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168 Oceanography vs. behaviour 6.A Choice of the last two optimal decisions In the simple model presented in section 6.2 (page 119 and following) it is possible to describe analytically the choice of the last two decisions. It helps in understanding how the optimisation algorithm works. Last optimal decision d # (θ, x, T − 1) The value function is 0 V (θ + ∆θ 0 , x + ∆x 0 1 , T ), | {z } Foraging, d = 0 V (θ, x, T − 1) = p · max d B @ V (θ − ∆θ 1 , x − ∆x 1 C , T ) A | {z } Swimming, d = 1 = p · max (θ + ! ∆θ0 ) · 1 {x+∆x 0 =0}, (θ − ∆θ 1 ) · 1 {x−∆x 1 =0} Now, x + ∆x 0 ≠ 0 because x ≥ 0 and ∆x 0 > 0. Thus ( V (θ, x, T − 1) = p · (θ − ∆θ 1 ) · 1 {x−∆x 1 =0} d # (θ, x, T − 1) = 1 Therefore, the optimal decision at time T −1 is swimming if x−∆x 1 = 0. It means that the larva will swim if it can reach the reef by choosing swimming. But, if x is not equal to ∆x 1 (it cannot reach the island), V equals zero for any decision. In this case it does not have any favourite decision for its last choice. Before the last optimal decision u # (θ, x, T − 2) Assuming the last decision was swimming (d # (θ, x, T − 1) = 1), the value function at T − 2 is 0 V (θ + ∆θ 0 , x + ∆x 0 1 , T − 1) , | {z } Foraging, d = 0 V (θ, x, T − 2) = p · max d B @ V (θ − ∆θ 1 , x − ∆x 1 C , T − 1) A | {z } Swimming, d = 1 = p · max (θ + ! ∆θ0 − ∆θ 1 ) · 1 {x+∆x 0 =∆x 1 }, (θ − 2∆θ 1 ) · 1 {x−∆x 1 =∆x 1 } As we cannot have at the same time x+∆x 0 = ∆x 1 and x−∆x 1 = ∆x 1 , it comes that: “ V (θ, x, T − 2) = p · (θ + ∆θ 0 − ∆θ 1 ) · 1 {x+∆x 0 =∆x 1 } + (θ − 2∆θ 1 ) · 1 {x−∆x 1 =∆x 1 } ”
Acknowledgements 169 Therefore: • If x = ∆x 1 − ∆x 0 , then d # (θ, x, T − 2) = 0 , the larva chooses to forage. When the larva chooses to eat at time T − 2, it optimises its energy resources value and is taken away from the reef by ∆x 0 . So, at time T − 1, it will be at the correct distance to come back to the reef (i.e. ∆x 1 ). • If x = 2∆x 1 , then d # (θ, x, T − 2) = 1, the larva decides to swim. Here again, this choice is natural as swimming brings the larva to a distance ∆x 1 from the reef. It will only have to swim once more at last time step to reach it. The explicit calculation of V (θ, x, t) becomes more and more complex as one goes backward in time. A computer code is developed in Scilab to find numerically all optimal decisions. Still, we have noted that solving Bellman’s equation gives very intuitive results at time T − 1 and T − 2. 6.B Acknowledgements The authors are thankful to J-P. Chancelier, from the CERMICS, for his help in coding the model in Scilab and to A. Lo-Yat, M. S. Pratchett, D. J. Gibson and two anonymous referees for their comments on the manuscript submitted to the Journal of Theoretical Biology. Regarding further development of the model, the authors are grateful to P. Marsaleix for its help in adapting Symphonie to the configuration used here, and most indebted to C. Dong for sharing his ROMS configuration. This work was accomplished with financial support from the French ministry of research (ACI MOOREA, quantitative ecology).
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À mon étoile.
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vi Résumé La plupart des organism
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viii Abstract Most demersal marine
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x Table des matières 1.3.3 Simple
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xii Table des matières 6.2.3 Examp
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xiv Liste des figures 4.6 Distribut
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Liste des tableaux 1.1 Potential or
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2 Introduction La survie des larves
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4 Introduction Limitation par le re
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6 Introduction Les courants déterm
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8 Introduction où m est le taux de
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10 Introduction biologique simple.
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12 Introduction Le milieu lui-même
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14 Introduction Figure I.6 Schémat
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16 Introduction obstacles technique
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18 Introduction Figure I.8 Prédict
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20 Introduction I.5 La biologie des
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22 Introduction du fonctionnement d
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24 Introduction L’objectif de cet
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26 Behaviour in models 1.1 Introduc
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28 Behaviour in models Using mean p
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30 Behaviour in models 1.3 Vertical
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32 Behaviour in models . . . bring
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34 Behaviour in models of measured
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36 Behaviour in models Additive dis
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38 Behaviour in models In situ stud
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40 Behaviour in models Operational
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42 Behaviour in models Effects on p
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44 Behaviour in models begins is no
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46 Behaviour in models 1.10 Conclus
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48 Larvae orientation in situ Diver
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50 Larvae orientation in situ . . .
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52 Larvae orientation in situ optio
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54 Larvae orientation in situ if >
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56 Larvae orientation in situ Table
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58 Larvae orientation in situ that
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60 Larvae orientation in situ 2.A A
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62 Behaviour of settling fishes at
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Chapter 4 Biophysical correlates in
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Methods 71 4.2 Methods 4.2.1 Sampli
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Methods 73 Three rotations were com
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Methods 75 test really focuses on w
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Results 77 Figure 4.3 Mean of insta
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Results 79 Figure 4.5 Perspective v
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Results 81 Table 4.1 Abundances of
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Results 83 Figure 4.8 Distribution
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Results 85 Figure 4.10 Concentratio
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Discussion 87 Figure 4.11 Compariso
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Discussion 89 But the most likely e
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Acknowledgements 91 This evidence a
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Chapter 5 Ontogenetic vertical “m
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The statistical analysis of vertica
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Methods 101 80, 55, 30 m (Figure 4.
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Methods 103 same station were likel
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Results 105 Figure 5.2 Univariate r
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Results 107 5.4.3 Ontogenetic shift
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Results 109 Figure 5.5 Violin plot
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Discussion 111 5.5 Discussion 5.5.1
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Discussion 113 expression of differ
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Chapter 6 Oceanography vs. behaviou
Page 135 and 136: Introduction 117 a good description
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Page 145 and 146: The balance between feeding and pre
Page 155 and 156: Oriented swimming and passive advec
Page 179 and 180: General discussion 161 such integra
Page 181 and 182: General discussion 163 6.5.2 Why se
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Page 185: General discussion 167 energeticall
Page 190 and 191: 172 Conclusion Un environnement pé
Page 192 and 193: 174 Conclusion global sur la connec
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Page 196 and 197: 178 Conclusion La différence entre
Page 198 and 199: 180 Conclusion Mieux décrire la di
Page 200 and 201: 182 Conclusion Comme tout comportem
Page 202 and 203: 184 Undescribed Chaetodonthidae Fig
Page 204 and 205: 186 Undescribed Chaetodonthidae Fig
Page 206 and 207: 188 Bibliographie [12] Cushing DH.
Page 208 and 209: 190 Bibliographie [39] Johnson MW.
Page 210 and 211: 192 Bibliographie [71] Paris CB, Co
Page 212 and 213: 194 Bibliographie [99] Lara MR. Mor
Page 214 and 215: 196 Bibliographie [127] Masuda R, S
Page 216 and 217: 198 Bibliographie [157] Holbrook SJ
Page 218 and 219: 200 Bibliographie [187] Bowen B, Ba
Page 220 and 221: 202 Bibliographie [218] Oxenford HA
Page 222 and 223: 204 Bibliographie [246] Wellington
Page 224 and 225: 206 Bibliographie [276] Sargent RC,
Page 226 and 227: 208 Bibliographie [308] Greenberg D
Page 228 and 229: 210 Remerciements solides pour ces
Page 230 and 231: 212 Remerciements tri des larves de
Page 232: 214 Remerciements Évidemment, tout
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