Goodbye K, Welcome MThe Interrelationship between

Life Span, Growth and Reproduction

Rainer FroeseIFM-GEOMAR

Kiel, Germany

The Interrelationship between Lifespan, Growth and Reproduction

• Relationship between lifespan and reproduction:– In stable populations, on average, every adult

produces ONE new adult during its lifetime

• Relationship between growth and reproduction:– In species with indeterminate growth, fecundity

increases proportional to body weight (GSI is constant)

• Relationship between lifespan and growth:– Currently none – Size at age is a function of maximum size and a

parameter (K) indicating how fast it is reached

Approaching the Problem

• Growth is constrained by oxygen• Lifespan is determined by mortality rate• Maximum size is determined by lifespan• K = 2/3 M• The most important point in life• Growth and maturity• Framework for lifespan, growth & reproduction• Implications for managment

Growth is Constrained by Oxygen

• Growth needs food and energy for assimilation

• The oxygen-temperature window restricts growth (Pörtner, Science 2004)

• Water-breathing is energy-demanding

• Physics determine gill-area

Gill Area vs Body Weight

0.1

1

10

100

1000

10000

100000

1000000

10000000

0.01 0.1 1 10 100 1000 10000 100000 1000000

Body weight (g)

Gill

are

a (

cm

2 )

Megamouth0.8 tons57 m2

Goby0.02 g0.2 cm2

log GA = 0.65 + 0.91 log BWr2 = 0.93

Data for 118 species of fishes, from FishBase 11/2006

Lifespan is Determined by Mortality Rate

• M is constant during adult life– Intrinsic mortality rate increases (wear & tear, mutations

acting late in life) – Extrinsic mortality decreases (with size and growing

experience in predator avoidance)

Size of prey vs predator

1

10

100

1000

1 10 100 1000

Length of predator (cm TL)

Len

gth

of

pre

y (c

m T

L)

1 : 1

4,453 cases for 1,743 species of fishes, data from FishBase 11/2006

Lifespan is Determined by Mortality Rate

• M is constant during adult life– Intrinsic mortality rate increases (wear & tear, mutations acting late in

life) – Extrinsic mortality decreases (with size and growing experience in

predator avoidance)

– Intrinsic and extrinsic mortality balance each other, resulting in constant adult mortality

– Confirmed by 168 studies fitting a linear regression to (log) numbers at age (data in FishBase 11/2006)

• If adult mortality is constant, then mean adult life expectancy E = 1/M (from continuous life table analysis)

Approaching the Problem

• Growth is constrained by oxygen

• Lifespan is determined by mortality rate

• Maximum size is determined by lifespan

Maximum Size is Determined by Lifespan

Based on growth data for 6,418 populations of fishes in FishBase 11/2006

1

10

100

1000

10000

0.1 1 10 100 1000

Maximum age (years)

To

tal

len

gth

(cm

)

smallestvertebrates

fastest growth warm water

largest water-breathers

longest lifespancold water

Maximum Size is Determined by Lifespan

1

10

100

1000

10000

0.1 1 10 100 1000

Maximum life span (years)

Max

imu

m l

eng

th (

cm)

Carcharhinus acronotus Pleuronectes platessa Stolephorus indicus

Empiricalwithin species:Linf = C tmax

0.41

Slope of Linf-tmax trade-off

The slope is not significantly different in

• Taxonomic groups (Class, Order, Families)

• Salinity (marine, freshwater, diadromous)

• Deep waters

• Climate zones (temperate, subtropical, tropical)

Approaching the Problem

• Growth is constrained by oxygen

• Lifespan is determined by mortality rate

• Maximum size is determined by lifespan

• K = 2/3 M

The von Bertalanffy Growth Equation

ttt kWHW

dt

dW 3

2

3)1( Ktt eWW

)1( Ktt eLL

Growth and Mortality

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24

Age (years)

Weight

Growth and Mortality

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24

Age (years)

Weight

max. growth rate

Winf

Growth and Mortality

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24

Age (years)

Weight

Probabilityof survival

max. growth rate

Growth and Mortality

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24

Age (years)

Weight

Probabilityof survival

max. growth rate

Expected weight

M/K = 3/2

M/K > 3/2Peak left and smaller

M/K < 3/2Peak right and smaller

Growth and Mortality

• Empirical slope of Linf-tmax trade-off is

0.41 (95% CL 0.35 – 0.48)

• Theoretical slope of Linf-tmax trade-off that results in overlap of maximum growth rate with maximum expected body weight is 0.45 (i.e., not significantly different from empirical value)

Growth, Mortality and Fecundity

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24

Age (years)

Weight

Probabilityof survival

max. growth rate

Expected weightExpected fecundity M/K = 3/2 maximizes fitness

Empirical Evidence for M/K = 3/2

0.01

0.1

1

10

100

0.01 0.1 1 10 100

M = 1.5 K

M o

bse

rved

1 : 1

Based on growth and mortality data for 272 populations of 181 species, FishBase 11/2006

The New Growth Equations

)1()1( )(667.03

)(2

0

0ttME

tt

t eLeLL

bttMbE

tt

t eWeWW )1()1( )(667.03

)(2

0

0

E

W

E

W

dt

dW flt 296.0max

More Evidence

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Age (E )

We

igh

t (W

/Win

f)

Fishes Bivalve Euphausiid Trees Newt Squid

33

)(2

)1(0

E

tt

t eWW

Approaching the Problem

• Growth is constrained by oxygen

• Lifespan is determined by mortality rate

• Maximum size is determined by lifespan

• K = 2/3 M

• The most important point in life

The Most Important Point in Life

LeLLopt 667.0)1( 3

265.1

WeWWopt 296.0)1( 33

265.1

EMM

topt 65.1648.1

2

)3ln(3

• Where growth is maximum

• Where expected fecundity is maximum

Growth and Maturity

1

10

100

1000

10000

1 10 100 1000 10000

L∞ (cm)

Lm

(cm

)

L∞

0.67 L∞

0.35 L∞

Maturity data for 672 populations of 351 species from FishBase 11/2006

low fecundity,low dependenceon environment

high fecundity,high dependenceon environment

Framework forLifespan, Growth and Reproduction

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

Relative age (t/E )

Rel

ativ

e si

ze

L/L ∞=0.667

W/W ∞=0.296

0.65 1.65 2.65

t opt

Framework forLifespan, Growth and Reproduction

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

Relative age (t/E )

Rel

ativ

e si

ze

spawners

L/L ∞=0.667

W/W ∞=0.296

old spawners

E

L/L ∞=0.83

0.65 1.65 2.65

t opt

W/W ∞=0.57

Framework forLifespan, Growth and Reproduction

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

Relative age (t/E )

Re

lati

ve

siz

e

spawnersearly spawners /

juvenilesjuveniles

L/L ∞=0.667

W/W ∞=0.296

old spawners

E

L/L ∞=0.35

L/L ∞=0.83

0.65 1.65 2.65

t opt

W/W ∞=0.57

Approaching the Problem

• Growth is constrained by oxygen• Lifespan is determined by mortality rate• Maximum size is determined by lifespan• K = 2/3 M• The most important point in life• Growth and maturity• Framework for lifespan, growth & reproduction• Implications for managment

Implications for Management

0.0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

Relative age (t/E )

Pro

ba

bili

ty o

f s

urv

iva

l (lt

)

spawnersearly spawners /

juveniles old spawnersjuveniles

0.65 1.65 2.65

Implications for Management

0.0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

Relative age (t/E )

Pro

ba

bili

ty o

f s

urv

iva

l (lt

)

spawnersearly spawners /

juveniles

current fishing

old spawnersjuveniles

0.65 1.65 2.65

t opt fishing

t opt+E fishing

Implications for Management

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4

Relative age (t/E )

Rel

ativ

e b

iom

ass

(B/N

o)

old spaw nersspaw nersearly spaw ners /

juvenilesjuveniles

current f ishing t opt f ishing

t opt +Ef ishing

no fishing

0.65 1.65 2.65

Conclusions• Growth is determined by mean adult life

expectancy, which is the inverse of the mortality rate

• At Lopt = 2/3 Linf, growth rate and expected fecundity are maximum

• Maturity is reached between topt -1 and topt, depending on life history strategy

• Ignoring the relationships between growth, mortality and reproduction may contribute to widespread failure of fisheries management

Thank You