An Optimization Model that Links Masting to Seed Herbivory

Glenn Ledder, gledder@math.unl.eduDepartment 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