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Keywords:

  • chaos;
  • life histories;
  • matrix models;
  • population dynamics;
  • sensitivity analysis

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. References
  • 1
    Elasticity and sensitivity analyses are used widely in evolutionary biology, ecology and population management. However, almost all applications ignore density dependence, despite the widespread assumption that density dependence is ubiquitous. We assess whether this matters by comparing density-dependent and density-independent elasticity analyses for the LPA model of Tribolium.
  • 2
    Density-independent elasticities of λ are a poor indicator of the effects of changes in demographic parameters on population size, even for populations at stable equilibrium. With non-equilibrium dynamics, the divergence can be particularly large. In the extreme, a change in a demographic parameter with a positive effect on individual fitness can reduce mean population size, so even the sign of a density-independent elasticity may be wrong. Elasticities of larval, pupal and adult numbers are not proportional to each other, neither are they proportional to elasticities of total population size.
  • 3
    A full density-dependent analysis is therefore vital when concerned with effects on population numbers, as in population management, pest control and prediction of population effects of toxins.
  • 4
    When examining the consequences for individual fitness of changes in demographic parameters, density-independent elasticities provide a more useful approximation to the density-dependent values. However, they fail to detect cases where non-equilibrium dynamics means that particular life histories gain an advantage by exploiting predictable periods when density dependence is relaxed.
  • 5
    This phenomenon can produce a marked change in the pattern of elasticities as a bifurcation is crossed. The corresponding changes in selection pressures may act to stabilize dynamics in some circumstances and destabilize them in others. There is no single answer to the question of whether selection should favour equilibrium or non-equilibrium dynamics.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. References

Elasticity and sensitivity analyses have proved to be very powerful tools in evolutionary biology, ecology and population management (Benton & Grant 1999; Caswell 2001). The standard analysis examines the effects of small changes in demographic parameters on population growth rate, λ. For a density-independent population this also quantifies their effects on individual fitness (De Kroon et al. 1986; Benton & Grant 1999; Caswell 2001). Extension of this analysis to include environmental stochasticity is straightforward (Benton & Grant 1996; Grant & Benton 2000). Elasticity analysis can also be carried out when population growth is density dependent, but here there are a number of potential complications that must be taken into account. Takada & Nakashizuka (1996) developed a model of a forest tree in which density dependence was a simple function of total population size. They showed that elasticities of population size were proportional to elasticities of λ evaluated at the equilibrium point. More generally, this proportionality will hold for any weighted sum of the abundance of age or stage classes, provided population regulation is a simple function of the same measure of population size (Takada & Nakajima 1992, 1998; Caswell 2001). However, this result (which Caswell refers to as ‘the Takada–Nakajima theorem’) holds only if population regulation is a function of a scalar measure of population size and the population is at stable equilibrium (Benton & Grant 2003; Grant & Benton 2000). Our examination of a number of population models has shown that the consequences of a change in a demographic parameter for individual fitness are not necessarily proportional to its effects on population size. One or both of these analyses may give very different results from a density-independent analysis (Grant 1997; Grant & Benton 2000; Benton & Grant 2003). Our terminology follows Grant & Benton (2000).

For populations at stable equilibrium, the fitness consequences of changes in demographic parameters may be assessed by calculating elasticities of λ for the projection matrix evaluated at the equilibrium point. For populations that experience stochastic environmental fluctuations or show non-equilibrium dynamics, we need to examine how changes in demographic parameters of an invading genotype or clone alter its ability to displace a resident. This is done using elasticities of the invasion exponent ϑ (Grant 1997; Benton & Grant 2000). A density-independent analysis on the observed mean projection matrix often gives a good approximation to this for populations subject to stochastic perturbations, but for models examined so far this approximation is much poorer when populations exhibit non-equilibrium dynamics (Benton & Grant 2000; Grant & Benton 2000) or experience occasional ‘catastrophic’ reductions in vital rates.

When we are concerned with the effects of changes in demographic parameters on population size, the potential errors that arise from ignoring density dependence become much more serious. The effect of a change in a demographic parameter on population size can be completely different from its effect on individual fitness, and from the results of a density-independent analysis. This is particularly true if population regulation is not a simple function of the total number of individuals in the population. For example, in the rather common situation where there is competition for a fixed number of territories, it is possible that a change in a demographic parameter that has a large impact on individual fitness may have no effect on equilibrium population size (Grant & Benton 2000). With overcompensating density dependence, a change in a demographic parameter that produces an increase in individual fitness may even lead to a decrease in mean population size. This occurs quite commonly when populations show non-equilibrium dynamics (Grant & Benton 2000) but can also occur for populations at a stable equilibrium (see Benton & Grant 1999, Box 4; Smith, Reynolds & Sutherland 2000; Benton & Grant 2002; Moe, Stenseth & Smith 2002). As many applications of elasticity analysis are to the management of populations (see Benton & Grant 1999 for a discussion) these results have important practical implications.

Our previous work has examined simple population models, and has served to map out the potential pitfalls in applying a density-independent elasticity analysis to populations that are, in reality, density dependent. To assess the extent to which these problems actually do occur we need to apply these methods to realistic models of real populations. Here we do this by applying the methods for the elasticity analysis of density-dependent populations to the ‘LPA model’ of flour beetle, Tribolium castaneum (Herbst.) (Dennis et al. 1995). These experimental populations are regulated by cannibalism of eggs and pupae by larvae and adults, and food is not limiting. Caswell (2001) gives a brief discussion of perturbation analyses of this model. For a single set of parameter values that gave a stable equilibrium, he found that the elasticities of λ and of equilibrium population size were very similar. In a companion paper (Benton & Grant 2003) we carry out a similar analysis for a mathematical model of laboratory populations of a mite, Sancassania berlesei (Michael) exposed to stochastic environmental variation.

Methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. References

The LPA model is a discrete time model of Tribolium spp. populations that tracks the numbers of larvae, pupae and adults using a time interval of 2 weeks. It does not consider eggs separately, as hatching time is much shorter than the time step of the model. Dennis et al. (1995) express the LPA model as a set of three difference equations:

  • Lt+1 = bAt exp(−ceaAt − celLt)((eqn 1))
  • Pt+1 = Lt(1 − µl)((eqn 2))
  • At+1 = Ptexp(−cpaAt) + At(1 − µa)((eqn 3))

L, P and A are the numbers of larvae, pupae and adults. The parameters of the model are explained in Table 1. Table 1 also includes the numerical estimates of the parameters obtained by Dennis et al. (1995) from data on the corn oil sensitive strain of Tribolium castaneum (Desharnais & Costantino 1985; Desharnais & Liu 1987). We can also express the same model in matrix form:

Table 1.  Parameters in the LPA model, biological meaning and numerical estimates obtained for all four replicate cultures analysed by Dennis et al. (1995)
ParameterBiological interpretationNumerical value
bAverage number of larvae recruited per time step in the absence of cannibalism11·6772
µaProbability of adults dying from causes other than cannibalism 0·1108
µlProbability of larvae dying from causes other than cannibalism 0·5129
ceaRate of cannibalism of eggs by adults 0·0110
celRate of cannibalism of eggs by larvae 0·0093
cpaRate of cannibalism of pupae by adults 0·0178
  • image(4)

In the remainder of this paper we denote the projection matrix in eqn 4 by the symbol D and its elements as dij. Following Dennis et al. (1995), we treat adult mortality rate (µa) as a control parameter. For the parameter values given in Table 1, the population displays a two-point cycle. As the adult mortality rate is increased, the behaviour changes first to a stable equilibrium point and then to quasiperiodic dynamics. In the original version of the LPA model, the age of adults is ignored and all adults are grouped into a single stage class. To allow us to examine the effects of age and stage on elasticities, in some of the analyses presented here we have extended the model by disaggregating the first four age classes of adults and grouping together only the older adults. This is simply a book-keeping exercise which has no effect on the dynamics of the model, as numbers of eggs produced and mortality rates are held constant across all adult age classes.

A detailed discussion of the calculation of elasticities for density-dependent models has been given elsewhere (Grant & Benton 2000). For the LPA model of Tribolium we can use the standard density-independent methods to calculate elasticities of λ for the elements of the projection matrix D. These indicate the effect of small changes in demographic parameters on individual fitness. We can carry out this analysis for populations at very low population density and also for populations at stable equilibrium. The usual formulae involving the left and right eigenvectors of D give us elasticities for the non-zero elements d13, d21, d32, and d33. We can also calculate elasticities for the six parameters of the LPA model. Three of these are simple functions of elasticities of the elements of D:

  • image((eqn 5))
  • image((eqn 6))
  • image((eqn 7))

Workers familiar with the standard elasticity analysis will find the elasticities for the elements of matrix D easier to understand. Those familiar with the LPA model will be more at home with elasticities for the parameters of this model. We therefore present both sets of elasticities in the results section, recognizing that this involves inclusion of some redundant information. We refer to elasticities for b rather than the transition d13, but this choice is arbitrary.

For a population that does not have a stable equilibrium, we need to assess the evolutionary consequences of changes in parameters using an invasibility analysis. More precisely, we calculate elasticities of the invasion exponent ϑ:

  • image((eqn 8))

ϑ is the rate at which a variant type invades a resident population that is on an attractor, calculated at densities that are low enough for the invader to make no contribution to density-dependent interactions; x denotes any of the model parameters or individual elements of matrix D (Grant 1997). For comparison we can calculate the elasticities of λ for the projection matrix evaluated at the (unstable) equilibrium point. We can also calculate elasticities of λ for the non-zero elements of matrix D for the observed mean projection matrix (cf. Grant & Benton 2000).

Elasticities of total population size can be calculated as:

  • image((eqn 9))

where N=L + P + A; but we can also calculate separate elasticity values of each of the stage classes in the LPA model, i.e.

  • image((eqn 10))
  • image((eqn 11))
  • image((eqn 12))

When the population has a stable equilibrium, application of these formulae is straightforward. When the population shows non-equilibrium dynamics or is subject to stochastic environmental variation, eqn 9 could be replaced by either:

  • image((eqn 13))

or

  • image((eqn 14))

Equation 13 is the elasticity of arithmetic mean population size and eqn 14 the elasticity of geometric mean population size. Similarly, there are two possible variants of eqns 10, 11 and 12 when the population is not at equilibrium, and these are distinguished using the same convention. In the Results section we compare the performance of these two different types of elasticity of mean population size. Elasticities of invasion exponents can be calculated from stochastic stage structure and reproductive value vectors (Caswell 2001). However, this is not possible for elasticities of mean population sizes, so to avoid artefacts caused by using two different numerical methods, all elasticities have been calculated by numerical differentiation, using an increment of 5% of the parameter values. Models were iterated for 200 generations to avoid transient dynamics, then invasion exponents and population sizes were averaged over 1000 generations using Mathematica (Wolfram Research, Champaign, IL, USA).

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. References

density-independent population growth

At very low population density, individuals rarely encounter each other so cannibalism does not occur. In consequence, the density-dependent mortality terms will be negligible. We can achieve this mathematically by setting L, P and A to zero. The projection matrix then becomes:

  • image(15)

With the parameter values in Table 1, this density-independent population grows rapidly (λ = 2·14) and we obtain the following matrix of elasticities:

  • image(16)

As no opportunities for cannibalism occur, the numerical values of the three cannibalism parameters (cea, cel and cpa) have no effect on population growth rate and their elasticities are equal to zero. The elasticities in eqn 16 describe the effect of changes in demographic parameters on population growth rate. The projection matrix is ergodic, so the population converges to a stable stage distribution. As a result, these elasticities also quantify the effect of changes in demographic parameters on the growth in numbers of each of the three stages. For density-independent populations, population growth rate measures fitness, so they also indicate the effect of changes in demographic parameters on individual fitness. If the adult mortality rate is higher (µa = 0·6, as used in the following example) then at low population density the elasticity for adult survival rate is lower than in eqn 16 and the other elasticities are correspondingly higher, as expected:

  • image((eqn 17))

a population with a stable equilibrium

The next simplest case is where the population has a stable equilibrium, which occurs when adult mortality rates are in the range µa≈ 0·36 to µa≈ 0·75. We illustrate this with a population in which µa is set to 0·6, and all other parameters have the values given in Table 1. The equilibrium point is at L= 110·8; P= 54·0; A= 42·3 and the elasticities of λ calculated from the projection matrix evaluated at the equilibrium point are:

  • image((eqn 18))

Cannibalism reduces the survival rates of eggs and pupae so the influence of adult survival rate on fitness is greater than when population size is low. As cannibalism does occur, the three cannibalism rates now have non-zero elasticities. The elasticities for the two density-independent mortality rates µl and µa are simple functions of the elasticities for d21 and d33 (see eqns 6 and 7) but are included here for completeness. The four elasticities for the elements of D sum to 1 as expected (De Kroon et al. 1986). The remaining five elasticities are supplementary to these and do not sum to one. Elasticities of λ for the equilibrium projection matrix are plotted in Fig. 1. As is generally true for equilibrium populations, elasticities of ϑ calculated using invasibility methods are identical to elasticities of λ, so this density-independent analysis at the equilibrium point indicates the effects of changes in demographic parameters on individual fitness (Grant 1997). As expected, and displayed in eqn 18, the fertility term, b, and the three survival rates, d21, d32 and d33, have positive elasticities of λ. The mortality parameters, µa, µl, cea, celand cpa, have negative elasticities of λ, as increasing a mortality rate reduces fitness. In drawing comparisons between positive and negative elasticities of λ, it is their absolute magnitude that is important. The two density-independent mortality rates and cannibalism of eggs by larvae have similar effects on λ; cannibalism of pupae by adults has a smaller effect and cannibalism of eggs by adults has the least effect. On the basis of this, we might predict that there should be stronger selection pressure in favour of defences against cannibalism of eggs by larvae than by adults, although most forms of defence are probably equally effective against both. The numerical value of the rate of cannibalism of eggs by larvae (cel) is lower than that for the other two cannibalism parameters, but this process has a greater effect on λ because larvae are more than twice as abundant as adults.

image

Figure 1. Elasticities of λ for a Tribolium population at stable equilibrium (µa = 0·6).

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These calculations have used a mathematical model and values of the parameters other than µa have been estimated by Dennis et al. (1995) from data on population fluctuations. However, if we were to measure demographic parameters directly in a population that was at equilibrium and then perform a density-independent elasticity analysis on the resulting observed projection matrix, we would obtain the same values for the elasticities, except of course for differences due to sampling variation. This latter approach is the one that is usually adopted when carrying out elasticity analysis on field populations. This analysis does assume that stochastic population variations around the equilibrium are negligible. In the presence of moderate-sized fluctuations the agreement is no longer perfect, but is usually reasonably good (Benton & Grant 2000; Grant & Benton 2000; Benton & Grant 2002); so when we are concerned with examining the effects of changes in demographic parameters on individual fitness in a population at equilibrium, the standard density-independent analysis serves us well, even if there are some stochastic perturbations of the equilibrium.

Many of the applications of elasticity analysis to field populations are concerned with population management (Benton & Grant 1999). Here, the presence of density dependence presents more of a problem. In the Tribolium system, population regulation does not involve a simple function of total population size. The different stages are not equal in their effects, with pupae not contributing to density dependence, and adults being responsible for more cannibalism per capita than larvae. The evolutionary consequences of a change in a demographic parameter are therefore not the same as the consequences for population size (cf. Grant & Benton 2000). Density-dependent interactions between larvae, pupae and adults mean that a small change in one demographic parameter does not, in general, have the same proportional effect on the abundance of each of the three stages or total population size; for some purposes we may need also to consider elasticities of each of the stages in addition to elasticities of total population size. Elasticities of the population size measurements L, P, A and N to changes in each of the model parameters are shown in Fig. 2 and the ratio of these to elasticities of ë are shown in Fig. 3. For this set of parameter values, the elasticities of total population size have the same signs as the elasticities of λ for the corresponding parameters but do not have the same absolute or relative values (shown most clearly by Fig. 3). To examine the differences in more detail, elasticities of λ for b, d21 and d32 are identical (Fig. 1) because only adults reproduce, and every individual spends single time steps as a larva and a pupa before becoming an adult. By contrast, these parameters have different effects on the abundance of larvae, pupae and adults and on total population size. Increasing fertility (b) has a greater effect on larval abundance than on other stages, because the increase in larval abundance will increase the intensity of cannibalism of larvae on pupae, and thus reduce the abundance of both pupae and adults. Increasing the survival rate of larvae (d21), for example, has a large effect on the abundance of pupae, but the additional adults that result cannibalize both eggs and pupae, so numbers of larvae and adults (and thus total population size) increase by a smaller amount. Larval and pupal survival rates (d21 and d32) have the same effect on the abundance of adults because pupae cannot cannibalize other stages. Even when the population is at a stable equilibrium, questions about the effects of demographic parameters on population size and stage distribution require a full density-dependent elasticity analysis.

image

Figure 2. Elasticities of total population size, N (◆), number of larvae, L (•), number of pupae, P (▾) and number of adults, A (▪;) for a Tribolium population at stable equilibrium (µa = 0·6).

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image

Figure 3. Ratio of elasticities of total population size, N, to elasticities of λ for a Tribolium population at stable equilibrium (µa = 0·6).

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a population that shows two-point cycles

The next most complicated pattern of behaviour is in the region of low adult mortality rates (< 0·36), when the population shows a two-point cycle. A population with the estimated value of µa (0·1108) shows this pattern of dynamics. Because the population is not at a stable equilibrium, we need to use elasticities of ϑ to assess the evolutionary consequences of changes in demographic parameters. These values are plotted in Fig. 4. As adult mortality rates are low, adults live and reproduce in several time steps. In consequence, the largest elasticity of ϑ is for adult survival and the elasticities for b, d21 and d32 are much smaller than those in eqn 18. For comparison, elasticities of λ for the projection matrix evaluated at the (unstable) equilibrium point are also presented in Fig. 4. For the individual elements of matrix D, we can also calculate elasticities of λ using the observed mean projection matrix. The equilibrium and mean matrices, and consequently the elasticities, are not identical. The difference is, however, small so the latter are not presented here. For this population, elasticities of ϑ are almost identical to the elasticities of λ calculated from the equilibrium matrix, so in this case the density-independent elasticity analysis on the observed mean projection matrix is quite a good approximation to the full invasibility analysis, as is often true when we are concerned with evolutionary consequences of changes in demographic parameters (cf. Grant & Benton 2000).

image

Figure 4. Elasticities of fitness measures for a Tribolium population that undergoes two-point cycles (µa = 0·1108). •, Elasticities of ϑ calculated using an invasibility analysis; ▾, elasticities of λ calculated for the (unstable) equilibrium projection matrix.

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There is, however, one marked difference between the density-dependent and the density-independent elasticities. The elasticity of ϑ for cannibalism of eggs by larvae is close to zero (−0·02), whereas the elasticity of λ is five times greater (−0·10). This difference is a consequence of the cyclical population dynamics. Most individuals are larvae during the period when the total number of larvae is high. The majority of eggs that are laid during this period are cannibalized by larvae. Increasing the rate of cannibalism of eggs by larvae does reduce the number of eggs that hatch still further, but this number is already very small, so the effect on individual fitness is negligible. During the intervening 2-week period, the density of larvae is low. Adults are the main predators of eggs and larval cannibalism is only a minor cause of egg mortality, so increasing cel has only a small impact on the number of eggs hatching. This illustrates the way in which optimal life histories can depend upon patterns of population dynamics, and shows the potential ability of life histories to ‘track’ population fluctuations by exploiting predictable periods of reduced intraspecific competition or cannibalism (cf. Van Dooren & Metz 1998; Diekmann, Mylius & ten Donkelaar 1999). This generates a prediction about the evolution of cannibalism in Tribolium. When populations are cycling, there will be little fitness cost to cannibalism of eggs by larvae. By contrast, when populations are stable, cannibalism of eggs by larvae will have a fitness cost, and might be predicted to lead to the evolution of defences against egg cannibalism. Obviously this analysis does not take into account the energy benefits that accrue to an individual from cannibalism on eggs (Mertz & Robertson 1970) or as an adaptation to poor environmental conditions (Via 1999), neither does it take into account the degree of kinship between the predator and the eggs being eaten. It would, however, be interesting to assess whether the intensity of cannibalism of eggs by larvae altered over time in the predicted direction in populations moved from conditions that produced cyclical behaviour to those that produced equilibrium conditions and vice versa.

What effects do changes in demographic parameters have on population size in a deterministic fluctuating population? As noted in the Methods section, we can calculate elasticities of each of the stages in addition to elasticities of population size, and we can use either geometric or arithmetic mean population size when calculating these elasticities (eqns 13 and 14). Figure 5 shows both types of elasticity of each of the three stages and of total population size. The two possible ways of calculating these elasticities give similar results for elasticities of adult and of total numbers. By contrast, the two sorts of elasticity of numbers of larvae and pupae have similar magnitudes but opposite signs. This is a consequence of the way that changes in vital rates modify the high and low points of the two-point fluctuations in larvae and pupae numbers. The two formulae give results for elasticities of larval numbers that are almost exact mirror images of each other (Fig. 5a). To see why, consider the elasticity of larval numbers for changes in the fertility term b. For these parameter values, the number of larvae present in the population alternates between 18 and 324. A 5% increase in the value of b changes these values to 15 and 349. This represents a reduction of 17% in numbers at the low point but an increase of 7% in numbers at the peak. With arithmetic mean larval abundance, the effect of the change in the high point predominates, leading to a large positive elasticity. With geometric mean larval abundance, then the greater percentage change in the numbers at the low point predominates, producing a large negative elasticity. Neither is more correct a priori. In applications such as pest control, where changes in peak population size are more important, then elasticities of arithmetic mean abundance are probably more useful. When influences on the lowest population sizes are most important, as in the management of endangered species, then elasticities of geometric population size or of some other measure such as the variance of population size or the 10th percentile of its distribution may be more appropriate.

image

Figure 5. Elasticities of measures of population size for population that displays two-point cycles (µa = 0·1108). •, Elasticities of arithmetic mean abundance; ▾, elasticities of geometric mean abundance. (a) number of larvae, L; (b) number of pupae, P; (c) number of adults, A; (d) total population size, N.

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The number of pupae shows similar magnitude alternation, in antiphase with larval numbers, so there is a similar difference in the results obtained depending upon whether eqns 13 or 14 are chosen to calculate elasticities for pupal numbers. However, adult numbers show much lower magnitude fluctuations as does total population size (because larval and pupal numbers fluctuate in antiphase). As a result eqns 13 and 14 give similar results for elasticities of numbers of adults and of total population size (Fig. 5c,d). We focus the remainder of our discussion of this example on adult and total population numbers, although for pest species in which larvae do the main economic damage one would want to focus on larval numbers.

Figure 6 shows that the pattern of elasticities of total population size is completely different from the pattern of elasticities of ϑ. The largest elasticity of ϑ is for adult survival rate (d33), and this parameter also has the greatest influence on adult population size (Fig. 5c). However, this parameter has very little influence on total population size (Fig. 5d). An increase in adult survival rate produces an increase in adult population size, but the heavy predation of adults on eggs and pupae reduces the combined number of larvae and pupae and largely negates the impact on total population size. A density-independent analysis on the mean matrix would suggest that an attempt to control this Tribolium population should focus on trying to reduce adult survival. However, the density-dependent analysis shows that this would have little effect on the total population, and that reducing adult fertility would be much more effective. Contrary to the suggestion made by Caswell (2001, section 18.2.1), elasticities of λ can be a very poor indicator of an appropriate control strategy. It is even possible that elasticities of λ could point towards a control strategy that has the exact opposite of the desired effect, as they can even have the opposite sign to elasticities of population size. Despite being a mortality rate, cel has a positive elasticity for both adult and total population numbers. The reason for this is that an increase in larval cannibalism on eggs increases still further the magnitude of the two-point cycles in larval and pupal numbers. The increases in the high points are much greater than the decreases in the low points, and the result is a greater number of individuals becoming adults and a greater average population size.

image

Figure 6. Ratio of elasticities of total population size, N, to elasticities of ϑ for a Tribolium population that displays two-point cycles (µa = 0·1108).

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a population that shows quasiperiodic dynamics

At high adult mortality rates, the LPA model shows quasiperiodic dynamics. Figure 7 shows elasticities of ϑ for a population with an adult mortality rate (µa) of 0·9 and elasticities of λ calculated for the observed mean projection matrix. The pattern of elasticities of fitness is similar to those for the population with a stable equilibrium presented in Fig. 1, although the low adult survival rate means that the elasticity for d33 is even lower than that in Fig. 1. Again, density-independent elasticities of λ calculated for the projection matrix evaluated at the unstable equilibrium perform poorly for some demographic parameters. There is some divergence between the two measures of fitness elasticity for cel as occurs with two-point cycles (cf. Figure 4). There is a much larger divergence for µa. There is also a large proportional difference for d33, but this is not obvious from Fig. 7 as the absolute magnitude of both values is small, so the occurrence of population fluctuations means that changes in adult mortality rate have much greater evolutionary consequences than would be apparent from an analysis that ignored the population fluctuations. A large adult population at time 1 results in a large number of larvae in time-step 2. The intensity of cannibalism is high and survival rates are low. Because their mortality rate is high, few adults survive to reproduce a second time. However, those that do survive to reproduce again benefit from the much lower numbers of larvae in time-step 3 and the survival rates of the offspring produced are correspondingly higher. The large elasticity of ϑ for adult survival reflects the much greater value of eggs produced at age 4 compared with those at age 3 (see the next section for a more detailed discussion). This in turn provides an example of how selection pressures on individuals (indicated by positive elasticities) could push a population from non-equilibrium to equilibrium dynamics. Consider a population with an adult mortality rate that places it on the stable side of the bifurcation to quasiperiodic dynamics. Elasticities for this population will be similar to the dotted line in Fig. 7. If the adult mortality rate is increased slightly, the population will move into the region of quasiperiodic dynamics, and the pattern of elasticities will switch to a pattern similar to the solid line in Fig. 7. This will result in selection for reduced adult mortality. If this selection leads to a reduction in adult mortality rate without altering other parameters, the population will revert to a stable equilibrium.

image

Figure 7. Elasticities of fitness measures for a Tribolium population with quasiperiodic dynamics (µa = 0·9). Symbols as in Fig. 4.

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Figure 8 shows the elasticities of the population size measures L, P, A and N for the same population, calculated using both arithmetic and geometric means, and Fig. 9 shows the ratio of elasticities of the arithmetic mean of N to elasticities of ϑ. As in the population that undergoes a two-point cycle, elasticities of L, P and A show qualitatively different patterns depending upon whether eqns 13 or 14 are used. However, for total population size, the two equations give elasticities with the same sign for all parameters, and a large difference in magnitude only for µa. Of particular note are the elasticities of total population size for adult survival probability d33. Both ways of calculating elasticities give small negative values indicating that an increase in the survival rate of adults produces a reduction in population size. Equivalently, the adult mortality rate has a positive elasticity, with the increased cannibalism of eggs and pupae producing a reduction in the abundance of immature stages that more than outweighs the increase in adult numbers. As in the previous example, the elasticities for population size show a very different overall pattern from the elasticities of ϑ and the two have opposite signs for d33 and µa. A density-independent elasticity analysis would again be an unreliable guide to the management of this population.

image

Figure 8. Elasticities of measures of population size for a Tribolium population with quasiperiodic dynamics (µa = 0·9). Other details as Fig. 5.

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image

Figure 9. Ratio of elasticities of total population size, N, to elasticities of ϑ for a Tribolium population with quasiperiodic dynamics (µa = 0·9).

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age cannot be ignored with non-equilibrium dynamics

The LPA model ignores the age of adults and combines all of them into a single stage class. In a population at stable equilibrium, offspring produced at any age contribute equally to fitness and to population size. However, when Tribolium populations are undergoing population cycles, the intensity of cannibalism varies from one time step to the next. As seen above, eggs laid during time intervals when cannibalism is intense will be ‘worth less’ in terms of their contribution to the fitness of an individual than those laid during time intervals when the intensity of cannibalism is lower. The aggregation of all adult age classes into a single stage class could therefore give misleading results. The extended model, which separates out the first four age classes of adults, allows us to examine the consequences of this. For a population at stable equilibrium, the elasticities of ϑ (or λ) for age-specific adult fertility decline with age. The elasticity for one age class is equal to that for the previous age class multiplied by the adult survival probability. The result is that a graph of elasticity against age is a straight line on a semilogarithmic plot (see Fig. 10a). For a population that displays quasiperiodic dynamics (µa = 0·9) there is some departure from this proportionality, but the high adult mortality rate means that elasticity of ϑ declines rapidly with age and the departures from proportionality have only a modest influence on the pattern of elasticities (Fig. 10a). For populations that show two-point cycles and have moderate adult mortality rates, elasticities of ϑ show a completely different relationship with age from the steady decline that occurs in populations at stable equilibrium (Fig. 10b). Elasticity for reproduction in the first adult age class is very low, as offspring born in this time interval will experience intense cannibalism. By contrast, elasticity for reproduction in the second adult age class is high as cannibalism is much less intense at this time. For the same reasons, elasticity for adult age class 3 is low and that for adult age class 4 is high. We have aggregated all subsequent age classes in Fig. 10, but this alternating pattern of high and low elasticities continues throughout the adult life. The absolute values of these elasticities are low, but this simply reflects the much higher elasticity values of other age classes. The relative magnitudes of the elasticities reflect the fitness consequences of reproduction at different ages. They therefore reflect the pattern of selection pressures on the distribution of reproductive effort throughout life, so when a population is undergoing two-point cycles, reproduction immediately after adult eclosion will offer little fitness benefit. By contrast, reproduction in the next time interval offers a very substantial fitness benefit and there will be considerable benefit in delaying maturity, assuming that the resources saved can be used to increase subsequent fecundity or reduce larval and/or adult mortality rates. There will also be selection pressures on the timing of subsequent reproduction, favouring the production of offspring in the time intervals when cannibalism is low. If these selection pressures produce a delay in maturity or other alterations in the timing of reproduction that increase the number of offspring during times of low cannibalism intensity, the result will be to increase further the magnitude of population cycling. This will, in turn, increase the risks of extinction of local populations. In this case, selection on individuals acts to further destabilize the dynamics, the opposite of the pattern in the previous section.

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Figure 10. Elasticities of ϑ for age-specific reproduction in an extended LPA model in which the first four age classes of adults are tracked separately. Age class 5 + represents individuals of age 5 and above. (a) •, Elasticities for a population with stable equilibrium (µa = 0·6); dtri;, elasticities for a population with quasiperiodic dynamics (µa = 0·9). Note logarithmic Y axis. (b) Elasticities for populations undergoing two-point cycles. •, Elasticities when µa = 0·1108; dtri;, elasticities when µa = 0·3.

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Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. References

The analyses presented here give some cautionary warnings to those carrying out elasticity analyses on density-dependent populations. They also provide some insights into the evolutionary biology and population dynamics of the Tribolium model system and into broader issues of the extent to which selection pressures on life histories may act to stabilize or destabilize population dynamics.

As would be predicted from a density-independent analysis, elasticities for fertility terms and juvenile survival are higher and elasticity for adult survival is low when adult survival rates are low. Elasticities for juvenile stages become less important when adult survival rate is high, a pattern that is rather general in cross-species comparisons (Saether & Bakke 2000). As in our previous work (Grant 1997; Benton & Grant 1999; Grant & Benton 2000), the density-independent analysis provides a reasonable approximation to elasticities for fitness in most, but not all, cases. The exceptions are that the density-independent analysis over-estimates the impact of cannibalism of eggs by larvae when populations are fluctuating and under-estimates the impact of adult mortality on fitness when adult mortality rate is high. Therefore the density-dependent analysis provides information on some subtle aspects of life-history evolution of Tribolium populations in density-dependent environments but does not alter drastically most of the conclusions that would be reached from a density-independent analysis. However, we know that in other cases the presence of non-equilibrium dynamics can completely alter the pattern of elasticities (Grant 1997; Grant & Benton 2000). We need to carry out density-dependent elasticity analyses of several more mathematical models of fluctuating populations parameterized using real data before we can make generalizations about the circumstances in which non-equilibrium dynamics and/or stochastic population fluctuations invalidate the use of density-independent elasticity analysis of fitness. The fact that non-equilibrium dynamics does, in some cases, significantly alter the pattern of elasticities confirms the conclusion of the ‘adaptive dynamics’ approach that the pattern of dynamics of a resident population can alter the ability of other phenotypes to invade (see, for example, Van Dooren & Metz 1998; Diekmann et al. 1999).

When one is interested in the effects of changes in demographic parameters on the number of individuals in the population, the conclusions are much less reassuring. It is clear that Takada & Nakashizuka's (1996) result that population size elasticities are proportional to fitness elasticities is not general. It holds true for their model only because the elements of the projection matrix are simple functions of total population size (see Benton & Grant 2000, 2003). It is theoretically convenient to assume that population regulation is a function of total population size, but it is not clear how frequently this will be true for real populations (cf. Mueller 1997). It is unlikely to be the case when there is niche separation between stage classes (as is often the case in invertebrates), or there are considerable differences in resource requirements or resource holding potential between stage classes (such that, for example, one adult ‘equals’ several juveniles). Certainly, for laboratory populations that have been studied in detail, such as the Tribolium system discussed here, the Drosophila system studied by Mueller and coworkers (Mueller 1988) and the mite Sancassania berlesei (Benton & Grant 2003) population regulation is not a function of total population size. For Tribolium, the standard density-independent analysis performs poorly, even for populations with a stable equilibrium. For populations with non-equilibrium dynamics, the divergence between the two analyses can be extreme. Changes in demographic parameters that have a positive effect on individual fitness may reduce mean population size, leading to the two sorts of elasticities having opposite signs. The divergence between fitness elasticity and population size explains (Desharnais, Dennis & Costantino 1990) the observation that selection does not invariably maximize population size in Tribolium. The results support further our previous assertion that a density-independent analysis is likely to be a misleading guide to population management and that a full density-dependent elasticity analysis should be carried out whenever possible (Grant & Benton 2000). In some cases, one will be reasonably sure that density dependence is not important as, for example, in the control of pest species early in an outbreak, but one cannot make this assumption for endangered species, which may be rare because of habitat loss and consequently show strong density dependence within the small habitat patches in which they still occur. At minimum, studies should include some analyses that include plausible patterns of density dependence. Caswell (2001, p. 615) has argued that ‘management prescriptions based on linear, time invariant demography are much more robust than anyone might have expected’. He presents (pp. 632–634) a single elasticity analysis of the LPA model, only examining the terms d21, d32, d13 and d33 in our formulation. He found that changes in demographic parameters have very similar effects on population size to their effects on population growth rate at low density. However, the parameters used include an adult survival rate of only 4%. As adult survival rate is so low, the elasticity of both λ and population size to changes in adult survival rate is very small. The other three elasticities calculated by Caswell are equal to each other for populations that are growing in a density-independent manner or are at stable equilibrium. If elasticity for d33 is close to zero, then the other three elasticities are close to 1/3 and the two sets of elasticities are inevitably very similar. As we have shown above, elasticities for density-independent Tribolium population growth are not the same as those for a population at equilibrium if the elasticity for d33 is appreciably larger than zero. For populations not at stable equilibrium the elasticities of population size to changes in d21, d32 and d13 are no longer equal to each other. The pattern of elasticities for the invasion exponent is not necessarily the same as that for elasticities of population size. The Takada–Nakajima theorem can be used only if its assumptions about the nature of population regulation hold. If they do not, as is frequently the case, it cannot be used safely even as an approximate guide to population management.

There has been considerable interest in the issue of whether selection favours the evolution of non-equilibrium dynamics in populations with stable equilibria or vice versa. One possible explanation for the apparent rarity of non-equilibrium population dynamics in field populations is that selection on individuals may favour the evolution of equilibrium dynamics (Hansen 1992; Gatto 1993; Ebenman et al. 1996). Evolution of a laboratory population of blowflies towards equilibrium dynamics has been identified tentatively from patterns of population fluctuations (Stokes et al. 1988), although laboratory experiments on Drosophila designed specifically to look for this failed to show any alteration in the pattern of dynamics over 45 generations (Mueller, Joshi & Borash 2000). Our results suggest that there is no single answer to this question, perhaps explaining the contradictory results of empirical studies. The analysis of the model in which adults are dissagregated by age shows clearly that the presence of non-equilibrium dynamics can generate selection pressures that serve to increase the magnitude of population cycles. The opposite is true for the elasticities of adult mortality rate in the population with quasiperiodic dynamics when all adult age classes are aggregated into a single stage. The presence of quasiperiodic dynamics favours reduced adult mortality and, if it is biologically possible for mortality rates to be reduced, doing so will move the parameter values towards the region of equilibrium dynamics. In this case, if the population is close to the bifurcation between stable and unstable behaviour, selection may be able to move the population back to a stable equilibrium. So whether selection favours the evolution of non-equilibrium or equilibrium dynamics depends upon the details of an organism's life history, the nature of density dependence and the constraints that there are on evolutionary change in these. Density-dependent elasticity analysis is a powerful tool in helping us to investigate this.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. References
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