DEB-IBM User Manual
Contents
1. To convert moles of structure into volume L 3 or volumetric length L we use My which converts moles to cubic centimetres a typical valve for this parameter is 4 mmol cm Kooijman 2010 2 2 4 Development and reproduction parameters M MB k Kr The fraction of mobilized energy not allocated to the soma 1 is allocated to development or reproduction DEB theory divides the life history of all species into three classes embryos juveniles and adults Embryos do not teed externally but use maternal reserves for growth and development at conception an embryo is composed of nearly only reserves A transition from embryo to juvenile marks a switch to exogenous feeding Neither embryos nor juveniles reproduce The transition trom juvenile to adult marks the start of investment into reproduction the actual reproduction may occur a little later In DEB theory these two transitions are made after a given amount of mobilized energy m has been allocated to maturity to transition from embryo to juvenile and mM to transition from juvenile to adult Unlike structure maturity has no mass or dimensions but rather is considered information maturity is quantified by the cumulative amount of reserves invested in it Like for soma individuals must pay costs associated with maintaining a given level of maturity These costs are taken proportional to the maturity level in mol of invested reserves the proportionality cons2. Figure 1 Input fields be run with just defining the eight scaled DEB parameters in the left column of input fields on the interface Fig 1 The scaled version of DEB is a simplification of the standard DEB model which uses Compound parameters that are functions of standard DEB primary parameters These compound parameters are often easier to extract from the data Kooijman et al 2008 Below we first provide a brief description of each of the 12 standard DEB parameters and then how these parameters are transformed to the scaled parameter set for the eight scaled DEB parameters used by default in DEB IBM screenshot taken from the NetLogo program Differences between species are primarily represented by these valves 37734434 26685E 6 Table 1 The 12 primary DEB parameters in dimension of mass and their associated proc esses Standard DEB Parameters Symbol surface area specific searching rate Feeding assimilation rate somatic YVE mol mol yield of structure on reserve l growth maintenance ade l somatic Jery mol d m surface area specific somatic maintenance growth maintenance i j somatic Jeu mol d m volume specific somatic maintenance l growth maintenance i reproduc maturity at birth q specific maturity maintenance reproduc i tion development ici reproduc duct ff fePoductonettcerey tion development 2 2 1 Feeding and assimilation related parameters Fn Jam
3. O99 my pet doto and thus organisms raised on ad libidum concentrations of varying quality food can assimilate and thus grow at different rates However we can use the zoom factor to estimate the value of p The base Zoom factor is the maximum volumetric length Ly of an organism in centimetres It is called the zoom factor because DEB theory makes several predictions for scaling of DEB parameters inter specifically with body size and the scaling of these parameters leads to many observable covariations such as growth rate respiration and life span van der Meer 2006 Kooijman 2010 chapter 8 In DEB theory maximum length is a function of three primary parameters maximum assimilation rate kappa and volume specific maintenance costs Dam I ED Because add my pet gives values for all the parameters in this equation other than p we can rearrange the formula to determine its valve Once the valve of pis determined all other conversions are straightforward For v andu we need to convert maturation threshold for birth and puberty to scaled maturity at birth and puberty by dividing by the surface area specific maximum assimilation rate U E pam And UB E Pam WNEr pam pu Jz K For the two compound parameters we just use the formulas consisting of pri mary DEB parameters i pu _ Ec v ky and e MT Ec De 3 2 2 Food dynamics In the basic version of the model
4. Us Notice that we now use scaled us and ut for parameters related to life stage transitions which are equal to mg and mg divided by Jam The length dynamics simplify to E 2 dt 3 gl 2 6 DEB IBM parameters We have now derived the equations used in DEB IBM If food conditions are intraspecific variation 0 2666 Figure 2 The interface of DEB IBM Elements of the Interface are buttons e g setup in put fields e g E_H_b sliders e g timestep choosers e g aging monitors e g count turtles and plots e g population density constant we only need eight parameters to run simulations To run DEB IBM at constant food conditions set the food dynamics chooser to constant and set scaled assimilation rate to the desired valve The input boxes for the eight parameters needed to run DEB IBM at constant food conditions are on the left most side of the interface Fig 1 The bottom four parameters in the col umn are primary DEB parameters K Kr and k The top four parameters include the two compound parameters g and ky in addition to the two life stage transition parameters U and UZ 3 The interface In Fig 2 you see how the interface looks like after opening the model in Net Logo 3 1 Running the basic model The first thing you should know is how to run the model In Fig 3 you see the most Important buttons on the interface The
5. weak homeostasis which means that at a constant food density the ratio of reserves to structure remains constant The derivation of reserve dynamics from this assumption is rather complex and explained in Kooijman 2010 p 37 In a simplified case of an organism which does not grow for reserve density to remain constant assimilation would have to equal mobilization However for growing organisms mobilization must be lower than assimilation due to dilution of reserve density via growth The dynamics resulting from the assumption of weak homeostasis is that mobilization of reserves will be proportional to reserve density with the proportionality constant depending on the ratio of energy conductance v and the length of the individual Higher values of energy conductance imply a lower resistance of transfer from reserves to structure along the reserve structure interface thus the higher the conductance the faster reserves are depleted and mobilized for use The maximum reserve density Em is given by the maximum surface area specific assimilation rate Jam and energy conductance v The allocation fraction parameter kappa x is the fraction of mobilized reserves which is allocated to somatic growth and maintenance while the remainder 1 x is allocated to development and reproduction 2 2 3 Somatic growth and maintenance parameters Jer Jeu Yve The fraction x of mobilized reserves is allocated to the soma i e t
6. Fast growing species need shorter time steps go once Figure 3 These buttons are needed to run the model 3 2 Optional submodels 3 2 1 Add my pet parameters A growing database of parameter sets for species can be found at httpo www bio vu nl thb deb deblab The parameter sets contained in the add my pet database are the primary DEB parameters in energy Thus we need to convert the primary DEB parameters from the add_my_pet database to those used in our implementation On the interface of DEB IBM are input boxes for five add _ my pet parameters Fig 4 By selecting on in the add _ my pet chooser and clicking the setup button DEB IBM automatically converts these five add _my pet parameters to the four DEB IBM parameters at the top of the scaled DEB IBM parameters column Fig 1 The bottom four are primary DEB parameters require no conversion from those listed in the add my pet database The code for the conversion is in the procedure con 16 vert parameters within the setup procedure In the following paragraph we explain how these parameters are calculated Noticeably missing from the add_my_ pet database is the maximum 2 Da nn parameters in Fig 1 surface area specific assimilation can be calculated parameter pam This is because p iS from these five pa Figure 4 The four top food type specific Some food types are rameters given in the of higher energetic value than others
7. but many other processes can also stabilize populations at high food carrying capacities such as inducible defences of the prey type multiple prey types interference competition environmental heterogeneity and stochasticity The importance of these mechanisms should be considered when modelling in a population context and the inclusion of one or more of these mechanisms may be required to replicate realistic population dynam ICS 30 References Grimm V Berger U Bastiansen F Eliassen S Ginot V Giske J Goss Custard J Grand T Heinz S Huse G Huth A Jepsen JU Jorgensen C Mooij WM Muller B Pe er G Piou C Railsback SF Robbins AM Robbins MM Rossmanith E Ruger N Strand E Souissi S Stillman RA Vab R Visser U DeAngelis DL 2006 A standard protocol for describing individual based and agent based models Ecological Modelling 198 1 15 126 Grimm V Berger U DeAngelis DL Polhill G Giske J Railsback SF 2010 The ODD protocol a review and first update Ecological Modelling 221 2760 2768 Hancock P A and E J Milner Gulland 2006 Optimal movement strategies for social foragers in unpredictable environments Ecology 87 2094 2102 Kooijman S A L M T Sousa L Pecquerie J Van der Meer and T Jager 2008 From food dependent statistics to metabolic parameters a practical guide to the use of Dynamic Energy Budget theory Biological Reviews 83 933 552 Kooijman S A L M 2010 Dynamic Ener
8. the scaled functional response f f_scaled in DEB IBM is set to 1 ad libidum feeding Figure 5 Feed ing related pa rameters as seen Feeding related parameters on the interface In the food conditions Keeping the food dynamics constant you can change the food supply by changing f scaled on the slider This can be done while the model is running without a new setup If you want to simulate a scenario with a dynamic food source you can change the food dynamics to logistic just Click on it In the built in scenario a logistically growing prey population is depleted by the DEB population via predation This is a very simple scenario and it may be more realistic to let the DEB animal feed on another DEB organ isms but this is too specific for this implementation For logistic prey dynam ics wo new parameters are needed for the DEB individuals Ja ANd Fy dynamics chooser con stant or logis tic can be se Fu is a primary DEB state variable for maximum surface area specific search rate ANd Jxam is maximum surface area specific ingestion rate J xam differs Jeam in that the ram I xam Vex The prey population is characterized by the state variable density X and two parameters describing population growth rate r and carrying ca pacity K respectively For details see the prey dynamics submodel in the ODD model description In the default model we assume feeding
9. Vex ME mol maturity at pubert SO PONE y tion development i The surface area specific searching rate Fm influences the functional response for a given prey type Earlier versions of DEB used the half saturation coefficient K which relates to FE via K J ram vex Fn Here ya is the yield of reserves on food or in other words the conversion efficiency of moles of food into moles of reserve in most bioenergetic models this is referred to as assimilation efficiency Dividing the surface area specific maximum assimilation rate Jzam this conversion efficiency gives you the surface area specific maximum ingestion rate Jxam The ratio between the maximum surface area specific ingestion rate and the surface area specific searching rate Fn gives you the half saturation coefficient K in a Holling type Il functional response this resoonse follows from the assumption that the full time budget of an organism is spent either searching or handling food In recent formulations of DEB theory z has replaced K as a primary parameter in the standard DEB model because it is more closely linked to the underlying mechanism For more mechanistic details and reasoning behind the feeding process see Kooijman 2010 p 25 2 2 2 Reserve dynamics parameters K v In DEB theory assimilated energy first enters a reserve before mobilized for somatic or develooment and reproduction purposes One of the assumptions of DEB theory is
10. all available energy into a final reproduction bout emergency reproduction How individuals respond to periods of starvation is likely driven by the fitness benefits associated with different strategies under the environmental conditions in which their genotype has evolved Even within species the response fo periods of starvation can vary depending on environmental conditions For example the energy allocation strategies of the pond snail depend on day length Zonneveld and Kooijman 1989 Below we give an example of how to modify reserve dynamics to an alternate starvation strategy For a more thorough discussion of possible starvation strategies see Kooijman 2010 p 118 One possible starvation strategy is for individuals to stop growth reproduction and the payment of maintenance costs when e lt L Ly and alter reserve dynamics to only mobilize enough energy for paying somatic maintenance This starvation strategy was found to be appropriate for pond snails kept under short day conditions 12 hrs light 12 hrs dark In the Unscaled version of the model this would be easy to implement just by setting mobilization pC Pu IL Note that we are dealing with the DEB in energy and pC is analogous to died However DEB IBM is scaled and uses compound parameters which re quire some rearrangement __PC _ pu e Pam Pam and 26 Eg gx pam v thus sc MEK 73 V Modifying the model is rather easy from here Th
11. individuals bioenergetics and life cycle The main task of users of DEB IBM will be to parameterize the model and to possibly revise the underlying NetLogo program In this manual we therefore first explain the DEB parameters used in DEB IBM and then give examples for how to revise the code The overall pur pose of this model and its implementation are explained in Martin et al Year whereas the model itself is described using the ODD protocol for de scribing individual based models Grimm et al 2006 2010 is available at ODD website 1 1 About NetLogo The model is implemented in NetLogo version 4 1 1 Wilensky 1999 NetLogo is a software platform specifically designed for implementing individual based and agent based models It includes powerful primitives procedures that allow users to implement even relatively complex models with relatively few lines of code and little or no previous programming experience Complete novices to NetLogo and programming can generally program efficiently in 2 3 days two textbooks on individual agent based modelling are in press Railsoack and Grimm in press Wilensky and Rand in press 1 2 About DEB Dynamic Energy Budget theory DEB theory Kooijman 2010 provides a set of rules that specifies the acquisition and use of energy in an organism and thereby the life history traits for the whole life cycle of a single organism More information about DEB can be found on http www bio vu nl
12. strategies overall do not result in widely different population dynamics Default reproduction stage class density E50 embryo M juvenile M adult d He cag a F hi s N N N i 0 i 283 Daphnia reproduction stage class density IN cut En i ip iP 1 Er 4 2 3 Starvation Starved individuals follow standard reserves dynamics until their scaled reserves e falls below their scaled length L Lm Under this condition individuals no longer mobilize enough reserve to the soma to pay somatic 25 maintenance costs and thus must alter energy allocation in some way Continued starvation beyond fhis condition requires some alteration of the reserve dynamics or ifs allocation By default in DEB IBM Individuals will no longer grow but divert just enough mobilized energy trom reproduction and develooment to the soma to pay maintenance costs The remainder of mobilized energy is then allocated to reproduction and development When scaled reserve density falls below KL Lm an individual no longer mobilizes enough energy to pay somatic maintenance costs and thus dies A technical description of the starvation submodel is given in the ODD model description However species differ in their response to starvation conditions For example an individual may stop allocation to reproduction altogether when starved reduce maintenance costs stop paying maturity maintenance burn structure to pay maintenance costs or allocate
13. takes place in a three dimensional environment e g aquatic filfer feeders However this can be modified to model feeding over two dimensional surfaces The pa rameter volume represents the size of the environment The feeding sub model is only connected to the standard DEB model via the dimensionless scaled assimilation rate f Therefore the units of X and volume can be user defined e g energy liter mg em cells per mm as long as they are consis tent with each other from latter only considers assimilated energy or ageing related parameters Figure 6 Pa rameters as seen 3 2 3 Ageing In the basic version of the model the on the interface ageing submodel is turned on which affect the Individual s age as described in ageing sub Kooijman 2010 and the ageing sub Mede model section of the ODD If the ageing submodel is turned off animals have a daily background mortality rate 3 2 4 Stochasticity In the standard DEB model the only Figure 7 parameter inherent which controls the source of stochasticity comes from the coefficient of varia ageing is no intraspecific submodel This can lead to extreme variation in DEB pa population fluctuations because life histories are exactly the same for all individuals which leads to synchronisation One likely reason natural systems do not always exhibit such drastic fluctuations is that stochas tic processes and heterogeneity amo
14. understand what processes are actually driving fluxes for each of the DEB state variables To facilitate a better understanding of DEB theory for those interested in population dynamic applications we below provide a brief introduction to the standard DEB parameters how the compound parameters used in DEB IBM relate to these parameters and how changes in the each of the parameters effects the life history of the modelled individuals In addition to this user manual and the ODD model description we recommend those new to DEB theory to first read hnitp www bio vu nl tho deb index html for a non technical introduction to the concepts of DEB theory and Kooijman et al 2008 and Kooijman 2010 for a more formal description For those already familiar with compound parameters and the scaled DEB model this section can be skipped 2 1 DEB notation In all text manuscript ODD model description and user manual we used standard DEB notation This notation may look somewhat cumbersome at the beginning but has a long history and is by itself highly consistent Therefore careful attention to the notation will spare users considerable time and confusion We recommend routinely using Table 1 in the ODD model description which contains a comprehensive list of all parameters dealt with in both the text and in the implementation of the model Quantities that are expressed as unit per structural volume are surrounded with for example Jeu
15. DEB IBM User Manual Dynamic Energy Budget theory meets individual based modelling a generic and accessible implemento tion Benjamin Martin Elke Zimmer Volker Grimm Tjalling Jager Dept of Ecological Modelling Helmholtz Center for Environmental Re search UFZ Permoserstrasse 15 04318 Leipzig Germany 2 Dept of Theoretical Biology Vrije Universiteit de Boelelaan 1085 NL 1081 HV Amsterdam the Netherland This manual explains how to use the model DEB IBM which is a NetLogo im plementation of a generic individual based model based on Dynamic Energy Budget DEB theory It also gives a quick overview of DEB theory and its basic parameters The rationale of the model and its implementation are also explained in Martin B Zimmer E Grimm V Jager T Year Dynamic Energy Budget theory meets individual based modelling a generic and accessible im plementation Journal Volume pages The NetLogo implementation and the complete model description following the ODD protocol can be found here NetLogo implementation ODD model description We recommend reading the article and the ODD model description first Leipzig Amsterdam December 2010 Corresponding author email btmarti25 gmail com Contents EET TNIV ONO tc nd sedges ese octets ah eee reeset EEE AON ts ocala ahi ieaieds 4 MIR Noob HN kele Oseira ae E dane heea aaa cecauaeaunesenaeates 4 A PAO A Dl nesagutesa nese A A E ae haaeeoens 4 ko leo
16. K E reproduction buffer 2 4 From standard to compound parameters As mentioned earlier it is offen convenient to work with compound parameters as they require less data to parameterize These compound parameters represent combinations of primary parameters that are grouped together in the differential equations for the standard DEB model In the current implementation we used two compound parameters energy investment ratio g and specific somatic maintenance rate ku The former g is derived from the standard primary parameters using units of mass My v p K J Am YVE or energy _ Ee v S T e pam Because maximum reserve density ee V we can think of g as the cost to create a unit of structure relative to the maximum reserve density which would be allocated to the soma Ee Kl Eu g ku is derived from the standard parameters using units of mass a Jem Ive My or energy Pu i EG The reason why we see my in the formulas for g and kin the mass parameterization and not in the energy parameterization of the standard DEB model is that Eg converts energy in reserves to growth in the dimension of length while yyg converts moles of reserves to moles of structure and My is needed to convert to structural length 2 5 From standard DEB to scaled DEB Using the two compound parameters g and k in place of the primary parameters we get a simplified version of the stan
17. ces of stochasticity in natural systems However the sources of stochasticity to be included in the model are likely to be system specific and should be carefully considered by the researcher 3 3 Plots Histograms and Monitors In the default model we include several output plots and histograms to monitor the population dynamics of the modelled species A plot of the population density a plot of their prey density and a plot of the stage class density with the x axis in days embryo juvenile and adult Additionally there are three histograms i e the frequency distribution of length L and scaled 19 reserve density e for both juveniles and adults Ultimately the user can record and plot any individual or population level variable of interest For Information on generating output data plots monitors and histograms see the NetLogo User manual and programming guide stage class density embryo M juvenile M adult food density population density size distribution juv e distribution adult e distribution Figure 8 The three default plots and three histograms displayed on the DEB IBM interface The plot stage class density shows the density of each life stage embryo juvenile and adult over time The plots food density and population density show the density of the food X and total population The size distribution histogram shows the distribution of lengths L of the population The histog
18. cite ta sate aeaatacstehtovnas A u eed sateen 4 TAHOW TOUS STAC IOC Cl uirceinean anii E NIN REA 5 A MDE DAME TO a E E 6 ZAND AOS Ne 6 22 ere DEE DAMES aserrea a e e 2 2 1 Feeding and assimilation related parameters En J ram VEX vennen 8 2 2 2 Reserve dynamics parameters K V sersa a a 9 2 2 3 Somatic growth and maintenance parameters Jer Jem VvE 9 2 2 4 Development and reproduction parameters M MP kj Kra 10 2 3 Standard DEB can be expressed IN mass or energy sensoren 10 2 4 From standard to compound parameters neons venn even evenveen 1 2 5 From standard DEB to scaled DEB vseotsactecsinwiccwubinsecaesnieybeeosdeminedetwisteoxiawdeosees 12 ZONE DDIM Cl ire a a AT E E AT A 14 SAE SAE OCE wetten E TT A ace sted EA E TEE 14 OUR OMIM O Mig DOC MOGO kesrin A E OE hanes 15 SZ GOHOMGISUDIMMO CE IS enei a a a rada 15 SZ A POG MY OS LOCI ING IONS aariin E 15 7 FOOCS NONE maioria e e a AAA EE 17 SA AGON a A G 17 SA O AOST ea E E ETE E E 18 323 POTS ALOG amn ANd MONITOR sariei neea A NEETI 18 Ze PROCS CS wanna obiit ea 19 1 BASIC COGS Si CO Ss tsi aca sutsouaatea shaun a N a a 19 ARC ec Hiel gele lele ige ING IMOG LE neue Sadeew 20 AN Ee Nn 21 Ae IRS OCMC IOM neri iE A E E T AE E EOR 21 AOS NONO Aenean beenie eh 24 ASO CNE Sanaa 26 1 Getting Started This manual describes how fo install and use the model DEB IBM a generic in dividual based model that includes general DEB theory as a submodel for the
19. dard DEB differential equations for reserves Ms ad D ad with Lk Jec J zam l 14 Be gte and vM D J zm e Is the scaled reserve density the scaled is in reference to the amount of reserves per unit of structure reserve density relative to the maximum amount of reserves per unit of structure Remembering that maximum reserve density Eu AE Vv we see that e is the total mass of the reserves divided by volume and maximum reserve density and will have a value between O and 1 The changes in DEB s three state variables Table 3 length maturity and reproduction buffer can then be calculated as follows d len J ru B hve i Sre dt My n l Jec kjMy for My lt ML else Mn 0 d oak K Jec k M for My gt M else Mr 0 As we mentioned earlier our model is implemented in the scaled version of DEB By this we mean that to remove the unit mol or Joule if using the energy parameterization of DEB we divide all state variables in moles Mr My and Mr by the maximum surface area specific assimilation rate J ram OF pan If working with energy to get scaled reserve Ug scaled maturity Uy and scaled reproduction buffer Ur Table 3 Dividing both sides of the three differential equations by Jz gives de fr Up fL Sc With see SL Hi gte V d g 1 x Sc kj Un for Uy lt U else Un 0 d t Ur I K Sc kiU fp for Un gt U else
20. e starvation strategy is coded in the calc dL procedure We need to modify the code to set dU_H or dU_LR to 0 depending on whether and individual is a juvenile or an adult and then alter the mobilization flux to its new formula An individual then dies when its scaled reserve density falls below zero to calc dL calculate change in structural length set dh CELA lt B gt A AAM rate GCG A LAZ ASO kM orate LY ite staled lt L 1V rate 7 o K M rate set dl 0 ifelse U_H lt U_P_H set dU_H 0 set dU _R Q set S_C k_M_rate kap g L 3 vrate set dUE SA SC if e scaled lt 0 die end 4 2 4 Spatial dynamics DEB IBM is non spatial However population dynamics can be influenced by the spatial distribution of resources The model can be made spatial by click ing on the settings button on the interface tab and setting the max xcor and max pycor coordinates to the desired size If you do so make sure to deactivate the primitive no display in the setup procedure Files with co ordinates of resource distribution or GIS files can be input into NetLogo to model real landscapes However to include spatial dynamics the DEB species of interest should likely have some movement or dispersal capability which requires including a dispersal submodel in the procedure tab Below we pre sent an example of how to include spatial dynamics into the current version of the model This example is mea
21. gy Budget theory for metabolic or ganisation Cambridge University Press Van der Meer J 2006 Metabolic theories in ecology Trends in Ecology and Evolution 21 136 140 Zonneveld C and S A L M Kooijman 1989 Application of a general energy budget model to Lymnaea stagnalis Functional Ecology 3 269 278
22. h at 0 1 if random number gt p a p b and random number lt p a p b p r move to patch at 1 0 if random number gt p a p b p r and random number lt psa p b par p r move to patch at 1 0 LE rancom number gt peat pb Pet pl and random number lt p a p b p r p 1l1 p ar move to patch at 1 1 EE random number gt Pe t sbi par F oak Po prar and random number lt p a p b p r p l p ar p br move to patch at 1 1 it random number gt p a F p b t p r A p l F p ar t Pb and random number lt p a p b p r pel p ar PDE p al move to patch at 1 1 if Pendom numbper S p a A PD p r F Pel p ar HP br pal and random number lt pea T po sea h Pl p ar t pbr tomes tp pl move to patch at 1 1 E randoms number 2s pAn aby 4 Pet pal es p ar PDE A Ospel move to patch at 0 0 Additionally to show the spatial distribution of prey density we can implement the following code in the go statement following the movement submodel procedure ask patches ask patches set pcolor scale color green X 2 0 This line of modified code does not affect the model run but scales the color of each patch to its level of food density see primitive scale color in color subsection of NetLogo programming guide Because food density is a patch state variable each patch has an independent food density which undergoes logistic growth Predation is local as DEB predators only reduce t
23. he density of prey in the patch they currently occupy 29 wy A ge Pal Figure 10 Results of a simulation in an 80 x 80 patch environment Parameters were taken from the add_my_pet database for Daphnia magna Food submodel parameters were set to submodel parameters of Jy 1 F 1 X 0 5 X 2 and volume 0 01 The simulation was run for 180 days The three panels on the left show stage class density mean food density of the patches and population density The view of the simulated arena on the right shows the spatial distribution of resources at day 230 of the simulation Notice that in Fig 10 the population initially shows large fluctuations but even tually these fluctuations dampen dramatically It is interesting to note that here population oscillations are much smaller than in the non spatial model Like many other models DEB population models typically exhibit the phe nomenon known as paradox of enrichment When carrying capacity of the food is much higher than the half saturation coefficient which in our feeding model is given by Jxam Fn The population will offen show large fluctuations and if the carrying capacity of the food is much higher the populations will even collapse The above example shows how the inclusion of spatial dynam ics and the movement behaviour of individuals can lead to a resolution of the paradox of enrichment However this is just one mechanism which can stabi lize populations
24. he non reproductive parts of the organism In the soma maintenance costs are paid first and the remaining energy is allocated to growth There are two basic categories of maintenance costs those which are surface area specific and those which are volume specific Surface area specific costs typically relate to heat loss of endotherms but can also represent other surface area related costs such as osmoregulation In the current implementation of the model we focus on ectotherms and we assume surface area specific costs to be negligible i e Jer 0 Volume specific maintenance rate Je represents the costs associated with maintaining and implementing somatic functions maintaining concentration gradients turnover of structure movement Because this is a volume specific rate volume related maintenance costs of a certain individual are obtained by multiplying Jzu by the individual s volume or structural length cubed L Thus in the absence of surface area related 10 costs an individual two times larger in volume or weight would have double the daily maintenance costs The remaining mobilized energy is converted to growth with an efficiency of yvg or in other words how many moles of structure are produced from one mol of reserve Usually in DEB theory we consider structure in units of volumetric length which is L V 3 see Kooijman 2010 p 10 for explanation structural volumetric length in comparison to measured physical length
25. in some way to address a specific research question These alterations may be either to adapt the standard DEB model to reflect the life history of the species of in terest ex modifying the reproduction submodel or adapt the model to ad dress a specific research question ex Including spatial dynamics or more complex prey dynamics Below we provide some examples of how the de fault model can be adapted In each example we show the major code changes needed to implement each model adaptation however the com plete code for each example is given on the website 21 4 2 1 Feeding In the standard DEB model the assimilation rate depends on the surface area of the predator and the density of the prey These two variables are often sufficient to describe feeding rates in controlled laboratory settings Usually however varying environmental conditions strongly influence foraging success For example light intensity turbidity and turbulence strongly influences encounter rate and capture success for most visual predators in aquatic environments Different types of habitat provide varying degrees of refuge for prey species thus influencing the foraging rate of predators These influences can be easily incorporated into DEB IBM via a mechanistic foraging submodel or a simple modification of f as a function of important environmental variables 4 2 2 Reproduction Differences between species are for the most part characterized by differences i
26. is maintenance rate per unit of volume while a symbol enclosed in indicates a quantity that is expressed per unit of surface area for example Jamis the surface area specific maximum assimilation rate The dots above J in Jem Jew and all other symbols indicate that the quantity is a rate per unit of time Because the use of DEB notation is not possible within the code of NetLogo we have to convert the notation into a code compatible notation The names of the parameters are given corresponding to the standard DEB notation as follows a rate which is in standard DEB identified by a dot above the letter is here identified with the extension _rate For instance energy conductance v is called v_rate_int The int portion refers to the fact that these are the initial or baseline parameters for a species Users can allow individuals to vary in their DEB parameters from the initial parameters in some way as we do for four of the DEB parameters see stochasticity section below Subscripts and superscripts in DEB notation are indicated by _ and A respectively Although NetLogo is not case sensitive we keep cases consistent with DEB notation Kooijman 2010 When a DEB parameter contains both a super and subscript the subscript goes first For instance scaled maturity at birth v2 is written U_HAb_int 2 2 Standard DEB parameters The most basic version of the model can DEB IBM parameters
27. ly reduces the prey density on that patch Finally global variables are typically parameters which can be used by either turtles or patches Globals can either be declared in the procedures tab or created on the interface tab in which case they are not declared in the globals own The DEB parameters which do not vary between individuals could have been made global variables but we chose to make them turtle variables so that users could allow individuals to vary in any DEB parameter with little programming effort Note that on the interface you can only use global variables no turtle or patch variables Therefore all eight DEB parameters on the interface Fig 1 are distinguished from the turtles variables by the suffix _int four of these parameters are then made to vary between individuals see section 3 2 4 The remainder of the code includes two major procedures setup and go The setup procedure involves all processes required to initialize the model In the setup procedure some initial individuals are created and their state variables and parameters are specified A detailed description of the Initialization is given in the ODD model description The go procedure runs the population model An overview of the model processes and their scheduling and a detailed description of each submodel are given in the ODD model description 4 2 Guide for adapting the model For most applications the default model will need to be adapted
28. n their set of DEB parameters However species also exhibit differences in behaviour which are important for population dynamics In the context of the DEB model the most notable variation in behaviour is the reproduction strategy of a species The default reproduction strategy in the model is for mature individuals to check if they have enough energy to reproduce if they do they produce one embryo Altering the reproductive strategy of the DEB individual to produce clutches of offspring requires q minor modification of the code Below we give an example of how to modify the reproduction behaviour of the DEB animal DEB theory assumes that mothers in good conditions higher scaled reserve density produce higher quality offspring offspring with higher scaled reserve density This has been observed for many species but there are exceptions Kooijman 2010 Thus in the standard DEB model mothers invest enough energy in an embryo so that when the embryo hatches U_H U_H_B its scaled reserve density will be equal to its mothers scaled energy density In the default version of DEB IBM mature individuals reproduce when they have enough energy to produce a single embryo with enough reserves to meet the condition noted above However the water flea Daphnia magna does not produce one offspring at a time but rather mature daphnids produce new broods every 2 3 days and the release of a brood coincides with molting Time between reproduction events fo
29. ng individuals prevent strong synchroni zation of life histories One way of incorporating stochasticity is to allow indi viduals to vary in some of their DEB parameters This method is justified be cause experiments offen find that repeated physiological measurements of individuals are less variable then those between individuals We followed the method outlined in Kooijman 1989 where individuals have a random component in the maximum surface area specific ingestion rate Jaan In our implementation of the scaled DEB model there is no tion If set to O there parameter Jam because we scaled it out of our model but changing J ram affects other parameter values indirectly Both valves of the life stage transition parameters will be affected because they are both scaled by J ran The maximum surface area specific ingestion rate will be influenced J xam J cam Yev Which further influences the half saturation coefficient K as K Jeram vex Fn and finally it affects g as J Is in the denominator of this formulation Thus variation in all of these parameters is included by multiplying for Jxam Or dividing for g U2 and uf by a scatter multiplier which is a log normally distributed number with user defined standard deviation cv Users can select the value of cv in the cv input box Entering a value of O results in all individuals having the same parameters Obviously there are many other sour
30. nt to only demonstrate how to technically link DEB IBM to spatial dynamics the demonstration model was not designed to answer any specific research question First you have to set the max xcor and max pycor coordinates to the desired world size For this you need to decide on what grid cell size would be appropriate for your question Note that within grid cells spatial relationshios are often ignored for example all individuals within a grid cell might compete globally for the resources within the grid cell Grid cell size 2 usually is chosen to represent typical distances of local Competition for further aspects of Choosing appropriate spatial and temporal scales see Grimm and Railsback 2005 and Railsback and Grimm in press We arbitrarily picked a 80 by 80 grid We then included a submodel in the go procedure after all the DEB procedures We implemented a simplified version of the movement heuristic Used in Hancock 2006 Once every day the individuals make a probabilistic decision whether to stay in their current patch or move to one of their eight neighbouring patches where the probability of staying on its current patch or moving to a neighbour patch is proportional to the relative amount of resources in each patch To accomplish this at the first time step of every day the food x on the eight neighbouring patches and in the turtle s current patch are summed The probability of the individual of moving
31. o little reserves left when the embryo reaches maturity needed to hatch or the embryo has to little energy to reach the maturity threshold for hatching the lower bound Is then set to the estimation of initial reserves used in the simulation This process repeats itself until the reserve density of the embryo s matches that of the mothers within some acceptable range of error In this simulation we allow 5 deviation between the embryo s and mother s reserve density to lay eggs hatch floor U R estimation set die 0 set mother id id set id who A set scatter multiplier e random normal 0 cv set lay egg 0 set repro time 0 set UR U R floor U_R estimation estimation end Notice that we also created two new state variables mother id and id This section of code sets the mother id of a new turtle to the id of the mother and the id of the new turtle to who a built in state variable of each turtle which is a unique identity number We create these state variables because Daphnia carry their broods internally thus if the mother dies so do her offspring We then have to modify the update procedure so that when a mother dies the program checks to see if she is carrying any embryos if she is They die too to update individuals update their state variables based on the calc_state variable proccesses ask turtles it die I and UH gt HP let m id id let offspring tu
32. oecies such as behaviour soace and predation There are two levels of use for DEB IBM The first requires only familiarity with the interface On the interface users can input the DEB parameters of their soecies and observe various population and individual variables such as population density size structure and reserve levels under various feeding conditions data of all diagrams can be exported via the diagrams context menus or Netlogo s export primitives This level of use requires no programming All information that users need to use the model are the DEB parameters of their species of interest Thus at this level the program allows users to learn how changes in metabolic parameters alter characteristics of individual life histories and population dynamics The second level of using DEB IBM is modifying the generic program to answer specific research questions or to adapt the model to specific species For example a researcher may be Interested in how the population dynamics of a species is influenced by changes in land use In this case the researcher would adapt the standard model to include space and movement behaviour of individuals with the DEB theory acting as the energetic model This more engaged use of the model requires users to be familiar with both the interface and procedure tabs Likewise soecies may show specific behaviours that are not captured by DEB theory these behaviours could be added to the generic model Fo
33. r Daphnia is dependent on temperature but Is independent of food Because we are considering a situation where temperature is constant we will assume that some internal clock triggers molting and subsequently reproduction at fixed intervals To accomplish this we need to give Daphnia individuals a new state variable repro time to keep track of time since the last reproduction event which increases by 22 each timestep remember timestep represents how many timesteps one day is broken up into We also need to create a global variable days between repro which is a parameter representing how many days are between reproductive events we will set this value to 2 5 days We update the reproduction part of the go procedure as follows if U_H gt U_H p set repro time repro time 1 timestep if repro time gt days between repro calc lay eggs if lay egg 1 calc embryo reserve investment lay eggs As we see above individuals only reproduce when their time since last reproduction is greater than the new parameter days between repro which represents the time between reproduction events Calc lay eggs is the next procedure which makes sure the individual has enough energy in the repro buffer to create at least one embryo If not repro time will be set back to O and the reproduction buffer remains unchanged The individual will then continue to accumulate energy in the reproduc
34. r this second level of using DEB IBM basic training in modelling and NetLogo are required Beginners in both fields would need obtaining some literacy in both fields for example by using the textbook of Railsback and Grimm in press 2 DEB parameters Our implementation of the DEB IBM is based on the scaled DEB model Kooijman et al 2008 and uses compound parameters These compound parameters are derived from the 12 primary parameters of the standard DEB model Implementing our model in the scaled version of DEB rather than the standard DEB model further simplifies the model by dividing the state variables reserve maturity and reproduction by the maximum surface area specific assimilation rate one parameter of the standard model is removed as well as the unit of either energy or mass standard DEB can be based in either trom the model Working with the scaled DEB model with compound parameters allows parameterizing the DEB model for a species without directly measuring energy or mass you cannot estimate energy parameters without measuring energies See Kooijman et al 2008 for a guide for parameterizing a DEB model While the general principles of DEB theory are relatively simple the formulas used in our implementation have been algebraically rearranged reduced using compound parameters and scaled Thus the resulting formulas used in DEB IBM may not seem intuitive For a novice to DEB theory it may be difficult to
35. rams juv e distribution and adult e distribution give the distribu tion of scaled reserve density e of juveniles and adults 4 Procedures In this tab the structure of the model can be modified and other aspects relevant to population dynamics can be included Here we discuss the basic structure of the code and suggest how to modify it 4 1 Basic code structure The first section of the code declares the variables and specifies which type of variables they are DEB individuals are referred to as turtles which is the NetLogo term for agent or individual Turtle variables are state variables characterizing the state of a certain turtle i e L Ug and Uy Additionally 20 because we allow some of the DEB parameters to differ between individuals we made the entire set of DEB parameters turtle variables Grimm et al 2010 In NetLogo the spatial arena consists of square grid cells called patches The default model is non spatial and therefore consists of only one patch updating the view of the model world or view is therefore deactivated in the program Patch variables are the state variables of a patch In our model the density of prey is a patch variable This allows to easily make the model spatially explicit by defining a grid of patches each with their own states e g prey and turtle density Local predator prey interactions are then easy to include e g feeding of DEB predators on a patch on
36. rtles with mother id m id and U_H lt U H b if any offspring ask offspring die die the mother then dies 24 if die 1 die end An important thing to remember is that all new state variables must be declared either as turtles own of global variables Here all variables except for days between repro which is a global variable that we declared in the interface are turtle variables Additionally in the setup procedure you have to tell turtles to set id who within the brackets following the hatch primitive where turtles are created Below we show comparisons of the population dynamics under logistic prey dynamics where on the top frame the population uses the Daphnia re production behaviour and the bottom frame is the default reproduction be haviour Daphnia parameters were taken from the add_my_pet database Figure 9 Density of 3 life stages embryo black juvenile blue and adult red under the default reproduction strategy reproduce when enough energy to create one embryo and the daphnia reproduction strategy release broods at fixed intervals 2 5 days In the default strategy eggs are laid externally and survival is not dependant on the mother s survival in the daphnia reproductive strategy if the mother dies while carrying embryos those embryos also die The predation submodel parameters of Jy 1 F 1 X 0 5 X 2 and volume 5 Below we see that the two reproduction
37. setup and go buttons By pressing the setup button you initialize the system for instance individuals are created and obtain values for their state variables for example their DEB parameters By pressing the go button the simulation starts the simulation will run until you press go again Pressing go once makes the program execute one timestep In the current implementation of the model the parameters for a species are set for individuals during the setup procedure Thus altering a parameter value in the interface will not result in a change in the DEB individuals parameter values unless the setup procedure Is run after the changes were made This could be altered to allow mid simulation modification of parameter values In the basic version of the model all of the standard DEB parameters are derived from the Add my pet database for the water flea Daphnia magna hito www bio vu nl tho deb deblab add_my_pet index php The timestep slider allows the user to control the timestep The valve selected on the slider bar represents how many timesteps a day is divided into Thus all of the DEB parameters which are input into the model should be daily rates Because the model is a discrete implementation of differential equations Euler method the timestep needs to be small enough for the equations to function properly How small a timestep needs to be is dependent on the parameter valves of a species
38. tant is the maintenance rate coefficient k with the units of d Total energy spent on maintaining maturity is proportional to maturity level When individuals reach puberty at MH M maturation is complete and Mm represents a maximum value of maturity Once an individual reaches puberty energy remaining after maintenance costs for maturity are paid are allocated into a reproduction buffer The reproduction buffer is depleted during reproduction the creation of offspring and is converted into embryonic reserves embryos are nearly 100 reserves with an efficiency equal to kr 2 3 Standard DEB can be expressed in mass or energy Seven of the 12 standard DEB parameters shown in Table 1 are expressed in dimension of mass mol However DEB can also be expressed in the 11 dimension of energy Joules In this case different notation is used for those seven DEB parameters Table 2 Table 2 The parameters and units of the energy or mass specific parameters in standard DEB The five standard parameters not listed are not specific to either energy or mass Table 1 A typical value to convert between energy and mass is 550 kJ mol and mass can be converted to volume via My Section 2 2 3 with a typical valve of 4 mmol cm Kooijman 2010 mol mot assimilation effi ciency X G Table 3 Symbols used for the state variables in the dimensions of mass and energy and in the dimensionless scaled DEB used in DEB IBM
39. thb By using the Individual Based Model IBM as implemented in NetLogo it is possible to investigate the population dynamics of a species following DEB theory by in putting the specific metabolic DEB parameters for the species of interest A continuously growing library of DEB parameters for all kinds of different species can be found on the above mentioned homepage in the DEB laboratory see add_my_ pet parameter section below 1 3 Installation NetLogo is free software and can be downloaded from 1 See also www railsback grimm abm book com hitp ccl northwestern edu netlogo versions for the operation systems Windows MacOS and Linux are available Installation is straightforward and usually does not take more than five minutes To run and use the model DEB IBM start NetLlogo and open DEB IBM nlogo which comes with the DEB IBM package that be downloaded trom DEB IBM webpage 1 4 How to use the model NetLogo comes with three tabs interface information and procedures The interface tab is where users can input the DEB parameters of their species alter environmental variables and observe individual and population level output of the model The procedures tab contains the NetLogo program or code implementing the DEB IBM model Here users can alter model structure create new variables to monitor add procedures for file output and include other aspects of importance to the population dynamic of their s
40. tion buffer for another 2 5 days and then reproduce to calc lay eggs set L_embryo L_O set U_E_embryo U_R kap_R set U_H_embryo 0 loop if U_H_embryo gt U H b 1 set lay egg 1 stop if e_scaled_embryo lt e_scaled set repro time 0 stop end Once the energy required to create one offspring is determined the individual will produce as many offspring as it has reserves for each with the initial reserves equivalent to the value determined using the bisection method In the calc embryo reserve investment procedure see http en wikipedia org wiki Bisection method The bisection method determines initial reserves via adaptive trial and error Each estimation is the mean therefore the name of this method bisection of upper and lower bounds set for the possible valves of initial reserves In the first estimation the 23 Upper bound is U_R kap_R this is because this is the highest valve a mother can invest in an offspring and a lower bound of 0 A simulation of the embryonic life stage is then run and if the embryo matures with too much energy remaining in its reserves when if reaches energy for birth the Upper bound is then set to the previous estimation We can do this because we know if the estimation was too large then all values larger than estimation will be too large and thus we can exclude those values from the range of possible values If the value set for initial energy results in t
41. to or staying on patch i of the nine patches is determined by Xi 9 De J Pr P Below we show the entire movement submodel procedure This procedure Is run for each individual We use p to denote probability followed by the coordinates of the patch of interest relative to the patch the turtle currently occupies a above b below r right and left and h here patch the individual is on Combinations of two letters denote the patches located diagonally from the current patch e g p ar is the patch above and to the right Then we choose the target patch by drawing a uniformly distributed random number from the interval 0 1 and assigning target patches according to intervals within O 1 that correspond to the target patches probability of being chosen i e Pr P to movement submodel ask turtles with U_H gt U HSB if ticks mod timestep 0 let scale sum x of neighbors x of patch here let p a x of patch at 0 1 scale let p b x of patch at 0 1 scale let p r x of patch at 1 0 scale let p l x of patch at 1 0 scale let p ar x of patch at 1 1 scale beb be x Of Patcher k sI scale let pak x Of p tch at l L scale let p bl x of patch at 1 1 scale let p h x of patch at 0 0 scale let random number random float 1 if random number lt p a move to patch at 0 1 if random number gt p a and 28 random number lt p a p b move to patc
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