The ladybird is holometabolous (having complete metamorphosis) and passes through the following developmental stages: egg, larva (with four instars), pupa and adult. The total developmental time is around 1–2 months. The egg stage lasts 15–20% of the preimaginal developmental time, the larva 55–65% and the pupa 20–25% (Honěk and Kocourek 1990; Dixon 2000).

Similarly to oviposition, the food quality and quantity affect the ladybird’s development time and final size. Hauge et al. (1998) found that Metopolophium dirhodum and Sitobion avenae were significantly better prey than Rhopalosiphum padi, resulting in faster development, likely higher survival and bigger adult ladybirds. Acyrthosiphon pisum was significantly better than Rhopalosiphum maidis in terms of developmental times and adult weight (Obrycki and Orr 1990). Similar results were found by (Omkar and Srivastava 2003) for developmental weight and size with the same order of species suitability as they found for total oviposition (see Section Oviposition / Prey type). However, when considering development and oviposition scales, it is noteworthy that these two studies (i.e. Hauge et al. 1998; Farooq et al. 2020) disagree on the suitability of S. avenae as food compared to R. padi. Farooq et al. loc. cit. note, however, that the host plant may affect the palatability and suitability of aphid species to ladybirds, and this has been shown for Aphis fabae (Shannag and Obeidat 2008).

If prey quality is constant, then the development and transition between the developmental stages is a function of the temperature and weather (e.g. Ricci et al. 2005). So, depending on the particular year or latitude, the population dynamics pattern could shift over time. A linear model approximates the course of development rate in the ecologically relevant temperature range and enables the calculation of two thermal constants. The first of these is the above-mentioned lower development threshold (LDT), also known as the basal temperature T_b (Trudgill et al. 2005), and this is the temperature below which development ceases. The other constant is the sum of effective temperatures (SET) or, in other words, the thermal time T, which is the number of day degrees [dd] above LDT for completion of a developmental stage.

Several studies have estimated the fundamental developmental values (LDT and SET) for ladybirds under different conditions (see Obrycki and Orr 1990; Formusoh and Wilde 1993; Omkar and Srivastava 2003; Katsarou et al. 2005), and (Hodek et al. 2012) for an overview of other species. Based on these studies, Skirvin et al. (1997) fitted sigmoid curves to obtain their model’s temperature-dependent development rates for the juvenile stages. However, the estimated values do not necessarily concur, and the methods of the studies and treatments differ significantly. There is, however, an agreement on the optimal temperature for development being approximately 30 °C (Xia et al. 1999; Aman et al. 2023). In addition, some studies provide extensive datasets that include the influence of additional factors, such as prey density or prey type, on development (e.g. Xia et al. 1999; Farooq et al. 2020). It seems clear that prey density will change the feeding rate of the larvae and, therefore, their developmental rate. However, the precise effect depends on the prey quality and quantity available.

Assuming constancy of food supply, the daily number of aphids consumed grows with the individual’s age, and the relationship is close to linear (Fig. 2).

For the sake of simplicity in implementation, the immobile prepupa stage of the larva is treated as a pupa. In the implementation, we emphasise several important development stages of the juvenile ladybird:

Hourly air temperatures are utilised to compute the accumulated degree days for each life stage. The developmental constants applied in this calculation are detailed in Table 1, and were estimated to fit the data from Xia et al. (1999).

In addition to temperature, the growth of the larvae is influenced by the density of aphid prey (Table 2). In extreme situations, larvae will die after three consecutive days of starvation. To account for prey quality differences, the prey quality scaler E_A will be applied, as it is for oviposition. This appears to be reasonable since, in almost all cases, growth rates seem to scale with quality of prey. The exception to this rule is that not all studies showed clear differences in larval growth between aphid species, despite these causing differences in fecundity (e.g. Kalushkov and Hodek 2004). However, in this case, all prey species compared were considered to be high-quality food.

For the sake of consistency, we exclusively used the data from Xia et al. (1999) to inform the development equations and parameters. We note that this assumes the prey values are scalable if prey with higher or lower prey quality than the cotton aphid A. gossypii is provided using the prey quality scaler E_A. The relationship is linear (the development is faster when the prey level is higher). Practically, this is implemented by the introduction of a prey-level dependence in SET (SET_prey; Equation 1).

SET _prey = SET × f_L (N_prey) × E_A Equation 1

where the function f_L (N_prey) factors the SET value, increasing the desirable threshold value when the prey amount is insufficient.

The values were again estimated from Xia et al. (1999) for each instar. Prey density is assumed to be indexed by the average number of prey eaten per day over the lifespan of the stage. This method increases the length of the stage depending on the food consumed. The levels of prey fed to each instar differed (Xia et al. 1999); hence, the values from the study needed to be translated to relative values of aphids/m² for each instar. To do this, a 2^nd order polynomial was fitted to the data (Equation 2).

f_L (N_prey) = a + bN_prey + cN_prey² Equation 2

where f_L (N_prey) is the relative developmental rate compared to the maximum, a, b, and c are constants, and N_prey is the number of aphids/m² (Table 9).

To implement this in the model, the mean aphid density per day experienced by the larva at each stage is updated daily, and the development time (SET) is scaled daily accordingly. Values above 45 (1^st and second instar), 80 (3^rd instar) and 120 (4^th instar) aphids/m² are assumed to be 45, 80 or 120 aphids/m², respectively in the calculation of mean aphid densities experienced (Table 3). This fixes maximum development rates to factor 1.0 of the maximum development rate at that temperature.

The lifetime of a female (L_f) in days is calculated after hibernation and is calculated stochastically by the Equation 3.

L_f = 4 × (u–0.5) × 10.77 + 147.75 Equation 3

where u is a random number from a Uniform distribution U (0,1).

L_f would normally not be needed, but would kill the ladybirds if they fail to lay their complement of eggs, e.g. if aphid numbers are too low.

Due to ladybirds’ role as natural enemies, the relationship between the coccinellids and their prey was one of the first areas of study. Therefore, considerable work has been focused on understanding ladybird prey-searching patterns, prey preference, and the influence of diet on ladybirds’ development, reproductive potential, and survival.

Noting that when a toxic aphid, Aphis sambuci, was eaten by C. septempunctata, having detrimental effects, it was suggested that physiological parameters, such as growth, reproduction and mortality, should be used as the decisive criterion for diet effects (Hodek 1956). The proposal was to discriminate between two principal kinds of food: essential that ensures oviposition and the completion of larval development, and alternative that serves only as a source of energy and thus prolongs survival in comparison with starvation (Hodek 1962, 1967, 1993). This approach has been widely accepted (Evans et al. 1999; Berkvens et al. 2010). For the list of essential prey in C. septempunctata see Table 4.

Studies of different strains of the same coccinellid species indicated important differences in main developmental life history parameters as a function of prey suitability (Michels and Flanders 1992). There is also a significant impact of prey type on mortality (Özder and Sağlam 2003) (Fig. 3).

The diet of the ladybirds can vary with season, depending on prey availability or their stage of development. For example, in early spring (before aphid emergence), C. septempunctata feeds on non-aphid arthropods, fungal spores, pollen etc. (Ricci et al. 2005). However, in June and July, its diet consists mainly of aphids. Likewise, the physical size of the ladybird influences the necessary prey density (Yang et al. 2022). The aphid density required by a small ladybird is increased because, due to its size, the ladybird is unable to catch larger aphid instars (in this case, fourth instars and adults); this might be more important with large aphid species (Sloggett 2008).

Coccinellids spend most of their time searching for prey (Table 5). However, their foraging activities are influenced by several indirect factors, including taxis, plant structure, plant surface, and diurnal periodicity. A decision-making framework proposed by Hodek et al. (2012) suggests that ladybirds navigate through a landscape, landing randomly on plants during foraging. Upon detecting prey, they initiate a localised search pattern, resulting in aggregation in areas with high prey density. Satiation typically ensues, which reduces their tendency to disperse and leads to a localisation of oviposition. Indirect support for this general decision-making framework comes from the observations of aphidophagous coccinellids that oviposit in areas of high prey density (e.g. Dixon 1959; Banks 2009).

Bianchi and Van Der Werf (2003) described the residence time in a prey patch as being proportional to the aphid density in their searching area (Equation 4).

t_r = r_c · A/SAI Equation 4

where t_r is the mean residence time (days), r_c is the residence time coefficient (days/aphids/m² search area), which is a scaling factor. A is the aphid density (aphids/m² ground area), and SAI is the search area index (m² of leaf area/m² of ground area) of the landscape element.

Literature shows that ladybirds are capable of detecting the substances that can guide them to the prey: honeydew (and sugar in sugar-treated plots; Evans and Swallow (1993)), aphid pheromones, and volatile compounds released by plants attacked by herbivores (Ninkovic et al. 2001). However, the minimum prey density capable of attracting C. septempunctata is considered to be ten aphids per m² (Honěk 1980).

After the dispersal, ladybirds become active and search for food through foraging behaviour. An adult ladybird requires ten aphids daily for sustenance. The nutritional needs of larvae vary across their four instars, with the maximum daily aphid consumption assumed to be 45, 45, 80, and 120 aphids for the first, second, third, and fourth instars, respectively. Larvae are limited to local foraging (see Section Movement / Planned implementation / Short-range movement); should they fail to find sufficient aphids within their allotted daily range, they may resort to cannibalism (Section Cannibalism / Planned implementation). Adults engage in local foraging (Section Movement / Planned implementation / Short-range movement) and long-distance foraging (Section Movement / Planned implementation / Long-range movement) and will commence egg-laying if they encounter an adequate supply of aphids.

The potential oviposition period is regulated by two parameters: the commencement month and the termination month. During this period, the female will attempt to commence oviposition only if it has successfully survived the winter. If these two conditions are met and the female successfully locates a suitable site for egg laying, a day counter will commence. Once the day counter reaches a predetermined value (defined as Minimum maturation period), the female will initiate egg-laying.

Each female can lay a maximum number of eggs in its lifetime. Stochastic variability around this value is added based on what is typical of laboratory studies. The actual lifetime egg number chosen depends on the assumed food quality that the ladybird consumes. Assuming the maximum value reached when feeding on Lipaphis erysimi is 1764 eggs, this is 2.3 times the number produced when feeding on A. gossypii, according to Omkar and Srivastava (2003). A scaler can then be introduced based on the daily prey consumed (E_A).

Female aphids only lay eggs when the aphid density at a specific location is above predetermined thresholds. This behaviour is influenced by the natural habitats and crops in which they reside, as these environments may have varying thresholds for egg production. Due to a lack of specific information, we assume these thresholds are consistent, but we will include stochastic variation to replicate variation in Table 7. Based on the average temperature from 0 to 50 °C, with a step size of 1 °C, and the prey level, Equation 7 will be used to calculate the daily number of eggs (N_e) to lay.

Ne=fT(⌊T⅃)×Rd Equation 7

where the T represents the daily average temperature in °C, and R_d is the oviposition rate (Equation 6).

f_T ([T]) is calculated by Equation 8 and Equation 9 below:

gT([T])=f2f1×f3×⌊T⅃-Ttf1f2-1×exp-⌊T⅃-Ttf1f2 Equation 8

fT(⌊T⅃)=gT(⌊T⅃)×CmaxmaxgT(⌊T⅃) Equation 9

where T_t = 11 °C is the temperature threshold below which f_T ([T]) = 0, f₁ = 18.22, f₂ = 3.11, and f₃ = 349.5. C_max = 45 is the maximum clutch size and max (g_T ([T])) is the maximum value from g_T ([T]) for the temperature range 12 to 50 °C.

R_d is derived from Equation 6 and parameters for daily egg production from Table 8, multiplied by the aphid scaler E_A to account for prey quality. For simplicity, we assume a refractive period of one day between egg-laying batches. Therefore, every second day, the number of eggs to be laid (N_e) is summed for the previous two days to determine the size of the egg batch to be laid at a certain location. Upon completing its egg-laying cycle, when all eggs are laid and unless already killed, the female is removed from the simulation, representing its death.

The model does not utilise information about the dependence between aphid density and the number of eggs laid by individual ladybirds. However, ladybirds are attracted by the aphids and spend more time in areas with higher aphid density. Consequently, the number of egg clutches is expected to increase in landscapes with high aphid densities, and we expect a linear relationship. This relationship will need to be verified during calibration.

We assume a minimum maturation period of four days before oviposition is possible, based on estimates from literature (Honěk 1980; Lanzoni et al. 2004).

Farming activities, such as intensive soil processing, can be connected to increased mortality. The list of activities that have non-zero mortality chance is provided in Table 12. Note that in ALMaSS, pesticide applications are often treated in detail, and the implementation of this will be considered in the application of the model on a case-by-case basis.

The temperature-related mortality implemented in our ladybird model used values derived from Xia et al. (1999). They provided parameters for a 2^nd order polynomial for the relationship between relative mortality and temperature for each life stage (Table 11; Fig. 7). Equation 10 is used to generate the temperature-based mortality (M_T) rates at 5 °C intervals of the temperature (T) for inclusion in the model.

M_T = d₁ + d₂T + d₂T² Equation 10

Parameters d₁, d₂ and d₃ provided by Xia et al. (1999) (Table 11) for the adult stage depend on adult’s age, expresssed in the form of 12 age classes. These 12 age classes encompass the maximum ladybird lifespan with three standard deviations added. Unfortunately, there is no clear statement about what maximum lifespan was used. It is also notable that provided adult lifespans are short, suggesting that diapause phase was not considered by Xia et al. (1999). Thus, to be able to use parameters from Xia et al. (1999) in Equation 10, the value for ladybird maximum ladybird must be assumed.

Farooq et al. (2020) recorded prey-type dependent mean lifespans of up to almost 60 days. This is in the middle of the range recorded by Omkar and Srivastava (2003) (44–85 days, dependent on sex and prey species). In contrast, Kalushkov and Hodek (2004) recorded a mean female longevity of 68–78 days on four different aphid species. Hence, initially, we will assume 72 days, this being easily divisible by 12; meaning that each age class from Table 11 spans six days. However, no additional penalty for low prey was applied to allow them to survive without aphid populations early in the year (see Section Foraging).

It will depend on the size and temperature extremes of the hibernacula clump. If the clump size is smaller than 3, the mortality chance is 0.14, while it is 0.07 when the clump size is not smaller than 3 (see details in Section Diapause and Hibernation / Planned implementation). These mortality chances are applied once at emergence. These values are lower than the mean of 48% from the literature because they are subsequently combined with parasitoid mortality and extremely low-temperature mortality to give the correct total mortality and range after the calibration.

To implement extremes of temperature, an extra mortality of 25% is applied during hibernation as a single mortality chance if extremes have occurred (Table 13). For this, the SCP is used as a minimum mortality threshold. SCP (t_extreme,min) is different for each month of the year and corresponds to (Nedved 1993), January to December: -15, -15, -10, -9, -5, -5, -5, -5, -5, -9, -12, -17 °C. Because ladybirds hibernate in groups and hide in insulated places, the dormant ladybirds are assumed to have effectively lower SCP (by 20%, calibrated value), if in a hibernaculum with more than five ladybirds. A maximum extreme mortality threshold is t_extreme,max = 40 °C for all the stages (including adult). This mortality is currently assumed to be in addition to the standard hibernation mortality, but this assumption may need to be evaluated during calibration. For non-hibernating ladybirds of all stages, extreme temperature mortality is assumed to be applied daily when the extremes occur.

To account for all other mortalities, an individual’s mortality is regulated by random variables drawn at each step of the simulation. Each stage will be assumed to have a specific daily chance of mortality, which will be a fitting parameter. Parasitoid and cannibalism mortalities are considered in their separate sections below.

Parasitism is an important factor that affects mortality and reproduction chances in coccinellids. Beetles are more vulnerable to a parasite while dormant. Main parasite that affects C. septempunctata is Dinocampus coccinellae (Schrank). D. coccinellae is a braconid wasp attacking adult ladybirds, primarily females (Cartwright et al. 1982), although this is not always consistent (e.g. Koyama and Majerus 2008). The parasitoids oviposit during late summer, but the eggs rest in the body cavity until the following spring and only develop when the host leaves the hibernating site and starts to feed (Ceryngier 2000). It seems that olfactory cues can contribute to the orientation of the wasp towards its prey (Pettersson et al. 2005). The ladybirds usually survive the parasites’ emergence but are weakened and usually die within a few days.

Parasitism rates in C. septempunctata have been recorded as high as 33% post-dormancy (Triltsch 1996; Ceryngier 2000), but may be very variable during the season (0 to 33%) (Triltsch 1996; Kızıltepe et al. 2009; Soleimani and Madadi 2015). Lipa et al. (1975) recorded 20% mortality by parasitoids and diseases including microsporidian species of Nosema coccinellae, Gregarina coccinellae and Gregarina ruskowski (fungi), and parasitoid Perilitus coccinellae (synonom Dinocampus coccinellae). Koyama and Majerus (2008) found 12.2% +/- 1.8% parasitisation by D. coccinellae in field-collected populations of ladybirds. Ceryngier (2000) showed that the parasitisation rate may differ depending on the location of the hibernation site. In Poland, the parasitisation rates of populations hibernating at the mountain tops (> 1.300 m a.s.l) were much lower (15 to 25%) than of those hibernating at the lower altitudes (800 m a.s.l.; > 70%).

In the model’s early version, parasitism dynamics are simplified. The effect of parasitism is included as an extra one-time stochastic mortality when the adults disperse from hibernacula. We assume an initial value between 0 and 33% with a central tendency. The parasitism mortality (M_p) is sampled from the uniform distribution (Equation 11).

M_p~U (0,0.33) Equation 11

Intra-guild predation and cannibalism by or on predatory coccinellids is very common, and the seven-spot ladybird is a characteristic species in that sense (Dixon 2000; Lucas 2013). These behaviours are influenced by several factors, including prey aggregation (Dixon 1987), richness and abundance of competitors (Rosen 1990), drastic changes in body size (almost five-fold during the immature life stages), and mobility of ladybirds during the life cycle. There is a relationship between the intensity of cannibalism and the abundance of prey (aphids). Thus, cannibalism is more common when aphid prey is scarce and decreases when the prey becomes abundant (Şengonca and Frings 1985; Agarwala and Dixon 1992; Lucas et al. 1998; Sato et al. 2009). At the same time, if the amount of aphid prey increases, the chance of intra-guild predation increases, probably due to predator competition (Nóia et al. 2008). The outcome of cannibalistic interactions is size-dependent; eggs and young larvae are more vulnerable to cannibalism than older larvae. Literature states that adult coccinellids are never successfully attacked (Hodek et al. 2012).

Four hypotheses were built to describe the reason for cannibalism in general and cannibalism of the ladybirds in particular (Lucas 2013):

Sibling egg cannibalism in ladybirds has been shown to minimise starvation risks prior to dispersal (Khan and Yoldaş 2018). Therefore, the coccinellid eggs often hatch synchronously, typically within 10 minutes, to avoid being eaten by siblings. Michaud and Grant (2004) investigated three aphidophagous Coccinellidae species that lay clustered eggs, Cycloneda sanguinea, Harmonia axyridis, and Olla v-nigrum, and found that 50–60% of eggs hatched synchronously, while 20–40% were cannibalised by siblings, depending on the species. Early-hatching larvae can consume the remaining eggs. When the larvae cannibalise conspecific eggs, they will develop faster, have lower mortality, and the duration of the first moult will be shorter (Pervez et al. 2022).

Table 14 summarises various laboratory studies on the cannibalistic behaviour of C. septempunctata. However, in these studies, the chances of cannibalisation events may have been overestimated. For example, Takahashi (1987) showed cannibalism increasing with the larval stage in mass-rearing experiments, which do not represent natural conditions. Therefore, the extent to which we can infer natural cannibalism from laboratory studies is hard to determine.

There is also potential for sexual dimorphism in regard to cannibalistic behaviour. Males of Adalia bipunctata consumed an average of 35.2 out of 40 eggs offered (88%), whereas females consumed significantly fewer, averaging 23.2 out of 40 eggs (58%, Table 14) (Agarwala and Dixon 1992).

Since we currently only consider one species, intra-guild predation is ignored. We adopted hypothesis III in the model, implementing cannibalism as a ‘last resort’ for a ladybird. Cannibalism occurs only when a ladybird cannot locate any aphids to consume, prompting it to cannibalize younger individuals. Regardless of the success of a cannibalisation attempt, the adult or larva subsequently moves in search of areas with higher aphid density. Each stage can only eat the less advanced stage: instar four can only eat instars one to three and eggs, instar three can only eat instars one to two and eggs, and so on. Cannibalism is only implemented for larval stages; adult cannibalism is ignored in the current model.

Nutritional demands drive cannibalistic behaviour in this model, necessitating the tracking of ladybird appetite to determine daily behaviour. Cannibalised individuals are assigned a nutritional value, which is assumed to be consistent within each life stage or larval instar. The nutritional value was estimated from the comparison between the mean weight of the ladybird stage and an aphid, see Table 15, based on Honek, Dixon, and Martinkova (2008) and calibrated values of sister species from the study of Sengonca and Arnold (2003).

Since the first studies (Dobrzhanskii 1922), experimental research has indicated a very wide plasticity in C. septempunctata (Obrycki and Tauber 1981; Phoofolo and Obrycki 2000; Hodek and Michaud 2008), enabling heterogeneous activity in different fractions of populations. This is especially true regarding the multivoltine vs. univoltine behaviour, which can differ even within various regions of Europe (see Hodek et al. (2012) for a thorough review). Generally, while populations inhabiting Northern and Central Europe are predominantly univoltine, the alternations in temperature, food abundance or length of day (experimental) can potentially trigger multivoltine behaviour (Jöhnssen 1930; Hämäläinen and Markkula 1972). However, naturally, the second generation is extremely rare in colder climates, with ladybirds mostly dying in the egg or larva phases. At the same time, the southern populations, such as those in southern-eastern France (Iperti 1966), exhibit partial multivoltinism (Bodenheimer 1943), and some isolated observations from southern Spain (Hodek and Okuda 1993) report bivoltine behaviour. Hagen (1962) described six patterns of voltinism for European coccinellids, ranging from single generation with hibernation to multiple generations with or without aestivation. C. septempunctata appears to possess sufficient plasticity to exhibit several of these patterns.

However, in cold and temperate climates like Central Europa, C. septempunctata spends 2/3 of its lifespan in hibernation. The adults stay at the diapause sites from July/August to April/May (which corresponds to the maximum flying activity in April and August; Boiteau et al. (1999)). However, the exact times are subject to geographical factors. Ladybirds normally choose to hibernate at lowland sites with warm microclimates (southern slopes and woodland margins, roadside), under dead leaves or various debris, in dry vegetation, singly or in small groups. Not all individuals hibernate, some stay active looking for food (Tauber et al. 1986). At the same time, the overwintering mortality in the hibernating ladybirds is recorded to be significantly lower than in the active ones (Barron and Wilson 1998; Carrillo et al. 2004; Koch et al. 2004).

It has been stated that for diapause, C. septempunctata often looks for prominent landscape features, e.g., peaks or large rocks. After gathering around a potential hibernation place and spending some time there (Hodek et al. 1993; Hodek et al. 2012), ladybirds use possibly olfactory stimuli as a cue (Al Abassi et al. 1998) to find other individuals already present at the hibernacula (Hagen 1962; Hodek and Honêk 2013). C. septempunctata can overwinter up to 2000 m altitude in Europe (Ricci et al. 2005). However, observations of large aggregations in rocks on mountains have been questioned because of the potential misidentification of the specimens (Hodek and Honêk 2013).

Aggregation is probably a protective strategy against predators and/or parasitoids. Adults can form clumps on substrates such as rock debris and grass (Honek et al. 2007; Hodek et al. 2012). The seven-spot ladybirds form small to medium-sized clumps (relative to other coccinellids) even if the adults are abundant (Honek et al. 2007), with clump size varying with population density (Fig. 8). Ladybirds tend to use the same hibernaculae in the same landscape year after year. The information transfer mechanism is not exactly known, but it could be via olfactory cues (Al Abassi et al. 1998) or thanks to individuals that survive to the second hibernation.

The literature identifies several factors that trigger diapause in ladybirds (Hodek and Čerkasov 1961). The change in photoperiod is considered the dominant factor, with temperature playing a considerable modifying role. Long day lengths have been shown to retard diapause (Obrycki et al. 2024). In addition, there are reports, albeit from other species, indicating that the physiological ageing of plants and consequent changes in aphid populations can influence diapause (Rolley, Hodek, and Iperti 1974).

Overwintering typically ends around April (around 110–120 day of year; Bianchi et al. (2007)). While some authors refer to a photoperiodic trigger to the emergence (Hodek et al. 1977), others state that no specific diapause-terminating stimulus has been identified (Tauber et al. 1986). Usually, overwintering ends before the emergence of aphids; thus, the first prey of C. septempunctata is necessarily different.

In this model, only the overwintering diapause is implemented, excluding aestivation diapause. Aestivation is disregarded due to its rarity in Northern and Western Europe, where we intend to apply the model.

The landscape features that ladybirds are looking for in the search for hibernation places are types of landscape connected with rocks or forest/orchard edges. Currently, we ignore the possible tendency of ladybirds to prefer southern slopes and mountainous areas.

The primary triggers for hibernation are photoperiod (explicit through the first and last aggregation day) and decrease in temperature and physiological ageing of the plants (implicitly through the aphid density). The ladybirds cannot stay active after the last day of aggregation. Ladybirds will initiate aggregation under the following conditions (Table 16):

From August to December, when the day length falls below a set value, d_F:

After a ladybird finds a suitable location for hibernating, it will do nothing but wait until the beginning of next year. At the beginning of next year, the clump size will be calculated in a 100 × 100 m² rectangle centring on the ladybird’s location. There are three categories based on the ladybird density in this local surrounding: category (1) with less than 10 ladybirds, category (2) with ≥ 10 of ladybirds but < 100 ladybirds, and category (3) with 100 or more ladybirds. Subsequently, the hibernation clump size will be determined using probabilities outlined in Table 17, which categorizes clump sizes into four distinct groups. Two smaller clump-size categories will result in a mortality rate of M_Hs during hibernation, while the mortality rate for the two larger clump-size categories will be M_Hl (Table 16). Aggregation movement is described in the movement section (Section Movement / Planned implementation / Long-range movement), and we assume that the target habitats are linear features such as forest edges, road verges or hedgerows.

The photoperiod also drives the end of overwintering. The aggregation will cease when the day length exceeds d_L minutes. The ladybird will have a 10% chance to wake up from hibernation when the day length is longer than t_E minutes if it passes the mortality test based on the clump size (see section Mortality / Planned implementation), and can switch to movement behaviour if the temperature allows (see section Movement / Planned implementation / Long-range movement).

There are two movement types planned for implementation: short-range movement (walking or short flights, considered as walking) and long-range movement (flying). Short-range movement simulates intensive search behaviour when the adult ladybird or larva scans the local surroundings in search of prey. It is performed in steps until either a maximum movement distance is reached or a suitable density of aphids is found. Here, Equation 13 (see below) is applied to calculate maximum movement distance. Long-range movement simulates the extensive search for a different type of landscape element or vegetation. This movement could be either part of aggregation or dispersal behaviour or a response to insufficient resources at the current location. Details for the implementation of both types of movement behaviours are presented below, with state variables provided in Table 19.