Introduction

Largely as a result of human development, the sizes of many wild animal populations have been falling.1 Much of this decline has resulted not from direct killing of wild animals by humans, but indirectly by fragmenting their habitats and cutting animals off from the resources that sustained their historical population sizes.2,3 This has immediate consequences for the welfare of individuals, who may face starvation, competition for resources, or conflict with humans as they try to access resources in their former habitat (e.g. urban pigeons4; fur seal5; black bear6). According to many ethical views, it is better to have more happy animals than fewer equally happy animals; but as a population grows, so does the number of individuals who may be adversely affected by further growth, requiring us to think carefully about trade-offs between the quantity and quality of welfare. A thorough understanding of individual welfare levels in a given population and how they are affected by changes in population density could eventually enable us to estimate the population size which would maximize the welfare experienced in that population.

Population modeling and density-dependence

The density of a population is defined by the ratio between the number of individuals and the amount of habitat or a limiting resource available to them; if a population grows or its habitat declines, population density increases. The population size (N) at which a balance between birth rate and death rate is achieved—its maximum sustainable size—is known as the “carrying capacity” (K) under given environmental conditions (Box 1).

Box 1 A population’s growth rate per generation (r) is equal to the difference between the number of births (b) and the number of deaths (d): r = b - d. The growth rate before any effects of density are taken into account is denoted r 0 . To the extent that r 0 exceeds zero, density-dependent effects must reduce age-specific survival or fecundity rates to achieve a stable population . The population size (N) at which r = 0 and the population remains stable long-term is known as the carrying capacity (K). These parameters come together in the logistic growth equation, where any excess of births over deaths shrinks to zero as N approaches K: dn/dt = r 0 N(K - N)/(K). While this equation captures how extremes of population density relate to population growth rates, it is certainly incomplete. For example, intermediate densities may have less consistent effects than the simple logistic function suggests, especially in populations facing strong density-independent stressors. Populations inhabiting an environment that changes much more rapidly than the time between their generations may never reach their carrying capacity at any given time, but may still encounter strong density effects during periods when N > K, as might occur during a drought or similar period of low productivity.

Death rates may rise and birth rates may fall as a result of stressors experienced by the animals in a population. These stressors can be density-dependent or density-independent. For example, competition for food can lead to starvation and limit the energy available for reproduction7, and crowding in early life can cause an animal to age prematurely8. Infectious diseases may also spread more rapidly in a dense population9.

Other causes of mortality are density-independent. In a river, water temperature or pollutants might kill fish at a rate that is independent of the number of fish in the river10. Indeed, density-independent factors can make an environment so hostile that the maximum stable population size is very low, and the surviving individuals face no meaningful competition for resources.

Whether a certain factor is density-dependent or density-independent can also vary according to the circumstances. For example, the direct threat posed to any individual by fierce winds and lightning during a storm is independent of density, but density-dependent mortality might follow in the wake of the storm if a limiting resource has been damaged. Population density within a certain range can also have positive effects for individual welfare by providing safety in numbers and more accessible social interactions. For example, having fellow newts around during their transition to maturity means that individuals of Triturus cristatus are less likely to undertake dangerous migrations to find mates11.

Welfare implications of population density

In a recent article , I proposed the concept of welfare expectancy as a way to link age-specific survival rates and welfare. Welfare expectancy is a sum of the lifetime welfare an individual born into a given population can expect to experience. Age-specific survival rates may also be correlated with age-specific welfare, suggesting that any density-related impairment of survival could greatly reduce welfare expectancy, especially when it affects juvenile animals.

A general expectation of declining welfare expectancy per individual as a population grows towards its carrying capacity raises the possibility that the population size which maximizes the total welfare of a population may be smaller than its maximum sustainable size. Whether this occurs depends on how many individuals the population would contain at its carrying capacity and how average welfare varies with population size (Box 2). The conditions are most likely to be met if the strongest negative effects of population density on welfare manifest at high densities (i.e., as N approaches K; Figure 1). This could occur under the following three conditions:

if lower densities primarily limit reproduction while higher densities affect survival rates. if density-dependent mortality disproportionately affects older animals at lower densities but young animals at high densities; or if density-independent effects are nearly sufficient to stabilize the population’s growth rate, so any negative effects of density are only felt near the carrying capacity.

In general, the shape of average welfare expectancy with respect to density depends on the particular ecology and life history of the population in question, especially whether density preferentially affects fecundity versus survival rates or the survival rates of young versus old individuals. If increased density is mainly accommodated by reduced fecundity or late-life survival, for instance, then average welfare might not decline so steeply as a population grows. Theory and reviews of demographic models for a variety of large vertebrates suggest that juvenile survivorship is usually the first casualty of high population densities12,13,14, which is one set of conditions under which we might expect the welfare-optimal population density to be lower than the population’s carrying capacity. Part of the explanation for this pattern in large vertebrates is the competitive advantage older animals typically have over younger animals who are smaller and less experienced. However, it is not yet known how broadly these age dynamics apply among the much more numerous smaller-bodied, “ fast-living ” animals, where there are examples of adult advantage15 and disadvantage16. More research is needed into the expected shape and mechanisms of density-dependence among their populations.

Box 2 Let W N represent the average welfare expectancy at population size N and let K represent the population size at carrying capacity. Then, the welfare-optimal population size will be less than the carrying capacity if W K-1 /W K > K/(K - 1). The welfare expectancy per individual in a population depends on the life expectancy from birth and the welfare they typically experience over that lifespan. If average welfare is positive, average welfare expectancy will correlate with life expectancy. If an increase in population density disproportionately harms the youngest animals, this can greatly reduce life expectancy.

Implications for welfare interventions

If we are completely uncertain about the absolute welfare level of animals in a population, then it follows that we are completely uncertain about whether their lives are dominated by pleasure or suffering. If their lives contain more goodness on average, then increasing the population size would be expected to increase the total amount of welfare experienced within the population as a whole. On the other hand, if their lives are primarily unhappy , then adding more unhappy individuals through population growth would actually subtract from the population’s total welfare. From a standpoint of complete uncertainty, then, the expected value of population growth or decline is symmetrical17. Whether or not population growth increases or decreases total welfare of the population depends on the balance of the positive and negative experiences of the individuals added by that growth.

However, the existence of a density-dependent decline in average welfare would set up an asymmetry in favor of relatively smaller populations, since increasing population density would reduce the probability that average welfare remains net-positive, giving population growth a lower expected value than population reduction. Of course, this heuristic assumes complete ignorance of the population’s welfare characteristics, which is rarely the case.