An Optimization Model that Links Masting to Seed Herbivory
Glenn Ledder, [email protected] of MathematicsUniversity 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 StructureX𝟎
Y𝒊
W𝒊X 𝒊
• Resource levels:
• Theoretical fitness: F = ⋯• Reproductive value = • First reproduction = year 0• Yearly survival probability = σ
Growth Xi = ψ ( Yi-1 )
AllocationYi = Y ( Xi )
ReproductionWi = W ( Xi – Yi )
The Optimization Problem
Specify adult yearly survival probability: σ Growth model: Xi = ψ (Yi-1 ) Reproduction model: Wi = W ( - Yi )Determine the allocation strategy ) that maximizes fitness F = ⋯
Growth ModelMathematical Properties: No input means no output ψ (0) = 0 Excess input is not wasted ψ′ ≥ 1 Additional input has diminishing returns ψ′ ≤ 0The specific function is determined by an optimization problem for the growing season.
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
s0
s0
(s0 )
Reproduction Model
We assume that reproduction value is diminished by startup cost and perfectly efficient seed herbivores with capacity . That is
Preferred-Storage Allocation: An Important Special Case
• The formula F = ⋯ is difficult to compute.
Preferred-Storage Allocation: An Important Special Case
• The formula F = ⋯ 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 , provided a threshold is exceeded:
• If , store • Otherwise, store everything.
Preferred-Storage Fitness𝒀
𝝍 ? 𝒀𝝍 𝒋+𝟏 (𝑿𝟎 )
𝝍 𝒋 (𝑿𝟎 )
𝑹 𝑱=𝝍 𝑱 (𝑿𝟎 )−𝒀
𝒀
𝒋= 𝑱
𝒋< 𝑱𝑹𝟐 𝑱=𝝍 𝑱 (𝑿𝟎 )−𝒀𝑹𝟑 𝑱=𝝍 𝑱 (𝑿𝟎 )−𝒀⋮
Preferred-Storage Fitness𝒀
𝝍 ? 𝒀𝝍 𝒋+𝟏 (𝒀 )
𝒀
𝒋= 𝑱
𝒋< 𝑱 𝑹𝟑 𝑱=𝝍 𝑱 (𝒀 )−𝒀⋮
In general, if ≤ , the life history is periodic with a period of j years.
.
𝑹𝟐 𝑱=𝝍 𝑱 (𝒀 )−𝒀𝑹 𝑱=𝝍 𝑱 (𝒀 )−𝒀
𝝍 𝒋 (𝒀 )
Optimal Preferred-Storage Strategy
PROBLEM:Determine the preferred-storage strategy
to maximize where J is determined by
≤ .
Optimal Preferred-Storage Strategy
SOLUTION:
1. Use calculus to find optimal storage amount for masting period J.
2. Use continuity to find optimal cut-off value for given J and .
3. Use algebra to find optimal masting period J* for given and .
Optimal Preferred-Storage Strategy
Masting occurs when annual reproduction is possible
Optimal Preferred-Storage Strategy
Masting occurs when annual reproduction is possible, but 2-year cycles are better:
Masting PeriodC +M
σJ=2
J=3
J=4
J=5
J=1
Increasing either the survival parameter or the fixed cost parameter increases the optimal period.
J = 5J = 4
J = 1
J = 3J = 2
Increasing the herbivory parameter increases the cut-off parameter continuously, but changes in storage parameter are discrete.
Allocation Parameters
Claim: The optimal preferred-storage strategy is optimal among all strategies. Established by dynamic programming:1. Let be the optimal preferred-storage strategy.2. Define 3. Define 4. Show that maximizes