Density Dependent Selection

Scientists today agree that environments differing in population abundance or fluctuation select for different strategies. This insight is, besides fast-slow continua, the second aspect that has been preserved from the r/K concept. An empirical example comes from the island of St. Kilda, Scotland, where Soay sheep (Ovis aries) with a darker coat had a higher survival rate at high densities compared to sheep with a lighter coat. Similarly, sheep with unscurred horns survived better at intermediate densities, while sheep with scurred horns did better at high densities. Other empirical examples for density-dependent selection come from Drosophila flies reared in the laboratory, tadpoles in temporary ponds, or guppies (Poecilia reticulata) in Trinidad.

The r/Kconcept suffered from not deriving an explicit (and correct) link between density-dependent selection and the life-history traits favored by selection. In modern models that consider density-dependent regulation, such a link is made specific: an increase in population density affects the life history which feeds back negatively on population growth. For example, an increase in population density could lead to a lower fecundity. Which trait or traits change and how they change exactly depends on the nature of the specific model.

The procedure used in such models is clear in principle: a population consisting of individuals with particular life-history traits will obey characteristic fluctuations that depend not only on the traits themselves but also on the resources available (resource use being perhaps a function of the life-history traits in question), as well as any environmental fluctuations. An alternative life history can invade if it enters the population in a way that eventually creates more descendants than what the original strategy was able to achieve. 'Eventually' here refers to the fact that the number of descendants left in one generation is not necessarily maximized; for example, if fast reproduction is essential, then a strategy that leaves few but early offspring may eventually win over others. Counting the 'eventual' offspring is made mathematically precise by calculating the so-called invasion theta, or Lyapunov exponent, of the linearized system describing the invasion of a population.

Calculating the Lyapunov exponent is not a trivial procedure, however: one can liken it to a tool that is so general that it sometimes resembles a hammer while at other times it provides the scissors needed to dissect a given problem. Put more precisely, the effect that density regulation has on the optimal life history has been shown to depend strongly on the exact type of life cycle and on the way stochasticity is incorporated. It would consequently be useful to know what shape this tool takes for biologically relevant questions, such as the traits that underlie fast-slow continua. Some progress has been made in this direction. An important paper by Mylius and Diekmann published in

1995 showed that there are conditions under which natural selection will always favor strategies that maximize r. an example is density regulation that increases the mortality of all age classes. Density-dependent juvenile mortality, on the other hand, implies that the correct measure of fitness is R0, the expected lifetime reproductive success. However, for many situations neither r nor R0 are correct measures of fitness and there is then no other escape than to have exact knowledge of the type of density regulation present - or at least make the assumption explicit - while deriving the invasion prospects by calculating the demographic consequences of each strategy.

That the mathematics of fitness measurements produces such nontrivial conclusions regarding individual fitness estimation is bad news for empiricists, who are rarely blessed with easy estimates of density regulation in the populations they study. In terms of ease of calculation, the two most practical measures of individual fitness are A, which is the discrete-time version of r (A = er if A and r are measured at equivalent timescales), and the lifetime reproductive success R0, which simply equals the number of offspring and is often abbreviated as LRS. A typical pragmatic approach is to draw conclusions based on detailed analysis of one measure or to calculate both A and LRS and compare the results. There are also examples where more detailed data on individual performance in density-regulated populations allow building tailor-made models, giving therefore much more power to draw inferences regarding optimal strategies in a particular species. However, it has also been argued that for many practical applications, simple measures such as the lifetime reproductive success perform quite well.

It is probably fair to say that theoretical work is currently better equipped to answer questions about a single trait, for example, age of first reproduction, than about suites of traits such as fast or slow life histories. Clearly, tradeoffs between current and future investment must lie behind such suites but we do not know enough about how they evolve exactly according to prevailing environmental conditions, including the different stochastic components of life cycles, as exemplified by those determining recruitment of birds into territorial populations. Some theoretical results exist where variable nonequilibrium dynamics select for slower rather than faster life history, and they clearly pose a puzzle to be solved in this context. The solution may lie in the fact that detailed knowledge of the density regulation really matters. Indeed, alternative assumptions concerning variability and temporal correlations of vital rates lead easily to opposite conclusions regarding the effect of variability on the 'speed' of a life history. Thus, it is unclear whether or under which conditions different types of density regulation evolve and how density regulation is related to life-history traits. Future work in this area might help to find explanations for observed life-history patterns.

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