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 λ 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.

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 h to reflect the mating system (e.g., polygyny). We show that the nonlinear component of the elasticity vanishes only when h/α = 1. In a strictly monogamous species (h = 1), elasticity to female survival is larger than elasticity to male survival when α < 1 (less females) and vice versa.  In a polygynous species (h > 1),  elasticity to female survival can be larger than that of male survival rate even when sex ratio is female biased (α > 1) but decreases when α  >> h. Our results explicitly show the dependence of the response of λ on the demography (α) and the mating system (h).