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COS 127-6
Frequency-dependence and elasticity of population growth rate: Implications for two-sex models

**Background/Question/Methods**

An age- or stage-structured population is frequency-dependent when vital rates depend on the relative frequencies of individuals of different ages or stages. A well-known example where frequency-dependence occurs is a two-sex model where, for instance, fertility depends on the sex ratio **α** (ratio of adult females to adult males) of the population. It is known that such populations can have a long-term growth rate ** λ** and a stable stage structure. Hence it is natural to ask how

**would respond to changes in vital rates. In density-independent models, elasticity of**

*λ***to a vital rate, i.e., the proportional change in**

*λ***, can be analytically evaluated from the projection matrix of vital rates. However, in frequency-dependent models it is not explicitly known how frequency-dependence affects the response of**

*λ***to perturbation of a vital rate. For instance, in two-sex model, how does sex ratio affects the elasticity of**

*λ***to male survival? Perturbation of a vital rate in such models has two effects: a linear effect through change in the value of the vital rate and a secondary (non-linear) effect through a change in the stage structure.**

*λ***Results/Conclusions **

We derive an explicit formula for the elasticity of ** λ** to a vital rate in frequency-dependent model. This elasticity is the sum of two components: first component is simply the linear elasticity evaluated from the equilibrium projection matrix as in density-independent models. The second component captures the effect of frequency-dependent vital rates through changes in stage structure. We use this result to derive elasticity to male and female survival rates in a two-sex model where male and female fertilities are modeled using a harmonic function that includes the average harem size

**to reflect the mating system (e.g., polygyny). We show that the nonlinear component of the elasticity vanishes only when**

*h***/**

*h***α**= 1. In a strictly monogamous species (

**= 1), elasticity to female survival is larger than elasticity to male survival when**

*h***α**< 1 (less females) and vice versa. In a polygynous species (

**> 1), elasticity to female survival can be larger than that of male survival rate even when sex ratio is female biased (**

*h***α**> 1) but decreases when

**α**>>

**. Our results explicitly show the dependence of the response of**

*h***on the demography (**

*λ***α**) and the mating system (

**).**

*h*