#### 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:

- (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:

- (16)

As no opportunities for cannibalism occur, the numerical values of the three cannibalism parameters (*c*_{ea}, *c*_{el} and *c*_{pa}) 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 (µ_{a} = 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:

- ((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 µ_{a}≈ 0·36 to µ_{a}≈ 0·75. We illustrate this with a population in which µ_{a} 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:

- ((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 µ_{l} and µ_{a} are simple functions of the elasticities for *d*_{21} and *d*_{33} (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, *d*_{21}, *d*_{32} and *d*_{33}, have positive elasticities of λ. The mortality parameters, µ_{a}, µ_{l}, *c*_{ea}, *c*_{el}and *c*_{pa}, 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 (*c*_{el}) 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.

These calculations have used a mathematical model and values of the parameters other than µ_{a} 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*, *d*_{21} and *d*_{32} 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 (*d*_{21}), 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 (*d*_{21} and *d*_{32}) 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.

#### 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 µ_{a} (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*, *d*_{21} and *d*_{32} 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).

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 *c*_{el} 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.

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 (*d*_{33}), 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, *c*_{el} 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.

#### 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 (µ_{a}) 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 *d*_{33} 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 *c*_{el} as occurs with two-point cycles (cf. Figure 4). There is a much larger divergence for µ_{a}. There is also a large proportional difference for *d*_{33}, 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.

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 µ_{a}. Of particular note are the elasticities of total population size for adult survival probability *d*_{33}. 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 *d*_{33} and µ_{a}. A density-independent elasticity analysis would again be an unreliable guide to the management of this population.

#### 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 (µ_{a} = 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.