Snapback Repellers in Chaos Theory and their Application in Ecology

Resource Budget Model

reserve

Si t

income byphotosynthesis

PS

investment to flowers

investmentto fruits

reproduction

reproductive threshold

pollen availability

resource depletion coefficient

ki

Pi t

LT

Si t 1 Si t PS if Si t PS LT

Si t PS 1 Pi t ki Si t PS LT otherwise

Yi t1 Yi t 1

kiPi t Yi t 1

if Yi 0

otherwise pollen availability

resource depletion coeff.

ki

Pi t

Yi t normalized reserve

Si t1 Si t PS if Si t PS LT

Si t PS 1 Pi t ki Si t PS LT otherwise

year

Y (t)

reproduce every year

year year

ki 0.7

ki 1.5

ki 4.3

Yi t Si t PS LT PSwith

Pollen Coupling

Pi t 1

N 1Y j t

ji

N

Pollination efficiency for a tree depends on the reproductive activity of others

Yi t1 Yi t 1

kiPi t Yi t 1

if Yi 0

otherwise

β=0.2 β=1.0 β=1.5

Pi(t) pollen coupling strength

Flowering intensity of the forest

pollen availability:

Phase Diagram

Clustering

Desynchronize

Coherent chaoticCoherent periodic

CoherentAnnual

β : po

llen

cou

plin

g st

reng

th

ki : resource depletion coefficient

(Satake and Iwasa 2000, JTB)

1 2 3 4

0

1

2

Annual

CoherentClustering

Desynchronized

To discuss the evolution,

We trace the whole life cycle, not just the number of seeds produced.

Lottery Competition

Forest is composed of a number of canopy trees, and new recruit is possible only when a canopy tree dies.

Life cycle of trees within a year

growth reproduction

deathrecruitment

Growth

Treesaccumulatesresources obtained by photosynthesis

Trees with large reserve produceflower and fruits.This depletes the reserve.

Reproduction

Seed crop ：

i t kiPi t Yi t Whole forest ：

t i t i 2

Trees with large reserve produceflower and fruits.This depletes the reserve.

Reproduction

Death of canopy trees

Annual mortality

A gap is created.Young individuals from seedscompete forthe gap.

FreeSite

An offspring from seeds produced in the year wins the site.

Recruitment

A new individual (haploid) inherits the trait from one of the parents.

It inherits the trait k of the i th tree with probability:

i t 1

2

i t t

Yi t

Y t i 1

j t t ji i 1

N

Small mutations

Adaptive Dynamics

Repeating these processes many times, what remains in the end is evolutionary advantageous.

Thank you for your attention!