Optimal life history strategy andsperm competition of dwarf males ofbarnacles○Sachi Yamaguchi*, Yuki Ozaki, Yoichi Yusa, Satoshi Takahashi( Nara Women’s University )*e-mail: sachi@lisboa.ics.nara-wu.ac.jp1

Barnacles’ Biology (1)- Barnacles are crustaceans that distribute widely in seasworldwide.
Octolasmis warwicki

Scalpellum stearnsii

Large hermaphrodite + dwarf ♂
attaches to crab

large ♀ + dwarf ♂
lives in deep sea

hermaphrodite

femaledwarf male↑We can not see dwarf males in this picture.
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Barnacles’ Biology (2)- Barnacles have three patterns of sexuality depending on thespecies. (Darwin, 1851)(a) Simultaneous hermaphrodite only(b) Large hermaphrodite + Dwarf male (Androdioecy)(c) Large female + Dwarf male (Dioecy)- Males are very small! ---> “dwarf male”--- Males attach to a large female or hermaphrodite.
--- Males living in shallow sea are larger than ones living in deep sea(Pilsbry, 1908).
--- Males grow up in some species, they do not in the others.
--- Males of androdioecious species tend to be larger than dioecious
ones (Yusa and Yamato, unpublished).

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Research Purpose
Dwarf males’ body size varies depending on species.
↓Differences of dwarf males’ life histories such as their growth rate

Differences in size of dwarf males among species arecaused by two factors.
1. The amount of sperm of large individuals, α
2. Food availability, βTo explain inter-specific differences in body size of dwarfmales, we calculate the optimal life history strategy thatmaximize lifetime reproductive success using Pontryagin’s
maximum principle.
4Life history of barnacles
Ex. Scalpellum stearnsii
molts

cypris larva

nauplius larva

becomes ♀？

♀
becomes
dwarf malereproduction

attaches to ♀

grows

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ModelLife history strategy of barnacles’ males: (u(t), c(t))- The proportion of the resources allocated to reproduction, u(t)to growth, 1-u(t)- The consumption speed of initial energy storage, c(t)- Assume that n dwarf males attach to a large individual.
- The resource flow (resource usage per unit time) r(t) is obtained byc(t) and p(t).

Consumption speedof energy storage

Feeding rate

-The resource flow r(t) is divided for two usages.
Resources allocated to reproduction: u(t)r(t)Resources allocated to growth: (1-u(t))r(t)6

- The consumption speed c(t) is larger, the energy storage
remainder e(t) becomes smaller.

Consumption speedof energy storage- As resource allocated to growth is larger, body size s(t) becomeslarger.

Resource allocated to growth- The feeding rate p(t) is proportional to the area of body size.

Feeding rate

Feeding efficiency
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- Instantaneous reproductive success of the dwarf male, φ(t),Number of eggs ofa large individual

The amount of spermof the dwarf male

Sperm competition

The amount of sperm of
all the dwarf males

The amount of sperm ofsurrounding large individuals

If large individuals are females, α equals 0.
If large individuals are hermaphrodites, α is positive.

Which u(t) and c(t) maximize the lifetime reproductive success of thedwarf male?
Optimal strategy (uopt(t), copt(t)) is calculated byPontryagin’s maximum principle.
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Pontryagin’s maximum principle (1)- We define Hamiltonian function.

Where v(t) is reproductive success until time t.

- Adjoint variables pe(t), ps(t) and pv(t) follow the equations below.

- Conditionse(t)>0, e(T)=0, v(0)=0, pe(T)=0, ps(T)=0, pv(T)=1
boundary condition

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Pontryagin’s maximum principle (2)- The optimal strategy (uopt(t), copt(t)) maximizes Hamiltonian at any time t.

0 ≦ uopt≦ 1, 0 ≦ copt≦ Cmax

- We solve simultaneous differential equations by Runge-Kutta method.
- We calculate ESS where the optimal life history strategy (uopt(t), copt(t))
coincides with the strategies of the other dwarf males and denote it by(u*(t), c*(t)).

10Optimization with boundary condition
The method of Lagrange multipliers
G =v(T) ：Objective function = Lifetime reproductive successControl interval: 0 ≦ t ≦ T
Initial state: (e(0), s(0), v(0))=(E, s0, 0)

given

Final state: (e(T), s(T), v(T))- without boundary condition- with boundary condition

Lagrange multiplier11

Result (1)Effect of the sperm amount of large individuals, αIn food-rich environments (β=1.00)
Resource allocation ratio toreproduction, u*(t)ui*(t)

di*(t)

Consumption speed ofenergy storage, c*(t)81.0Body size, s(t)

si(t)
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α=100
α=10

α=0
α=1000.50

α=0

α=104

α=100

α=100

8

α=0

4

360t

0

360t03- Dwarf males use up energy storage as fast as possible.
- Body size increases with α.
- Strategy u*(t) tends to “bang-bang control” as α becomes large.

612tResult (2)Effect of the feeding efficiency, βIn dioecious species (α=0)
Resource allocation ratio toreproduction, u*(t)di*(t)
ui*(t)

1.0

Consumption speed ofenergy storage, c*(t)88β=0.05

Body size, s(t)si(t)β=1.00β=0.30β=1.00β=1.00440.5

β=0.30β=0.30β=0.05

00

360t

0360t

β=0.0503- Dwarf males grow in shorter time when β becomes smaller.
- They do not grow and use all the resources to reproduction
when β is very small.

613tResult (3-1)
Compound effect of the amount of sperm of large individuals, α, andthe feeding efficiency, β3-1. Three patterns of the consumption speed of energy storage, c*(t)α pattern 3100

Pattern 1: c*(t) = Cmax → 0

8060
pattern 2

Pattern 2: c*(t) = Cmax → Const.(>0)

pattern 140

Pattern 3: c*(t) = E/T

20
00.05 0.2

0.4

0.6

0.8

1.0

β- The temporal pattern of the consumption speed of energy storage
substantially depends on the feeding efficiency, β.
14Result (3-2)
Compound effect of the amount of sperm of large individuals, α, andthe feeding efficiency, β3-2. Contour graph of the final body size, s(T)
α100
6.080

- When β is large, the final bodysize increases with α and β.

8.0
11.0

- When β is at an intermediatelevel, the final body size depends
only on β.

12.0

60
10.0

40

7.0

20

3.0
1.0

0
0.05 0.2

9.0

5.0- When β is very small, dwarf
males do not grow at all.

2.0 4.0

0.4

0.6

0.8

1.0

β

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Summary- small β, all α-- No growth.
-- Dwarf males use all the resources to reproduction.

- large β, α=0-- Simultaneous growth and reproduction (“intermediate growth”)- large β, α>0-- Comparing toα=0, dwarf males grow larger.
-- They stop growth earlier.
-- Strategy u*(t) tends to “bang-bang control” as α becomes large.

- Males’ body size becomes larger at large β than at small β.
16Discussion (1)- small β, all α : Food-poor environments-- Dwarf males attain higher reproductive success by investing all
the resources to reproduction than growth.

- large β, α=0: Food-rich environments, Dioecious species-- Sperm competition is weak for dwarf males.
-- Dwarf males have advantage to fertilize eggs for a longer time.
-- Large amount of sperm lowers efficiency.
---> “intermediate growth” (simultaneous growth and reproduction )- large β, α>0: Food-rich environments, Androdioecious species-- Sperm competition is severe for dwarf males.
-- To counter large hermaphrodites, dwarf males have advantage inmaking many sperm after becoming larger quickly.
---> approaches to “bang-bang control” (all (u=1) or none (u=0) strategy)
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Discussion (2)Compared between the field data and our model’s result…

Length of dwarf male (mm)Yusa and Yamato, unpublished data
Androdioecious
species

Dioecious
species

210

Hermaphrodite
Female（N = 56)（N = 27)
Sex of large individuals---> Our model explains the field data that dwarf males inandrodioecious barnacles are larger than dioecious ones.

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