling time was lower while the attack rate and potential of mortality was higher at higher temperatures which might be associated with the higher moving activity, metabolic rate, energy demands, and food intake by the predator B. mali6,7. Interestingly, the effectiveness of the predator varied significantly at lower temperatures, specifically between 10 °C and 20 °C, depending on the stage of prey. For eggs as prey, the potential for prey mortality increased steadily with rising temperatures. In contrast, when preying on T. putrescentiae males, the instantaneous attack rate initially showed a slight increase before declining, exhibiting fluctuations until 25 °C, after which a marked increase was observed as temperatures rose to 35 °C. The fluctuations in attack rate might be associated with the variable effects of temperature on the relative mobility of the predator and the prey and the proportion of successful attacks14. It must be stressed that the effectiveness of the predator against mobile prey not only depends on its ability to attack and subdue a prey but also on the behaviour and defensive ability of the prey. The temperature might have differently influenced the physiology and behaviour of B. mali females and T. putrescentiae males as well as the outcomes of their interactions. When endangered, T. putrescentiae emits alarm pheromones and attempts to escape81, However, the extent to which temperature impacts pheromone production and the prey’s behavior, especially in relation to varying prey densities, remains unclear.

In our study, we found that an elevated instantaneous attack rate or increased potential for prey mortality at a given temperature did not always result in a simultaneous reduction in handling time or an increase in the maximum predation rate, making it difficult to interpret the actual impact of the predator on the T. putrescentiae. To address this issue, we also calculated the functional response ratio proposed by Cuthbert et al.26, for both the attack rate and potential of mortality. This parameter clearly showed that the impact of B. mali on both the eggs and males of T. putrescentiae intensified with rising temperatures, peaking at 35 °C.

Our findings suggest a high potential for the predatory mite B. mali to reduce the population of T. putrescentiae at higher temperatures. We determined a strong impact of temperature on predator’s efficiency as predator action accelerated under warming and increased prey consumption. The functional response type did not change with increasing temperatures; however, it changed with changing the prey stage. Although the findings provide valuable insights into the potential effectiveness of B. mali against T. putrescentiae at varying temperatures and prey stages, the scope of the study may limit its applicability to real-world scenarios. It must be stressed that under natural conditions, this predator inhabits various substrates, such as soil, litter, and decaying plant material, in which not only temperature but also humidity can affect predator and prey interaction in various ways. Moreover, substrates can vary in complexity, creating various opportunities for prey to hide and avoid predation73. Thus, further studies should explore the common effects of different levels of humidity and temperature as well as the role of habitat structure and prey behaviour on the functional response of B. mali and the stability of the predator-prey system.

Methods

Mite culture

The primary culture of T. putrescentiae reared on instant dry bakers’ yeast and wheat bran in equal parts by weight, was obtained from the mass rearing of the Department of Plant Protection, Warsaw University of Life Sciences, Warsaw, Poland25. Adults of T. putrescentiae were carefully selected and reared in glass Petri dishes measuring 90 mm in diameter, with a mixture of yeast and wheat bran in the same 50/50 ratio, to obtain 24 h eggs according to the methodology suggested by Jena et al.25. The colonies were kept in darkness at 26 °C and 95 ± 5% in a Sanyo Environmental Test Chamber (Panasonic MLR-350).

The stock population of B. mali was obtained from mass-rearing of the predator, which was maintained within wheat bran and fed on different stages of T. putrescentiae in the climatic room of the Department of Plant Protection at Warsaw University of Life Sciences, Warsaw, Poland. The rearing unit consisted of foam platforms, drenched in water and covered with foil within larger containers as described by Michalska et al.41,52,53. The cultures of B. mali were maintained in the Panasonic Environmental Test Chamber (MLR-352-PE), at a temperature of 23 °C, with a photoperiod of 16 h of light and 8 h of darkness and a relative humidity of 85 ± 5%.

Functional response experiment

The experimental setup included a Plexi-glass cage with a circular hole of 8 mm diameter, a piece of filter paper affixed to the bottom of the cell, and a glass coverslip of 18 mm × 18 mm attached to the top of the cell using paraffin wax25. The female cohorts were prepared following the methodology described by Jena et al.25. The colony was fed with the mixed life stages of T. putrescentiae reared on yeast 24 h before choosing the female predators. Female predators were randomly chosen from the colony and exposed to varying densities, either 10, 20, 40, 60, 80, 120, or 160 eggs or 1, 2, 3, 5, 7, 10, or 15 males of T. putrescentiae at six different temperatures, 10 °C, 15 °C, 20 °C, 25 °C, 30 °C, and 35 °C, with a relative humidity of 85 ± 5% RH and a photoperiod of 16:8 h in an incubator (MIR-154-PE) for one day (24 h). To select the specific density of prey, pilot tests were conducted on the predation rate of B. mali female on T. putrescentiae eggs or males over one day. The separation of T. putrescentiae eggs from other life stages was achieved by sieving the rearing colonies through a 100 μm mesh screen and transferring eggs to the cell of the cage using a fine paintbrush25. The selection of T. putrescentiae males was done manually from the mixed population and placed in the cages with care to avoid any harm. Wet filter paper was initially placed around the hole to prevent the males from escaping, which was then replaced with a cover slip once the desired densities were achieved. After the 24 h exposure period, the predators were removed, and the consumption of eggs or males was noted, excluding any that remained. Cages where a live predator was not recovered due to loss or death were excluded from the analysis. Each egg density was replicated twenty times at each temperature, while each male density was replicated fifteen times at each temperature.

To examine the impact of temperature and prey density on the consumption of T. putrescentiae eggs or males by B. mali, we applied Generalized Linear Models (GLM) with a Poisson probability distribution. To further analyze the results, Tukey’s linear contrast was employed as a post hoc test.

The analysis of the functional response data was conducted in two phases. Initially, we focused on identifying the specific type of functional response followed by an estimation of the parameters associated with the functional response82,72. The functional response type was identified by applying the generalized functional response equation developed by Real64. The modified Holling disc equation, as proposed by Real64 (Eq. 1), was as follows82:

$$\:\:{\text{N}}_{\text{a}}=\:\frac{\text{a}\text{T}{\text{N}}_{0}^{(\text{q}+1)}}{1+\text{a}{\text{T}}_{\text{h}}{\text{N}}_{0}^{(\text{q}+1)}}$$
(1)

where Na is the number of prey eaten, N0 is the initial number of prey densities provided, a is the predator’s instantaneous attack rate or searching efficiency, Th is the handling time, T is the time length of the assay, and q is the scaling component that determines the shape of the curve. The functional response curve can be of different types: Type I, which is a linear relationship (q = 0 and Th = 0), Type II, characterized by a hyperbolic curve (q = 0, Th > 0), and Type III, displaying a sigmoid curve (q > 0, Th > 0).

After identifying the appropriate shape of the functional response, the functional response parameters, i.e., instantaneous attack rate (a), handling time (Th), and the potential for prey mortality (α), were determined by fitting them to appropriate models. The data was then fitted to equations proposed by Roger65(Eq. 2), Hassell14 (Eq. 3), and Cabello et al.22 (Eq. 4), using non-linear least squares regression, as the prey that was depleted during the experiment was not replenished:

$$\text{Roger}\:\text{Type}\: \text{II}:{\text{N}}_{\text{a}}={\text{N}}_{0}\bigg[1-\text{exp}\bigg\{-\text{a}\text{P}\bigg(\text{T}-\frac{{\text{T}}_{\text{h}}{\text{N}}_{\text{a}}}{\text{P}}\bigg)\bigg\}\bigg]$$
(2)
$$\text{Hassell}\:\text{Type}\:\text{III}:{\text{N}}_{\text{a}}={\text{N}}_{0}\bigg[1-\text{exp}\bigg\{-\frac{\text{b}{\text{N}}_{0}}{1+\text{c}{\text{N}}_{0}}\bigg(\text{T}-\frac{{\text{T}}_{\text{h}}{\text{N}}_{\text{a}}}{\text{P}}\bigg)\bigg\}\bigg]$$
(3)
$$\text{Cabello}\:\text{et}\:\text{al.}\:\text{Type}\:\text{III}:{\text{N}}_{\text{a}}=\:{\text{N}}_{0}\:\bigg[1-\text{exp}\left\{-\frac{{\upalpha\:}{\text{N}}_{0}}{1\:+{\text{T}}_{\text{h}\:}(\text{exp}\left(-{\upalpha\:}\right)-1){\text{T}}_{\text{h}}}\left(\text{T}-\frac{{\text{T}}_{\text{h}}{\text{N}}_{\text{a}}}{\text{P}}\right)\right\}\bigg]$$
(4)

where Na is the number of prey eaten, N0 is the initial number of prey density offered, a is the predator’s instantaneous attack rate, Th is the handling time, P is the number of predators used, T is the time length of the assay, α is the potential of mortality of the predator, and b and c are the constants that relate a and N0 in Type III functional response as a= \(\:\frac{b{N}_{0}}{1\:+c{N}_{0}}\). In our experiment, P = 1 and T= 1 day.

We determined the values of parameters a, Th, and α at all tested temperatures using a non-linear least square regression approach. The confidence intervals (± 95% CI) were calculated for these parameters, with significant differences between means indicated by non-overlapping intervals (P < 0.05). To calculate the Confidence Intervals (CI), we employed the permutation test method as described by Ernst83. Additionally, we analyzed the proportion of prey eaten by the predator at varying densities using Generalized Linear Models (GLM) with a gamma probability distribution. Further, to compare and analyze the functional response parameters, a, α, and Th at six temperatures for both eggs and males as prey, the FRR was estimated by using either the attack rate (a) or potential of prey mortality (α) divided by the handling time (Th)25,26. Furthermore, the maximum predation rate84 was determined which is defined as the maximum number of prey that a predator consumes during a particular time frame. It was estimated by dividing the duration of the assay, T (day) by the handling time, Th (day). To assess whether FRRs and predation rates varied across tested temperatures, one-way Kruskal-Wallis’s rank sum tests were conducted. Post hoc comparisons were made using the Dunn test with Bonferroni corrections25. All statistical analyses were conducted using R version 4.3.085. However, to estimate the functional response type and parameters, “FRAIR: An R Package” was applied82.