Dynamic programming A gentle introduction using zooplankton behaviour as an example

Outline Lecture 1: A gentle introduction to Dynamic Programming: Behavioural and life-history decisions in zooplankton as an example (Øyvind Fiksen) Lecture 2: An advanced application of dynamic programming: Life history evolution in cod (Christian Jørgensen)

Depth + ÷ Growth Pelagic vertical gradients: a classical dilemma Light (risky) Dark (safe) + ÷ Fish feeding efficiency

Diel vertical migration

Size-structured patterns of distribution Increasing size Increasing depth

Multiple predators – complicates the trade-off.. Fish Large zooplankton Small zooplankton

Pseudocalanus in Dabob Bay Ohman 1990

Predator regimes in shallow and deep areas Ohman 1990

Flexible DVM behaviour in Pseudocalanus Ohman 1990

An experiment with Daphnia magna Loose & Dawidowicz 1994

State dynamics in discrete time Mass gained in time interval New body mass New body mass - alternative notation Reproduction

Optimal habitat selection and allocation of energy Eggs Growth Backward iteration Risk * * *

Computer pseudo-code DEFINE TERMINAL FITNESS(STATE,H) DO TIME = H-1, 1, -1 DO STATE = MINSTATE, MAXSTATE DO HABITAT = 1,N_HABITATS DO ALLOCATION = 1, N_ALLOCATION Find NEW_STATE(HABITAT, ALLOCATION) Find REPRODUCTION(HABITAT, ALLOCATION) Find SURVIVAL(HABITAT,ALLOCATION) Find FITNESS=SURVIVAL*[FITNESS(NEW_STATE,T+1) + REPRODUCTION] IF(FITNESS>MAX_FITNESS) THEN STORE HABITAT*(STATE,TIME) STORE ALLOCATION*(STATE,TIME) ENDIF ENDDO ALLOCATION ENDDO HABITAT ENDDO STATE ENDDO TIME State dynamics (physiology) & mechanics Evaluate consequences of actions in terms of fitness Loops

The dynamic programming equation Maximise fitness = find the behavioural and life history decision that maximises the sum of current and expected future reproduction: Future fitness (new state, time)Eggs Fitness (size, time) Survival

Optimal behaviour and life history Optimal strategy depending on environment, body mass, time and implicitly, expectations of future conditions These matrixes of the best strategy can be applied in forward projections with IBMs or state-structured population models

Optimal depth selection: data and model Model Data from Loose & Dawidowicz 1994

Behaviour and life-history decisions interact 0.01 fish/L with DVM 0.01 fish/L restricted from DVM Low fish density High fish density

Real dilemma: when access to safety is restricted.. Sakwinska & Dawidowicz 2005 L&O

..decrease size at first reproduction! Sakwinska & Dawidowicz 2005 L&O

Conclusions Dynamic programming is excellent in clarifying the role of state in behavioural ecology and life history theory It is good at –integrating proximate constraints, physiology, ecological mechanics and physics with evolutionary theory –asking ‘What if’-questions and make predictions It is not suitable for density- or frequency-dependent traits