Modeling the demographic effects of endocrine disruptors

Abstract

In this article we describe a series of strategic models of populations and individuals subject to challenge by endocrine disruptors. These models are not designed to be fitted to detailed data on specific species but rather are intended to provide general insights on the relative importance of different demographic mechanisms in the population context. Therefore, the models contain the minimum necessary biological detail, but in recompense they are highly accessible to mathematical analysis. We show that, over a range of models with contrasting biological detail, population viability is controlled by the number of female offspring that result from the average female's lifetime reproductive activity. Thus, male fertility changes have little effect at the population level until they become severe enough to reduce this average female output. We argue that in many circumstances endocrine disruptors are likely to produce directly deleterious effects on female fecundity at levels far below those required to reduce male fertility to dangerously low levels. Finally, we formulate a simple model of individual energetics that we argue can form the basis of a strategic discussion of the likely sensitivity of female demographic parameters to chemically induced changes in physiological function.

Figure 1

A population with no density dependence. (A) The time history of adult female density and adult male-to-female ratio for a population with a 1:1 sex ratio (ρ = 0.5), half-saturation male-to-female ratio R_h = 0.2, juvenile survival S_j = 0.1, and normalized fecundity βτ= 10, initialized with 0.1 females and 0.005 males per unit area. (B) Normalized long-run growth rate, λτ, as a function of BF^*τ with μτ as a parameter. The dotted line shows the boundary between growing and decaying populations (λ= 0).

Figure 2

A population with density-dependent egg hatching. (A) The time history of adult female density and adult male-to-female ratio for a population with ρ = 0.5, R_h = 0.2, μ_fτ = μmτ, S_j = 0.1, βτ = 10, and R_h = 5, intitialized with 0.1 females and 0.005 males per unit area. (B) Normalized steady-state adult female abundance, as a function of BF^* with μ as a parameter.

Figure 3

Steady-state variation with male parameters under model 2. (A) Normalized steady-state abundance of adult females as a function of half-saturation sex ratio (R_h) for a population with R* = 1, and B/μ_f = 4. (B) Normalized steady-state adult female abundance, as a function of μ_m/μ_f for a population with B/μ_f = 4 and R_h = 0.1.

Figure 4

Simple individual growth model. (A) Carbon weight against time for an individual with ζ = 0.09, κ = 0.1 growing in a constant environment, implies a value of φ as marked. (B) Asymptotic fecundity against assimilation scale (φ) with ζ as a parameter for an individual with κ= 0.1.