Modelling size-structured populations David Boukal (IMR, Bergen, Norway) FishACE Methods Course, Mallorca, 2-3 May energy budget models 2. ecological dynamics of structured populations 3. evolution of age and size at maturation (tomorrow)

2 intricate reality Individuals vs. ecosystems: What do we do?

3 1. Energy budget models lsize-structured populations: many individual properties depend on size

4 Size/physiological structure lindividual life history driven by size: l feeding capacity (gape size) l digestive capacity (stomach size) l fecundity (gonad size) l predation risk (body size) l maintenance (body size) lw = weight... ‘typical’ exponents are b=1 (processes ~ body volume: maintenance, fecundity) b=2/3 (processes ~ body surface, gape size: feeding) l‘general’ scaling laws (allometries) F(w) ~ c w b

5 Slight diversion: von Bertalanffy growth lvon Bertalanffy growth simple derivation: isomorphic (‘bubble’) growth ( W  L 3 ) ingestion ~ gape size ( ~ L 2 ~ W 2/3 ) maintenance ~ body volume ( ~ L 3 ~ W) Age (d) Length (mm) Daphnia (Kooijman, 1993)

6 How to model individual life histories? lDynamic Energy Budget (DEB) models: l individuals = ‘engines’ converting food into body mass & offspring l individual state characterized by a few variables: structural biomass (soma), reserves,... l energy allocation described by a set of (simple) rules l three main processes: growth, maintenance and reproduction ?

7 Energy allocation rules net production models net assimilation models food reproduction structural biomass (somatic) maintenance maturity maintenance storage faeces  1-   -rule (Kooijman 1993) food reproduction structural biomass (somatic) maintenance reserves faeces  1- 

8  -rule (Lika & Kooijman 2003) time ‘maturation investment’ A mat starvation max juv. structural biomass reserves SB mat max adult Energy allocation rules II: maturation net assimilation models (+ some net production models) net production models (L. Persson & Co) ‘gonads’ maturation ~ threshold surplus energy repro ~ dtto gain... from storage, as (1 -  ) proportion loss... maintenance of matur. investment, (starvation) maturation ~ threshold surplus energy repro ~ threshold relative energy gain... from net intake, as (1 -  ) proportion loss... maintenance + starvation starvation gonads

9 Energy allocation III:  -rule (Kooijman, 2000) Age, d Length, mm Cum # of young Length, mm Ingestion rate, 10 5 cells/h O 2 consumption,  g/h Respiration  Ingestion  Reproduction  VB growth

10 Energy allocation rules IV lnet allocation models l Kooijman (1993, 2000), Nisbet et al. (2000), Ledder et al. (JMB 2004),... l usually assume  constant over lifetime (juveniles ‘prepare’ for repro)  difficulties when studying evolution of size at maturation l starvation requires additional assumptions & equations (in continuous time: repro/growth shuts down immediately / gradually) lnet production models l Lika & Nisbet (JMB 2000), Ledder et al. (JMB 2004), de Roos & Persson & co... l usually assume switch in   easier to study evolution of size at maturation... but Lika & Nisbet (2000):  < 1 constant, repro starts at threshold reserves l starvation also requires some thought l(my) conclusions: DEB models can be VERY complicated use something sensible & not too complex avoid  -rule ?

11 2. Dynamics of size-structured populations lsize-structured populations: individuals are usually not the same

12 How to model structured populations? lage-specific life histories (LHs) l e.g. juveniles  adults (time lags) lstage-specific LHs l e.g. distinct larval stages (insects) lsize/physiological structure (fish!) l age and individual state decoupled matrix models (Caswell 2001), delay differential equations, PDEs (Forster-McKendrick) and IBMs PDEs (size rather than age), numerical approximations, IBMs, DEB models, physiologically structured population models (PSPMs)

13 PSPMs & role of ecological feedback density dependence: dense populations  individuals grow slowly  reproduce less (if fecundity linked to size) ... equilibration(?) Individual state: e.g. length and energy reserves Processes: vital rates (feeding, maintenance, energy allocation) reproduction mortality environment can be anything … typically: resource density, density of conspecifics … levolution: AD easy to implement due to full feedback loop ldensity dependence via food supply, not restricted to juveniles

14 Simple model of (planktivorous) fish life cycle lsize-structured fish population l individuals characterized by irreversible/reversible mass (incl. gonads) l individual growth is density- and size-dependent (no fixed age-size relationship) l pulsed reproduction & maturation at fixed size lpopulation dynamics = sum of individual life histories reproduction growth feeding ‘herring & copepods’ lunstructured resource in deterministic, closed system

15 Size/physiological structure lindividual life history driven by size: lnet production model & indeterminate growth (i.e. also after maturation) l‘general’ scaling laws (allometries) vital rates scaling with body size/weight w: F(w) ~ c w b

16 Size- & state-dependent life history irreversible mass (‘bones and vital organs’) growth Persson et al. (1998) reproduction reversible mass (‘reserves+gonads’) starvation threshold max. juvenile condition max. adult condition metabolic maintenance limit starvation maturation general life history pattern (indeterminate growth & pulsed reproduction)

17 Mathematical formulation (Persson et al. 1998) Individual level processes (x = irreversible mass; y = reversible mass; R = resource density)

18 Mathematical formulation (Persson et al. 1998) Individual level dynamics (x = irreversible mass; y = reversible mass; R = resource density) Population level dynamics = sum of individual life histories resourcedR/dt = r(K-R) -  i I (x i,R) consumerdN i /dt = -  (x i,y i ) N i reproductive pulses with period T  added newborn cohort (i=0): N 0 =  i F(x i,y i ) N i (+ resets of reversible mass) dx i /dt = [  (x i,y i,R) E g (x i,y i,R)] + dy i /dt = [(1-  (x i,y i,R)) E g (x i,y i,R)] + l consumer population composed of a number of discrete cohorts (i  I) l cohorts characterized by physiological state (x i, y i ) and number of individuals N i l typical approach: age-based cohorts (computational load)

19 Role of competition for a shared resource lsize-dependent competition abilities  min. requirements for the shared resource: 3 qualitative types of outcome  = scaling of size-dependent attack rate A(w)~b w  standardized mass, w (g) minimum resource density  = 1.1  = 0.8  = 0.5 adults can outcompete juveniles juveniles can outcompete adults

20 Population dynamics - role of exponent  l  low: single-cohort cycles (recruit-driven cycles) l  intermediate: equilibrium dynamics l  high: non-recruit-driven cycles juveniles adults resource 6-year single-cohort cycle juveniles adults resource 8-year cycle (not single-cohort) juveniles adults resource fixed-point dynamics (annually)

21 Population dynamics - role of environment l[once more] an example: recruit-driven, single-cohort cycles cycle length l can change only in discrete steps l decreases with mortality and increases with maturation size juveniles adults resource 6-year single-cohort cycle Cohort cycle length decreases with mortality 4yr 3yr 2yr 1yr 5yr

22 Conclusions lPSPMs l cover phenotypic plasticity via environmental feedback loop (e.g. common food source) l based on individual-level, mechanistic rules l(sort of...) easy to incorporate other ecological phenomena l e.g. cannibalism, ontogenetic niche shifts, spatial variation in resources … l price to pay I: more complex extensions are computationally very intensive l price to pay II: the models require a number of parameter values (detailed life histories), key ones not a priori clear lsetup amenable to study of evolutionary responses l quantitative genetics, adaptive dynamics (examples tomorrow) l deterministic system: exhaustive check of parameter (sub)space doable (x IBMs, dyn.prog.) l price to pay III: usually heavy computational load (scan of parameter space)