A comprehensive dynamic growth and development model of Hermetia illucens larvae
Murali Padmanabha, Alexander Kobelski, Arne-Jens Hempel, Stefan Streif
AA comprehensive dynamic growth and development model of
Hermetia illucens larvae
Murali Padmanabha, Alexander Kobelski, Arne-Jens Hempel, Stefan Streif * Automatic Control and System Dynamics Lab, Technische Universitt Chemnitz,Chemnitz, 09107, Germany* [email protected]
Abstract
Larvae of
Hermetia illucens , also commonly known as black soldier fly (BSF) havegained significant importance in the feed industry, primarily used as feed foraquaculture and other livestock farming. Mathematical model such as Von Bertalanffygrowth model and dynamic energy budget models are available for modelling the growthof various organisms but have their demerits for their application to the growth anddevelopment of BSF. Also, such dynamic models were not yet applied to the growth ofthe BSF larvae despite models proven to be useful for automation of industrialproduction process (e.g. feeding, heating/cooling, ventilation, harvesting, etc.). Thiswork primarily focuses on developing a model based on the principles of the aforementioned models from literature that can provide accurate mathematical description ofthe dry mass changes throughout the life cycle and the transition of development phasesof the larvae. To further improve the accuracy of these models, various factors affectingthe growth and development such as temperature, feed quality, feeding rate, moisturecontent in feed, and airflow rate are developed and integrated into the dynamic growthmodel. An extensive set of data were aggregated from various literature and used forthe model development, parameter estimation and validation. Models describing theenvironmental factors were individually validated based on the data sets collected. Inaddition, the dynamic growth model was also validated for dry mass evolution anddevelopment stage transition of larvae reared on different substrate feeding rates. Thedeveloped models with the estimated parameters performed well highlighting itsapplication in decision-support systems and automation for large scale production.
Hermetia illucens , commonly known as the black soldier fly (BSF), is an insect specieswhich is widely studied for the high nutrition value of its larvae. Studies [1–4], showcasethese nutritional values and its suitability as a source for animal feed and human food.Several studies, [5] and [6–9] amongst the recent, also indicate their application forrecycling food and bio waste. These studies clearly demonstrate the potential of
Hermetia illucens in addressing the approaching food scarcity while reducing theresource usage for their production. Irrespective of the potential applications of
Hermetia illucens , for their (mass) production, it is necessary to study: (1) theunderlying biological processes such as assimilation, respiration, morphological changes,etc.; (2) the fundamental resource prerequisites such as feed composition, growingenvironment conditions, etc.; (3) the resulting growth dynamics that exhibits theAugust 14, 2020 1/20 a r X i v : . [ q - b i o . Q M ] A ug arious stages of the larval growth in response to the supplied resources; and (4) theinteraction between the larvae and its environment (microbiome, substrate, etc.) andbiological effects that trigger certain events (e.g. fleeing from substrate due to low O concentration etc.).This insect species originates from tropical South American climate zones and thusrequires warm and humid environment. Such conditions were verified in the researchstudies: [10] highlighted the threshold temperatures and thermal requirements; [11]compared the development rates over different temperature ranges; and [12] studied theeffect of humidity on the egg eclosion and adult emergence. The influence of diet, itsmoisture content and the temperatures were studied together to showcase its importancein the development of the larvae [13–16]. Another study [17], proposed and showed theeffects of pH levels of the substrate (feed), in which larvae are grown, on the larvaldevelopment. From these studies, one can conclude the importance of the environmentalconditions (temperature, humidity, etc.), the substrate conditions (moisture, pH, etc.)and the feed composition for the growth and development of the larvae.A thorough literature survey revealed only time-invariant static models that describecertain biological processes of the BSF larvae. The authors of [10] suggested a model todescribe the development rate as a function of temperature and, similarly, a model forcalculation of metabolic rate as a function of temperature was presented in [18]. In caseof [19], a logistic model was suggested for modelling the larval growth in response to theair flow rate. Also, a more recent work [20] suggested the use of a Richards model to fitthe larval growth. These models from literature are mostly static models and do notadequately describe various time-dependent dynamical aspects of larvae production suchas resource dynamics, environment dynamics, etc. Also, it can be observed that themotivation behind the above mentioned literature was to improve the growth and hencethe large scale production of BSF larvae. In order to fully utilize such models forperforming simulation studies, reactor design, process design, automation, control andresource optimization, it is also necessary to appropriately formulate them as dynamicmodels. The main aim of this work is to develop a suitable mathematical model thatadequately describe the effect of environmental conditions on the larval metabolism; thelarval growth describing the evolution of its dry mass over time; and finally, thetransition of development stages between larvae and pupae.The following sections provide a detailed approach taken to develop the models,analyze the data and obtain the model parameters. Firstly, a detailed explanation ofthe experimental setup is provided. Then, a dynamic model describing the growth anddevelopment of the larvae is presented. The experiments performed for the estimation ofparameters are described followed by the model parameter estimation. Finally, theresults of the models are compared with the actual measurement data and the quality offit is determined for the models. In this work, data for model development, parameter estimation and model validationare mainly obtained from literature and experiments performed in this study. Details ofthe experiments performed and the data source are also provided. The followingsections provide the details of the mathematical models developed in this work and theprocedure followed for estimating the model parameters.
The studies on the production of larvae in an artificial controlled environment in thiswork are conducted in a custom built production unit [21] that can provide theAugust 14, 2020 2/20ecessary growing condition and simultaneously perform measurements of variousparameters (e.g. air and substrate temperature, CO and O concentrations, humidity).The controlled environment has a volume of 75 L and holds a growing tray of dimension22 cm x 32 cm x 5 . and O concentrations; andhumidity in air and moisture in growing medium are recorded by the sensors integratedwithin. Similarly, information related to the states outside the production unit, e.g.,temperature, humidity and CO concentration of external air source, are logged usingdata loggers. Further details regarding the production environment can be found in [21](see Section 2.1.4 and 3.7). To study the dependency of substrate or feed moisture, a larval growth experiment wasperformed. In this experiment, ten small containers of height 8 cm and diameter 5 cmwere filled with 10 g of dry feed and varying amounts of water, from 0 to 40 g, wereadded. Then, 20 larvae of about 8 days old with a starting weight of 2 mg were addedto each container. A small net was placed over each container to prevent the larvaefrom escaping while allowing air exchange. All containers were then placed inside theproduction unit with air temperature set at 29 ◦ C and air ventilation at a rate of7 . − . The weight of each container was checked daily. Any changes in containerweight, considered mostly due to evaporation, was supplemented to keep the moistureconstant. The final fresh and dry weight of the larvae (dried for 6 h at 70 ◦ C in an airdryer) was measured at the end of the experiment (on 8th day).
The main focus of this work is to obtain a model that not only describes the evolutionof dry mass over a given period in response to various environmental factors but also inaddition to capture the drop in larval dry mass due to the maturation process that BSFlarvae undergo during their last larval instars. Furthermore, it is also necessary toobtain information related to the development phases of the larvae that can be used forstreamlining the production process. This description can be assistive in determiningharvesting strategies such as harvesting for maximum larval dry weight or for obtainingpupae for rearing adults.The most commonly used models to describe the growth of biological organism,among others, are the von Bertalanffy growth model [22] and dynamic energy budget(DEB) [23] model. The former model describes the growth empirically while the latter isbased on mechanistic description using the concepts of energy reserves and volume.Despite having a simple model structure, the von Bertalanffy model can be used tomodel the dry weight/size change over time. However, no inference can be obtainedregarding the current development phase of the larvae or the drop in dry weight duringmaturation. The DEB model in comparison, uses states (energy density and structuralvolume) that are either difficult or not directly measurable. Also, it is not evident ifusing this model, information pertaining to the development phases could be obtained.Therefore, in this work, a new model is developed based on the mass balance approachand uses concepts such as asymptotic maximum size proposed for use with vonAugust 14, 2020 3/20ertalanffy model [24] and the concept of maturity reserves used in DEB model. Thefollowing section provides some background to the fundamentals of larval growth and anoverview of the model development based on these fundamental principles. Table 1 listsall the symbols used in this work for developing the models.
Table 1. List of symbols used in the description of the models.Symbol Description Unit B dry dry mass per larva [g] B wet wet mass per larva [g] B eff non structural assimilates in larva [g] B str structural mass of the larva body [g] T Σ development sums of larvae from neonates to prepupa [h] B feed total feed (dry mass) available in the growing medium [g] T med temperature of growing medium in production unit [ ◦ C] W med total water in the growing medium [kg] W med% moisture concentration of substrate [kg kg − ] C air CO concentration of air in production unit [kg m − ] O air O concentration of air in production unit [kg m − ] H air absolute humidity of the air in production unit [kg m − ] A air air flow rate to the larvae production unit [l min − ] φ B ing flux of feed from substrate into the larva [g s − ] φ B excr flux of non digested feed back to substrate [g s − ] φ B assim feed converted into energy and spent to digest the ingested feed [g s − ] φ B mat assimilates spent towards building of new structure [g s − ] φ B maint assimilates spent for maintenance of existing structure [g s − ] φ B eff effective assimilates available from the ingested feed for growth and maintenance [g s − ] φ B metab total assimilates spent for metabolic activities [g s − ] k α excr fraction of ingested feed excreted out [-] k α assim fraction of ingested spent for digestion [-] (cid:15) inges efficiency of the ingested feed [-] k inges specific ingestion rate of larva [g g − s − ] k maint specific rate of maintenance and maturity of larva [g g − s − ] k dev ts conversion factor to obtain development sums in hours [s − ] k T Σ development point at which the assimilation process starts to cease [h] k T Σ development point at which the assimilation process ends [h] k T Σ development point at which the larval development phase ends [h] k B asy asymptotic size of the larvae in dry mass [g] k T L lower boundary temperature for Arrhenius equation [K] k T ref reference temperature for Arrhenius equation [K] k T H upper boundary temperature for Arrhenius equation [K] k T AL Arrhenius temperature for the lower boundary temperature k T L [K] k T A Arrhenius temperature for the reference temperature k T ref [K] k T AH Arrhenius temperature for the upper boundary temperature k T H [K] k r max T maximum observed development rate in response to temperature (Logan-10 model) [s − ] k r base T minimum development rate observed above the lower temperature boundary (Logan-10 model) [s − ] k r ρ T development rate change per degree change in temperature (Logan-10 model) [ ◦ C − ] k T base lower temperature boundary above which the development is observed (Logan-10 model) [ ◦ C] k T max lethal maximum temperature for larval survival (modified Logan-10 model) [ ◦ C] k δ T width of the high temperature boundary (modified Logan-10 model) [ ◦ C] k r max dm maximum development rate of the larvae in response to feed density/availability [s − ] k B half dm feed density/feeding rate fow which the development rate is half [g g − d − ]continued on next page ...August 14, 2020 4/20 able 1. (continued ...) Symbol Description Unit k r max gm maximum growth rate of the larvae in response to feed density/availability [g s − ] k B half gm feed density/feeding rate for which the development rate is half [g g − d − ] k r max W maximum growth rate in response to feed moisture concentration [g d − ] k W med C1 lowest feed moisture below which the growth ceases [g g − ] k W med C2 feed moisture above which the ingestion rate can reach maximum [g g − ] k W med C3 feed moisture above which the diffusion of oxygen/air exchange starts to cease [g g − ] k W med crit feed moisture above which the larvae begins to die [g g − ] k r max A maximum observed growth rate in response to airflow rate [g s − ] k A half air flow rate for which the growth rate is reduced to half [l min − ] The collective processes that define the growth and development of an organism areknown to be the metabolism. The important processes of the metabolism in anorganism —here abstracted—are assimilation, maintenance, growth, maturity. Usingthe mass/energy balance approach, these abstracted metabolic processes can be used todescribe the growth and development rate of an organism. General consideration in thisapproach is that the afore mentioned abstract processes can either use energy or massfor developing the model. Since mass is easy to measure in-situ both in laboratory andin production compared to measuring energy, in this work models are derived based onthe mass.Growth, which describes the increase in structural volume or mass in an organism,requires ingestion of feed. This feed ingested by larva is converted into assimilatesthrough the process of assimilation consuming a portion of the assimilates for thisprocess. The assimilates are converted into structural mass towards growth andmaturity. Maturity, which is an indicator for larval development, consumes assimilatesthroughout the larval stage. Maintenance respiration that keeps the organism alive alsoconsumes some of the assimilates. Using Forrester diagram, the flow and paritioning ofbiomass and energy through the afore mentioned processes are presented in Fig 1.With this context and background, growth or the rate change of the larval dry massis represented using mass balance model asd B dry d t = (cid:124) (cid:123)(cid:122) (cid:125) φ Beff φ B ing − φ B excr − φ Bmetab (cid:122) (cid:125)(cid:124) (cid:123) φ B assim − φ B mat − φ B maint , (1)where φ B ing is feed flux from substrate into the larvae, φ B excr is the flux of non digestedfeed back to the substrate, φ B assim is the feed converted into energy necessary forassimilation of the ingested feed, φ B maint is the assimilates converted into energy forbasal maintenance of existing structure, φ B mat is the assimilates spent for growth andmaturity (responsible for accumulating new structure) and φ B metab represents all theassimilates consumed for metabolic activities.The effective assimilates available for growth and maintenance φ B eff can beexpressed as φ B eff = (1 − k α excr − k α assim ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:15) inges φ B ing , (2)where k α excr and k α assim are respectively the fractions of feed excreted and spent in theprocess respectively and (cid:15) inges corresponds to the efficiency of the digested feed andprovides information related to the quality of the feed.August 14, 2020 5/20 nvironment LarvaSubstrate B feed W med B eff B str C air O air H air T med ingestionexcretioningestionexcretion maturity/growthmaintenancerespirationevaporation/condensationconvection/radiation conduction/radiation B wet − B wet B dry T air B maint Fig 1. Mass and energy flow between the larva, substrate and the growingenvironment.
The rectangles represent the different states and the arrows indicate theflow of mass and energy (fluxes) between these states. Biomass and water in thesubstrate enters and exits larvae by ingestion and excretion. Gas exchange as a result ofmetabolic respiration takes place between the larva and the environment. Assimilatedbiomass and reserves B eff is further converted into structure towards the larval maturity B str and energy B maint necessary for maintenance of the structure. The statesrepresented in dashed lines indicate that they are not directly measurable unlike thelarva wet and dry mass B wet and B dry respectively. A part of the B maint is converted toheat, a byproduct of metabolism, and is lost to the substrate increasing its temperature T med .Furthermore, maturity and maintenance expenses ( φ B mat and φ B maint ) cannot bedistinguished since these two processes are active during the entire development phaseof the larvae [16]. Due to this reason as well as the terms being difficult to separatelymeasure, they are combined into one flux term. These flux components correspondingto the assimilation and maintenance are considered proportional to the weight/size ofthe organism [22]. Therefore, Eq (1) can be rewritten in terms of dry weight asd B dry d t = (cid:15) inges k inges B dry − k maint B dry , (3)where k inges and k maint are the specific assimilation rate and specific growth andmaturity maintenance respectively (g feed g − larvae s − in dry matter). The Eq (3) isof the form similar to the general form of von Bertalanffy model given in Eq (5) of [22]with m = 1 for insects as suggested in [25]. This model given by Eq (3) is rudimentary,describing only partitioning of the biomass across different biological processes. Toachieve the model goals described previously, it is also necessary to model variousfactors that regulate the rate of flow of the mass and energy fluxes across differentprocesses. Modelling of these factors are presented in the following sections. Growth process of the larvae takes place in stages which is commonly known as instarswith the
Hermetia illucens larvae undergoing a total of 6-7 instars [26, 27]. The larvaeundergo these development changes when the conditions necessary for growth aresuitable. This development, however, seems to be not dependent on the size of thelarvae. This can be inferred from the data presented in [13], where the larvae completedAugust 14, 2020 6/20heir development despite not reaching nearly half their maximum size. Therefore, analternate mechanism is required to model the developmental stages of the larvae.The use of temperature sums or degree days to track the developmental stages fortracking the growth of an organism, including plants and insects, can be seen inliterature [28]. In such applications, temperature sums served as an indicator to thetotal energy that organisms received in their lifetime. However, this might only work forcases where the resources such as food, air concentration and heat are not limited.Therefore, not only temperature but also other environmental conditions such as feeddensity, air concentration etc., needs to be considered to obtain virtual reference to thetotal energy received. In this work, such indicator for total energy is obtained by usingthe sum of suitable growing conditions that the larvae receives for its development in itslifetime. This sum, referred to as development sums in this work, can therefore bewritten as a function of all factors that affect the development asd T Σ d t = rc F ( B feed ) rc W ( W med ) rc A ( O air ) rc T ( T med ) k dev ts (4)where rc F , rc W , rc A and rc T are the rate constants that describe the influence of feed,water, air concentration, and temperature on the development rates respectively and k dev ts constant to convert the development rate in development hours. This Eq (4) ineffect provides a new dimension similar to age but calculated based on the rateconstants of influencing factors. Modeling of these influences are presented in the nextsection but first the changes to the feeding and maturation process in response to thedevelopment sums are presented. Feeding and growth stage
Larval structural growth is a result of constantassimilation—the main function of the larvae is to accumulate enough assimilates andmass—lasting up-to the 6th instar [26]. According to the results published in [18], thelarvae assimilates the feed at highest rates during the 1st to 4th instar. This graduallydrops from 4th to 6th instar and assimilation finally ceases before 7th instar. It is alsoobserved in the works of [18], that the mouth parts of the larvae undergo morphologicalchanges suggesting changes in feeding behavior.With respect to the model considered in this work, this transition into non-feedingstage indicate that the assimilation is maximum in the early larval stages, graduallydecreases with increase in mass, and finally ceasing when the feeding stage is completed.This transition of the assimilation process over the development stages, indicated by T Σ ,can be described as r B assim ( T Σ ) = r assim max ( B dry ) if T Σ < k T Σ r assim max ( B dry ) (cid:18) T Σ − k T Σ k T Σ − k T Σ (cid:19) if k T Σ ≤ T Σ < k T Σ T Σ > k T Σ , (5)with r assim max ( B dry ) = 1 − B dry k B asy , where k B asy is the maximum asymptotic mass of thelarvae, k T Σ is the transition point until which the larvae feeds at a maximum rate and k T Σ is the point beyond which the feeding comes to a halt. The function r assim max represents the ingestion/feeding potential of the larvae in relation to it its current sizeand the maximum size it can reach when infinitely fed. With this Eq (5), a relationbetween the development sums and the size dependent ingestion is established. Maturation stage
In the final larval instar stage, the accumulated assimilates andreserves (e.g. fats) are further spent in developing the parts necessary to reach theAugust 14, 2020 7/20aturity and transform into a pupae. This maturity process was studied in [29] whichindicates a drop in the dry mass and, in specific, the crude fats during the transitionfrom prepupae to pupae. However, maturation ceases at the end of this transformationand then the pupal stage begins. This maturity allocation, can be modeled as a ratethat allocates the assimilates and reserves to the maturation process. Allocation tomaturity can be also seen in the modeling approaches of DEB [30] (see Section 2.4). Inthis work, such scheduling of assimilates to maturation is done such that the maturityprocess continues further after the feeding phase and until the larvae turns into a pupae.This continued allocation describes the drop in mass and indicates the change in bodycomposition. Therefore, the maturity allocation is described as r B mat ( T Σ ) = (cid:40) T Σ < k T Σ T Σ ≥ k T Σ , (6)where k T Σ indicates the end of prepupal or beginning of pupal stage. For the model totrack pupal development, the rate of maturity allocation shall be replaced with a nonzero value since the pupae undergoes further metamorphosis consuming reserves. Factors that affect the growth and development of the larvae considered in this workinclude temperature, feed density, feed quality, moisture and air concentration. Each ofthese parameters influence the larval development through various biological processes.An attempt is made to model these influences through mechanistic and analyticalmodels both from literature and developed based on the analysis of aggregatedliterature and experiment data.
Temperature
Temperature has a direct effect on all the biochemical reactions thattake place in the larvae and thus affecting its growth and development rate. The effectof temperature on the growth of larvae can be modeled using Arrhenius equation [31].Corrections to the metabolic rates for the temperatures beyond the upper and lowerboundaries can be applied as r T ( T med ) = k r ref T exp (cid:18) k T A k T ref − k T A T K (cid:19)(cid:18) (cid:18) k T AL T K − k T AL k T L (cid:19) + exp (cid:18) k T AH k T H − k T AH T K (cid:19)(cid:19) (7)with T K = T med in K, where k T A , k T AL and k T AH are Arrhenius temperatures at thereference k T ref , lower boundary k T L and upper boundary k T H temperatures respectivelyand k r ref T is the known reference rate.Another analytical model proposed in [32] (see Eq (10) of [32]), referred to in thiswork as Logan-10, also describes the growth rate in response to the temperature as wellconsidering the effects of denaturation and desiccation at high temperatures. TheLogan-10 model was modified such that for temperatures beyond the upper threshold,the resulting growth is zero instead of a negative growth. The resulting modifiedLogan-10 model is given as r T ( T med ) = k r max T (cid:18) k γ exp ( − k ρ T ( T med − k T base )) + exp (cid:18) − k T max − T med k ∆T (cid:19)(cid:19) − , (8)with k γ = (cid:16) k rmaxT − k rbaseT k rbaseT (cid:17) , where k r max T is the maximum observed rate (s − ), k r base T is the minimum rate at the temperature above the lower threshold, k ρ T rate change inAugust 14, 2020 8/20esponse to the temperature, k T max is the lethal maximum temperature and k ∆T is thewidth of the high temperature boundary layer. The Logan-10 model has one lessparameter compared to Arrhenius model presented in Eq (7) and can be intuitivelyapproximated from the available data. In this work, both these models will be evaluatedand the corresponding parameters will be estimated. Feed Density
The feed flux assimilated by the larvae is effected by few factors,considered important in this work due to its application, such as feed availability andchange in feeding behavior due to the modifications to the mouth parts of the larvae inits final instars. It is common in literature to model the change in ingestion rate due tosubstrate availability using a type II function. Monod presented an adaptation of thisfunction to model the growth of bacterial cultures in [33]. This Monod equation isadapted in this work as r F ( B feed ) = k r max dm B feed B feed + k B half dm , (9)where B feed is the feed density in the substrate/growing medium, k r max dm is themaximum development rate (s − ) at highest feed density and k B half dm is the feeddensity resulting in half of the maximum rate.Similarly, Eq (9) can be rewritten to also model the growth rate of the larvae as r F grw ( B feed ) = k r max gm B feed B feed + k B half gm , (10)where k r max gm is the maximum growth rate (g s − ) of the larvae and k B half gm is the feeddensity resulting in half of the maximum growth rate. Feed Moisture
Based on the data presented in [19], the authors suggest that themoisture of the feed (kg water in kg wet feed) has a first order effect. However, in thatstudy, data was only available for the moisture content of 48-68 %. Another early workstudied the development of different flies, including
Hermetia illucens , under differentsubstrate moisture conditions [34]. In this work, the authors concluded that thedevelopment increased with increase in moisture content between 30-70 %, while, at 20,80 and 90 % there was no development observed. The moisture experiment performedas part of this work, covered the feed moisture in the very low and high concentrations.Based on these results, the feed moisture has different influences on the larval growth.Firstly, with lower moisture, the feed may not be ingestible and thus result in slowergrowth and high mortality rate at very low moisture levels. Secondly, with increasingmoisture, feed could be assimilated better resulting in better growth. Finally, at highermoisture concentrations, intake of oxygen might be reduced resulting in slower growthand higher larval mortality.Based on these observations, the influence of water content in feed on the larvalgrowth can be modelled as r W ( W med% ) = k r max W r W assim ( W med% ) r W resp ( W med% ) , (11)where k r max W is the maximum growth rate, and the influence of moisture content on theAugust 14, 2020 9/20ssimilation rate and respiration rate r W assim and r W resp respectively are modelled as r W assim ( W med% ) = W med% ≤ k W med C1 W med% − k W med C1 k W med C2 − k W med C1 if k W med C1 < W med% < k W med C2 W med% ≥ k W med C2 , (12) r W resp ( W med% ) = W med% ≤ k W med C3 W med% − k W med crit k W med C3 − k W med crit if k W med C3 < W med% < k W med crit W med% ≥ k W med crit , (13)where k W med C1 is the lowest water content in the feed below which growth ceases due toreduced feed ingestion rate, k W med C2 is the moisture concentration above which theingestion rate is maximum, k W med C3 is the water concentration above which diffusion ofair into substrate and thus the larvae drops, and k W med crit is the highest waterconcentration above which oxygen diffusion ceases. Air concentration
Effect of aeration in the growing environment influences thelarval growth through the availability of O necessary for respiration. A studyperformed in [19] that compared the larval growth at different air flow rates and thusthe available O concentration used a logistic model to describe the data. r A ( A ) = k r max A (cid:18) (cid:18) A − k A inf k A trans (cid:19)(cid:19) − , (14)where k r max A is the maximum rate, k A inf is the infliction point and k A trans is the slope.However, in [35] the authors despite highlighting the logistic model, have used a type IIfunction to model the development rate based on the O concentration. In this work,the influence of O concentration is considered as a resource necessary for theunderlying biological processes and therefore modeled as r A ( A ) = k r max A AA + k A half , (15)where k r max A is the maximum growth rate under certain high O concentrations (Airflow rate) and k A half is the airflow rate for which the growth rate is reduced by half. The considered factors affecting the growth of larvae and the movement of mass andenergy between larva, growing medium and the environment is summarized in Fig 2using Forrester diagram. The B str , representing the structural mass of the larva, has itsinfluences on most of the rate flows as seen in Fig 2.With the knowledge of factors affecting the different biological processes, here, weapply the obtained rate functions Eq (5), (6), (8), (10), (11) and (15) to thecorresponding mass and energy flow. The combined rates and states represent thelarvae growth asd B dry d t = (cid:15) inges rc F grw rc W rc A rc T r B assim k inges B dry (cid:124) (cid:123)(cid:122) (cid:125) effective assimilation − rc F grw rc A rc T r B mat k maint B dry (cid:124) (cid:123)(cid:122) (cid:125) maturity-maintenance . (16)Further, rearranging the terms of the above equation, for simplification, results ind B dry d t = ( r assim (cid:15) inges k inges − r mat k maint ) B dry , (17)August 14, 2020 10/20 nvironment LarvaeSubstrate B feed W med B eff − B wet B dry B str C air r mat O air T air H air T med r resp r resp r mat B maint r assim r assim Fig 2. Mass and energy transfer.
The flow of mass and energy between thesubstrate or growing medium, larva body and the environment in response to variousstates and environment conditions are represented using the valves that regulate thisflow. Influence of the states on the rate are indicated using dashed lines. Influence ofthe states on the rates are not explicitly indicated when the flow takes place betweenthose corresponding states. Rates r assim , r mat , and r resp , represent the assimilation,maturity-maintenance and respiration as a function of various states that influence theflow of biomass and energy.with r assim = r B assim rc F grw rc W rc A rc T models the factors affecting the assimilationprocess rate and r mat = r B mat rc T rc A rc F models the factors affecting the maturityand maintenance. The rate constants rc F , rc F grw , rc W , rc A and rc T are obtained bynormalizing the rate functions r F , r F grw r W , r A , and r T presented in Eq (8) to (11)and (15) respectively. Implementation of switching functions
All functions modelled in this work ascases or switching functions, Eq (5), (6), (12), and (13), are realized as logistic functionsfor a smooth transitioning between the conditions. The parameters defining theboundary conditions are replaced in the logistic function, for example, for Eq (5) as r B assim ( T Σ ) = (cid:18) (cid:18) − (cid:18) T Σ − k T Σ inf k T Σ − k T Σ (cid:19)(cid:19)(cid:19) − (18)with k T Σ inf = k T Σ + 0 . k T Σ − k T Σ ). Models presented in this work are mostly nonlinear and therefore, nonlinear leastsquares data fitting method was used for parameter estimation. This was formulated asan optimization problem with the objective of finding the parameter that minimizes thesum of square of errors as min p (cid:88) i ( f ( p , X i ) − Y i ) , where p represents the parameters to be estimated, f ( p , X ) represents the model, Y isthe measured data and i represents the measurement samples. This parameterestimation problem was implemented in MATLAB using the lsqcurvef it function in aAugust 14, 2020 11/20ulti-search framework to explore possible solutions within the specified boundaryvalues of parameters. Simulation of dynamic models were performed using the ode Table 2. Data sets and their source used for model validation and parameter estimationDataset ID Source Description Application
T1, T2 Fig 3 of [10] Development time of BSF on different diets atdifferent temperatures Validation and parameterestimation for Eq (7) and (8)T3, T4 Table 1 of[11]F1 Table 2, 3 of[13] Development time and dry weight respectively of BSFlarvae under different feeding rates Validation and parameterestimation for Eq (9) and (10)F2-F4 Table 2, Fig 1of [36] Development time and dry weight respectively of BSFlarvae under different feed and feeding ratesM1 Fig 4 of [19] Larvae growth/dry weight change under differentsubstrate moisture content Validation and parameterestimation for Eq (11) to (13)M2 This workM3 Table 2 [34]A1 Fig 2 of [19] Larvae growth/dry weight change under different aer-ation rate Validation and parameter estima-tion for Eq (14) and (15)G1 Fig 1 of [29] Larvae growth/dry weight change over the develop-mental phases Validation and parameterestimation for Eq (5), (6) and (17)D1, D5 Fig 2 of [13] Larvae growth/dry weight change over the develop-mental phases under different feeding ratesD2-D4 Fig 2 of [13] Larvae growth/dry weight change over the develop-mental phases under different feeding rates Validation of Eq (17)
In this section, firstly, the performance of the individual rate functions Eq (7) to (11),(14) and (15) describing the influence of external factors on growth and development arepresented. Secondly, performance of the combined dynamic model representing thegrowth Eq (17) and development Eq (4) are presented, highlighting the validity of therate functions Eq (5) and (6) for assimilation and maturation respectively. Finally, thedynamic growth and development model are validated using additional datasets.
A total of four data sets (T1-T4) representing the influence of temperature on larvaedevelopment was obtained from [10, 11]. These four data sets represent the temperaturedependency under four different feed types and was used to obtain the parameters formodels Eq (7) and (8). Fig 3 shows the results of the parameter estimation using theArrhenius model (7) and Fig 4 for the modified Logan-10 model Eq (8). Both modelsperform well in describing the data with good quality of fit ( R > ◦ C. ModifiedLogan-10 model has overall better quality of fit for both normalized and actual datasets. For temperatures below 15 ◦ C, Eq 7 provides a better fit at the expense of oneadditional parameter. (a) (b)
Fig 3. Temperature influence on development rate using Arrhenius model.(a)
Parameters estimation using Arrhenius model (7) and normalized data. (b)
Development rate estimation using the parameters estimated for data set obtained byaveraging all (T1-T4) data sets. Model fit represented as avg shows the performance ofthe final model. (a) (b)
Fig 4. Temperature influence on development rate using modifiedLogan-10 model. (a)
Parameters estimation using modified Logan-10 model (8) andnormalized data. (b)
Development rate estimation using the parameters estimated fordata set obtained by averaging all (T1-T4) data sets. Model fit represented as avgshows the performance of the final model.
Results published in [13] was used to obtain data set F1 and [36] for data sets (F2-F4)representing the development and growth rate under different feeding densities and feedtypes. The feed density defined in these works use gram dry mass of feedavailable/provided per larvae per day (g d − per larva) during the feeding periods.These data sets were used to obtain the development rates and growth rates using themodel Eq (9) and Eq (10) as shown in Fig 5 and Fig 6. The models describe accuratelyAugust 14, 2020 13/20 R = 0 .
97) for the data set F1 due to the availability of measurement for uniformlydistributed feed densities. In case of F2-F4, the data set also includes both batch fed (a) (b)
Fig 5. Feed availability on development and growth. (a)
Larvae developmentrates at varying feed availability. (b)
Larvae growth rates at varying feed availability.Model fit avg represents the results of the average model scaled to the maximumobserved development rate from the 4 data sets.and continuous fed experiment measurements resulting in scattered measurements andthus lower quality of fit. Growth rate model, on the contrary, performs better indescribing all data sets F1-F4 with R > .
92. The parameters for these models areobtained by combining the normalized data sets F1-F4. The resulting model from thiscombined data is shown with the dashed line (indicated as avg) as in Fig 6. From theseresults it could be concluded that these models can be used to compute the growth anddevelopment rates under different feeding rates. (a) (b)
Fig 6. Feed availability on development and growth (Normalized). (a)
Larvae development rates at varying feed availability. (b)
Larvae growth rates atvarying feed availability. Mode fit avg represents the results of the average model forthe combined data sets.
To evaluate the model presented in Eq (11), for influence of moisture on the growth, atotal of three data sets M1-M3 were obtained. M1 and M3 are results published in [19]and [34] respectively. Data sets M1 and M2 does not contain measurements for theentire moisture concentration range but M3 provides data for the range from 20% to90% as seen in Fig 7(a). The proposed model is capable of describing the growth for theconsidered data sets and also the observations from the moisture experiment coincidesAugust 14, 2020 14/20ith [34] for the higher moisture concentration. On the contrary, higher developmentrate was observed for moisture at 80% in [37]. Further investigation withcomplementary data sets may be necessary to identify the boundaries for highermoisture concentrations. (a) (b)
Fig 7. Moisture and airflow on growth rate. (a)
Effect of substrate/feedmoisture on the growth rate. (b)
Effect of airflow rate on the growth rate in closedproduction. Model-1 represents the Monod model Eq (15) and Model-2 represents thelogistic model Eq (14).
Only one literature was found that included the effect of airflow on the growth of thelarvae in [19]. In that work, closed 750 mL bioreactors with different aeration rates wereused to study the larval growth rates. As expected, in closed environment, growth wasslow at lower aeration rates and increased gradually with increasing aeration rates andfinally saturates. Model Eq (15) and (14) were evaluated and the results are presentedin Fig 7(b). From these results, one can see that both models can describe the growthunder various aeration rates. However, Eq (15) provides better results for the availabledata sets. Further studies might be necessary to obtain the growth response to differentflow rates under varying moisture concentrations to identify correlation between them.
The larvae growth model presented in Eq (17), describing the evolution of dry mass ofthe larvae, and the development model presented in Eq (4) was validated based on thedry weight measurements presented in literature [13] and [29].Dry mass of the
Hermetia illucens from eggs to adult, presented in [29], was used toobtain estimates of the parameters marking the important stages k T Σ , k T Σ , and k T Σ .Based on the biomass conversion efficiency for chicken feed provided in [8, 13],parameters for dry mass distribution to metabolism, excretion and growth are alsoestimated to 0.24%, 0.62%, and 0.11% respectively. The performance of the modelbased on the estimated parameters is presented in Fig 8. Initially, the change of larvaldry mass from the 1st larval instar to the 5th instar is regulated by the asymptotic sizeof the larva (in dry mass) as seen between 0-320 h marked by k T Σ . As the larvaeapproaches its last instar, ceasing of the ingestion process marked by the morphologicalchanges such as modification of mouth parts and darkening of the skin are identified bythe k T Σ and k T Σ . The final transition from prepupae to pupae is marked at the end of k T Σ , indicating the end of larval growth and start of pupal stage.Furthermore to validate the model for different data sets, data set from [13] wasused. Using the data sets D1 and D5, model parameters were further adjusted for theAugust 14, 2020 15/20
100 200 300 400 500 600020406080 (a) (b)
Fig 8. Larvae growth, development and biomass partitioning . (a) Larvae drymass evolution over time (b)
Partitioning of assimilates and dry mass over developmentphases. The development sums where the assimilation and maturity transition, areindicated by the horizontal markers labelled k T Σ , k T Σ , and k T Σ .new setup and using these new parameters the performance of the model was validatedfor all data sets D1-D5. The results as seen in Fig 9, highlights the performance of themodel by providing the dry weight evolution as well as the indication of the differentLarval development stages. Fig 9. Validation of larvae growth and development model
Models arevalidated based on the data sets D1-D5 as published in [13]. The vertical line indicatesthe time point where about 50% of the larvae are transformed into prepupae. The circleon the corresponding model fit indicate the k T Σ time point when development of larvaeare completed.As seen from Fig 9, the R for the data sets D1,D2 and D5 are > .
91 butcomparatively lower for D3 and D4. This is purely due to the variance in the finalrecorded weight for D3 and D4. To support this inference, we can also observe that forD3, D4 and D5 there were larvae with higher mass on the same day when 50% prepupaewere observed. The Eq (17) also models higher assimilates allocation to maturity forhigher feed density, indicating that more reserves are available in larvae growing inhigher feed density. This can be seen in the drop in dry mass when larvae transforms toprepupae. This drop, as explained by the model and also as observed from the data ishighest for D5 and lowest for D1.
From these observation one can conclude that the dynamic model developed in thiswork serves the two main intended goals: (1) model the larval growth through change ofdry mass B dry over its development stages under different growing conditions; and (2)August 14, 2020 16/20odel the transition of larval development through the development sums T Σ underdifferent growing conditions. These results were achieved using simple model structureswhich plausibly and consistently explained various data from the literature. Finally, themodel parameters estimated in this work are summarized in Table 3. Table 3. Estimated model parameter valuesParameter Est. value Parameter Est. value Parameter Est. value k r ref T(7) . ∗ k T A (7) k T AL (7)
60 000 K k T AH (7)
40 667 .
275 K k T ref (7) .
92 K k T L (7)
285 K k T H (7) .
96 K k r max T(8) . ∗ k r base T(8) . ∗ k ρ T(8) . ◦ C − d − k T max (8) . ◦ C k ∆T(8) . ◦ C k B half dm(9) . − k r max dm(9) . ∗ k B half gm(10) .
005 32 g d − k r max gm(10) ∗ k W med C1(11) .
329 kg kg − k W med C2(11) .
69 kg kg − k W med C3(11) .
76 kg kg − k W med crit(11) .
833 kg kg − k r max A(15) . ∗ k A half (15) . − kg − k B half dm(9) . − k r max dm(9) ∗ + k B half gm(10) . − k r max gm(10) ∗ + k α excr (2) . k α assim (2) . (cid:15) inges(2) . k inges(17) . × − g g − s − k maint(17) . × − g g − s − k T Σ .
35 h k T Σ . k T Σ . k B asy (5) .
115 g ∗ values are normalized. + parameters re-estimated for data set D1 and D5. Based on comprehensive data sets aggregated from literature, various factors that affectthe growth and development of larvae were analysed. Models were developed based onliterature and based on the analysis of data to accurately and plausibly describe theinfluence of different environmental conditions such as temperature, feed density, feedmoisture, and airflow rate on growth and development rates of BSF larvae. Building onthe principles of mass balance, von Bertalanffy and DEB models, a novel dynamicmodel describing the growth of
Hermetia illucens larvae was developed. Concept ofdevelopment sums was proposed to establish a relationship between growth anddevelopment. The comprehensive dynamic model was obtained consisting of twodifferential equations, larval dry mass B dry and development sums T Σ , and combiningthe different rate equations. Model parameters were estimated for all the proposedmodels based on extensive data sets from different literature and the models werevalidated with R > .
90 with only few exceptions. The resulting dynamic modeldescribing the growth and development of the
Hermetia illucens larvae was validated,using parameters obtained from only a subset of data, on all available data sets. Thedynamic model, proposed and validated in this work, consistently explained: the changeof larval dry mass over time; transition of development phase and its effect on growth;and influence of external factors on the larval growth and development. Performance ofthe model could be further improved with newer and larger data sets that could revealother mechanisms or biological processes not explored in this work. Extension of thiswork with considerations for energy and resource efficient production of
Hermetiaillucens larvae in large scale production environments will be the future goal.August 14, 2020 17/20 cknowledgments
This research was partially funded by ESF grant number 100316180.
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