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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.