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Populations are characterized by variability among individuals, correlated both with their intrinsic qualities and with environmental effects on their performance (Pfister & Stevens 2003). In variable environments, previous environmental conditions can influence life-history traits and performance of individuals at future times (Lindstrom 1999; Beckerman et al. 2002). The relationship between birth environment and these delayed life-history effects can lead to cohort effects (Beckerman et al. 2003). Cohort effects are population-level responses to common environmental conditions within generations, and have been described across a wide range of taxa including mammals and birds (e.g. Albon, Clutton-Brock & Guiness 1987; Lindstrom 1999). Variation in fitness among cohorts has recently prompted increased attention to the differential contribution of successive cohorts to the dynamics of fluctuating populations (Beckerman et al. 2002).
Details about how cohort effects influence population dynamics can be particularly important where models are used to guide management. For example, age-structured matrix models and associated elasticity analyses have been used to develop conservation strategies for endangered populations as well as control methods for invasive and pest species (e.g. Shea & Kelly 1998; Benton & Grant 1999). Because elasticity analyses quantify the relative importance of a given matrix element to population growth rate, they generally provide more information than do sensitivity analyses (Caswell 2000), suggesting that management should focus on those demographic parameters with the largest elasticity values (e.g. Caswell 2000; De Kroon, Groenendael & Ehrlén 2000). None the less, variation in cohort-specific vital rates due to temporal variation in the environment may result in population growth rates that vary over a series of good and bad years. This variation in growth rates may yield matrices with elasticity values that also vary from one year to the next. Additionally, substantial variation in environmental conditions may generate variation in survival and fecundity that can be correlated (Beckerman et al. 2002; Coulson, Gaillard & Festa-Bianchet 2005). Both temporal variation in population growth rates and covariation between vital rates can substantially affect elasticity values and thus the decision as to which vital rate should be targeted as well as the preferred timing of management.
The European stoat Mustela erminea (Linnaeus), a small (< 350 g), fast-moving and wide-ranging (home ranges to 200 ha) mustelid (Murphy & Dowding 1994), was first introduced to New Zealand in 1884, and is now widespread and common. In forests of southern beech Nothofagus spp., the dynamics and productivity of stoat populations are related to the 3–5 year masting cycle of the beeches and its consequent effects on the abundance of introduced feral house mice Mus musculus (Linnaeus). Temporary population irruptions of stoats usually follow the heavy beech seed falls that stimulate an equally short-lived increase of mice (Choquenot & Ruscoe 2000), and introduced ship rats Rattus rattus (Linnaeus) (Blackwell, Potter & Minot 2001; Dilks et al. 2003). The cycle set off by each seed fall is complete within 18–24 months, and is followed by a period of relative scarcity of mice, rats and stoats. King (2002) hypothesized that the observed fluctuations of stoats in New Zealand beech forests are driven by cohort effects as a consequence of low survival rates of first-year stoats in post-seedfall years.
The temporary irruptions of stoats are often associated with severe predation on endangered native birds and bats of New Zealand beech forests (e.g. O'Donnell & Phillipson 1996; Wilson et al. 1998; Basse, McLennan & Wake 1999; Dilks et al. 2003; Pryde, O'Donnell & Barker 2005). In managed populations of stoats, the period of greatest risk to native fauna is during the summer of the irruption, because the additional predation by the large number of young stoats is not buffered by the extra rodents (White & King 2006). If a substantial proportion of this large cohort of stoats survives into the following year, as in unmanaged populations, the period of high risk may continue over winter and possibly into the next summer (Murphy & Dowding 1994). To offset the impact of stoats on endangered native fauna in beech forests, it is therefore essential to develop effective control strategies that anticipate and prevent post-seedfall irruptions of stoats.
Our objective was to understand the contributions of successive cohorts to variation in population growth rates of fluctuating populations of stoats in New Zealand beech forests. We developed four different matrices, one each for stoat populations living through: (1) seedfall years when mice are increasing; (2) post-seedfall years when both mice and stoats are at peak numbers; (3) crash years when mice populations have declined to low densities; and (4) transitional years before the next seedfall year, when both mice and stoats are at low densities. We then linked the matrices into cycles and investigated how the relative importance of different vital rates changed as the cycle progressed. This approach allowed us to study the effects of group-level variability on the annual growth rates of populations in temporally variable environments, and to contrast a sequence of scenarios with different population growth rates. Understanding the importance of different cohorts is critical for understanding the dynamics of fluctuating populations, to develop effective conservation strategies for fluctuating populations of endangered species, and to control invasive or pest species.
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Our matrix modelling approach quantifies the variation in the growth rates of stoat populations in New Zealand beech forests among years. The stoat populations we modelled, subject to year-round trapping throughout the beech mast cycle, increased significantly during seedfall and transitional years and decreased during post-seedfall and crash years. Despite substantial uncertainties, the model's results match the observed seasonal density indices for the populations we sampled (King 1983; Fig. 1).
Our analyses produced three important results. First, fluctuations in growth rates of populations of stoats in New Zealand beech forests are primarily driven by (1) the high reproductive success of females born 6 months before a beech mast (i.e. the seedfall year cohort) combined with (2) the low survival and fertility of their young, the females of the post-seedfall cohort. Second, variation in population growth in fluctuating populations of stoats in beech forests is consistently most sensitive to changes in survival rates, not fertility. Finally, the relative contributions from survival and fertility to variation in population growth depend on the year of the cycle and the number of years within a cycle.
Our λs calculated both for individual matrices and for matrices combined into cycles of different length are consistent with a broad literature across the extensive geographical range of stoats (summarized by King & Powell 2007) that documents dynamic population irruptions when prey populations increase, frequently followed by local extinction and recolonization. Most wild stoat populations do not exhibit truly cyclic dynamics because prey populations in most places fluctuate randomly or unpredictably. The same is true for the stoat populations that we modelled. Figure 1 shows that intervals between seed falls varied from 2 to 4 years or more (King 1983). Recent evidence suggests that rising temperatures are promoting increased seed production as well as (occasionally) heavy seed falls in successive years (Dilks et al. 2003; Richardson et al. 2005). Consequently, we expect that future estimates of annual rates of population growth in beech forest stoat populations will vary across, and even beyond, the span of 0·58–1·98 that we calculated from our matrices based on historic data. Of particular concern from a conservation perspective is the possibility of consecutive seedfall years that lead to tremendous growth of stoat populations (λ = 1·98). Thus, our analyses should be considered a baseline for estimating predicted ecosystem-wide changes in community structure following changes in environmental conditions.
Three previous models of stoat population dynamics in beech forests have been based on the parameter values published in King (1983). Our models, using the entire, original data set, generally support them, but with some modifications. Barlow & Barron (2005) concluded that if culling is done only once during a beech mast cycle, it should concentrate on the post-seedfall year and avoid the crash year. Blackwell et al. (2001) concluded that stoats could control rodent populations during crash and transition years, and implied that releasing rodent populations from predation by stoats between mast years might lead to higher rodent populations after the next mast and, hence, higher stoat populations. Choquenot (2006) explored the interplay between the costs of monitoring environmental cues predicting periods of high stoat density with the benefits of concentrating control efforts at times of highest risk to threatened species.
Our results show clearly that the observed fluctuations in growth rates of stoat populations in New Zealand beech forests are the consequences of a cohort effect. The major population decrease during post-seedfall conditions is caused by the unusually low survival and poor reproduction of first-year stoats, because they reach independence in unusual numbers just as the formerly abundant mice are disappearing (King 2002). The long breeding cycle of stoats requires females to survive not just to, but 1 year past the next increase in food supplies (King et al. 2003b). Summer trapping only brings forward by some months the deaths of many females of the post-seedfall cohort (King & McMillan 1982) that have very little chance of successfully weaning their first litter at age 1 year. We note, however, that large males born in seedfall years that survive to their first mating season at age 1 year should be dominant over smaller males (Erlinge 1977), may breed with many females of any cohort, and may contribute more offspring to future generations than do the females of their own cohort.
The consequence is that females born during peak food availability contribute less to future generations than do females of other cohorts. Figures 2–5 all show low elasticities for first-year females during post-seedfall years. This result is consistent with those of analyses of comparable population cycles between predators and prey in the northern hemisphere. For example, reproductive values of northern forest owls were consistently higher for individuals born during the phase when prey were not at peak densities (Brommer, Kokko & Pietiäinen 2000). The cohort effects we quantified are likely to occur in many other systems with cyclic, or widely fluctuating, population dynamics with large variation in growth rates. Examples may include the snowshoe hare Lepus americanus (Erxleben) – Canada lynx Lynx canadensis (Kerr) cycle (Krebs, Boutin & Bounstra 2001), the possible moose Alces alces (Linnaeus) – wolf Canis lupus (Linnaeus) cycle on Isle Royal (Post et al. 2002), and population cycles for species of the Tetraonidae (Moss & Watson 2001). Understanding the relationship between environmental conditions and cohort-specific contributions to population growth in these cyclic dynamics is essential both to understand selection of life-history traits and contribute towards management.
Variation in population growth was consistently most sensitive to changes in survival rates, particularly survival of 0–1-year-old stoats. That sensitivity changed, however, not only with the phase of the cycle but also with the starting year for the cycle and the number of years in the cycle. These results are expected and not new in principle (Caswell & Trevisan 1994), but are not commonly documented. One expects high reproductive or survival rates to have different effects early vs. late in a cycle. Figures 3–5 show that populations coming out of the crash phase are predicted to be relatively insensitive to variation in vital rates during any future phase of the cycle. In contrast, populations in most other phases of the cycle appear quite sensitive to survival of 0–1-year-old in crash years. Thus, targeting control efforts towards the crash year should have major effects on future population growth, including the next population irruption.
The sensitivity of population growth to first-year survival was also reported for stoat populations in Britain (McDonald & Harris 2002) and appears typical for species with similar life-history strategies (Tuljapurkar & Caswell 1997). None the less, cross-population generalizations may be meaningful only across similar environments and limiting factors (Coulson et al. 2005). In their study of the dynamics of stoat populations in Britain, McDonald & Harris (2002) pooled data over a wide geographical area and over several years, masking annual variation in the environment and making direct comparison with our results difficult.
Matrix models have been used widely to evaluate the link between vital rates and variation in population growth. The method assumes, however, that temporal fluctuations in the demographic structure of the population do not have a substantial impact on variation in lambda (Caswell 2000). In many natural populations, however, this assumption is violated. Failure to allow for the contribution of covariation between vital rates can therefore result in misleading conclusions when evaluating the differential contributions of vital rates to population growth. For example, Coulson et al. (2005) found that covariation between vital rates in ungulates accounted for up to 50% of the variation in population growth. In fluctuating populations, this problem can be addressed by linking matrices into cycles (Caswell & Trevisan 1994).
When we linked matrices into cycles, elasticities for each vital rate in each phase of the cycle changed, in part because the stable age distribution and thus the demographic structure of the population differed with each phase of the cycle. Each phase of the cycle presented the following year with an unstable age distribution, resulting in the observed dependency on whatever the present phase of the cycle is. Vital rates that appear important when viewing the matrix of one phase in isolation, or when comparing two phases not connected into a cycle, may not be important when viewed as part of the entire cycle. For example, results from individual matrices indicated first-year fertility to be exceedingly important during seedfall years and adult survival in post-seedfall years. In contrast, those particular vital rates do not stand out when viewing the entire cycle.
That each phase of the cycle presents the next phase with an unstable age distribution is also critical to understanding λ for the whole cycle (Caswell & Trevisan 1994). Simply multiplying the λs for the matrices, or multiplying the matrices for the cycle and then calculating λ, produces false values for λ. For example, multiplying the lambdas for the matrices in a 3-year cycle (and taking the cube root to yield annual growth) produces a value of 0·61, while calculating the λ for the product matrix of the three matrices produces a value of 0·95; the true λ for the 3-year cycle is 0·85. Because of the constantly unstable age distribution, the only way to calculate λ for a cycle is to multiply the matrix for each phase by the incoming population vector and continuing to multiply year by year by year until enough cycles have been produced to be able to calculate λ accurately from the projected population growth. Knowing the matrix for each phase of a cycle is not enough. One must link them to understand their dynamics.
Linking matrices into cycles allowed us to derive three important management recommendations. First, management targeting survival of stoats will consistently be more effective than management targeting fertility. Second, while intensive stoat control during peak years is likely required to minimize predation effects on the native fauna, control measures during post-seedfall and crash years (i.e. when stoat populations are at low densities), may be effective in limiting future irruptions of stoats, at least where immigration is minimal. Third, understanding fluctuating populations requires attention to all phases of the fluctuations, each within the context of the other phases. These generalizations are relevant both for conservation of endangered prey populations and for control of pest species in other systems whose populations fluctuate naturally. In such systems, management strategies based on elasticities calculated using mean, invariant vital rates are unlikely to translate into effective management recommendations.