Population density in August (birds/km2)

100 studies of terrestrial and marine mammal populations where density dependence was detected.

Delayed density dependence has been recorded in winter mortality of snowshoe hares in the Yukon and in overwinter mortality of red grouse in Scotland (Watson and Moss 1971) (Fig. 8.11). For the hares the delay appears to have been due to a lag of 1-2 years in the response of predator populations to changing hare numbers (Trostel et al. 1987), while for the grouse the delay came from density responding to food conditions in the previous year (see Section 8.8.3).

8.5 Applications of Causes of population change can be divided into (i) those that disrupt the popula-regulation tion and often result in "outbreaks," and which can be either density dependent or density independent; and (ii) those that regulate and therefore return the population to original density after a disturbance (Leirs et al. 1997). These are always density dependent.

Knowledge of regulation may be useful for management of house mice (Mus domes-ticus) plagues in Australia. In one experimental study (Barker et al. 1991), mice in open-air enclosures were contained by special mouse-proof fences. The objective was to create high densities, thus mimicking plague populations, in order to test the regulatory effect of a nematode parasite (Capillaria hepatica). It turned out that they could not test the effect of the parasite because other factors regulated the population and thus obscured any parasite effect. The replicated populations declined simultaneously. Why did this happen? By dividing up the life cycle into stages they found that late juvenile and adult mortality was strongly density dependent but that other stages, including fertility and newborn mortality, were not. This allowed them to discount causes that would affect reproduction and focus more closely on what was happening amongst adults, in particular the social interactions of mice.

Other studies suggest that mouse populations in Australia may be regulated by predators, disease, and juvenile dispersal (Redhead 1982; Sinclair et al. 1990). Under conditions of superabundant food following good rains, the reproductive rate of females increased faster than the predation rate, and an outbreak of mice occurred. The implication of these results for management is that if reproduction could be reduced, for example through infections of the Capillaria parasite, then predation may be able to prevent outbreaks even in the presence of abundant food for the mice.

8.6 Logistic model In Chapter 6, we derived geometric and exponential growth models. In 1838, Pierre-of population Francois Verhulst published a paper (Verhulst 1838) that challenged the assumption regulation of unlimited growth implicit in these models. Verhulst argued that the per capita rate of change (dN/Ndt) should decline proportionately with population density, simply due to a finite supply of resources being shared equally among individuals. If each individual in the population gets a smaller slice of the energy "pie" as N increases, then this would prevent them from devoting as much energy to growth, reproduction, and survival than would be possible under ideal conditions. As we saw in Chapter 6, changes in demographic parameters lead to corresponding changes in the finite rate of population growth Xt or its equivalent exponential rate rt, where t denotes a specific point in time. Other factors, such as risk of disease, shortage of denning sites, or aggressive interactions among population members, might also cause the rate of population growth to decline with population size. The simplest mathematical depictions of such phenomena are commonly termed "logistic" models.

There are numerous ways to represent logistic growth. For simplicity, we will focus on population growth modeled in discrete time, which is often a reasonable approximation for species that live in a seasonal environment. One of the most commonly used forms is called the Ricker equation, in honor of the Canadian fisheries biologist, Bill Ricker, who first suggested its application to salmon stocks (Ricker 1954):

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