Parameter Identification in Partial

Integro-Differential Equations for Physiologically

Structured Populations

Sylvia Moenickes∗,1, Otto Richter1, and Klaus Schmalstieg1

1Technische Universitat Braunschweig Carolo Wilhelmina, Institut fur Geookologie, Abt. Umweltsystemanalyse∗Corresponding author: Langer Kamp 19 c, s.moenickes@tu-bs.de

Abstract: Continuous dynamic models,e.g. Comsol based simulations, play – be-sides statistical or iterative methods – amayor role in theoretical ecology; in forecast-ing and management, but also in systemsanalysis. Ecological issues generally com-prise highly interacting agents and/or un-known side effects. We here show how com-bining direct simulation with Comsol withsimple optimization tools in Matlab helps inresearch in ecology. We analyse the dynam-ics of a physiologically structured populationunder changing temperature and food condi-tions in two ways. At first we conduct virtualexperiments aiming at efficient experimen-tal design for parameter identification, after-wards we accomplish simulations for processanalysis in real life experiments for the thefreshwater shrimp Gammarus pulex.

Keywords: Parameter estimation, physio-logically structured populations, experimen-tal design

1 Introduction

Climate change surely has an impact on eco-logical systems and forecasting their reactionis a major goal in nowadays ecological re-search, [2] and [3]. Dynamic models can be areliable help in these attempts if they includethe governing processes and are correctly pa-rameterized. When it comes to ecologicalsystems comprising highly coupled processeslike dynamics of populations in their envi-ronment one might choose between compil-ing a range of well analyzed dependencieswith the risk of unjustified omission of oth-ers or integrative modelling with the risk ofmistaken process attribution. In this paperwe show how preceding simulation and sub-sequent optimization tools can help in de-

signing effective experiments for sound pro-cess analysis and parameter identification.We do so for the dynamics of physiologi-cally structured populations of the freshwa-ter shrimp Gammarus pulex under changingtemperature and food conditions.

2 System componentsand equations

We suppose that the population dynamics ofG. pulex is mainly affected by its own pop-ulation density, food supply and tempera-ture. By this reason the following constitut-ing equations are assembled:

Individual growth of an animal followsthe von Bertalanffy equation, [6], which as-sumes that anabolic and catabolic processesscale with the animal’s surface and volumerespectively:

dw

dt= g(w) = γw

23 − ρw

with γ [µg−23 d−1] and ρ [µg−1 d−1] being

rate constants and w weight [µg]. Whenfood F [1] is limited, the equation modifiesto

dw

dt= g(F,w) = γ

F

F + Fhw

23 − ρw

with the half saturation constant Fh [1]. Be-sides, when temperature is other than opti-mal, the equation is complemented by

dw

dt= g(F, T,w) = Φ(T )(γ

F

F + Fhw

23 − ρw)

with Φ(T ) a temperature response functionfollowing O’Neill, [5]:

Φ(T ) = k(Tmax − T

Tmax − Topt)pe

pT−Topt

Tmax−Topt

Excerpt from the Proceedings of the COMSOL Conference 2008 Hannover

where p = 1400W 2(1+

√1 + 40

W )2, with W =(q10−1)(Tmax−Topt) and shape parametersq10, Tmax, Topt and possibly Tmin compris-ing a number of divergent physiological pro-cesses related to temperature.

However, an overall, and weight struc-tured, population is described by a weightdensity function n(w, t) and its dynamic fol-lows

∂n(w, t)∂t

= −∂g(F, T,w)n(w, t)∂w

+ λ∂2n(w, t)

∂w2

−p−(F, T,w, . . .) + p+(F, T,w, . . .)

Here λ∂2n(w,t)∂w2 describes stochasticity of

physiological processes; p− and p+ are birthand death terms.

Mortality might be due to a numberof external effects like temperature T andfood F , again expressed in a way similar totheir influence on growth, but it might alsobe due to internal effects like weight w andage, e.g. expressed by t. We come back tothat in section 4.

Birth is expressed as the total numberof offspring B(F, T, t, . . .) distributed oversmall weights w with a density functionΠ(w) to p+(w, t, . . .) = B(F, T, t, . . .)Π(w).The total number of offspring is the resultof an integration of population density overweight, where fertility may depend on weightw, food F and temperature T , e.g.

B(F, t) =∫ wmax

wmin

rmaxF

F + Fhw

23 n(w, t)dw

A number of additional modifications ofthe population density model itself will betaken into consideration later on. For thetime being there are two system compart-ments left: food and temperature.

The latter can be given explicitly, as adata set of true weather conditions or as afunction of time which controlled the tem-perature during the experiments.

In an experimental setting food F mightalso be controlled, for example it may bekept in abundance, however in nature is iscoupled to the population as follows:

dF

dt= L(t, T )

−ImaxΦ(T )F

F + Fh

∫ wmax

0

w23 n(w, t)dw

where Imax is efficiency of food uptake andL(t) a possible source term combining litterfall and microbial processing. Note that foodis given in an ordinary differential equationwhereas population is given in an partial dif-ferential equation.

3 Parameteridentification

One major challenge in ecology is to de-fine system boundaries properly, another isto identify predominant processes correctly.Modern ecological experiments aim at in-tegral studies combined with mathemati-cal tools rather than conducting countlesshighly specific laboratory studies for singleparameters.

We appreciate Comsol multiphysics asa tool with which quickly to do both, ex-clude, include and modify processes on theone hand and have multiple visualisations ofthe results on the other hand. However, af-ter creatively defining hypotheses about theecological system under consideration, theirvalidity should be tested against experimen-tal data by statistical analysis. The facilityto combine Comsol multiphysics scripts withMatlab appeared appealing to us as we al-ready used Matlab for data evaluation.

Experimental data on weight structuredpopulation dynamics generally consist of fre-quencies of classified mean weights ni for acouple of times tj . Optimization tools areconfronted with the minimization of the sumof squared residuals

L(θ) =∑

i

∑j

(ni(tj)− ni(tj , θ))2

based on simulation results ni(tj , θ) for agiven set of parameters θ. Goodness of fitwas evaluated using model efficiency, [1],

ME = 1−∑

j

∑i(ni(tj)− ni(tj))2∑

j

∑i(ni(tj)− n(tj))2

with n(tj) being the mean of the meanweights at time tj .

4 Application inexperimental design

Before conducting integral ecological stud-ies it is important to make sure that within

determinated boundaries the required pro-cesses and related parameters can be iden-tified. We here show how well thought-outexperimental design can be achieved basedon virtual experiments and can help reducecosts of real life experiments. Synthetic datawere acquired via probing direct simulationresults on four monitoring times (analogu-ous to a real experimental set up describedin the last subsection) and adding a normallydistributed error respresenting measurementerrors.

4.1 Reducing experimentaleffort

The dependence of anabolic and catabolicprocesses on temperature was expressed asO’Neill function. Earlier experiments deter-mined the related parameters via piles of an-imals fed and kept at constant but differingtemperatures being thoroughly monitored intheir growth, [4]. Our first experiment showsthat one (possibly subdivided however) pop-ulation kept at time dependent temperaturemight suffice to determine all of the param-eters appearing in the O’Neill function andthe anabolic and catabolic rates γ and ρ. Inthis experiment we created a virtual realitythat was based on true weather conditions,that is true temperatures measured duringthe experiment described in the last subsec-tion. In the same vain, we sampled our vir-tual reality at the same monitoring times aswas done in the experiment to be explained.

Figure 1 displays the close match be-tween data and optimization results, alsoreflected by a model efficiency of 0.9785,compared to 0.9921 for the model based onthe given parameters (Remember that thedifference to 1 is due to the normally dis-tributed error that was added). However,the table below compares given and foundparameters. We find that given parametersare not always within standard deviation ofthe found parameters and suppose that withtemperature data varying in the full rangethe O’Neill function covers model efficiencywould be closer to one.

γ ρ Topt q10

direct 0.16 0.04 17.5 1.70optimized 0.32 0.06 18.3 2.58std. dev. 0.02 0.05 0.05 0.10

Figure 1: Weight structured populationdensities taking into account temperature

dependent growth, here the optimizedparameter set

4.2 Piloting multi-componentexperiments

Both the traditional type of experiments tofind O’Neill parameters as well as the oneexplained above are based on food kept inabundance. In natural settings food supplyvaries and is governed by the dynamics ofboth the population itself and all the actorsin the food web. Insufficient food supply re-sults in a decrease of anabolic processes andfinally in weight loss of the affected animal.Experiments conducted with a temporarilyinsufficient food supply allow for a simulta-neous estimation of the half saturation con-stant Fh which quantifies this effect. How-ever, it is difficult to fully control food sup-ply in an experiment. We here show thatit is possible to estimate Fh even with onlyrough knowledge of the course of food sup-ply by estimating a food supply function, orto be more exact, its parameters at the sametime. On the one hand we can optimize theparamters in an explicit expression, e.g.

F (t) = F0e−(t/t1)

p1 (1− e−(t/t2)p2 )

or those of the food dynamic given in thesecond section. The scenario described withthe above equation is that food is givenonly once in the beginning of the experi-ment, but it becomes accessible after mi-crobial turnover and is depleted after sometime.

Figure 2: Weight structured populationdensities taking into account temperature

dependent growth and dynamic food supply,here the optimized parameter set

Figure 2 shows the close match betweensimulation results and virtual data. Note thedifferences in the underlying dynamic com-pared to figure 1. The population densityshifts to smaller weights in the first moni-toring, displaying an overall loss in weight,increasing weight around the second moni-toring, just to shift back to smaller weightsuntil the last monitoring. The efficiency ofthe model parameterized via optimization isequal to that parameterized with given val-ues (0.9947). Given and found parametersvary less than in the previous example.

Fh p1 p2 t1 t2direct 4.7 20 25.0 82 98optimized 4.27 18.4 25.1 84.0 97.9std. dev. 0.4 0.12 0.83 0.07 0.29

We suppose that the crucial differencebetween the two optimizations discussed sofar is the range of possible values of theenvironmental component that are adoptedduring the experiments. The course of thefood supply applied here also belongs tothe experiment to be described below, how-ever this course comprises ultimate maxi-mum and minimum and is therefore moresuitable for optimization.

4.3 Process analysis

What remained quite unlifelike until now isthat no mortality was taken into account.Mortality might depend on the animal’s ageand on environmental factors as tempera-ture and food. Age is unknown for animalscollected from natural environments, and atime-dependence or a constant value is sup-posed for mortality. As soon as tempera-

ture and food vary with time and as soonas they do so in such a way that optimalconditions for survival are lost, an analysisof experimental data must include assump-tions on lethal processes. We here createda virtual experiment with constant, food de-pendent and temperature dependent mortal-ity and tried to find back their proportions.We supposed that all parameters discussedso far are given and only mortality is to beexplained. We supposed an inverse O’Neill-like mortality µT (1 − Φ(T )) with mortalityrate µT and a parameterization identical tothe one included in growth and a food de-pendency of µF

Fµ

Fµ+F with mortality rate µF

and a shape parameter Fµ. Together with aconstant mortality of µ0 we have a survivalrate σ of

σ = (1−µ0)(1−µT (1−Φ(T )))(1−µFFµ

Fµ + F)

such that

p−(F, T ) = (1− σ)n(w, t).

Results differed depending on whether tosuppose knowledge of single process shapeparameters (Fµ, Topt and q10), or not, thelatter is called ’all’ in the table below, theformer is called ’rates only’.

µ0 µF µT

direct 0.0050 0.0010 0.0010rates only 0.0040 0.0012 0.0041all 0.0045 0.0000 0.0045

Figure 3: Weight structured populationdensities taking into account temperature andfood effects, especially on mortality, optimized

parameter set

It was not possible to find back the trueinfluence of starving related death withoutprevious knowledge of Fµ, compare an esti-mation of µF of 0.0000 to 0.0012 while directsimulation input was 0.0010. Besides, the

role of temperature related death was over-estimated, again we suppose that a lack oftemperature amplitude is one possible rea-son. A model efficiency of 0.99 was foundand is illustrated in figure 3

4.4 Coupled systems analysis

Year long experiments in nature comprisemonitoring of newborn animals, e.g. unpub-lished data by Schneider et al. Model equa-tions then comprise p+(F, T, . . .). Other ex-periments distinguish between female andmale animals and focus on differing rates foranabolic and catabolic processes and mor-tality rates. Any of these modifications re-sults in multi-experiment analysis. Virtualtests performed so far show a fairly accept-able reproduction of the population densitiesas long as environmental components wereexcluded or were very pronounced. How-ever, parameter attribution in real life cir-cumstances is difficult as all possible en-vironmental components, like temperatureand food supply, will have to be monitoredcollaterally.

4.5 Application in real lifeexperiments

In data so far unpublished Suhling et al. an-alyzed the influence of increasing temper-ature as one effect of climate change onmean weight of G. pulex. They includedfour temperature regimes, one being trueweather conditions, the others differed tothis by two, four and six degree Celsius re-spectively. A single food reservoir was fur-nished and mean weight was monitored dur-ing little more than sixty days.

We supposed that all of the processes de-scribed in the previous subsections are rel-evant. Differences in dynamic behaviourbetween the four regimes would then befully explained by temperature dependencyof both anabolic processes and death. Asfood stock will be depleted depending on therate of anabolic processes we suppose thatfood dynamic shape parameters may varybetween the four temperature regimes.

Figure 4: Optimized parameter set forexperimental data with outside temperature

Including all of the processes given inthe previous subsection we found impressivemodel efficiencies of 0.98 when optimizingall parameters for the outside temperatureregime, see figure 4. The overall dynamic ofthe measured and modelled population den-sity is identical and close to what has beenmodelled during virtual experiments in sec-tion 4.3. However, the bigger the tempera-ture difference to the latter, the smaller themodel efficiency, see figure 5 and the tablebelow. The series of shrinking, growth andshrinking is amplified with rising tempera-ture, and death is rarer than modelled.

+0oC +2oC +4oC +6oCME 0.98 0.79 0.687 0.535

Not shown here but even worse are modelefficiencies when only food dynamic parame-ters are left to be optimized to the tempera-ture regimes of plus two, four and six degreeCelsius.

We therefore conclude that another tem-perature dependent process is hidden in theexperiment. Our final clue to this assump-tion is that even direct simulation of the ex-perimental set up with the parameters foundfor basis temperatures did not show the samedynamic as revealed by the experiments.

Figure 5: Optimized parameter set forexperimental data with +2oC,+4oC, and +6oC

respectively

5 Conclusion

Combining direct simulation with Comsolwith simple optimization tools in Matlabhelps in research in ecology in two ways. Onthe one hand it is an efficient way to analysecoupled processes found in integral experi-

ments, on the other hand it can guide re-searchers to design efficient experiments forsingle processes in varying contexts.

References

[1] J. E. Nash and J. V. Sutcliffe, Riverflow forecasting through conceptual mod-els part i a discussion of principles, Jour-nal of Hydrology 10 (1970), 282–290.

[2] Camille Parmesan, Ecological and evo-lutionary responses to recent climatechange, Annual Review of Ecology, Evo-lution, and Systematics 37 (2006), 637–669.

[3] Manfred Pockl, Bruce Webb, andDavid W. Sutcliffe, Life history andreproductive capacity of gammarus fos-sarum and g. roeseli under naturally fluc-tuating water temperatures: asimulationstudy, Freshwater biology 48 (2003), 53–66.

[4] D. W. Sutcliffe, T. R. Carrick, and L.G.Willoughby, Effects of diet, body size, agand temperature on growth rates in theamphipod gammarus pulex, Freshwaterbiology 11 (1981), 183–214.

[5] Shripad Tuljapurkar and Hal Caswell,Structured-population models in ma-rine, terrestrial, and treshwater systems,vol. 21, Chapman & Hall, 1997.

[6] L. von Bertalanffy, Quantitative laws inmetabolism and growth, Quarterly Re-view of Biology 32 (1957), 217–231.

Acknowledgements

The authors would like to thank PD Dr.Frank Suhling for freely sharing with us boththe data he aquired by the experiments de-scribed above and his knowledge in ecology.