Population-level Assessment of Risks of Pesticides to Birds and Mammals in the UK

Abstract

It is generally acknowledged that population-level assessments provide a better measure of response to toxicants than assessments of individual-level effects. Population-level assessments generally require the use of models to integrate potentially complex data about the effects of toxicants on life-history traits, and to provide a relevant measure of ecological impact. Building on excellent earlier reviews we here briefly outline the modelling options in population-level risk assessment. Modelling is used to calculate population endpoints from available data, which is often about individual life histories, the ways that individuals interact with each other, the environment and other species, and the ways individuals are affected by pesticides. As population endpoints, we recommend the use of population abundance, population growth rate, and the chance of population persistence. We recommend two types of model: simple life-history models distinguishing two life-history stages, juveniles and adults; and spatially-explicit individual-based landscape models. Life-history models are very quick to set up and run, and they provide a great deal of insight. At the other extreme, individual-based landscape models provide the greatest verisimilitude, albeit at the cost of greatly increased complexity. We conclude with a discussion of the implications of the severe problems of parameterising models.

Appendix A: Mathematics of the two-stage life-history model

Suppose the performance of juveniles differs from that of adults, but that there are no differences among juveniles or among adults. In this case we need to know five parameters: juvenile and adult mortality rates, age at first breeding, number of offspring produced at each breeding attempt, and interval between breeding attempts. Let subscripts a and j denote adults and juveniles, let μ represent instantaneous mortality rate, t time period, and n the number of offspring produced at each breeding attempt. Thus t _j is age at first breeding, and t _a is the interval between breeding attempts, and μ _a and μ _j are adult and juvenile mortality rates. Note that the relationship between annual survival and instantaneous mortality rate, μ, is given by

$$\hbox {Annual survival} = \hbox{e}^{-\mu}$$

(A.1)

$$\hbox {And the probability of surviving from birth to first breeding} = {\rm e}^{-\mu_{\rm j}t_{\rm j}}$$

(A.2)

The population growth rate, measured as r, is then determined from the equation

$$1={1\over 2} n{\rm e}^{-(\mu_{\rm j}+r)t_{\rm j}}+{\rm e}^{-(\mu_{\rm a}+r)t_{\rm a}}$$

(A.3)

(Calow and Sibly, 1990). This is the Euler–Lotka equation for the two-stage model. Note that r is measured per unit time and that the calculation requires the assumption of a stable age distribution, although in practice this assumption may not be restrictive (Sibly and Smith, 1998). Further details of the two-stage model including methods for calculating confidence intervals for r can be found in (Lande, 1988; Sibly et al., 2000a).

An alternative parameterisation in terms of survivorships S _j and S _a is sometimes useful:

$$1={1\over 2} nS_{\rm j}{\rm e}^{-rt_{\rm j}}+S_{\rm a}{\rm e}^{-rt_{\rm {a}}}$$

(A.4)

An interesting extension of the above is to make transitions between the stages depend probabilistically on other factors such as mating success or ability to obtain territories. There are many advantages to analysing such types using matrices, the standard reference being (Caswell, 2001).