WWolff C.-J.Soeder FR.Drepper (Eds.)

Ecodynamics Contributions to Theoretical Ecology Proceedings of an International Workshop, Held at the Nuclear Research Centre, Jülich, Fed. Rep. of Germany, 19-20 October 1987

With 116 Figures

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Contents

Par t I Evolution

Evolut ion: W h y the Whole is Greater T h a n the Sum of the Parts B y P . M . Al len ( W i t h 11 Figures) 2

The Evolut ion of Spatially Inhomogeneous Populations B y J . M . McGlade 31

Intraspecific Competi t ion and Evolution B y F . Bugge Christiansen ( W i t h 4 Figures) 40

Fundamental Components in Ecology and Evolution: Hierarchy, Concepts and Descriptions B y S . H . Cousins ( W i t h 3 Figures) 60

Simulated Evolution of Primit ive Organisms B y G . Duchateau, J . A . Meyer, P. Tarroux, P. Vincens, and G . Weisbuch ( W i t h 4 Figures) 69

Dynamics of Developmentally Constrained Populations B y M . Kerszberg ( W i t h 3 Figures) 77

Differential Evolut ion of Pesticide Resistance in Predators and Prey B y J . D . van der Laan and P. Hogeweg ( W i t h 4 Figures) 86

Collective Intelligence in Evolving Systems B y H . - P . Schwefel \ 95

Par t II Marine and Lake Ecodynamics

Biological-Physical Interactions i n the Sea: Marine Ecodynamics B y A . Okubo ( W i t h 1 Figure) .* 102

Understanding the Balt ic Sea: Systems Ecology in Theory and Practice B y F . Wulff ( W i t h 7 Figures) 113

Ult imate Causes of Vertical Migrat ion in Zooplankton: A n Evaluation by Evolutionary Game Theory B y W . Gabriel and B . Thomas (Wi th 2 Figures) 127

X

Trophy-Balanced Turnover Velocities in a Two-Species System of Competing Daphnids: A Test of the "Energy Residence Time" Concept B y W . Geller (Wi th 1 Figure) 135

Ecodynamic Changes i n Suburban Lakes in Ber l in ( F R G ) Düring the Restoration Process After Phosphate Removal B y G . K l e i n ( W i t h 6 Figures) 138

A Comparison of F ish Size-Composition i n the N o r t h Sea and on Georges Bank B y J . G . Pope, T . K . Stokes, S .A . Murawski , and S.I. Idoine ( W i t h 3 Figures) 146

Generic Models of Continental Shelf Ecosystems B y W . Silvert ( W i t h 1 Figure) 153

Par t III Ecosystems Analysis

O n Quantifying the Effects of Formal and F ina l Causes in Ecosystem Development B y R . E . Ulanowicz and A . J . Goldman ( W i t h 3 Figures) 164

Avoidance of Ecological Risk i n Opt imal Exploi tat ion of Biological Resources B y C . Wissel ( W i t h 7 Figures) 181

Deterministic Modell ing of the Combined Act ion of Light and Heat Stress on Microbia l Growth B y E . Fiolitakis, J . U . Grobbelaar, C . J . Soeder, and E . Hegewald ( W i t h 2 Figures) 202

Different Responses in Ecosystems to Environmental Stress B y E . Gutierrez, R. Guardans, and M . A . Canela ( W i t h 2 Figures) . . 211

A n Example of Modell ing in Ecotoxicology B y K . Mathes, V . M . Schulz-Berendt, and G . Weidemann ( W i t h 4 Figures) 220

Spatial Simulation of Population Dynamics i n the Evaluation of Foraging Theory in Complex Ecological Systems B y F . Perez-Trejo ( W i t h 2 Figures) . . . \ 229

Models for the Management of Renewable Resources Embedded i n Complex Marine Systems B y P . A . Shelton ( W i t h 4 Figures) 235

Par t I V Forest Ecosystems

Ecodynamics as a Function of Ecosystem Structure: Forest Dieback as an Example B y H . Bossel ( W i t h 14 Figures) 244

XI

Ecosystem Theory Required to Identify Future Forest Responses to Changing C 0 2 and Climate B y A . M . Solomon (With 4 Figures) 258

Par t V Modelling Approaches

The Role of Theoretical Research in the Design of Programmes for the Control of Infectious Disease Agents B y G . F . Medley (With 2 Figures) 276

Microinteractive Predator-Prey Simulations B y W . F . Wolff (With 5 Figures) 285

Biological Attractors, Transients, and Evolution B y S.P. Blythe and T . K . Stokes ( W i t h 9 Figures) 309

Unstable Determinism in the Information Production Profile of an Epidemiological Time Series B y F . Drepper (With 4 Figures) 319

Random Elements in a Population Model Based on Individual Development B y N . van der Hoeven ( W i t h 4 Figures) 333

Measures of Spatio-Temporal Irregularity B y A . V . Holden, J . Brindley, and R . M . Everson 343

Index of Contributors 351

Ultimate Causes of Vertical Migration in Zooplankton: A n Evaluation by Evolutionary Game Theory

W. G a b r i e l a n d B . T h o m a s 1

Max-Planck-Institute for Limnology, Department of Physiological Ecology, Postfach 165, D-2320 Plön, Fed.Rep.of Germany and I n s t i t u t für Entwicklungsphysiologie der Universität Köln,

Gyrhofstr. 17, D-5000 Köln, Fed.Rep.of Germany

INTRODUCTION

D i e l v e r t i c a l m i g r a t i o n i s a widespread but not w e l l understood be-haviour i n freshwater and marine Zooplankton (Pearre 1979a and 1979b). Many taxa avoid the warm and food r i e h upper waters during day, but at dusk they swim long d i s t a n c e s upward and stay i n the upper water l a y e r s during n i g h t . Around s u n r i s e they descend again and stay i n c o l d e r and food scarce waters during day. At l e a s t cladocerans can not compensate f o r the food shortage during day by i n c r e a s i n g feeding r a t e s and by s t o r i n g food during night (Lampert and T a y l o r 1985). The e x t r a swim-ming f o r m i g r a t i o n seems not t o be very c o s t l y i n terms of energy con-sumption. Most zooplankters c a r r y t h e i r eggs w i t h them. As egg develop-ment time i s i n v e r s e l y p r o p o r t i o n a l t o temperature, a lowered tempera-t u r e increases generation time d r a s t i c a l l y but may reduce metabolic c o s t s .

To f i n d u l t i m a t e causes f o r the migratory behaviour, the mentioned disadvantages have to be o f f s e t by f i t n e s s components, which increase because of v e r t i c a l m i g r a t i o n . Several hypotheses have been proposed: metabolic advantages or b e t t e r u t i l i z a t i o n of resources (McLaren 1963 and 1974, Kerfoot 1970, E n r i g h t 1977, E n r i g h t and Honegger 1977), avoid-ance of s t a r v a t i o n ( G e l l e r 1986), and avoidance of V i s u a l predators (Zaret and S u f f e r n 1976, Wright et a l . 1980). The v a l i d i t y and r e l a t i v e importance of these arguments can be t e s t e d only by q u a n t i f y i n g the r e l ­a t i v e s t r e n g t h of the v a r i o u s s e l e c t i v e f o r c e s . There are good reasons to assume migratory behaviour has a strong g e n e t i c component (Weider

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1984, G l i w i c z 1986). Therefore, a most promising framework f o r such a problem i s the concept of e v o l u t i o n a r i l y s t a b l e s t r a t e g i e s (=ESS) (May-nard Smith and P r i c e 1973, Maynard Smith 1982, Thomas 1984).

G a b r i e l and Thomas (1988a) developed an ESS-model on v e r t i c a l migra­t i o n of Zooplankton which i s able to e x p l a i n the observed coexistence of two s i m i l a r D a p h n i a s p e c i e s , one migrates w h i l e the other does not migrate ( S t i c h and Lampert 1981). A f t e r a s h o r t d e s c r i p t i o n of the model parameters we w i l l d e r i v e equations f o r a d i s c u s s i o n of the u l t i ­mate causes of v e r t i c a l m i g r a t i o n of Zooplankton.

MODEL PARAMETERS

A d e t a i l e d d i s c u s s i o n and d e s c r i p t i o n of the model i s given i n G a b r i e l and Thomas (1988a). M i g r a t i n g and non-migrating behaviour are t r e a t e d as two d i s t i n c t s t r a t e g i e s . The model c a l c u l a t e s the food up-take and i t s conversion i n t o s u c c e s s f u l r e p r o d u c t i v e Output f o r both s t r a t e g i e s . F i t n e s s i s c a l c u l a t e d i n terms of i n t r i n s i c growth r a t e on a time s c a l e of 24 hours. The i n t e r a c t i o n of Zooplankton w i t h i t s algae food i s considered i n d e t a i l according t o the w e l l s t u d i e d feeding physiology of Zooplankton. The payoffs depend on the f o l l o w i n g parame­t e r s which a l l are known from l a b o r a t o r y experiments and f i e l d s t u d i e s :

t : egg development time, A : a l g a l d e n s i t y , N : p o p u l a t i o n d e n s i t y of Zooplankton, p : p r e d a t i o n r i s k f o r Zooplankton by o p t i c a l l y

o r i e n t a t e d predators l i k e f i s h , r : p a r t i a l i n t r i n s i c growth r a t e of algae, T n: r e l a t i v e l e n g t h of n i g h t ( i n p a r t s of 24 hours), ß : conversion e f f i c i e n c y of food uptake t o s u c c e s s f u l r eproductive

Output ( m o r t a l i t i e s other than considered under p are taken i n t o account),

g : maximal g r a z i n g r a t e of Zooplankton.

Some of these parameters, e s p e c i a l l y t and ß, vary s t r o n g l y f o r the d i f -f e r e n t s t r a t e g i e s .

Due t o s e l f - i n t e r a c t i o n through food c o m p e t i t i o n and frequency depen-

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dent p r e d a t i o n r i s k , the payoff t o the two s t r a t e g i e s i s dependent on t h e i r r e l a t i v e f r e quencies. In f a c t , the payoff d i f f e r e n c e i s a n o n - l i n e a r f u n c t i o n .

BASIC EQUATIONS TO STUDY ULTIMATE CAUSES

To study the u l t i m a t e causes of v e r t i c a l m i g r a t i o n l e t us assume t h a t a p o p u l a t i o n c o n s i s t s only of non-migrating ethotypes and l e t us ask f o r c o n d i t i o n s a l l o w i n g a s u c c e s s f u l i n v a s i o n of m i g r a t i n g etho­types. This i s only p o s s i b l e when the payoff f o r m i g r a t i n g ethotypes i s l a r g e r than f o r non-migrating ones. Therefore, we can d e r i v e a minimal c o n d i t i o n f o r v e r t i c a l m i g r a t i o n by s e t t i n g the payoff d i f f e r e n c e equal t o zero (at r e l a t i v e frequency of non-migrating Zooplankton arbiträrily c l o s e t o x g = l ) . From the payoff equations given by G a b r i e l and Thomas (1988b), we have

S u b s c r i p t s i s used f o r the ( s t a t i o n a r y ) s t r a t e g y of non-migrating Zoo­plankton and s u b s c r i p t v f o r the s t r a t e g y of v e r t i c a l m i g r a t i o n . The food-uptake during n i g h t (=n) and day (=d) i n the upper water l a y e r s i s denoted as a n and a d , r e s p e c t i v e l y , and i s a f u n c t i o n of Zooplank­ton d e n s i t y , daylength, a l g a l growth r a t e , maximal f i l t r a t i o n r a t e of Zooplankton, and a v a i l a b i l i t y of a l g a l food. A slow-breeding c o r r e c t i o n f a c t o r w has t o be a p p l i e d t o measure the f i t n e s s disadvantage of pro-longed generation time ( f o r d e t a i l e d d i s c u s s i o n see G a b r i e l and Thomas 1988) :

The food uptake i s c a l c u l a t e d from the i n t e r a c t i o n of Zooplankton and algae. At low food l e v e l s the ingested food i s p r o p o r t i o n a l to the ac-t u a l food c o n c e n t r a t i o n . This r e s u l t s i n a food uptake during day of

(1) 0 = (1 - p)ß sa d - 1.5pß s(a n + a d ) t s + ß g a n -

(2) w = (1 + ß va nt v) / (1 + ß v ^ s )

(3) a d = g A [ e x p { ( r p - gx f iN)(1 - T n)} - l ] / ( r p - gx sN)

and d u r i n g n i g h t ( f o r non-migrating Zooplankton) of

(4) a n = A e x p { ( r p - gx gN)(1 - T n ) } [ l - exp{-gNT R}]/N

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At a l g a l c o n centrations above a c e r t a i n l i m i t ( A l i m ) the food uptake during day ( = a d) and nigh t ( = a n) i s independent of the a c t u a l a l g a l d e n s i t y but only a f u n c t i o n of t h i s l i m i t i n g c o n c e n t r a t i o n l e v e l . For high food concentrations a d and a n are then simply given by

= gA, • (1 - T ) ( f o r A > A n . ) d ^ l i m v n' v l i m ' a„ = gA-, • T ( f o r A > A n . ) . n ^ l i m n x l i m 7

For any S i t u a t i o n , v i z . f o r any given parameter s e t , we can now solve equation (1) i n order t o c a l c u l a t e t h r e s h o l d v a l u e s . Whether these t h r e s h o l d values are maximal or minimal values depends t r i v i a l l y on the d e r i v a t i v e of the payoff d i f f e r e n c e w i t h respect t o the parame­t e r i n question. To d i s c u s s u l t i m a t e causes we so l v e equation (1) f o r the r a t i o between the s t r a t e g y dependent conversion e f f i c i e n c i e s ß. From the model equations, i t f o l l o w s t h a t ß g i s always p o s i t i v e l y c o r r e -l a t e d w i t h the non-migration s t r a t e g y and any incre a s e i n ß v i s i n favour of v e r t i c a l m i g r a t i o n . Therefore, the t h r e s h o l d r a t i o ß v/ß g i s a minimum value: v e r t i c a l m i g r a t i o n can never be an ESS f o r values below t h i s t h r e s h o l d . In the f o l l o w i n g we w i l l d i s c u s s the consequences of t h i s t h r e s h o l d r a t i o which can be c a l c u l a t e d from equation (1) :

(6) (V ßs>thres = C 1 + V a n " P<Van + 1' 5 V 1 + ac/an> > 3 / w •

RESULTS AND DISCUSSION

Lets f i r s t c o nsider the simplest case of very low m o r t a l i t y caused by v i s u a l l y o r i e n t a t e d predators l i k e f i s h ; i . e . p=0. The t h r e s h o l d r a t i o (6) then g i v e s the value t h a t compensates f o r the lower food uptake and the prolonged development time of v e r t i c a l l y m i g r a t i n g Zooplankton. The value f o r t h i s t h r e s h o l d r a t i o i s then

( ßv / ß s ' t h r e s = [ 1 + a d / a n ] / w ( f o r p=0)

and i s reduced f u r t h e r i n the case of high food c o n c e n t r a t i o n to the simple expression

( ß v / ß s ) t h r e s = ( Tnw ) - 1 ( f o r p=0 and A > A l i n )

because of ac j / a

n= T n " 1 " " 1 ' A t l o w e r f o o d c o n c e n t r a t i o n s (A<A l i m) t h i s

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i s a l s o a v a l i d approximation under the c o n d i t i o n t h a t gNT n<<l and (r-gN)(1-T n)<<1; t h i s means b i o l o g i c a l l y t h a t the i n t e r a c t i o n of algae and Zooplankton i s not too f a r from a steady-state c o n d i t i o n and t h a t the m o r t a l i t y imposed on the algae by Zooplankton i s not too l a r g e .

The necessary advantage f o r v e r t i c a l l y m i g r a t i n g Zooplankton i n ß v , the e f f i c i e n c y of Converting food uptake i n t o s u c c e s s f u l reproduction, can be lowered d r a s t i c a l l y f o r p>0. For high food concentrations we get from equations (6) and (5)

<Wthres = [ 1 - P{1 - T n + 1.5 t s ) ] / w T n .

This i s again approximately t r u e a l s o at low food concentrations under the above mentioned c o n d i t i o n s .

I f p r e d a t i o n pressure i s high enough, v e r t i c a l m i g r a t i o n i s favoured even f o r ß v<ß g. For f u r t h e r i n c r e a s i n g p, the necessary ßv~value t o e s t a b l i s h v e r t i c a l m i g r a t i o n as an ESS becomes sm a l l e r and sm a l l e r . From equation (6) i t can be seen t h a t there i s an upper l i m i t f o r p

P t o l = ( a d + V 1 - V/^/'V 1 + 1«5ts> + 1-5ants) above which the t h r e s h o l d r a t i o would become negative. This means that at p r e d a t i o n pressure above t h i s value the s t r a t e g y of non-migrating Zooplankton can never be an ESS i r r e s p e c t i v e of the value f o r ß v , the conversion e f f i c i e n c y f o r m i g r a t i n g ethotypes. (This l i m i t on p i s d i s -cussed by G a b r i e l and Thomas (1988b) as the t o l e r a b l e p r e d a t i o n pressure f o r the non-migratory s t r a t e g y . ) In Figures 1 and 2 the t h r e s h o l d r a t i o s are shown f o r values of p v a r y i n g from p=0 t o P = P t o l # T n e v a r e p l o t t e d as f u n c t i o n s of t y / t s , and r e f l e c t v a r i o u s mean temperature d i f f e r e n c e s between upper and lower water s t r a t a which are experienced by migrating and non-migrating Zooplankton. From the f i g u r e s i t i s immediately c l e a r t h a t i t would be very d i f f i c u l t t o compensate f o r the disadvantage of v e r t i c a l m i g r a t i o n without the added advantage of predation avoidance. I t seems t o be impossible t o overcome t h i s disadvantage simply by i n -voking a more favourable metabolic s t a t e . But there might be a Chance to compensate f o r the disadvantage of v e r t i c a l m i g r a t i o n even at low pre d a t i o n pressure (small p) i f j u v e n i l e m o r t a l i t y f o r non-migrating Zooplankton i s very h i g h compared t o mi g r a t i n g ones.

Another i n t e r e s t i n g aspect a r i s e s from the attempt t o q u a n t i f y the

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Tw/T<

Fig . 1 and F i g . 2 : Threshold r a t i o of the conversion e f f i c i e n c y ß v/ß s of food i n t o succesful reproduction depending on the r a t i o t v / t s of egg development time of both s t r a t e g i e s . The predation pressure i s varied from 0 to 0.1 i n F i g . l and from O.i to 0.121 i n Fig.2. The broken l i n e s i n d i c a t e equal conversion e f f i c i e n c i e s ß v=ß c. (The fol l o w i n g parameter values are used : g = 0.55,

= 0.35, N = 1, T n = 0.4, t s = 5, A = 0.5 A l i n . )

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necessary compensation j u s t f o r the reduced food uptake d u r i n g v e r t i c a l m i g r a t i o n . Let us assume equal conversion e f f i c i e n c i e s f o r t h i s purpose. Therefore, we put ( ß

v / ßs ) t h r e s = 1* F u r t n e r m o r e l e t u s assume equal

developmental time f o r both s t r a t e g i e s which i s e q u i v a l e n t t o a n e g l i g i -b l e temperature d i f f e r e n c e between upper and lower water s t r a t a . There­f o r e , l e t us put t g = t v which i m p l i e s w=l. With equation (6) we can now c a l c u l a t e a p r e d a t i o n pressure P c o m t h a t compensates only f o r the lower food uptake of m i g r a t i n g ethotypes:

p _ = (1 + 1.5(a /a, + l ) t ) _ 1 . rcom N N n' a ' s '

For h i g h food c o n c e n t r a t i o n s (and a l s o approximately f o r low food con­c e n t r a t i o n s ) we get

p _ = (1 + 1.5t / T j " 1 ( f o r A>Alim) . rcom x s 7 n 7 v 7

I t may be c o u n t e r i n t u i t i v e t h a t t h i s value i s s t i l l dependent on t g , the egg development time f o r non-migrating Zooplankton. But the energy s t o r e d i n the eggs and o v a r i e s i s l o s t i n the case of p r e d a t i o n and t h i s energy i s a f u n c t i o n of the moulting p e r i o d which i s roughly e q u i v a l e n t t o t g . Therefore, a t lower temperatures i n the upper waters, f i s h p r e d a t i o n can compensate more e a s i l y f o r the reduced food supply due t o v e r t i c a l m i g r a t i o n . This i s completely independent of the temperature i n the bottom waters.

SUMMARY

The model equations demonstrate t h a t i t i s extremely u n l i k e l y t h a t me­t a b o l i c advantages by themselves are the u l t i m a t e causes of d i e l v e r t i c a l m i g r a t i o n i n Zooplankton. Various s e l e c t i v e f o r c e s i n t e r a c t i n a complex way t o s e l e c t t h i s behaviour, which only seems t o be disadvantageous at the f i r s t glance. R e l a t i v e s t r e n g t h and i n t e r a c t i o n of s i n g l e components of these s e l e c t i v e f o r c e s can be q u a n t i f i e d by a p p l y i n g e v o l u t i o n a r y game theory. This concept a l l o w s one t o c a l c u l a t e boundary c o n d i t i o n s f o r the i n v a s i o n of v e r t i c a l l y m i g r a t i n g ethotypes i n t o a p o p u l a t i o n c o n s i s t i n g o nly of non-migrating Zooplankton. A most u s e f u l q u a n t i t y t o study u l t i ­mate causes of t h i s b ehavioural phenomenon i s the t h r e s h o l d r a t i o of con­v e r s i o n e f f i c i e n c y of food i n t o reproduction. From t h i s a n a l y s i s , we sug-gest t h a t the r e d u c t i o n of predator-induced m o r t a l i t y i s one of the most important s e l e c t i v e f o r c e s i n f l u e n c i n g v e r t i c a l m i g r a t i o n .

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Acknowledgement

We thank L.J. Weider f o r improving the manuscript.

LITERATURE CITED En r i g h t , J . T. 1977. D i u r n a l v e r t i c a l m i g r a t i o n : adaptive s i g n i f i c a n c e

and t i m i n g . Part 1. S e l e c t i v e advantage: A metabolic model. Limnol. Oceanogr. 22:856-872.

En r i g h t , J . T., and H, W. Honegger. 1977. D i u r n a l v e r t i c a l m i g r a t i o n : adaptive s i g n i f i c a n c e and t i m i n g . Part 2. Test of the model: D e t a i l s of t i m i n g . Limnol.Oceanogr. 22:873-886.

G a b r i e l , W. and B. Thomas. 1988a. V e r t i c a l m i g r a t i o n of Zooplankton as an e v o l u t i o n a r i l y s t a b l e s t r a t e g y . Am.Nat. ( i n p r e s s ) .

. 1988b. The i n f l u e n c e of food a v a i l a b i l i t y , p r e d a t i o n r i s k , and metabolic c o s t s on the e v o l u t i o n a r y s t a b i l i t y of d i e l v e r t i c a l m i g r a t i o n i n Zooplankton. V e r h . I n t e r n a t . V e r e i n . Limnol.23 ( i n p r e s s ) .

G e l l e r , W. 1986. D i u r n a l v e r t i c a l m i g r a t i o n of Zooplankton i n a temper-ate great l a k e (L. Constance): A s t a r v a t i o n avoidance mechanism? Arch.Hydrobiol./Suppl. 74:1-60.

G l i w i c z , M. Z. 1986. Predation and the e v o l u t i o n of v e r t i c a l m igration i n Zooplankton. Nature 320:746-748.

Kerfoot, W. B. 1970. B i o e n e r g e t i c s of v e r t i c a l m i g r a t i o n . Am.Nat. 104: 529-546.

Lampert, W., and B. T a y l o r . 1985. Zooplankton g r a z i n g i n a eutrophic l a k e : i m p l i c a t i o n s of d i e l v e r t i c a l m i g r a t i o n . Ecology 66:68-82.

Maynard Smith, J . 1982. E v o l u t i o n and the theory of games. Cambridge U n i v e r s i t y Press, Cambridge.

Maynard Smith, J . , and G. R. P r i c e . 1973. The l o g i c of animal c o n f l i c t s . Nature 246:15-18.

McLaren, I . A. 1963. E f f e c t s of temperature on the growth of Zooplank­ton and the adaptive value of v e r t i c a l m i g r a t i o n . J.Fish.Res.Bd.Can. 20:685-727.

. 1974. Demographic s t r a t e g y of v e r t i c a l m i g r a t i o n by a marine copepod. Am.Nat. 108:91-102.

Pearre, S. J r . 1979a. On the adaptive s i g n i f i c a n c e of v e r t i c a l migra­t i o n . Limnol.Oceanogr. 24:781-782.

. 1979b. Problems of d e t e c t i o n and I n t e r p r e t a t i o n of v e r ­t i c a l m i g r a t i o n . J o u r n a l of Plankton Research 1:29-44.

S t i c h , H.-B., and W. Lampert. 1981. Predator evasion as an explanation of d i u r n a l v e r t i c a l m i g r a t i o n by Zooplankton. Nature 293:396-398.

Thomas, B. 1984. E v o l u t i o n a r y s t a b i l i t y : s t a t e s and s t r a t e g i e s . Theor. Po p . B i o l . 26:49-67.

Weider, L. J . 1984. S p a t i a l heterogeneity of genotypes: V e r t i c a l mig­r a t i o n and h a b i t a t p a r t i o n i n g . Limnol.Oceanogr. 29:225-235.

Wright, D., W. J . O'Brien, and G. L. Vingard. 1980. Adaptive value of v e r t i c a l m i g r a t i o n : A Simulation model argument f o r the predation hypothesis. Pages 138-147 i n W.C. K e r f o o t , ed. E v o l u t i o n and ecolo­gy of Zooplankton communities. Univ. Press of New England, Hanover, New Hampshire.

Zaret, T. M., and J . S. S u f f e r n . 1976. V e r t i c a l m i g r a t i o n i n Zooplank­ton as a predator avoidance mechanism. Limnol.Oceanogr. 21:804-813.