We saw in Chapter 1 how the IUCN classifies animals or plant populations in terms of their risk of extinction. Thus, species that are identified as critically endangered are those that face an extremely high risk of extinction (greater than 50% over the next 10 years or three generations), endangered species face a very high risk (greater than 1 in 5 over the next 20 years or five generations) and vulnerable species a high risk (greater than 1 in 10 over the next 100 years). In practice, it is rarely possible to calculate these risks explicitly and various surrogates are used (e.g., a population may also be classified as critically endangered if it has declined in abundance by more than 80% over the last decade or three generations, or if population size of less than 50 mature individuals). However, IUCN provides no advice on how species in these different categories should be managed, although it is implicit that any species that falls within one of the ‘threatened’ categories requires management to increase its population rate of increase.
10.5.1 Management approaches
Regulatory frameworks are often established for species that are subject to exploitation, and this is particularly true of fish stocks. In most developed countries this framework is based on what are known as Harvest Control Rules (HCRs): management procedures that rely on the relationship between a species’ current abundance and a series of biological reference points. If a population is found to have fallen below a particular reference point this triggers management action that is aimed at helping the population to recover (Punt and Donovan, 2007). Increasingly, these management procedures also try to take account of the uncertainties that are associated with estimates of abundance, population structure, environmental change and the way in which management action will actually affect exploitation. A classic example of this is the Revised Management Procedure (RMP) developed by the International Whaling Commission to manage the commercial exploitation of large baleen whales, although the Commission has never actually used this procedure.
An HCR that has been implemented is the one used to determine the Potential Biological Removal (PBR) that may be allowed under the US Marine Mammal Protection Act (MMPA). The development of the PBR formula was relatively simple because the MMPA has a single clear objective: to prevent populations from depletion. A population is defined as depleted if it is below the maximum net productivity level, or 50-70% of a historic population size thought to represent the carrying capacity of the environment (Wade, 1998). However, this objective, on its own, is insufficient to allow the evaluation of rival HCRs because it does not specify a time frame. Clearly, an HCR that always sets a zero harvest will achieve the objective of the MMPA, but there are many other HCRs that would allow some harvest, but prevent depletion or allow depleted populations to
recover rapidly. In fact, rival HCRs were evaluated using two criteria (Wade, 1998): there should be a 95% probability that any population, regardless of its current size, will no longer be depleted after 100 years, and there should be a similar probability that a population that is not currently depleted will remain so for at least 20 years.
In practice, application of either the PBR or the RMP is likely to result in low harvest rates of less than 2% of the population. PBR essentially provides a rule of thumb that takes account of uncertainty in the available estimates of abundance, which can be used when there is very limited information on the species being managed. It allows an estimate of the allowabletake of animals from a population. Interpretation of a take is straightforward if this involves either direct mortality through harvesting or, as is more usually the case, indirect mortality (e.g., as a result of entanglement of animals in fishing gear). However, a take can be defined as the number of animals that suffer disturbance if this results in an increased risk of mortality or reduced fecundity. Takes of this kind were not considered in the development of the PBR. The deliberate or accidental capture of an individual not only results in certain death, it also removesallfuture offspring that may have been born to that individual. Although changes in behaviour could have similar consequences, these will probably be rare and there will always be a degree of uncertainty associated with the predicted effects.
Nevertheless, the central methodology used in devising the PBR is still applicable to takes that are associated with disturbance. At the core of this methodology is an assessment of the impact of the take on the underlying (stochastic) growth rate of the population. If an approach like PBR is to be used to regulate takes that involve disturbance, it is necessary to develop ways of calculating how changes in behaviour impact on population growth rate. As we have seen in Chapter 9, there are a number of mathematical and statistical frameworks that could be used to achieve this. In Chapter 12 we consider how these frameworks might be implemented.
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B
Figure 10.1: Estimates of true pup production from two models of grey seal population dynamics fit to pup production estimates from 1984-2007 in four regions (see Thomas and Harwood, 2008 for details). Input data are shown as circles, while the lines show the posterior mean bracketed by the 95% credibility interval.
(A) Extended density dependent survival with no movement model (B) Extended density dependent fecundity with no movement model
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Figure 10.2: Estimates of adult population size from two models of grey seal population dynamics fit to pup production estimates from 1984-2007 in four regions (see Thomas and Harwood, 2008 for details). Lines show the posterior mean bracketed by the 95% credibility interval.
(A) Extended density dependent survival with no movement model (B) Extended density dependent fecundity with no movement model
CHAPTER
11
MATHEMATICAL AND STATISTICAL FRAMEWORKS FOR MODELLING POPULATION DYNAMICS
11.1 Introduction
In this chapter, we give an overview of the mathematical frameworks available for modelling the population dynamics of marine mammals. We also provide a more detailed description of a statistical framework for specifying discrete stage, discrete time models and their link to both demographic and population size data, as well as an overview of methods for fitting these models.