Control Mechanisms of Diel Vertical Migration: Theoretical Assumptions

J. theor. Biol. (2001) 210, 305}318doi:10.1006/jtbi.2001.2307, available online at http://www.idealibrary.com on

Control Mechanisms of Diel Vertical Migration: Theoretical Assumptions

BO-PING HAN,*-?A AND MILAN STRAS[ KRABA?

*Institute of Oceanography, Chinese Academy of Sciences, Qingdao 266071, People1s Republic of China,-Institute of Hydrobiology, Jinan ;niversity, Guangzhou 510632, Peoples1 Republic of China and

?Biomathematical ¸aboratory, Czech Academy of Science and Biological Faculty, ;niversity of SouthBohemia, Branis\ ovskaH 31, 370 05 C[ eskeH Bude\ jovice, Czech Republic

(Received on 17 October 2000, Accepted in revised form on 5 March 2001)

We explore control mechanisms underlying the vertical migration of zooplankton in the watercolumn under the predator-avoidance hypothesis. Two groups of assumptions in which theorganisms are assumed to migrate vertically in order to minimize realized or e!ectivepredation pressure (type-I) and to minimize changes in realized or e!ective predation pressure(type-II), respectively, are investigated. Realized predation pressure is de"ned as the product oflight intensity and relative predation abundance and the part of realized predation pressurethat really a!ects organisms is termed as e!ective predation pressure. Although both types ofassumptions can lead to the migration of zooplankton to avoid the mortality from predators,only the mechanisms based on type-II assumptions permit zooplankton to undergo a normaldiel vertical migration (morning descent and evening ascent). The assumption of minimizingchanges in realized predation pressure is based on consideration of DVM induction only bylight intensity and predators. The assumption of minimizing changes in e!ective predationpressure takes into account, apart from light and predators also the e!ects of food andtemperature. The latter assumption results in the same expression of migration velocity as theformer one when both food and temperature are constant over water depth. A signi"cantcharacteristic of the two type-II assumptions is that the relative change in light intensity playsa primary role in determining the migration velocity. The photoresponse is modi"ed by otherenvironmental variables: predation pressure, food and temperature. Both light and predationpressure are necessary for organisms to undertake DVM. We analyse the e!ect of each singlevariable. The modi"cation of the phototaxis of migratory organisms depends on the verticaldistribution of these variables.

( 2001 Academic Press

1. Introduction

Zooplankton Diel Vertical Migration (DVM)behavior is typical for both marine and fresh-water species. The time and range of DVM aredistinct for di!erent species and under di!erent

AAuthor to whom correspondence should be addressed.Institute of Hydrobiology, Jinan University, Guangzhou510632, People's Republic of China.E-mail: [email protected].

0022}5193/01/110305#14 $35.00/0

external conditions, including the presence ofother organisms. The most usual pattern is forthe zooplankton to ascend to water surface atsunset and to descend to deeper water duringsunrise. Other patterns observed are reversed mi-gration, migration produced by moonshine, andascension up to 2 hr before sunset. The phenom-enon of DVM has been known for more thana century. The "rst hypothesis for its explanationwas put forward by Weismann (1887): following

( 2000 Academic Press

306 B.-P. HAN AND M. STRAS[ KRABA

an optimal or preferred light intensity drives theanimals to swim up when the light intensity di-minishes and to swim down when the underwaterintensity rises. This hypothesis assumes a type ofphysiological mechanism of light response ofwhich little is known until now (Ringelberg,1993). The other light-driven hypotheses arebased on driving migration by changes of lightintensity, either changes of absolute values orrelative changes. Ringelberg (1980) and Haney(1993) studied relative changes, amongst otherpossibilities. Dodson (1990) summarized naturalobservations from a number of lakes with "shand adequate food supply, showing that the ex-tent of migration of Daphnia depends linearly onSecchi Disc transparency of the lakes. Therefore,light is one of the leading proximate forcesconsidered driving DVM.

Other hypotheses can be divided into threegroups: those based on proximate, immediatelyand directly acting forces or cues: light, food andtemperature; those based on ultimate, evolu-tionarily conditioned forces: predator avoidanceand metabolic advantage hypotheses; and thosebased on systems consequences of DVM. We willnot discuss here the third group. Kerfoot (1985)gives a comparative analysis of the adaptivevalue of the di!erent hypotheses. Among theevolutionary hypotheses, the strongly experi-mentally supported predator (or mortality)avoidance hypothesis originating from Kozhov(1963) dominates; this is also mentioned byMacLaren (1963) and investigated by Brooks &Dodson (1965). The energy (metabolic) advant-age hypothesis was "rst formulated by MacLaren(1963, 1969) for marine copepods and investi-gated by Kerfoot (1970). The leading assumptionwas based on observations of the negative tem-perature dependence of copepod size and egg-numbers, so that migration into colder layers wasassumed to result in energetic and consequentlydemographic advantage of migrating animals.However, the observed and experimental dataexert a doubt on this hypothesis both for marinecopepods (Ohman, 1990; Bollens & Frost, 1991)and freshwater cladocerans (Stich & Lampert,1984; Dawidowicz & Loose, 1992b).

Stich & Lampert (1981) found in Lake Con-stance that two species of Daphnia have di!erentstrategies of vertical migration. Although the

non-migrating D. galeata has a much higher birthrate than the migrating D. hyalina, the latter isnumerically dominant, as D. galeata su!ers highmortality near the surface. Therefore, the migra-tion}non-migration is a question of the demog-raphic budget: how far is the decreased growthrate connected with the cost of migration anddecrease of energy gain due to shorter time spentin the upper layers rich in food balanced by thegain due to reduced mortality.

The experimental support of the predatoravoidance hypothesis came "rst from observa-tions by Zaret & Su!ern (1976) who showed thatDVM is a prevention mechanism of animals toavoid predation risk from predators. Gliwicz(1986a) supported this hypothesis by demonstrat-ing a gradual increase in the amplitude of DVMconnected with an arti"cial augmentation ofpredators. The strongest evidence can be at-tributed to the inducible and reversible DVM inexperiments with "sh and chemical exudates*kairomones from "sh (Dodson, 1988; Tjossem,1990; Lampert, 1993; Loose et al., 1993; DeMees-ter et al., 1995). Loose & Dawidowicz (1994)showed in laboratory experiments that belowa critical level the amount of "sh present log-linearly determines the depth of migration. Thepredator-avoidance hypothesis answers the ques-tion why zooplankton follows a normal DVMbehavior. However, how the exact performance,timing and depth of migration are regulated re-mains little known. The experimental and "eldobservations show that a number of environ-mental variables control the detailed pattern ofDVM, but predation pressure, light intensity,food condition and temperature seem to bedominating. For understanding the physiologicalor behavioral mechanisms underlying DVM it isnecessary to assess the roles of environmentalvariables in controlling DVM (Van Gool& Ringelberg, 1995; Han & Stras\ kraba, 1998).

Gabriel & Thomas (1993) summarize theoret-ical models of DVM. In this paper, we consider ascrucial the explicit model formulation of anexpression relating the direction and velocity ofDVM to causal variables. In this respect, Gliwicz& Pijanowska (1988) de"ned the migration velo-city as a positive function of predation pressureand food gradient between epilimnion and hy-polimnion, without giving a speci"c form of this

CONTROL MECHANISMS OF DVM 307

function. Anderson & Nival (1991) constructeda model assuming the determination of DVM bylight and food. However, the models by the twoauthors lacked the inclusion of predation pres-sure as the most important variable.

In this contribution we aim at the theoreticalformulation of the ways in which the major vari-ables: "sh, light intensity, food and temperaturea!ect DVM. Evolutionary consequences andquantitative di!erences in the reactions of speciesand clones, as well as the dynamics of popula-tions due to predation and growth are onlymarginally considered in this paper, particularlyin Section 4. The focus is on control mechanismsof migration velocity.

2. Realized predation pressure and e4ectivepredation pressure

According to the predator-avoidance hypothe-sis, the surface layers are more nutritionally andthermally advantageous but dangerous. Deeperwater exerts severe disadvantage on the migra-tory organisms because of cold water andreduced food concentration, but is safer againstvisual predation. The retreat of organisms fromwater surface to deep water in daytime is causedby visual predation. During the night, even ifpredators are present, darkness makes surfacewater safe. Thus, both light and predators arenecessary for DVM.

To formulate the predation pressure by visualpredators more exactly, we de"ne the predationpressure without regard to light as relativepredation abundance (P), which is basicallydetermined by "sh abundance or exudate con-centration from "sh. The product of relativepredation abundance and light intensity refers torealized predation pressure (P

r), i.e. only when

light is present to enable the predator to recog-nize the prey, the relative predation abundancecan lead to the predation mortality of prey. Thefollowing relationships hold:

Pr"IP, (1)

I"I1(t)I

2(z), (2)

I2"e~az, (3)

where z is the water depth from water surface,a the attenuation coe$cient of water, I

1(t) the

time-dependent light intensity at water surface(Kirk, 1994). Realized predation pressure is time-and depth-dependent. For di!erent zooplanktonspecies, however, the predation pressure, which isactually sensed, may be di!erent. In order todistinguish from realized predation pressure, wede"ne the part of realized predation pressure,which really a!ects migratory organisms, ase!ective predation pressure (P

e). The e!ective

predation pressure depends on the activity ofpredators and the sensitivity of prey to the pred-ators. The sensitivity of prey to the predators isusually associated with their physiological state,which is basically determined by food conditionand temperature in#uencing growth. The empiri-cal evidence indicates that zooplankton are morelikely to migrate when food is abundant (Huntley& Brooks, 1982; Dagg, 1985; Johnsen & Jakob-sen, 1987; Pijanowska & Dawidowicz, 1987).When food is scarce, migratory organisms followthe strategy of &&better fed than dead''. In otherwords, when food is limited, zooplankton tend topay more attention to food than to "sh pred-ation. The negative e!ect of low temperature isusually evaluated by means of temperature lim-itation of organism growth. In a systematic ex-periment, Van Gool & Ringelberg (1997) foundthat predators sensitized the individual reactionof D. galeata]hyalina to change in light inten-sity. The experiments by Calaban & Makarewicz(1982) suggested that the avoidance of cold tem-perature could limit the intensity of DVM. Basedon this consideration, we express the e!ectivepredation pressure by organisms in a water col-umn as

Pe"IP

fkf#f

¹

¹max

, (4)

where ¹max

is the maximum temperature in thewater column, k

fthe half saturation constant of

zooplankton for food, and f the food concentra-tion. Food uptake is assumed to follow satura-tion, Monod equation, and the temperaturedependence is for simplicity assumed to linearlyincrease up to the maximum temperature in thewater column. According to eqn (4), e!ective


predation pressure is mainly determined by real-ized predation pressure but modi"ed by fooddistribution and temperature distribution in thewater column. Usually, surface water has thermaland nutritional advantages over deep water. Or-ganisms like to stay in surface water but feela higher predation risk than in deep water. Forpractical use of e!ective predation pressure, weneed to de"ne quantitatively relative predationabundance. In general, relative predation abund-ance mainly depends on "sh abundance and theirdistribution. Loose (1993) investigated the day-time average depth of zooplankton distribution(we call later day depth) under di!erent "sh dens-ities in a water column and found that when onlyone "sh is present, the e!ect of "sh on DVM isdetectable and the daytime average depth in-creases with the "sh density. At extreme levels of"sh density, a saturation of the e!ect on meanday depth was observed. Here, we present rela-tive predation abundance as

P(z)"N

p(z)

Nk#N

p(z)

, (5)

where Np(z) is the "sh density at a depth of z and

Nk

is a constant density, which may vary fordi!erent species. The meaning of the equation isthat the predation pressure depends on the den-sity of "sh in a nonlinear, saturation fashion andthat the relative predation abundance P(z) rangesbetween 0 when "sh are absent and 1 when "shdensity is very high. Low N

kwill indicate a "sh

intensively zooplanktivorous, with high values ofP(z) reached already at low "sh densities.

3. Control mechanisms of vertical migration

The pattern of DVM can be expressed bya "rst-order advection}di!usion equation. Al-though di!usive processes produced by turbu-lence or other mixing mechanisms can a!ect thepro"le of DVM, vertical migration is not a pass-ive process but is achieved by active swimming.In a systematic observation of swimming behav-ior of Daphnia hyalina by Dodson et al. (1997),the Daphnia rose by fast upward swimming, notby upward moderate swimming (with hop) in thepresence of "sh. The moderate swimming wasnearly horizontal and fast swimming was nearly

vertical. Thus, ignoring the di!usive and otherslow mixing e!ects is reasonable (Han & Stras\ -kraba, 1998). In order to reveal the pattern ofDVM, therefore, we need to express quantitat-ively the migration velocity of zooplankton. Inthe following, based on the predation*avoid-ance hypothesis we investigate two types of as-sumptions to formulate the migration velocity.Type-I assumptions are associated with the min-imum realized predation pressure or the min-imum e!ective predation pressure, type-IIassumptions concern the minimum change ofrealized predation pressure or minimum changeof e!ective predation pressure of zooplankton.

3.1. TYPE-I: MINIMUM OF REALIZED

PREDATION PRESSURE

A possible mechanism of DVM is that organ-isms move to minimize realized predation pres-sure. We assume that organisms, which stay ata depth z, will move to z#Dz at time t#Dt. Themigration velocity at time t can be determined byminimizing the realized predation pressure, i.e.

Min Pr"I(z#Dz, t#Dt)P(z#Dz),

Dz"<Dt, (6)

<)<max

,

where <max

is the maximum swimming velocity.We have the velocity of vertical migration in theform

<"!

L(Pr)/Lz

Dt L2(Pr)/Lz2

. (7)

A detailed formulation is given in the appendix.Using eqns (2) and (3), the velocity of migrationcan be rewritten as

<"!

(LP/Lz!aP)Dt (L2P/Lz2#Pa2!2a LP/Lz)

. (8)

The resulting equation indicates that the migra-tion velocity depends on the distribution of rela-tive predation abundance, i.e. on the "sh densityat di!erent depths and on the optical property ofwater, i.e. on the attenuation coe$cient (a). If the


attenuation coe$cient increases, water becomesless transparent and less light penetrates to lowerdepths, which limits the recognition of zooplank-ton by "sh. As the migration velocity is, accord-ing to eqn (8), independent of light intensity thatchanges daily, the migration of zooplankton in-duced by the expression of velocity is not a nor-mal diel vertical migration, because there is nodaily change in the migration direction.

3.2. TYPE-I: MINIMUM OF EFFECTIVE

PREDATION PRESSURE

Similar to the assumption in Section 3.1, an-other possible mechanism underlying migrationis that organisms move in order to minimizee!ective predation pressure. Organisms migrateat a speed determined by minimizing e!ectivepredation pressure, i.e.

Min Pe"P

e(z#Dz, t#Dt), (9)

where Peis given by eqn (4). The migration velo-

city can be formulated in this case as

<"!

LPe/Lz

L2Pe/Lz2

. (10)

By the use of light intensity function expressed byeqn (2), the migration velocity can be rewritten as

<"!

(L/Lz)(I2P ( f/(k

f#f ))¹)

Dt (L2/Lz2) (I2P ( f/(k

f#f ))¹)

(11)

Obviously, this velocity is time-independent, likein the previous assumption. The migration at thespeed expressed by eqn (11) is not a normal dielvertical migration. Thus, both type-I assump-tions investigated can induce organisms to mi-grate to safe water zone, but do not producea daily migration with zooplankton descendingto deep water in the morning and ascending towater surface in the evening.

3.3. TYPE-II: MINIMIZING CHANGES IN REALIZED

PREDATION PRESSURE

Migratory organisms stay in the safe zone be-fore realized predation pressure appears. Anotherpossibility is that there exists a certain optimum

background level of realized predation pressure,which the migratory organisms "nd conducivefor survival. Their reaction to any changes inmigratory stimuli will be to move to restore therealized predation pressure to this optimum level.Mathematically, this means that

d(Pr)

dt"

L(Pr)

dt#

L(Pr)

Lz<"0. (12)

We obtain an expression of swimming velocity inthe following form:

<"!

(1/I) LI/Lt(1/I)LI/Lz#(1/P) LP/Lz

. (13)

By the use of eqns (2) and (3), we have a simpli"edexpression of migration velocity,

<"1aI

LILt

11!(1/aP) LP/Lz

. (14)

In this case, the speed of vertical migrationdepends on the relative change in light intensity,the relative gradient of relative predation abund-ance and the attenuation coe$cient of water.When (1!(1/ap) LP/Lz)'0, organisms have adownward velocity in the morning and an up-ward velocity in the afternoon. This assumptionresults in organisms following normal diel verti-cal migration (see Fig. 1).

3.4. TYPE-II: MINIMIZING CHANGES IN EFFECTIVE

PREDATION PRESSURE

In general, migrating organisms avoid watersurface in daytime because of the risk from pred-ators and return to water surface due to surfaceadvantage in temperature and nutrition. There-fore, the timing and pattern of DVM is "nallydetermined by the trade-o! between surface ad-vantage and surface predation pressure. Now, weanalyse the possible pattern of vertical migrationby assuming that organisms move up or down tominimizing changes in e!ective predation pres-sure. The di!erence compared to the type-Iassumptions is that organisms do not follow thelayer in which the predation pressure isminimum, but the layer in which the predationpressure is optimal. Biologically, this is a more

FIG. 1. The graphical explanation of the assumption ofminimum change of realized predation pressure. Full lineindicates the distribution of realized predation pressure attime t, dotted line indicates that at the next time t#Dt.Arrows show the migration directions: (a) in the morningand (b) in the evening.


plausible assumption, as it does not require thatorganisms recognize in each instant of time thedanger of being preyed upon, but are driven bythe relative changes in light intensity which areeasily detectable. Expressed mathematically:

dPe

dt"

LPe

dt#

LPe

Lz<"0. (15)

We have the migration velocity in the followingform:

<"!

LPe/Lt

LPe/Lz

. (16)

Realized predation pressure is always higher insurface water than in deep water. As both foodconcentration and water temperature usuallyreach their maximum values at the surface, thee!ective predation pressure decreases with in-creasing depth, i.e. LP

e/Lt)0. Hence, it is the

sign of LPe/Lz (rate of change of e!ective pred-

ation pressure) which determines the direction ofmigration. In the model, the only time-dependentfactor which in#uences the e!ective predationpressure is the light intensity. In the morning, aslight intensity increases, the velocity determinedby eqn (16) is positive. This velocity indicates thatorganisms move downward. Conversely, in theafternoon, the organisms have negative velocityand move up. Therefore, the assumption of min-imizing changes in e!ective predation pressureresults in normal diel migration of organisms. Inorder to show clearly the role of di!erent factorsin determining migration velocity; we furtherformulate the migration velocity by substitutingeqns (2) and (3) into eqn (16) as follows:

<"

!

(1/I) LI/Lt(( f/(kf#f ))P¹)

P(L/Lz) (( f/(kf#f ))¹)#(LP/Lz!aP)( f/(k

f#f ))¹

.

(17)

4. Discussion

4.1. CHARACTERISTICS OF TYPE-II CONTROL

MECHANISMS OF DVM

We formulated two types of control mecha-nism of vertical migration and gave the corre-sponding expressions for migration velocity, atwhich organisms undertake DVM in a watercolumn. Although the two types of assumptionsallow organisms to move up or down to avoidpredation, only when following type-II assump-tions, organisms undertake normal diel verticalmigration. We reject type-I assumptions and ac-cept type-II assumptions as possible mechanismsby which organisms follow diel vertical migra-tion. A systematic simulation of DVM based onthe type-II assumptions was implemented andthe detailed results are given in Han & Stras\ -kraba (1998). The simulations proved that thetype-II assumption could be applied to the


observations available. In this discussion, wecon"ne our interest to theoretical analysis of thetype-II assumptions.

In the de"nition of e!ective predation pressure,we incorporated the e!ects of food and temper-ature. Therefore, the assumption of minimizingchanges in e!ective predation pressure is moregeneral than the assumption of minimizing cha-nges in realized predation pressure. When settingthe food and temperature in the water column tobe constant, the expression of velocity eqn (17)can be simpli"ed to that based on the assumptionof minimum rate of change in realized predationpressure.

A signi"cant characteristic of type-II assump-tions is that migration velocity is directly propor-tional to the relative change in light intensity.When relative predation abundance is constanttoo, eqn (14) can be simpli"ed to

<"(1/aI) LI/Lt. (18)

This expression of migration velocity is thesame as that at which the isolume moves daily inthe water column.

4.2. THE PHOTOTAXIS TO THE RELATIVE CHANGE

IN LIGHT INTENSITY

In recent years, little attention was paid to thephysiological mechanisms of DVM. As argued byRingelberg (1993), because normal diel migrationis always connected with light cycles, organismsneed a photoresponse mechanism to lightchange. Until now, it seems that responses to therelative change in light intensity may be the basisof this physiological mechanism. The observa-tions in the "eld (Haney & Hall, 1975; Ringelberget al., 1991; Ringelberg, 1993) and in the laborat-ory (Ringelberg, 1964; Dann & Ringelberg, 1969;Haney et al., 1990) supported this assumption.The response of organisms to the relative changein light intensity is usually modi"ed by otherenvironmental factors such as predation pres-sure, food and temperature. In experiments byVan Gool & Ringelberg (1997, 1998), the migra-tion velocity (called displacement velocity by theauthors) was linearly related to the relativechange in light intensity. At di!erent predatorlevels expressed by "sh kairomone concentration

and di!erent food abundance, the slope of linearrelationship increased with predation level andfood abundance. The observations support type-II assumptions. In both the type-II assumptionsdiscussed above, the relative change in lightintensity plays a primary role in determining thedirection and the velocity of vertical migration.Richards et al. (1996) simulated DVM by the useof three light-induced assumptions, namely, thepreferred light intensity assumption, minimumchange of light intensity assumption, and thestimulus-velocity assumption. In the last twoassumptions, the migration velocity is a linearfunction of the relative change in light intensity.Although the model by Richards et al. (1996)does not include predation pressure, which isa pre-requisite for DVM, it can show that theresponse to the relative change in light intensitycan cause DVM behavior. When predation pres-sure is constant over all depths, our "rst type-IIassumption, i.e. minimum change of realizedpredation pressure o!ers the same expression ofmigration velocity as that based on minimumchange of light intensity. Our theoretical analysis,modeling based on the type-II assumption (Han& Stras[ kraba, 1998) and the experimental evid-ence (Ringelberg, 1964; Haney et al., 1990;Ringelberg et al., 1991) support the claim that thephotoreaction of animals to the relative changein light intensity plays a central role in the con-trol of DVM. Why do organisms not respond tolight intensity or to the absolute change rate oflight intensity but respond to the relative changein light intensity? In both the type-II assumptionsproposed above, the response of animals to e!ec-tive predation pressure from visual predators isrelated to realized predation pressure, which isde"ned as the product of light intensity and pred-ation pressure. Predators hunt prey by sight.Thus, realized predator pressure de"nes the por-tion of relative predation abundance, by whichzooplankton is really preyed on. To escape frompredators zooplankton have to recognize andevaluate the existence of realized predationpressure by some ways which may have evolvedgenetically and physiologically. Therefore, theor-etically, two response mechanisms have to actsimultaneously: one is to detect the existence ofpredators and the other to test the light condi-tions. If these two response mechanisms really

FIG. 2. The e!ect of potential predation pressure on themigration velocity under three di!erent attenuation coe$-cients: a

1'a

2'a

3.


exist, they should have co-evolved. For avoid-ance of predators, just the photoresponse to lightintensity is not su$cient, because organisms needto know in which direction and at what speed tomove. The two problems can be solved whenzooplankton relate their swimming speed to therelative change in light intensity. In this state-ment, light can be viewed as both the ultimateand proximate cause. Nevertheless, direct controlof swimming rate by the relative change in lightintensity allows zooplankton to adjust to lightconditions, including irregular light changes dueto weather change.

4.3. THE EFFECTS OF PREDATION PRESSURE,

TEMPERATURE, FOOD AND THE ATTENUATION

COEFFICIENT OF WATER ON PATTERN OF DVM

The reaction of migratory organisms to lightintensity plays a primary role in DVM, but ismodi"ed by other factors such as predation, foodand temperature. This conclusion has been con-"rmed by the experiments (Ringelberg, 1993;Haney, 1993). We "rst demonstrate with themodel the e!ect of predation pressure. Whenfood and temperature are constant in the watercolumn, the assumption of minimum rate ofchange of e!ective predation pressure results inthe same value of migration velocity as thatbased on the assumption of minimum change ofrealized predation pressure. We discuss the e!ectof predation pressure in the form of the assump-tion of minimum change of realized predationpressure.

4.3.1. Predator abundance

Based on the assumption of minimum changerate of realized predation pressure, the e!ect ofpredation pressure on the photoresponse to therelative change in light intensity (E

P) can be ex-

pressed as

Ep"

11!(1/aP) LP/Lz

. (19)

This result shows that the e!ect of predation pres-sure on the photoresponse to the relative changein light intensity is related to the relative gradientof predation pressure. If the relative predationabundance is constant over the water column, the

migration velocity is independent of the magni-tude of relative predation abundance, as the sec-ond, gradient term in the denominator of eqn (19)is zero.

A common distribution of relative predationabundance in natural aquatic environments isthat predation pressure decreases with waterdepth. As a special case, we assume that pred-ation pressure has a form of linearly decreasingfunction of water depth, whose slope (gradient) isa negative constant, denoted as !S

p(S

p*0).

The e!ect of relative predation abundance can berewritten as

Ep"

11#S

p/aP

. (20)

Thus, the e!ect of predation pressure not onlydepends on relative predation abundance butalso on the optical property of water. The e!ectdecreases with the attenuation coe$cient.A number of experiments and observations indi-cate that the intensity of migration is closelyrelated to predator abundance (Bollens &Frost, 1989a, b; Neill, 1990; Loose, 1993; Hays,1994). In the observation by Dodson (1990), themigration intensity expressed as the mean migra-tion depth of population linearly increases withwater clarity. Figure 2 shows the dependence ofthe e!ect of relative predation abundance on

FIG. 3. The e!ect of food concentration on the migrationvelocity under three di!erent half saturation constants:k3'k

2'k

1.


migration velocity with di!erent attenuation co-e$cients, i.e. for the same density of "sh, the e!ectof predation pressure is higher in the water withlow attenuation coe$cient than in those withhigh attenuation coe$cient.

It is notable that the predation pressuremodi"es not only the value of the migrationvelocity but also the direction of migration.When predation pressure decreases linearly withwater depth, i.e. S

p'0, E

pis positive and does

not change the direction of migration. If Sp(0,

i.e. the deep water is more dangerous than surfacewater. When (1#S

P/aP)(0, the predation pres-

sure forces organisms to change their swimmingdirection.

4.3.2. Food concentration

To demonstrate the e!ect of food by using theassumption of minimum change of e!ective pred-ation pressure, we set relative predation abund-ance and temperature as constant for all waterdepths. The migration velocity is

<"!(1/I) (LI/Lt)f

(kf/(k

f#f )) Lf/Lz!af

. (21)

If food is fairly scarce, i.e. food level tends tozero, the velocity of migration is almost equal tozero. Therefore, scare food scarcity can inhibitDVM. This result was con"rmed by the experi-ments by Dagg (1985), Johnsen & Jakobsen(1987) and Flik & Ringelberg (1993). Huntley& Brooks (1982) found that the migration ampli-tude for Calanus paci,cus is correlated with foodabundance, and hungry animals stay near thesurface both day and night. When food is abun-dant in deep water and is therefore su$cient tomeet the daily physiological need, Pijanowska& Dawidowicz (1987) found that organisms stayin deep water in daytime to avoid predation riskand stay in deep water or just distribute random-ly at night. This result cannot be understoodfrom the assumption of minimum change of e!ec-tive predation pressure. For example, when foodis abundant but constant, migration velocityfrom eqn (21) is proportional to relative change inlight intensity and indicates a normal DVMinstead of a random distribution of population.Most often, there is abundant food in surface

water and food concentration decreases withwater depth. As an example, we give this distribu-tion as a linearly decreasing function of waterdepths, which has a constant slope of !S

f(S

f*0). The e!ect of food can then be for-

mulated as

Ef"

fSf

kf/(k

f#f )#af

(22)

Half saturation constant of food (kf) is a para-

meter re#ecting the sensitivity of animals to foodabundance. We expect that k

fis di!erent for

di!erent species. Figure 3 shows the e!ect of foodconcentration on the migration velocity underdi!erent half saturation constants. For the samefood abundance, the e!ect decreases with theincrease in the half saturation constant. In otherwords, species adapted to low food concentrationhave a higher sensitivity to predation pressurethan those that have a high food saturation con-stant. This is because if they have high foodrequirements, they prefer to satisfy them in spiteof predation danger. As food concentrationincreases, E

fapproaches 1/a.

4.3.3. ¹emperature

If both the food and predation pressure are setdistributed evenly, we can employ the assump-tion of minimum change of e!ective predation

FIG. 4. The dependence of the migratory velocity onrelative change in light intensity under constant temperatureand food concentration in water column: a

1'a

2'a

3.


pressure to analyse the e!ect of temperature.From eqn (17), we have

<"!

1I

LILt

¹

L¹/Lz!a¹. (23)

Therefore, the migration velocity is dependent ontemperature distribution in the water column.This statement is consistent with the experi-mental data of Calaban & Makarewicz (1982). Indimictic waterbodies, typical distributions areuniform in spring and summer strati"cation, i.e.L¹/Lz)0. Thus, the distribution of temperatureonly a!ects the value of migration velocity butnot its migration direction. During summer strat-i"cation, temperature is more or less evenly dis-tributed in the epilimnion and hypolimnion anddecreases with water depth in the metalimnion.The e!ect of temperature on the migration velo-city is di!erent in the three temperature layers.

Temperature has a signi"cant in#uence on thephysiological rates of zooplankton. Experimentalresults demonstrate that low temperature in deepwater leads to disadvantage for migrating organ-isms (Ocrutt & Porter, 1983; Stich & Lampert,1984; Loose & Dawidowicz, 1994). It is predict-able that temperature distribution can in#uencethe amplitude of migration. Calaban & Makare-wicz (1982) found that the magnitude of verticalmigration might be reduced by an avoidancereaction to cold temperatures commonly ob-served in temperate lakes during summer strati"-cation. Gerristen (1982) observed the durationand frequency of upward swimming. When tem-perature rises, animals swim up more frequentlyin direct proportion to the rate of change intemperature. As shown in eqn (23), an increasingtemperature results in an increase in migra-tion velocity. Therefore, low temperature in deepwater can limit the amplitude of migration.During temperature strati"cation, thermoclinemay set a lower boundary of DVM (Calaban& Makarewicz, 1982; Dawidowicz & Loose,1992a; Haney, 1993; Loose, 1993). Compared tothe e!ect of temperature in the metalimnion, thelower boundary set by thermocline may be more#exible (Dodson, 1990). This view can be under-stood when there is a minimum tolerable temper-ature of organism growth. However, it should be

argued that the e!ect of temperature on DVM iscomplicated, for the e!ective predation pressureby migratory organisms depends not only onmigratory animals themselves but on the pred-ators whose feeding and other actions are alsoin#uenced by temperature. When temperature islow, the predation activity greatly decreases andsimilarly the amount of activity of chemical mat-ter exuding from "sh decreases. At this time, thepredation pressure de"ned by "sh abundancemay be overestimated. Unfortunately, all system-atic experiments on the e!ect of temperature onlypay attention to migratory animals and not topredators (Dawidowicz & Loose, 1992a).

4.3.4. ¹he attenuation coe.cient of water

The attenuation coe$cient of water directlyrelates with the average daytime depth of DVM(Gliwicz, 1986b; Gliwicz & Pijanowska, 1988;Dodson, 1990). Dodson (1990) used the data forall species and ages of Daphnia from literature tocalculate the amplitude of migration. He foundthat the correlation for Secchi Disk depth and themigration intensity is very signi"cant. This resultcan be interpreted on the basis of the dependenceof migration velocity on the attenuation coe$-cient of light [see eqn (18)]. Increased attenuationcoe$cients of light intensity lead to a decrease inmigration velocity (Fig. 4). In shallow lakes,where "shes distribute over all water depths, the


sediment becomes more a safe position in thewater column. As the predators hunt by sight,they tend to inhabit the layers close to the sur-face, i.e. the distribution of predators would de-pend on the attenuation coe$cient of water. Indeep water bodies, the distribution of "shesreaches usually only up to some critical depth,below which predation pressure can be ignored(Stras\ kraba, 1974). In any sense, the real safe zonemust be related to light intensity in water col-umns. Visual predators also probably need someminimum light intensity to recognize prey. Fordi!erent species, the minimum light intensity mayvary. Below the minimum light intensity, thepredation pressure placed on animals may de-crease abruptly. Thus, the minimum light inten-sity probably de"nes a critical day depth, whichis directly related to the attenuation coe$cient ofwater. The critical light level seems to be a prefer-red light intensity (Forward et al., 1984).

Light intensity is the most important factorin#uencing DVM, its distribution under water isdependent on the attenuation coe$cient of water.The e!ect of the other environmental factors isclosely related with the attenuation coe$cient ofwater. As an example, E

P, the e!ect of predation

pressure on the photoresponse to relative changein light intensity, decreases with increasing at-tenuation coe$cient of water. In other words,under the same light and predation conditions,zooplankton will migrate faster in clear water(a low attenuation coe$cient of water) than inless clear water.

4.4. THE TIMING AND PATTERN OF DVM

DVM is always related to the natural lightcycle, the exact timing in relation to sunrise andsunset may vary. In the light of the stimu-lus}velocity hypothesis (Dann & Ringelberg,1969), the stimulus for upward swimming and theintensity of the nocturnal migration is de"ned bythe relative change in light intensity. The begin-ning of rapid upward evening ascent correspondsto the surpassing of a threshold level of the rela-tive change in light intensity, called rheobase.Both in the laboratory and in the "eld, this thre-shold for Daphnia closely approximates to0.0017 s~1 (Ringelberg, 1964; Haney & Hall,1975; Ringelberg, 1991). The experiment of

Haney et al. (1990) with Chaoborus larvaeshowed a similar threshold. In both the type-IIassumptions, only when the relative change inlight intensity is zero, organisms stop moving;therefore, a threshold is required for organisms toinitiate migration. In other words, the onset ofDVM is controlled by a "xed physiologicalmechanism which is probably "xed for each spe-cies, but the inter-speci"c variation cannot beruled out. The timing and pattern of DVM aredetermined by the balance condition betweentwo contradicting factors: surface thermal andnutritional advantage and predation pressure.This tradeo! is implicit in the assumption ofminimum change of e!ective predation pressure.

5. Concluding remarks

The leading idea is that DVM results from anultimate cause, maximization of survival of zo-oplankton population under the force of environ-mental pressure of both biotic and abiotic nature,and with the physical constraints of organismdue to their evolutionary history. The biotic con-trol is both top}down and bottom}up, by "shpredation and by food sources. However, indi-vidual zooplankton organisms have no or limitedmeans to follow "sh and food distribution in thewaterbody. Some evolutionary simple cues havedeveloped which are followed by zooplankton inaddition to feeling directly the presence of "shand adequacy of food. Because the distribution offood and predators in di!erent water bodies israther variable, great plasticity in zooplanktonreactions is possible. The cues are mostly of phys-ical nature, light and temperature being amongthe most important. Due to the cyclic nature ofconditions in waterbodies during the day, thelight changes are the most important. The use ofrelative changes of light intensity as a proximatedriving force seems to solve the di!erences ofdi!erent waterbodies, di!erent light conditionchanges in one water body due to changes indissolved matter, abiotic and biotic particles anddi!erences of solar radiation intensity duringa year at higher latitudes. This abiotic cue isrelated to the evolutionary, ultimate cause:avoiding predation on the one hand and satisfy-ing food requirement on the other. These bioticand abiotic factors are combined together in our


model by assuming minimizing predation pres-sure faced by zooplankton (i.e. maximizing sur-vival of the migratory animals). In this theoreticalexamination aimed at formalizing the complexinterplay of variables controlling zooplanktonDVM, we explore control mechanisms underly-ing the vertical migration of zooplankton in thewater column under the predator-avoidance hy-pothesis. Two groups of assumptions in whichthe organisms are assumed to migrate verticallyby following the minimum realized or e!ectivepredation pressure (type-I) and the minimizingchanges in realized or e!ective predation pres-sure (type-II), respectively, are investigated; onlythe mechanisms based on type-II assumptionspermit zooplankton to undergo a normal dielvertical migration (morning descent and eveningascent). The assumption of minimizing changesin e!ective predation pressure takes into account,apart from light and predators also the e!ects offood and temperature. A signi"cant characteristicof the two type-II assumptions is that the relativechange in light intensity plays a primary role indetermining the migration velocity. The photo-response is modi"ed by other environmental vari-ables: predation pressure, food and temperature.

What we have to ask is how far the presentmodel does cover the whole story. The answer isnegative, as there are a number of questions re-maining. First, have all decisive variables beenincluded? Depending on what we call decisive,this task was ful"lled by the model. Second, dif-ferent species have di!erent rates, mechanisms,and other characteristics that need to be quanti-"ed for the species in question. The model showsthe parameters which are to be determined. Itwas pointed out, that the migration is a pay-o!between population gains and losses (e.g., Lam-pert, 1993). It is not without cost for the organismto migrate, to have access to higher food concen-tration only for part of the day and to change itshabits. The population characteristics of di!erentspecies like growth and predation rates underdi!erent circumstances are important for popula-tion survival. Therefore, the consequences ofmigration or non-migration, of migration toa certain depth and at certain times and predatordensities will be decisive for the speci"c realiz-ation of the model. The models of Iwasa (1982)and Gabriel & Thomas (1993) show one way to

treat the question of strategic decisions of thepopulation. We can see that the control by thesurrounding is decisive not only for the indi-vidual, but also for the population as a whole.Gliwicz (1986a) demonstrated by his observa-tions in Tatra Mountain lakes the initiation ofzooplankton migration. Although doubts wereraised if the reason was not successively increas-ing predation pressure of "sh, this observationsas well as a number of experiments such as Bol-lens & Frost (1991) suggest strongly that DVMcan be induced in an evolutionarily insigni"canttime, and can be therefore ranged as phenotypi-cal adaptation. It is probable that DVM canagain cease if there is no predation pressure(although we know no direct observation).Therefore, migration is from its origin through itsdetailed performance to its end a populationcontrol problem, and can be therefore withadvantage treated by methods of control theory.

In nature we need many more observations onsimultaneous behavior of "sh, zooplankton andphytoplankton, concentrating on situationswhere there is probability of measuring the rel-evant variables and processes indicated by themodel. Identifying the systems consequences ofDVM will be the goal far ahead.

The support from the Granting Agency of the CzechRepublic (Project No. 204/94/1672) and of the Minis-try of Education, Youth and Sports of the CzechRepublic (Project No. VS96086) to M. Stras\ kraba,from Chinese Academy of Sciences (The 100-talentproject, No. 131), Chinese Natural Science Founda-tion (Project No. 39900022) and the Yingdong Heeducation foundation (Project No. 71020) to B.-P.HAN, are acknowledged.

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APPENDIX

Minimum Pr"I(z#<Dt, t#Dt)P(z#<Dt),

(A.1)

dPr

d<"

LI(z#<Dt, t#Dt)Lz

DtP(z#<Dt)

#

LP(z#<Dt)Lz

DtI(z#<Dt, t#Dt)"0.

(A.2)

Using "rst-order Taylor series, we get:

I(z#Dz, t#Dt)"I#LILz<Dt#

LILt

Dt, (A.3)

LI(z#Dz, t#Dt)Lz

"

LILz

#

L2ILz2<Dt#

L2ILzLt

Dt,

(A.4)

P(z#Dz)"P#

LPLz<Dt, (A.5)

LP(z#Dz)Lz

"

LPLz

#

L2PLz2<Dt. (A.6)

Substituting eqns (A.3)}(A.6) into eqn (A.2) andignoring the second-order errors yields

<"!

L(Pr)/Lz

Dt L2(Pr)/Lz2

(A.7)

Using the light function de"ned by eqns (A.2)and (A.3) in context, we obtain

<"!

(dP/dz!aP)Dt(d2P/dz2#Pa2!2adP/dz)

. (A.8)

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Control Mechanisms of Diel Vertical Migration: Theoretical Assumptions