resource allocation, hyperphagia and compensatory growth

doi:10.1016/j.bulm.2004.03.008Bulletin of Mathematical Biology (2004)66, 1731–1753

Resource Allocation, Hyperphagia and CompensatoryGrowth

WILLIAM S. C. GURNEY∗

Department of Statistics and Modelling Science,University of Strathclyde,Glasgow G1 1XT,UK

Fisheries Research Services,Marine Laboratory,POBox 101,Aberdeen, AB11 9DB,UKE-mail: [email protected]

ROGER M. NISBET

Department of Ecology, Evolution and Marine Biology,University of California,Santa Barbara, CA 93106,USA

Organisms often shown enhanced growth during recovery from starvation, andcan even overtake continuously fed conspecifics (overcompensation). In an earlierpaper (Ecology 84, 2777–2787), we studied the relative role played by hyperphagiaand resource allocation in producing overcompensation in juvenile (non-reproductive) animals. We found that, although hyperphagia always producesgrowth compensation, overcompensation additionally requires protein allocationcontrol which routes assimilate preferentially to structure during recovery. Inthis paper we extend our model to cover reproductively active individuals anddemonstrate that growth rate overcompensation requires a similar combination ofhyperphagia and allocation control which routes the part of enhanced assimilationnot used for reproduction preferentially towards structural growth. We comparethe properties of our dynamic energy budget model with an earlier proposal, due toKooijman, which we extend to include hyperphagia. This formulation assumes thatthe rate of allocation to reserves is controlled by instantaneous feeding rate, andone would thus expect that an extension to include hyperphagia would not predictgrowth overcompensation. However, we show that a self-consistent representationof the hyperphagic response in Kooijman’s model overrides its fundamentaldynamics, leading to preferential allocation to structural growth during recoveryand hence to growth overcompensation.

c© 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rightsreserved.

∗Author to whom correspondence should be addressed.

0092-8240/04/061731 + 23 $30.00/0 c© 2004 Society for Mathematical Biology. Published byElsevier Ltd. All rights reserved.

1732 W. S. C. Gurney and R. M. Nisbet

1. INTRODUCTION

Many organisms grow faster during recovery from starvation than duringconstant exposure to the same food environment (Kennedy, 1953; Wilson andOsbourne, 1960; Weatherly and Gill, 1981; Bradley et al., 1991; Russell andWooton, 1992). Under some circumstances this effect can be so intense that theanimal exhibits overcompensation—that is a starved and re-fed animal is largerthan one grown continuously in the re-feeding environment (Miglavs and Jobling,1989b; Yu et al., 1990; Haywardet al., 1997).

Two individuals of the same size can only sustain differing growth rates in thesame environment if their assimilation or maintenance rates differ. Although main-tenance rates in starved animals are sometimes smaller than in well fed equiv-alents, this effect is generally both small and short lived (Miglavs and Jobling,1989a). However, large increases in ingestion rate (hyperphagia) occur in a varietyof mammals and salmonid fish recovering from starvation (Miglavs and Jobling,1989b; Weigle, 1994; Blum, 1997; Friedman, 1998; Jobling and Johansen, 1999).For a number of mammalian species, the biochemical mechanisms underlying thelinkage between body adiposity and compensatory hyperphagia are well under-stood (Blum, 1997), and several workers have suggested that a similar mechanismoperates in salmonid fish (Bull and Metcalfe, 1996; Jobling and Johansen, 1999).

Despite numerous observations of compensatory hyperphagia, and an extensiveliterature on individual energy budget models [e.g.,Kooijman(1993, 2000), Mullerand Nisbet(2000), Nisbet et al. (2000)], there has been relatively little theoreticalanalysis of the effects of hyperphagia on energy budgets. The work described inthis paper extends an investigation byGurneyet al. (2003) examining the relation-ship between hyperphagia, energy allocation and overcompensation in juveniles.They concluded that overcompensation cannot arise from hyperphagia alone, butwas frequently produced by the combination of hyperphagia and an allocation strat-egy which routed assimilate preferentially to growth during recovery. They notedthat allocation controlled by assimilation rate so as to render reserve status inde-pendent of assimilation rate produced the required effect.

In this paper we extend the model analysed byGurneyet al. (2003) to includereproduction, thus enabling it to be applied to adult females as well as to juve-nile individuals. We show that same combination of hyperphagia and allocationcontrol which produces overcompensation in juveniles also produces this outcomein adults—with the additional possibility of overcompensation in cumulativereproduction.

In cases where differing assimilation and maintenance allometry producesgrowth saturation, we show that a single bout of compensatory hyperphagiacan produce an individual which is larger than one grown continuously but hassmaller reserves. We examine the implications of this for an individual growingin fluctuating food conditions and show that our default model would predictstructural growth accompanied by reserve depletion continuing until reserves are

Resource Allocation, Hyperphagia and Compensatory Growth 1733

completely exhausted. We propose a modified allocation model which preservesthe key properties of that suggested byGurneyet al. (2003) while also leading tomorereasonable predictions in varying food conditions.

Finally, we contrast the model proposed in this paper with the dynamic energybudget model proposed byKooijman(1993, 2000). We begin by demonstratinghow Kooijman’s model can be self-consistently extended to include compensatoryhyperphagia and then examine the behaviour of this extended Kooijman modelin starvation–recovery conditions. We show that the extended Kooijman modelshares many of the compensation properties of the model proposed in this paper,and suggest that this happens because the extend definition critically increases thesimilarity of the two models.

2. HYPERPHAGIA AND OVERCOMPENSATION IN JUVENILES

The model described byGurneyet al. (2003) divides the individual’s carbonmass into two components—reserves,R, representing that part which can bemetabolised to meet basal costs under starvation, and structure,S, representingthat part which is inviolable. FollowingGurney and Nisbet(1998) they denote therate of carbon assimilation (net of feeding rate dependent costs) and the rate ofexpenditure on basal maintenance byA andM respectively. In line with their viewof body carbon, they divide the assimilate stream into one component (‘protein’)which can be used either to build structure or to meet maintenance costs andanother (‘fat’) which can only be used to meet immediate or future maintenancecosts. They denote the proportion of protein in the assimilate stream byp and theproportion of assimilated protein used to build structure byk, thus yielding theallocation scheme shown inFig. 1.

They adopt as state variables the structural carbon mass(S) and the fraction oftotal body mass composed of metabolisable tissue(X). Thepayoff from adopting afractional, as opposed to absolute, measure of reserve status is that during constant

Figure 1. Resource allocation in juveniles.A is the rate of carbon assimilation net offeeding rate dependent costs andM is the basal maintenance rate.R and S represent theindividual’s metabolisable and non-metabolisable carbon mass.p andk respectively arethe fraction of protein in the assimilate stream and the fraction of assimilated protein routedto structural growth.


food growth,

X ≡ R

R + S(1)

approaches a stationary value,X∗, although R changes continuously.They further assume that the basal maintenance rate(M) is proportional to the

total carbon mass, so that

M = µS/(1 − X), (2)

and that assimilation rate scales allometrically with structural mass, with a fooddependent constant of proportionality(α) and a multiplier(λ) to represent a hyper-phagic response, that is

A = αλSβ. (3)

Under these assumptions, the individual state dynamics are

d S

dt= kp A, (4)

d X

dt=

[A(1 − X)

S

] [(1 − X) − µS

A− kp

]. (5)

The payoff from the choice of the reserve fraction as the second state variable isthat for an individual withβ = 1 growing in constant food this quantity rapidlysettles to a stationary value. Moreover, they argue that separation of timescales nor-mally implies that the equivalent quasi-stationary value provides a good estimateof reserve fraction for individuals withβ �= 1.

They first investigated this model without a hyperphagic response(λ = 1), andshowed that routing a constant fraction of ingested protein to structural growth(k = constant) yields sub-optimal growth in high food environments and unneces-sary reserve exhaustion (implying death) in impoverished ones. Dynamic controlof protein allocation overcomes this difficulty and they showed that setting

k = 1

p

[1 − Xd − µS

A

], (6)

holds the (quasi-)stationary reserve fraction constant atXd independent of assim-ilation rate. Clearly this can only be achieved over a range of assimilation ratessuch that the requiredk ∈ [0, 1]. Within this range its effect is to route a constantfraction of net production to structural growth.

To introduce a hyperphagic response into their model, they wrote

λ = Xd/X, (7)

thus implying that the individual’s carbon assimilation rate in any given food envi-ronment (i.e., value ofα) increases (decreases) whenX < Xd (>Xd). A combina-tion of analysis (for the caseβ = 1) and numerical realisations showed that such


a response always implies growth compensation after a bout of starvation. How-ever, overcompensation was found to depend on the combination of a hyperphagicresponse such as equation (7) andprotein allocation control such as equation (6)—the effect of which is to route assimilate preferentially to structure during recovery,thus prolonging the compensation period.

3. ADULT ENERGETICS

3.1. Energy allocation and dynamics. The model discussed in this paperextends the framework set out byGurneyet al. (2003) to include reproduction.In addition to the individual’s structural and reserve carbon masses (S and R) wenow consider the cumulative total assimilate allocated to egg production, whichwe (somewhat arbitrarily) divide into structural and reserve components (ES andER). We assume that egg structure is manufactured from the protein part of theassimilate stream and that egg reserves are provisioned from the individual’sreserve stock. We useε to denote the fraction of the protein assimilate streamrouted to eggs, andφ to represent the ratio of reserve to structure in the resultingeggs.

The resulting pattern of resource allocation is shown inFig. 2. We recognise thatthis representation would require modification if our model were to be applied toembryonic development; but argue that the present context merely demands carefulaccounting of the material required for reproduction.

We choose as our state variables the individual’s structural mass,S, its reservefraction X [cf. equation (1)] and the cumulative carbon mass of eggs

E ≡ ER + ES. (8)

Figure 2. Resource allocation in adults.R andS represent theindividual’s metabolisableand non-metabolisable carbon mass andES and ER represent the equivalent componentsof the cumulative total allocation to eggs.A is the net carbon assimilation rate andMthe basal maintenance rate. The fraction of protein in the assimilate and the fraction ofassimilated protein routed to eggs arep and ε respectively. The ratio of fat to proteinin eggs isφ and the fraction of assimilated protein not used for eggs which is routed tostructural growth isk.


With this choice of state variables, inspection ofFig. 2 tells us immediately thatthe dynamics of cumulative egg mass and structure are

d E

dt= (1 + φ)εp A, (9)

d S

dt= k(1 − ε)p A. (10)

To formulate a dynamic description for the reserve ratio(X) we first define

θ(ε) ≡ 1 − εp(1 + φ) (11)

to represent the proportion of the assimilate stream not being used for reproductionand then see fromFig. 2 that the only differences from the situation leading toequation (5) (Fig. 1) are first that the incoming assimilate stream has the allocationto egg structure deducted before any other allocations are made, and second thategg reserves are supplied direct from the individual’s reserve stock. These changesalter the reserve fraction dynamics to

d X

dt=

[A(1 − X)

S

] [θ(ε)(1 − X) − µS

A− kp(1 − ε)

]. (12)

The formulation used byGurneyet al. (2003) for the hyperphagic response [equa-tion (7)] allows unlimited increases in appetite which are clearly unbiological. Wemodify their formulation slightly and write

A = αλSβ whereλ ={

Xd/X if X > Xd/λm,

λm otherwise,(13)

thus limiting the proportional appetite increase relative toX = Xd to λm .

3.2. Allocation control. The model defined by equations (9)–(13) contains twoparameters (ε andk) which play a similar role to the parameterk in the juvenilemodel described byGurneyet al. (2003). If ε is fixed, then at low ingestion ratesthe animal will reproduce itself to death while ifk is fixed then it will grow itselfto death.

We follow Gurneyet al. (2003) in assuming that these parameters are controlledby the instantaneous rate of assimilationA. We further assume that the defaultmode of operation is that a fixed fraction of protein(ε0) is allocated to eggs andthe remainder is divided between structure and reserves so as to hold the quasi-stationary reserve ratio constant at the defended levelXd . Toachieve this we definethe value ofk needed to achieve control as

kc = 1

p(1 − ε0)

[θ0(1 − Xd) − µS

A

]whereθ0 ≡ θ(ε0), (14)


Figure 3. The control functions. Immediate allocation of assimilated protein to reproduc-tion is described by the functionε [equations (16) and (17)]. Ingestate not routed to repro-duction is divided between structure and reserves as described by the functionk [equations(14) and (15)]. The abscissa can be interpreted as the ratio of the timescales for structuralmass gain(S/A) and carbon retention(µ−1).

and then set

k =

1 if µS/A < θ0(1 − Xd) − p(1 − ε0)

0 if µS/A > θ0(1 − Xd)

kc otherwise.(15)

As we cansee fromFig. 3, the effect of equations (14) and (15) is to shut downstructural growth(k → 0) when the assimilation rate is insufficient to support aquasi-stationary reserve ratio of at leastXd . We shall assume that even when struc-tural growth is shut down, the animal will try to keep reproduction going unless itthereby imperils survival. Equation (13) implies that if the quasi-stationary reserveratio falls belowXd , compensatory appetite increase occurs. We assume that whilesuch compensation is possible the animal continues to allocate the default fractionof protein(ε0) to eggs. However, if the assimilation rate falls sufficiently for thequasi-stationary reserve ratio to be less thanXd/λm no further appetite increasesare possible and so the only way to inhibit further falls inX is to decrease repro-duction. To describe such a decrease we defineεc as the value ofε needed to holdthe quasi-stationary reserve ratio constant atXd/λm thus:

εc = λm

p(1 + φ)(λm − Xd)

[1 − Xd

λm− µS

A

](16)

and then write

ε =

ε0 if µS/A < θ0(1 − Xd/λm)

0 if µS/A > (1 − Xd/λm)

εc otherwise.(17)


Figure 4. The quasi-stationary reserve fraction in constant food. Equation (18) is shown asa function of(µ/α)S1−β , whichcan be interpreted as the minimum maintenance rate overtheactual ingestion rate.

3.3. Growth and reproduction in constant food. We consider an individual obe-ying equations (9)–(17) growing in a constant food environment (that is withαheld constant). Routine (if tedious) algebra shows that such an individual’s quasi-stationary reserve fraction,X∗, is

X∗ =

[θ0 − p(1 − ε0)]Xd

θ0Xd + (µ/α)S1−βif θ0(1 − Xd) − p(1 − ε0) >

µ

αS1−β

Xd if θ0(1 − Xd) >µ

αS1−β

> θ0(1 − Xd) − p(1 − ε0)θ0Xd

θ0Xd + (µ/α)S1−βif θ0(λm − Xd) >

µ

αS1−β > θ0(1 − Xd)

Xd/λm if (λm − Xd) >µ

αS1−β > θ0(λm − Xd)

1 − µS1−β

αλmif λm >

µ

αS1−β > (λm − Xd).

(18)

We illustrate this result inFig. 4, from which we see that a (quasi-)steady stateonly exists if the maximum possible assimilation rate for an animal with structureS exceeds its minimum possible maintenance rate, that is

λmαSβ > µS. (19)

By default, a fractionθ0 of the assimilate stream is allocated to growth and main-tenance [cf. equation (11)]. Simultaneous growth and reproduction are thus onlypossible if the default allocation to growth and maintenance exceeds the mainte-nance cost of an animal with the default reserve fraction, that is

θ0αSβ >µS

1 − Xd. (20)


If default assimilation and allocation are insufficient for an animal withX = Xd tomeet its maintenance requirements, structural growth stops. Reproduction, how-ever, continues so long as reduced reserves and enhanced assimilation enable main-tenance costs to be met, that is so long as

λmαSβ >µS

1 − Xd/λm. (21)

If food is so scarce (α so low) that even at maximum enhancement, the assimilationrate, while sufficient to maintain current structure, is insufficient to meet mainte-nance withX = Xd/λm, the animal shuts down growth and reproduction andreduces its reserve stock until it can just maintain itself. Food abundance whichdoes not allow the current structure to be maintained implies continued reservedepletion and eventual death.

For most species we expectβ < 1, which implies that the process of growthunder constant food (i.e., constantα) causes the value of(µ/α)S1−β to increase.Thus an individual growing at the asymptotic reserve ratio(Xd) will reduce theproportion of assimilated protein allocated to structure as it grows and thus, ask → 0, tend asymptotically to a food dependent maximum size

S∗∞ =

[α

µθ0(1 − Xd)

] 11−β

. (22)

We note that, at the asymptotic size, the animal’s cumulative allocation to eggscontinues to increase at a rate

E = (1 + φ)ε0 pα(S∗∞)β. (23)

4. HYPERPHAGIA AND GROWTH COMPENSATION

4.1. Starvation and recovery. We consider two individuals withE(0) = 0,S(0) = S0 and X (0) = Xd . The control individual grows continually in foodwhich is sufficiently abundant for it to maintainX = Xd . We denote its structuralmass at timet by S∗(t). By contrast, the experimental individual is starved untilits reserve ratio reachesX0 and then reintroduced to the same food environment asthe control. We denote its reserve ratio and structure at timet by X (t) and S(t)respectively.

For the special caseβ = 1 we can follow the analysis for juveniles inGurneyet al. (2003). We see from equation (12) that the time taken for the experimentalindividual’s reserve ratio to reachX0 is

τ = 1

µln

[1 − X0

1 − Xd

]. (24)


During this time, the control individual’s structural mass reachesS∗(τ ) while theexperimental individual’s remains constant. So the deficit ratio, defined asDτ ≡S(τ )/S∗(τ ), is

Dτ = exp[−pk∗α(1 − ε0)τ ], (25)

which, substituting for the starvation timeτ from equation (24) and for the quasi-stationary fraction of protein allocated to structure(k∗) from equation (14), can beshown to be equivalent to

Dτ =[

1 − X0

1 − Xd

] αµ

(θ0(1−Xd )− µα)

. (26)

From equation (12) wecan show that if an individual starts with structureS(0) andreserve ratioX0 and grows continuously thereafter, then the asymptotic ratio of itsstructure to that of an equivalent individual whose initial reserve ratio isXd is

C∞ =[

1 − X0

1 − Xd

] 1−XdXd

. (27)

We haveovercompensation, that is our starved and re-fed individual is asymptoti-cally larger than the control, if and only if the deficit after starvation is smaller thanthe compensatory gain after re-feeding, that is if

C∞ > Dτ . (28)

Substitution from equations (26) and (27) shows thatthis is true if and only if

θ0α

µ(Xd(1 − Xd)) < 1. (29)

If we setε0 = 0, thus implying that all assimilate is used for growth and mainte-nance(θ0 = 1), we recover the result for juveniles given byGurneyet al. (2003). Inboth adults and juveniles, the tendency to overcompensation increases as the ratioα/µ decreases. In addition, adults tend towards overcompensation as the propor-tion of assimilate routed to growth and maintenance decreases. Both effects stemfrom the same cause, namely that an animal which expends most of its normalassimilation stream on reproduction and maintenance will show a greater marginalincrease in growth rate during a bout of hyperphagia than one which could alreadygrow fast in the given food environment.

We illustrate overcompensation for an exponentially growing adult inFig. 5(a).For these parameters the value of the LHS of inequality (29) is 0.53, and wesee strong overcompensation in both structural carbon mass and cumulativereproduction.


Figure 5. Starvation and recovery. All runs showtwo identicalindividuals, both grownuntil t = 0 in sufficient food (α = 1) to allow growth with X = Xd . The control(dotted trajectories) grows continuously withα = 1 while the treatmentindividual (solidtrajectories) is starved untilt = 0.8 and then reintroduced to food withα = 1. Frame (a)shows exponentially growing individuals [β = 1, ε0 = 0.03, S(0) = 0.1]. Frame (b)shows individuals with saturating growth [β = 0.8, ε0 = 0.05, S(0) = 0.1] starting farbelow saturation. Frame (c) shows individuals starting at the default asymptotic structuralmass [β = 0.6, ε = 0.03, S(0) = S∗∞]. For all runs the other model parameters are:p = 0.8, µ = 0.3, λm = 3, φ = 9, andXd = 0.3.

In Fig. 5(b) we show the effects of a similar treatment on a individual withβ = 0.8 and an initial structural mass which is small compared to its defaultasymptotic mass[S(0) � S∗∞]. Over theperiod of the simulation we see significantovercompensation in both structural mass and cumulative reproduction. However,since the reserve fraction returns to its quasi-stationary value(Xd) by the end ofthe simulation, the treatment animal’s structural mass will eventually asymptote atthe same value(S∗∞) as the control’s. Their rate of reproduction will then be thesame although the treatment individual’s lifetime reproduction will be very slightlygreater than the control’s because of the bout of compensatory hyperphagia.

Our final simulation, Fig. 5(c), shows two individuals whose initial structuralmass is the default asymptotic value(S∗∞). As we would expect, the control staysat this mass and reproduces at the rate given by equation (23). However, the treat-ment individual’s bout of compensatory hyperphagia causes its structural mass toincrease. We can see that this effect is generic by substituting equations (13), (15)


and (17) into equation (10) to find

S =

p(1 − ε0)A if µS1−β/α < θ0(1 − Xd)

−p(1 − ε0)

0 if µS1−β/α > θ0(1 − Xd)αXdθ0(1 − Xd)

XSβ − µS otherwise.

(30)

At the constant food asymptoteS = S∗∞ andX = Xd , so on reintroduction to food,the individual hasS = S∗∞ andX < Xd thus implyingS > 0.

During recovery, the animal’s reserve fraction rises but does not return toXd .Instead it asymptotes at a new equilibrium valueX∗ which is related to the newasymptotic sizeS∞(X∗), by the condition that the rate at which assimilate is allo-cated to growth plus maintenance exactly equals current maintenance expenditure,that is

S∞(X∗) =[

αXd

µX∗ θ0(1 − X∗)] 1

1−β

. (31)

4.2. Cyclic food. Our next simulation shows an individual withβ = 0.7, so itsgrowth saturates in constant food [seeFig. 6(a)]. This individual is raised in acyclic food regime consisting of alternating equal periods of starvation(α = 0)

and re-feeding(α = 1) with a cycle timeτ = 0.3. In Fig. 6(b) we show the growthof an individual whose dynamics is described by our default model [equations (9)–(17)], and see behaviour which is a natural extension of that shown inFig. 5(c).Each bout of hyperphagia induces structural growth accompanied by a reduction inreserve ratio. Even though the maximum appetite increase is only a factor of two(λm = 2), the process of structural mass increase and reserve decrease continuesuntil reserves are exhausted before the end of a starvation period and the animaldies.

This clearly points to a flaw in our default model, but the difficulty is easilyunderstood. The allocation behaviour defined by equations (14) and (15) has twoproperties which are essential to the prediction of overcompensation—first thatallocation is controlled by current assimilation rate and second that at high assim-ilation rates resource is allocated preferentially to structural rather than reservegrowth. So long as the reserves are not too depleted this hypothesis is defensible,but it cannot be true if reserve depletion threatens survival. In such circumstances,the individual must behave in the way hypothesised byBroekhuisenet al. (1994)and allocate its assimilate stream preferentially to reserves.

To incorporate this policy switch into our model, we assume, rather arbitrarily,that the reserve fraction at which the organism considers its survival endangeredcorresponds to the quasi-stationary reserve fraction which would be established ina constant food regime just too impoverished to permit any reproduction(Xd/λm).We replace equation (15) with a new formulation, which retains the properties of


Figure 6. Saturating growth in cyclic food. Three runs with an individual having assimila-tion allometric constantβ = 0.7, maintenance rate scaleµ = 0.5, compensation parame-tersXd = 0.4, λm = 2, and reproduction parametersε0 = 0.01,φ = 9. Frame (a) showsgrowth in constant food(p = 0.5, α = 1) with X (0) = Xd . Frames (b)and (c) showgrowth in a food regime with starvation(α = 0) and re-feeding(p = 0.5, α = 1) alter-nating with a cycle timeτ = 0.3. Frame (b) shows an individual with dynamics specifiedby equations (9)–(17). Frame (c) shows an individual with growth allocation dynamicsspecified by equation (32) instead of (15).

the original providedX ≥ Xd/λm but switches off structural growth whenX fallsbelow this level, thus

k =

1 if µS/A < θ0(1 − Xd) − p(1 − ε0) and X ≥ Xd/λm

0 if µS/A > θ0(1 − Xd) or X < Xd/λm

kc otherwise.(32)

Although Broekhuisenet al. (1994) assumed that reserve recovery was priori-tised over all else, we continue to assume that allocation to reproduction is gov-erned by instantaneous assimilation rate as defined by equation (17).

In Fig. 6(c) we show the growth behaviour predicted by this modified model.Initially the periods of hyperphagia pump the structure up and the reserve fractiondown, just as before. However, asX begins to fall transiently belowXd/λm therate at which this process proceeds begins to slow, and it stops altogether onceXnever rises above the critical level.


To calculate the new asymptotic structural mass(Sτ∞) we note that structuralgrowth will stop when the reserve fraction just fails to reach the override level(Xd/λm) at the end of the re-feeding period. We see from equation (12) that in thiscase the reserve fraction at the end of the starvation phase is

Xmin = 1 − (1 − Xd/λm) exp(µτ/2). (33)

At the asymptotic size, reserve recovery during re-feeding is linear, so the totalmaintenance expenditure over the feeding cycle is approximately

∫cycle

µSτ∞1 − X

dt ≈ µSτ∞τ(1 − Xd

λm

) (1+exp(µτ/2)

2

) . (34)

At the asymptotic size, the total allocation to growth and maintenance over thecycle must equal the total maintenance expenditure, that is

θ0λmα(Sτ∞)β τ

2=

∫cycle

µSτ∞1 − X

dt (35)

and hence

Sτ∞ ≈

[θ0

λm

2

α

µ

(1 − Xd

λm

) (1 + exp(µτ/2)

2

)] 11−β

. (36)

Comparing this with equation (22) shows that starvation–recovery cyclingincreases the asymptotic size of the individual relative to a continuously fedequivalent(Sτ∞ > S∗∞) provided that

(λm − Xd)(1 + exp(µτ/2)) > 4(1 − Xd). (37)

5. HYPERPHAGIA IN KOOIJMAN’S DEB MODEL

Kooijman(1993, 2000) proposed a dynamic energy budget model (hereafter theKDEB model) which shares a number of properties with the model defined byequations (9)–(17). This model is more compactly described inMuller andNisbet(2000) and a variant with a more plausible representation of the low food responseis given byFujiwara et al. (2004).

To facilitate comparison with the early literature we retain the notation used inKooijman(1993), in which quantities with dimension 1/t have identifiers with anover-dot and quantities with dimensions 1/V and 1/V

23 have identifiers enclosed

in square or curly brackets respectively. UnfortunatelyKooijman (2000) uses


different notation, butAppendix Bcontains a table showing the mapping betweenthe two notations.

The state variables of this model are structural biovolume,V , and reserve energydensity,[E]. Energy is assumed to be assimilated into a reserve compartment ata rate{ Am}V

23 f , where{ Am} is the area-specific maximum assimilation rate, and

f ∈ [0, 1] is the scaled functional response. The model’s fundamental postulate isthat energy utilisation is arranged so as to control the ratio of actual to maximumpossible reserve energy density([E]/[Em ]) at the value of the scaled functionalresponse( f ). We follow Muller andNisbet (2000) in using the model variant inwhich normal storage dynamics are assumed to operate until the implied utilisationrate is insufficient to cover maintenance, but we assume that when this occurs, utili-sation is increased to cover maintenance. Returning their non-dimensional descrip-tion of this process to dimensional form and modifying its low reserve behaviourappropriately yields

d[E]dt

=

{ Am}V

13

(f − [E]

[Em])

if[E][Em] ≥ m[G]

{ Am} V13

{ Am}V

13

f − m[G] otherwise.(38)

The second key assumption of the KDEB model is that under ‘normal’ conditionsthe flow of utilised energy is divided between reproduction and growth plus main-tenance; with a fractionκ going to the latter and 1− κ to the former. A furtherimportant assumption is that the individual’s biovolume cannot shrink, so ‘normal’conditions are defined as those allowing the default fraction of utilised energy allo-cated to growth and maintenance to at least cover maintenance. These assumptionsimply [Kooijman, 1993, equation (4.3)] that

dV

dt=

1

κ[E] + [G](

κ{ Am}V23

[E][Em] − m[G]V

)if

[E][Em] ≥ m[G]

κ{ Am}V13

0 otherwise.(39)

To facilitate comparison of the KDEB model with the model discussed earlierin this paper, we describe reproduction by the cumulative total energy allocated tothis process, which we denote byRc. Wenote thatRc includes energy required tomaintain the reproductive apparatus—a contribution called ‘maturity maintenance’in Kooijman’s formulation.

Equations (38) and (39) together withKooijman[1993, equation (3.8)] show that

d Rc

dt=

(1 − κ)[G][E]κ[E] + [G]

({ Am}[Em] V

23 + mV

)if

[E][Em]V 1

3

≥ m[G]κ{ Am}

{ Am} [E][Em]V

23 − m[G]V if

m[G]κ{ Am} >

[E][Em]V 1

3

≥ m[G]{ Am}

0 otherwise.(40)


To introduce compensation (making a variant we hereafter refer to as the eKDEBmodel) we assume that during a bout of hyperphagia the animal’s searching behavi-our (and hence scaled functional response,f ) remains constant, while its handlingtime decreases, thus increasing its size-specific maximum uptake rate{ Am}. Wethus define an appetite factorλ([E]) and write

{ Am} = { Am0}λ([E]). (41)

By analogy with equation (13) we wantλ to have the value unity when[E]/[Em ] isat its control value( f ) and to increase (decrease) when the reserve status is below(above) this value. A convenient function with these properties, which also limitsthe resultant appetite increase to a maximum value,λm, is

λ([E]) ≡

( [Em ] f

[E])ξ

if

( [Em] f

[E])ξ

< λm

λm otherwise,(42)

where the parameterξ controls the strength of compensation. The inclusion ofthe scaled functional response( f ) in the appetite factor is potentially controversialsince it implies that control is exercised both by the internal state of the organismand by its external environment. However, we note that this echoes a property ofthe original KDEB model in which reserves are regulated at a level proportionalto f .

We first examine the behaviour of this model with the compensation strengthparameterξ = 1. In Fig. 7(a) we compare KDEB and eKDEB model predictionsfor an individual raised in constant food untilt = 0 when it is starved for a periodbefore being re-fed. We see that making the maximum uptake rate scale({ Am})proportional to f/[E] has considerably altered the starvation–recovery behaviourof the KDEB model. Where the KDEB model predicts continued allocation ofenergy to growth and reproduction (and hence rapid reserve depletion) until[E]drops below a threshold value, the eKDEB model predicts an immediate switchinto a maintenance-only regime, thus implying notably higher reserve status at theend of a given starvation period. Similarly, the KDEB model predicts that structuralgrowth does not restart until[E] exceeds a threshold value, whereas the eKDEBmodel predicts an essentially instantaneous growth restart.

It is clear fromFig. 7(a) that extending the KDEB model to represent compen-satory hyperphagia allows it to predict growth compensation. We now ask whetherit can predict overcompensation. The results inFig. 7(b) and7(c) suggest that (atleast withξ = 1) it cannot. Back substitution of equation (41) with ξ = 1 intoequation (39) shows thatthis is a systematic result. The asymptotic size at whichthe trajectory is aiming is unchanged by the compensatory hyperphagia. The onlychange in the dynamics of structural growth, other than the altered timing of its ces-sation and resumption, is thus the weak growth enhancement produced by reducing


Figure 7. Starvation and recovery in the eKDEB model. Each frame shows normalised

length l ≡ V13 (m[Em]/{ Am0}), normalised reserve energy densitye = [E]/[Em] and

normalised reproduction energyrc ≡ Rcm/{ Am0} against normalised timemt , for twoindividuals, withκ = 0.5 and[G]/[Em] = 0.25, being grown in a medium implyingf = 0.7. Frame (a) compares trajectories for starved and re-fed individuals, one (solid)obeying the eKDEB model [equations (38)–(42) with λm = 2 andξ = 1], the other(dotted) obeying the KDEB model [equations (38)–(41) with λ = 1]. Frame(b) contrastsindividuals obeying the eKDEB model(λm = 2, ξ = 1), one starved andre-fed (solid),the other grown continuously (dotted). Frame (c) contrasts individuals obeying the eKDEBmodel (λm = 2, ξ = 1) and started at their asymptotic length, one starved and re-fed(solid) and the other grown continuously (dotted).

the stored energy density, while preventing that reduction from inhibiting energyallocation.

Although the foregoing suggests that the eKDEB model is incapable of predict-ing overcompensation withξ = 1, it also suggests that this may not be true ifξ > 1, since the nominal asymptotic size [essentially, the first term in the roundbrackets in equation (39)] then increases with decreasing[E]. This means thatreducing[E] can produce both significant enhancement of the rate of structuralgrowth, and significant positive structural growth in an individual already at theconstant food asymptotic size. InFig. 8(a) and8(b) we show that this indeedimplies that if ξ > 1, it is possible for a starved and re-fed individual to equalor exceed the size of its continuously fed equivalent, and that starvation and re-feeding and individual at or near its constant food asymptotic size can result in a


Figure 8. Strong compensation in the eKDEB model. Each frame shows normalised length,reserve energy density and cumulative reproduction energy (cf.Fig. 7) for individualsobeying the extended KDEB model withκ = 0.5, [G]/[Em] = 0.25, λm = 2 andξ = 3being fed with a medium implyingf = 0.7. Frame (a) compares two individuals startedwell below asymptotic length, one starved and re-fed (solid), the other grown continuously(dotted). Frame (b) shows a similar experiment with both individuals started at the asymp-totic length for a continuously grown individual. One individual (solid) is starved andre-fed, the other (dotted) is grown continuously. Frame (c) shows longer trajectories, one(dotted) for an individual grown continuously and the other (solid) for a individual subjectto cyclic starvation and re-feeding with a cycle periodTc = 1.5/m.

size asymptote larger than that possible in constant food conditions. We note how-ever that, in contrast to the behaviour exemplified byFig. 5, the eKDEB modeldoes not predict overcompensation in cumulative reproductive output.

If overcompensation makes it possible for a single bout of starvation and re-feeding to produce an asymptotic size larger than that achieved in constant food,wemight expect cyclic feeding to produce a growth curve asymptoting well abovethe value for an individual fed continuously at the re-feeding level. InFig. 8(c) weshow the results of a simulation in which an individual obeying the eKDEB modelwith ξ = 3 is subject to a cyclic starvation–re-feeding regime with a cycle periodof one and a half times the characteristic starvation time(m−1). Wesee behaviourbroadly reminiscent of that shown inFig. 6. Below the continuous-feeding asymp-totic size, the structural growth rates of the continuous and cyclically fed indi-viduals are very similar; however, the cyclically fed individual continues to grow,


eventually reaching an asymptote almost twice as large as its continuously fedcousin. The reserve energy density initially shows a downward trend, but eventu-ally shows a stable cycle with an average value significantly lower thanf . Themost significant difference from our earlier findings (Fig. 6) is that the cumulativereproductive output is smaller for the cyclically fed individual than for its continu-ously fed cousin.

6. DISCUSSION

In this paper we examine two models which include reserve controlled assimi-lation in their description of the energetics of individual growth and reproduction,and are thus capable of predicting post-starvation growth compensation driven byhyperphagia. One model is an extension of that proposed byGurneyet al. (2003)to explain compensatory growth in juveniles. We refer to this as the AA (assimila-tion allocation) model. The other is an extension of a well known dynamic energybudget model due toKooijman(1993). We refer to this as the eKDEB model.

The AA model [equations (9)–(13), (17) and (32)] assumes that unless the ani-mal’s reserve fraction(X) becomes critically low, allocation is controlled by theinstantaneous rate of assimilation, with the primary aim of holdingX at a value(Xd) which is independent of assimilation rate. The effect of this control mecha-nism is to route assimilate preferentially to reproduction and growth during a boutof hyperphagia, thus prolonging its duration and increasing the consequent growthcompensation.

The eKDEB model [equations (38)–(42)] shares with its progenitor [the KDEBmodel—Kooijman(1993)] the assumption that all incoming assimilate is stored ina reserve compartment from which allocation is controlled so that the ratio of actualto maximum possible reserve energy density relaxes to the current ratio of actual tomaximum possible assimilation rate (the scaled functional response). However, aswediscuss below, describing hyperphagia in a self-consistent way produces criticaldifferences in control behaviour between this model and its progenitor.

Provided that the AA model’s allometric constantβ �= 1, these models share acomplete repertoire of behaviour under constant food conditions:

• Any animal which can grow after reaching reproductive maturity eventuallyreaches an asymptotic size at which the default allocation to growth andmaintenance just meets current maintenance costs.

• The asymptotic size is a monotone increasing function of the availability offood (i.e., ofα in the AA model andf in the eKDEB model).

• At the asymptotic size, the flow of resource to reproduction is proportionalto the assimilation rate.

When the growth of a starved and re-fed animal is compared with that of an equiv-alent individual grown continuously, the two models yield noticeably differing pre-dictions. For an individual which is starved at a size small compared to its constant


food asymptotic size in the re-feeding environment the AA model predicts strongcompensation in both growth and reproduction, with overcompensation in boththese quantities being likely in cases where the re-feeding environment permitsonly relatively slow growth. By contrast, with a comparable hyperphagic response(ξ = 1) the eKDEB model shows weak compensation in growth and almost none inreproduction, with most of the compensatory assimilation being expended on rapidreplenishment of reserves. However, when the hyperphagic response is strong(ξ = 3) the eKDEB model can predict overcompensation in growth, althoughnumerical experiments suggest that overcompensation in reproduction is lesslikely.

Whenstarvation is imposed at or near the constant food asymptotic size in the re-feeding environment the two models again yield significantly differing predictions.The AA model always predicts that the animal settles to a new asymptotic sizelarger than the constant food value but with lower reserves. The eKDEB modelwith a hyperphagic response which is linear inf/[E] (i.e., ξ = 1) predicts thatthe animal’s post-starvation asymptotic state is exactly the same as its constantfood value, with the compensatory assimilation being used mainly for rebuildingreserves. With a strong hyperphagic response, the eKDEB model predicts a ‘pump-up’ effect similar to that of the AA model (although quantitatively weaker), butwith the animal’s reserve density eventually returning to its constant food value.

In cyclic food, the two models again show behavioural differences. The AAmodel initially shows each cycle producing a pump-up in size accompanied bya pump-down in reserves, but this process is eventually halted, and the animalbrought to a new asymptotic state, by the low reserve allocation override. By con-trast the eKDEB model with linear hyperphagic response predicts an asymptoticsize which exactly matches the constant food value in the re-feeding environment.With a strong hyperphagic response the eKDEB model predicts size and reservechanges which are similar to those predicted by the AA model—albeit consider-ably weaker.

Gurneyet al. (2003) argued that overcompensation requires the combination ofa hyperphagic response and an allocation scheme which has the effect of rout-ing the hyperphagically enhanced flow of assimilate preferentially to structuralgrowth. They further argued that while this effect is readily produced by an alloca-tion scheme controlled by current assimilation rate, it is hard to envisage a similaroutcome from a scheme controlled by reserve status. The work on the AA model,reported earlier in the paper, shows that these conclusions carry over almost com-pletely to a model which additionally describes reproduction, and is thus applicableto adults.

In view of these findings it comes as something of a surprise that Kooijman’smodel (Kooijman, 1993), one of whose central assumptions is that allocation iscontrolled by reserve status, can, when extended to include hyperphagia, predictovercompensation—albeit under rather more restricted conditions than the AAmodel. The key to understanding why this happens lies in the rather special nature


of the extension we have proposed. To prevent the hyperphagic response affectingthe constant food properties of the model we propose that appetite is controlled bythe ratio of the current scaled functional response,f , to theratio [E]/[Em ], whichrelaxes tof under constant food conditions. This implies that in the compensated(eKDEB) model the actual allocation rate is partially controlled by the food abun-dance. Indeed in the case whereξ = 1, provided[E]/[Em ] is not so low thatλ isclipped atλm, equation (39) becomes

dV

dt=

1

κ[E] + [G](κ{ Am0}V23 f − m[G]V ) if f ≥ m[G]

κ{ Am0}V

13

0 otherwise,(43)

showing that the rate of allocation to biovolume is directly proportional to thescaled functional response.

Ouroverall conclusion is thus that growth rate overcompensation in adults, as injuveniles, is unlikely to be produced by hyperphagia alone, but requires in additionan allocation control mechanism which routes resource preferentially to structuralgrowth under high assimilation conditions. We further suggest that allocation con-trolled by assimilation rate is most likely to provide the requisite dynamic.

Thequestion of overcompensation in asymptotic size, and thus in lifetime repro-ductive output, is more complex. In the assimilation allocation (AA) model anyhyperphagic bout which starts at a size near the constant food asymptote will pro-duce an increase in asymptotic size and hence lifetime reproduction. By con-trast the extended Kooijman model normally predicts an asymptotic size after ahyperphagic bout which is no larger than the constant food value in the re-feedingenvironment, the only exception to this conclusion being when the hyperphagicresponse is strong enough to produce growth rate overcompensation.

ACKNOWLEDGEMENTS

We thank Masami Fujiwara, Bruce Kendall and Bas Kooijman for valuable dis-cussions. This research was supported by a grant from the US Environmental Pro-tection Agency’s Science to Achieve Results (STAR) Estuarine and Great Lakes(EaGLe) programme through funding to the Pacific Estuarine Ecosystem IndicatorResearch (PEEIR) consortium, US EPA Agreement #R-882867601. It has not beensubjected to any EPA review and therefore does not necessarily reflect the views ofthe agency, and no official endorsement should be inferred.

APPENDIX A: SCALING THE KDEB MODEL WITH HYPERPHAGIA

Wedefine scaled time and state variables

e ≡ [E][Em] τ ≡ mt l ≡ V

13

(m[Em]{ Am0}

)rc ≡ m Rc

{ Am0}(A.1)


and hence re-express the dynamic equations of the KDEB model as

de

dτ=

λ

l( f − e) if e >

gl

λλ f

l− g otherwise

(A.2)

dl

dτ=

g

3(g + κe)

(κλe

g− l

)if e ≥ gl

κλ0 otherwise

(A.3)

drc

dτ=

(1 − κ)ge

g + κe(λl2 + l3) if e ≥ gl

κλ

λel2 − gl3 ifgl

κλ> e ≥ gl

λ0 otherwise

(A.4)

where

g ≡ [G][Em] λ = min

(f

e, λm

). (A.5)

APPENDIX B: KDEB MODEL NOTATION IN KOOIJMAN (2000) VS.KOOIJMAN (1993)

Definition Kooijman (1993) Kooijman (2000)

Structural biovolume V VReserve density [E] [E]Area-specific assimilation rate { Am} { pAm }Maximum reserve density [Em] [Em]Maintenance rate coefficient m [ pm]Growth efficiency [G] [EG]Energy partitioning factor κ κ

Scaled functional response f f

REFERENCES

Blum, W. F. (1997). Leptin: the voice of the adipose tissue.Hormone Res. 48, 2–8.Bradley, M. C., N. Perrin and P. Calow (1991). Energy allocation in the clodoceranDaph-

nia magna Strauss under starvation and refeeding.Oecologia 86, 414–418.Broekhuisen, N. W., W. S. C. Gurney, A. Jones and A. Bryant (1994). Modelling compen-

satory growth.Funct. Ecol. 8, 770–782.Bull, C. D. and N. B. Metcalfe (1996). Regulation of hyperphagia in response to varying

energy deficits in over-wintering juvenile Atlantic salmon.J. Fish Biol. 20, 498–510.


Friedman, J. M. (1998). Leptin, leptin receptors and the control of body weight.Nutr. Rev.56, 38–46.

Fujiwara, M., B. E. Kendal and R. M. Nisbet (2004). Growth autocorrelation and animalsize variation.Ecol. Lett. 7, 106–113.

Gurney, W. S. C., W. Jones, A. R. Veitch and R. M. Nisbet (2003). Resource allocation,hyperphagia and compensatory growth in juveniles.Ecology 84, 2777–2787.

Gurney, W. S. C. and R. M. Nisbet (1998).Ecological Dynamics, New York: OxfordUniversity Press.

Hayward, R. S., D. B. Noltie and N. Wang (1997). Use of compensatory growth to doublehybrid sunfish growth rates.Trans. Am. Fisheries Soc. 126, 316–322.

Jobling, M. and S. J. S. Johansen (1999). The lipostat, hyperphagia and catch-up growth.Aquaculture Res. 30, 473–478.

Kennedy, G. C. (1953). The role of depot fat in hypothalmic control of food intake in therat.Proc. R. Soc. B 140, 578–592.

Kooijman, S. A. L. M. (1993).Dynamic Energy Budgets in Biological Systems, Cambridge:Cambridge University Press.

Kooijman, S. A. L. M. (2000).Dynamic Energy and Mass Budgets in Biological Systems,Cambridge: Cambridge University Press.

Miglavs, I. and M. Jobling (1989a). Effects of feeding regime on food consumption, growthrates and tissue nucleic acid in juvenile Arctic charr,Salvelinus alpinus with particularreference to compensatory growth.J. Fish Biol. 34, 937–957.

Miglavs, I. and M. Jobling (1989b). The effects of feeding regime on proximate body com-position and patterns of energy deposition in juvenile arctic charr,Salvelinus alpinus.J. Fish Biol. 35, 1–11.

Muller, E. and R. M. Nisbet (2000). Survival and production in variable resource environ-ments.Bull. Math. Biol. 62, 1163–1190.

Nisbet, R. M., E. B. Muller, K. Like and S. A. L. M. Kooijman (2000). From molecules toecosystems through dynamic energy budgets.J. Anim. Ecol. 69, 913–926.

Russell, N. R. and R. J. Wooton (1992). Appetite and growth compensation in the Europeanminnow,Phoxinus phoxinus, following short periods of food restriction.Environ. Biol.Fishes 34, 277–285.

Weatherly, A. H. and R. W. Gill (1981). Recovery growth following periods of restrictedrations and starvation in rainbow troutSalmo gairdneri Richardson.J. Fish Biol. 18,195–208.

Weigle, D. S. (1994). Appetite and the regulation of body composition.FASEB J. 8,302–310.

Wilson, P. N. and D. F. Osbourne (1960). Compensatory growth after under-nutrition inmammals and birds.Biol. Rev. Camb. Philos. Soc. 37, 324–363.

Yu, M., F. E. Robinson, M. T. Clandin and L.Bodnar (1990). Growth and body compositionof broiler chickens in response to different regimes of feed restriction.Poultry Sci. 69,2074–2081.

Received 19 January 2004 and accepted 29 March 2004

resource allocation, hyperphagia and compensatory growth

Documents