Download An Optimization Model that Links Masting to Seed Herbivory

An Optimization Model
that Links Masting to Seed
Herbivory
Glenn Ledder, [email protected]
Department of Mathematics
University of Nebraska-Lincoln
Background
• Masting is a life history strategy in which
reproduction is deferred and resources
hoarded for “big” reproduction events.
Background
• Masting is a life history strategy in which
reproduction is deferred and resources
hoarded for “big” reproduction events.
• A tree species in Norway exhibits masting
with periods of 2 years or 3 years based on
geography. Any theory of masting must
account for periodic reproduction with
conditional period length.
Background
• Masting often occurs at a population level.
For simplicity, we assume either that
individuals are isolated or that coupling is
perfect, removing the issue of synchrony.
Background
• Masting often occurs at a population level.
For simplicity, we assume either that
individuals are isolated or that coupling is
perfect, removing the issue of synchrony.
• The Iwasa-Cohen life history model predicts
both annual and perennial strategies, but not
masting.
Biological Question
• What features of a plant’s physiology and/or
ecological niche can account for masting?
Biological Question
• What features of a plant’s physiology and/or ecological niche
can account for masting?
Fundamental Paradigm
• Natural selection “tunes” a genome to achieve
optimal fitness within its ecological niche.
Biological Question
• What features of a plant’s physiology and/or ecological niche
can account for masting?
Fundamental Paradigm
• Natural selection “tunes” a genome to achieve optimal fitness
in its ecological niche.
Simplifying Assumption
• Optimal fitness in a stochastic environment is
roughly the same as optimal fitness in a fixed
mean environment.
Model Structure
X𝟎
Growth
Xi = ψ ( Yi-1 )
X𝒊
Allocation
Yi = Y ( Xi )
Reproduction W𝒊
Wi = W ( Xi – Yi )
Y𝒊
• Resource levels:
before allocation = X𝒊
after allocation =Y𝒊
• Reproductive value =W𝒊
• First reproduction = year 0
• Yearly survival probability = σ
• Theoretical fitness: F = W𝟎 + σW𝟏 + σ𝟐W𝟐 + ⋯
The Optimization Problem
Specify
adult yearly survival probability:
Growth model:
Reproduction model:
σ
Xi = ψ (Yi-1 )
Wi = W (Xi - Yi )
Determine
the allocation strategy 𝒀(Xi ) that maximizes fitness
F = W𝟎 + σW𝟏 + σ𝟐W𝟐 + ⋯
Growth Model
Mathematical Properties:
No input means no output
Excess input is not wasted
Additional input has diminishing returns
ψ(0) = 0
ψ′ ≥ 1
ψ′ ≤ 0
2
The specific function is
determined by an optimization
problem for the growing season.
1.5
 (s0 )
1
s0
0.5
0
0
0.2
0.4
0.6
s0
0.8
1
Reproduction Model
We assume that reproduction value is diminished
by startup cost 𝑴 and perfectly efficient seed
herbivores with capacity 𝑪. That is
𝑹 − 𝑴 − 𝑪,
𝑾 𝑹 =
𝟎,
𝑹>𝑴+𝑪
𝑹<𝑴+𝑪
Preferred-Storage Allocation:
An Important Special Case
• The formula
F = W𝟎 + σW𝟏 + σ𝟐W𝟐 + ⋯
is difficult to compute.
Preferred-Storage Allocation:
An Important Special Case
• The formula
F = W𝟎 + σW𝟏 + σ𝟐W𝟐 + ⋯
is difficult to compute.
• Fitness calculations for preferred-storage
allocation strategies require computation of
finitely-many growing seasons and 2
reproduction calculations.
Preferred-Storage Allocation
Assume that the plant “prefers” to store a fixed
amount 𝒀, provided a threshold 𝑳 ≥ 𝒀 is
exceeded:
• If 𝑿 ≥ 𝑳, store 𝒀;
use 𝑹 = 𝑿 − 𝒀 for reproduction.
• Otherwise, store everything.
𝑿,
𝒀 𝑿 =
𝒀,
𝑿<𝑳
𝑿≥𝑳
Preferred-Storage Fitness
𝒀
𝝍
𝝍𝒋+𝟏 𝑿𝟎
𝝍𝒋 𝑿𝟎
𝒋=𝑱
𝒀
𝐗 > 𝐋?
𝒋<𝑱
𝒀
𝑹𝑱 = 𝝍𝑱 𝑿𝟎 − 𝒀
𝑹𝟐𝑱 = 𝝍𝑱 𝑿𝟎 − 𝒀
𝑹𝟑𝑱 = 𝝍𝑱 𝑿𝟎 − 𝒀
⋮
Preferred-Storage Fitness
𝒀
𝝍
𝝍𝒋+𝟏 𝒀
𝝍𝒋 𝒀
𝒋=𝑱
𝒀
𝐗 > 𝐋?
𝒋<𝑱
𝒀
𝑹𝑱 = 𝝍𝑱 𝒀 − 𝒀
𝑹𝟐𝑱 = 𝝍𝑱 𝒀 − 𝒀
𝑹𝟑𝑱 = 𝝍𝑱 𝒀 − 𝒀
⋮
In general, if 𝝍𝒋−𝟏 𝒀 < 𝑳 ≤ 𝝍𝒋 𝒀 , the life
history is periodic with a period of j years.
𝑭 𝑿𝟎 = 𝑾 𝑿𝟎 − 𝒀 +
𝝈𝒋
𝑾
𝒋
𝟏−𝝈
𝝍𝒋 𝒀 − 𝒀 .
Optimal Preferred-Storage Strategy
PROBLEM:
Determine the preferred-storage strategy
𝑿,
𝒀 𝑿 =
𝒀,
𝑿<𝑳
𝑿≥𝑳
to maximize
𝝈𝑱
𝑾
𝑱
𝟏−𝝈
𝑭 𝒀, 𝑳; 𝑿𝟎 = 𝑾 𝑿𝟎 − 𝒀 +
𝝍𝑱 𝒀 − 𝒀
where J is determined by
𝝍𝑱−𝟏 𝒀 < 𝑳 ≤ 𝝍𝑱 𝒀 .
Optimal Preferred-Storage Strategy
SOLUTION:
1. Use calculus to find optimal storage amount
𝒀𝑱 for masting period J.
2. Use continuity to find optimal cut-off value
𝑳𝑱 for given J and 𝒀𝑱 .
3. Use algebra to find optimal masting period
J* for given 𝒀𝑱 and 𝑳𝑱.
Optimal Preferred-Storage Strategy
Masting occurs when annual
reproduction is possible
𝑾 𝝍 𝒀𝟏 − 𝒀𝟏 > 𝟎
Optimal Preferred-Storage Strategy
Masting occurs when annual
reproduction is possible, but 2-year
cycles are better:
𝝈
𝑾 𝝍𝟐 𝒀𝟐 − 𝒀𝟐 > 𝑾 𝝍 𝒀𝟏 − 𝒀𝟏 > 𝟎
𝟏+𝝈
Masting Period
C +M
J=5
J=4
J=3
J=2
J=1
σ
Increasing either the survival parameter 𝝈 or the fixed cost
parameter 𝑪 + 𝑴 increases the optimal period.
Allocation Parameters
J=5
J=4
J=3
J=2
J=1
Increasing the herbivory parameter 𝑪 increases the cut-off
parameter 𝑳 continuously, but changes in storage
parameter 𝒀 are discrete.
Claim: The optimal preferred-storage
strategy is optimal among all
strategies. Established by dynamic
programming:
1.
2.
3.
4.
Let 𝒀∗ 𝑿 be the optimal preferred-storage
strategy.
Define 𝑭𝟎 (𝑿) = 𝑾 𝑿 − 𝒀∗ (𝑿) +
𝝈𝑭𝟎 𝝍(𝒀∗ (𝑿)
Define 𝚪 𝒀; 𝑿 = 𝑾 𝑿 − 𝒀 + 𝝈𝑭𝟎 𝝍(𝒀)
Show that 𝒀∗ 𝑿 maximizes 𝚪.