Beyond the Mean: Sensitivities of the Variance of Population Growth

Abstract

Populations in variable environments are described by both a mean growth rate and a variance of stochastic population growth. Increasing variance will increase the width of confidence bounds around estimates of population size, growth, probability of and time to quasi-extinction. However, traditional sensitivity analyses of stochastic matrix models only consider the sensitivity of the mean growth rate. We derive an exact method for calculating the sensitivity of the variance in population growth to changes in demographic parameters. Sensitivities of the variance also allow a new sensitivity calculation for the cumulative probability of quasi-extinction. We apply this new analysis tool to an empirical dataset on at-risk polar bears to demonstrate its utility in conservation biology We find that in many cases a change in life history parameters will increase both the mean and variance of population growth of polar bears. This counterintuitive behaviour of the variance complicates predictions about overall population impacts of management interventions. Sensitivity calculations for cumulative extinction risk factor in changes to both mean and variance, providing a highly useful quantitative tool for conservation management. The mean stochastic growth rate and its sensitivities do not fully describe the dynamics of population growth. The use of variance sensitivities gives a more complete understanding of population dynamics and facilitates the calculation of new sensitivities for extinction processes.

Keywords: conservation; extinction; polar bears; population growth; population viability; sensitivity; stochastic matrix model; variance of population growth.

Figure 1

A schematic illustration of the different effects of changing the mean versus the variance of a distribution. A: An example distribution of $\frac{\text{log}\mathrm{\Lambda }(t)}{t}$. In the limit of large t, the mean of this distribution converges on a, and the variance to υ. B: Distribution A after changing only the mean C: Distribution A after changing only the variance D: Distribution A after an increase to both mean and variance

Figure 2

Polar bear life cycle diagram (from Hunter et al. 2010)

Figure 3

Habitat-specific sensitivity of a and υ to changes in cubless adult survival

Figure 4

Elasticity of a and υ to perturbation of the means of parameters

Figure 5

Elasticity of a and υ to perturbation of the variance of parameters

Figure 6

Elasticity of P_q(t), probability of quasiextinction before time t, to changes in the means (top panel) and variances (bottom panel) of vital rates. Note that elasticity is maximal at 〈T〉, which has value (from left to right) 226.8, 53.1 and 22.6 years respectively. From left to right, values of a are 0.0203, −0.0868, −0.203

Figure 7

Sensitivity of P_q(t), probability of quasiextinction before time t, to habitat-specific changes in vital rates. From left to right, values of a are 0.0203, −0.0868, −0.203. Sensitivities shown are all for the case t = 〈T〉, to the nearest year, (from left to right) 227, 53 and 23 years respectively.