Periodic Matrix Models Research Articles

Periodic matrix models for seasonal dynamics of structured populations with application to a seabird population.

For structured populations with an annual breeding season, life-stage interactions and behavioral tactics may occur on a faster time scale than that of population dynamics. Motivated by recent field studies of the effect of rising sea surface temperature (SST) on within-breeding-season behaviors in colonial seabirds, we formulate and analyze a general class of discrete-time matrix models designed to account for changes in behavioral tactics within the breeding season and their dynamic consequences at the population level across breeding seasons. As a specific example, we focus on egg cannibalism and the daily reproductive synchrony observed in seabirds. Using the model, we investigate circumstances under which these life history tactics can be beneficial or non-beneficial at the population level in light of the expected continued rise in SST. Using bifurcation theoretic techniques, we study the nature of non-extinction, seasonal cycles as a function of environmental resource availability as they are created upon destabilization of the extinction state. Of particular interest are backward bifurcations in that they typically create strong Allee effects in population models which, in turn, lead to the benefit of possible (initial condition dependent) survival in adverse environments. We find that positive density effects (component Allee effects) due to increased adult survival from cannibalism and the propensity of females to synchronize daily egg laying can produce a strong Allee effect due to a backward bifurcation.

Identifying Critical Control Points in the Wild Oat (Avena fatua) Life Cycle and the Potential Effects of Harvest Weed-Seed Control

Wild oat is a problematic weed species that requires new management techniques in the face of herbicide resistance; harvest weed-seed control (HWSC) may be an option. Wild oat demographic information was collected in long-term, rotational field studies in Lacombe, AB, Canada, in 2006 and 2007, and a periodic matrix model was parameterized using management extremes (no IPM, no herbicide to high IPM, and full herbicide). Population growth rates were calculated for each treatment and year. Prospective (elasticity) and retrospective (LTRE) analyses were conducted alongside a rearrangement of the model equation in which population growth rates were designated and the required proportion of newly shed seed survival that gives that growth rate was solved for. All populations had λ > 1 or increasing populations. Elasticity analyses indicated that λ was most-highly elastic to the overwinter seedbank (Esw = 1), followed by seedling survival, fecundity, and survival of newly shed seed (0.63 to 0.86 across treatments). The latter may be the most-accessible vital rate for management of herbicide resistant populations. LTRE exposed the stochasticity of wild oat population growth rates between years and their ability to take advantage of lapses in control. Decreasing the proportion of newly shed seeds (snew) that survives was the most-effective and available control strategy until reduced to 0.1 to 0.3 when the summer seedbank becomes more critical. When averaged across treatments, > 80% of newly shed seed must be eliminated to stop the population from growing, resulting in a stable population, but not a decline. Because of preharvest shattering, HWSC will likely not be effective enough alone to cause wild oat populations to decline. New management techniques for wild oat control that can be used in combination with HWSC and integrated weed management strategies are needed.

Unravelling the annual cycle in a migratory animal: breeding-season habitat loss drives population declines of monarch butterflies.

Threats to migratory animals can occur at multiple periods of the annual cycle that are separated by thousands of kilometres and span international borders. Populations of the iconic monarch butterfly (Danaus plexippus) of eastern North America have declined over the last 21years. Three hypotheses have been posed to explain the decline: habitat loss on the overwintering grounds in Mexico, habitat loss on the breeding grounds in the United States and Canada, and extreme weather events. Our objectives were to assess population viability, determine which life stage, season and geographical region are contributing the most to population dynamics and test the three hypotheses that explain the observed population decline. We developed a spatially structured, stochastic and density-dependent periodic projection matrix model that integrates patterns of migratory connectivity and demographic vital rates across the annual cycle. We used perturbation analysis to determine the sensitivity of population abundance to changes in vital rate among life stages, seasons and geographical regions. Next, we compared the singular effects of each threat to the full model where all factors operate concurrently. Finally, we generated predictions to assess the risk of host plant loss as a result of genetically modified crops on current and future monarch butterfly population size and extinction probability. Our year-round population model predicted population declines of 14% and a quasi-extinction probability (<1000 individuals) >5% within a century. Monarch abundance was more than four times more sensitive to perturbations of vital rates on the breeding grounds than on the wintering grounds. Simulations that considered only forest loss or climate change in Mexico predicted higher population sizes compared to milkweed declines on the breeding grounds. Our model predictions also suggest that mitigating the negative effects of genetically modified crops results in higher population size and lower extinction risk. Recent population declines stem from reduction in milkweed host plants in the United States that arise from increasing adoption of genetically modified crops and land-use change, not from climate change or degradation of forest habitats in Mexico. Therefore, reducing the negative effects of host plant loss on the breeding grounds is the top conservation priority to slow or halt future population declines of monarch butterflies in North America.

Each life stage matters: the importance of assessing the response to climate change over the complete life cycle in butterflies

As ectothermic organisms, butterflies have widely been used as models to explore the predicted impacts of climate change. However, most studies explore only one life stage; to our best knowledge, none have integrated the impact of temperature on the vital rates of all life stages for a species of conservation concern. Besides, most population viability analysis models for butterflies are based on yearly population growth rate, precluding the implementation and assessment of important climate change scenarios, where climate change occurs mainly, or differently, during some seasons. Here, we used a combination of laboratory and field experiments to quantify the impact of temperature on all life stages of a vulnerable glacial relict butterfly. Next, we integrated these impacts into an overall population response using a deterministic periodic matrix model and explored the impact of several climate change scenarios. Temperature positively affected egg, pre-diapause larva and pupal survival, and the number of eggs laid by a female; only the survival of overwintering larva was negatively affected by an increase in temperature. Despite the positive impact of warming on many life stages, population viability was reduced under all scenarios, with predictions of much shorter times to extinction than under the baseline (current temperature situation) scenario. Indeed, model predictions were the most sensitive to changes in survival of overwintering larva, the only stage negatively affected by warming. A proper consideration of every stage of the life cycle is important when designing conservation guidelines in the light of climate change. This is in line with the resource-based habitat view, which explicitly refers to the habitat as a collection of resources needed for all life stages of the species. We, therefore, encourage adopting a resource-based habitat view for population viability analysis and development of conservation guidelines for butterflies, and more generally, other organisms. Life stages that are cryptic or difficult to study should not be forsaken as they may be key determinants in the overall response to climate change, as we found with overwintering Boloria eunomia larvae.

Increased population growth rate in invasive polyploid Centaurea stoebe in a common garden

Biological invasions are inherently demographic processes, but trait differences between native and introduced genotypes are rarely linked to population growth rates. Native European Centaurea stoebe occurs as two cytotypes with different life histories (monocarpic diploids, polycarpic tetraploids); however, only tetraploids have been found in its introduced range in North America. In a common garden experiment using artificial populations, we compared the demographic performance of the three geo-cytotypes in the presence and absence of a specialist herbivore using periodic matrix models. We found no difference in population growth rate between the two European cytotypes and no significant effects of herbivory in all geo-cytotypes. However, there was a pronounced increase in population growth rate for North American compared with European tetraploids due to increased seed production and juvenile establishment. These results suggest that genetic drift or rapid evolution, rather than pre-adaptation through polyploidy may explain the invasion success of tetraploids.

Demographic effects of fire on two endemic plant species in the longleaf pine-wiregrass ecosystem

Fire can have dramatic effects on the vital rates of plant species and has been used successfully for management in a number of ecosystems. However, the demographic response of species to fire in fire-dependent ecosystems is variable, making it important to study the effects of fire on rare and threatened species. We quantified the effects of fire on Astragalus michauxii and Pyxidanthera brevifolia, two rare endemics of the longleaf pine-wiregrass ecosystem of the southeastern USA, by means of periodic matrix models to project the effect of fire frequency on population growth. In contrast to many species in the longleaf pine-wiregrass ecosystem, fire had short-term negative effects on both species, causing reductions in survival, size, flowering, and fruit production. Relative to the three-year fire intervals to which the study populations are currently exposed, more frequent burning is projected to cause population decline, with the most dramatic effects under annual burning. Although the current longleaf pine-wiregrass ecosystem is fire dependent and has experienced frequent fire for at least several thousand years, we propose that the two endemic species may be remnants from a past vegetation assemblage that experienced less frequent fire and thus may be adapted to longer fire-return intervals compared to other species currently in the ecosystem. Despite the short-term negative effects of fire on the vital rates of these species, longer-term benefits such as reduction of woody encroachment and litter removal may be important for the ultimate success of the species.

Temperature effects on gametophyte life-history traits and geographic distribution of two cryptic kelp species.

A major determinant of the geographic distribution of a species is expected to be its physiological response to changing abiotic variables over its range. The range of a species often corresponds to the geographic extent of temperature regimes the organism can physiologically tolerate. Many species have very distinct life history stages that may exhibit different responses to environmental factors. In this study we emphasized the critical role of the haploid microscopic stage (gametophyte) of the life cycle to explain the difference of edge distribution of two related kelp species. Lessonia nigrescens was recently identified as two cryptic species occurring in parapatry along the Chilean coast: one located north and the other south of a biogeographic boundary at latitude 29–30°S. Six life history traits from microscopic stages were identified and estimated under five treatments of temperature in eight locations distributed along the Chilean coast in order to (1) estimate the role of temperature in the present distribution of the two cryptic L. nigrescens species, (2) compare marginal populations to central populations of the two cryptic species. In addition, we created a periodic matrix model to estimate the population growth rate (λ) at the five temperature treatments. Differential tolerance to temperature was demonstrated between the two species, with the gametophytes of the Northern species being more tolerant to higher temperatures than gametophytes from the south. Second, the two species exhibited different life history strategies with a shorter haploid phase in the Northern species contrasted with considerable vegetative growth in the Southern species haploid stage. These results provide strong ecological evidence for the differentiation process of the two cryptic species and show local adaptation of the life cycle at the range limits of the distribution. Ecological and evolutionary implications of these findings are discussed.

The relative importance of sexual and clonal reproduction for population growth in the perennial herb Fragaria vesca

AbstractThe relative importance of sexual and clonal reproduction for population growth in clonal plants is highly variable. Clonal reproduction is often more important than sexual reproduction but there is considerable interspecific variation and the importance of the two reproductive modes can change with environmental conditions. We carried out a demographic study on the woodland strawberry (Fragaria vesca), a widespread clonal herb, at 12 sites in Switzerland during 2 years. Study sites were selected in two different habitats, i.e., forest and forest edge. We used periodic matrix models to estimate annual population growth rates and carried out prospective analyses to identify life cycle components that influence population growth rates most. Retrospective analyses were applied to study how the two different habitats affected population dynamics. Furthermore, we tested whether trade‐offs between sexual and clonal reproduction occurred. There were large differences in annual population growth rates between sites and large within‐site differences between years. Results of the prospective analyses clearly indicate that clonal reproduction is the dominant reproductive pathway whereas sexual reproduction is rather insignificant for population growth. Compared to forest habitats, forest edge habitats had higher population growth rates in the first year but smaller growth rates in the second year. We attribute these differing habitat effects to different water availabilities during consecutive years. No trade‐offs between sexual and clonal reproduction were found. In conclusion, population growth of F. vesca relies heavily on clonal reproduction. Furthermore, reproduction and survival rates of F. vesca depend highly on spatio‐temporal variation of environmental conditions.

Sensitivity analysis of periodic matrix population models

Periodic matrix models are frequently used to describe cyclic temporal variation (seasonal or interannual) and to account for the operation of multiple processes (e.g., demography and dispersal) within a single projection interval. In either case, the models take the form of periodic matrix products. The perturbation analysis of periodic models must trace the effects of parameter changes, at each phase of the cycle, on output variables that are calculated over the entire cycle. Here, we apply matrix calculus to obtain the sensitivity and elasticity of scalar-, vector-, or matrix-valued output variables. We apply the method to linear models for periodic environments (including seasonal harvest models), to vec-permutation models in which individuals are classified by multiple criteria, and to nonlinear models including both immediate and delayed density dependence. The results can be used to evaluate management strategies and to study selection gradients in periodic environments.

First Estimate of the Intrinsic Rate of Increase of a Millipede:Polydesmus angustus(Polydesmida: Polydesmidae) in a Seasonal Environment

AbstractAlthough dramatic millipede outbreaks have been reported worldwide, there is no data available on the intrinsic rate of increase (rm) of these arthropods. This parameter was estimated in the west-European millipede Polydesmus angustus Latzel, a seasonally-breeding species with mixed 1-yr and 2-yr life cycles within populations. Individuals from 20 broods were reared throughout their life cycle under seasonal conditions simulated in the laboratory (monthly mean temperatures and naturally varying photoperiods), to determine age-specific survival and fertility in annual (1-yr life cycle) and biennial (2-yr life cycle) females. The finite rate of increase (λm) was calculated using a periodic matrix model and rm was found to be 5.85 per year. This estimate suggests that outbreaks of annual polydesmidan species in greenhouses or under favorable field conditions can be generated by a small number of fertilized females in 2 yr. The relevance of seasonal conditions to measure rm per year in long-lived arthropod species that breed seasonally is emphasized. Estimates such as that obtained in P. angustus integrate most seasonal adaptations of the species and are ecologically plausible in the absence of extrinsic sources of mortality. Finally, the results show that biennial females represent a very small proportion of individuals in the stable population structure, but analysis of short-term dynamics indicates that they may be more successful as colonists than annual females.

Effect of land use and climate change on the future fate of populations of an endemic species in central Europe

The identification of optimal management strategies for a given species is a major challenge of species conservation. It becomes especially challenging when the environmental conditions are expected to change in the future, and the optimal management applied today may differ from the management that is optimal under the changed conditions (e.g. due to climate change).This study evaluates prospect of a rare plant species endemic to semi-natural grasslands in central Europe, Gentianella praecox subsp. bohemica. The number of populations of this species has declined rapidly in the last 60years; currently, a conservation action plan has been established in the Czech Republic, where most populations of this species occur. This study uses periodic matrix models to compare different management regimes under different scenarios of climate change and to identify the optimal management in each case.Without management, populations of the species are not able to survive. Flowering individuals can occur for a long time after the cessation of management, but the extinction of the population is inevitable within several decades. Without management, even very large populations (1000 flowering individuals) will go extinct in less than 50years. Total extinction (including seed bank) will follow several years after observation of the last flowering plant. The most suitable management is mowing and disturbance (by harrowing), which is also the best method for restoration of threatened populations. Mowing is less suitable, but it is fully sufficient for large prospering populations. When managed, even small populations (10–15 flowering individuals) are able to survive. When management is applied, future climate change may have a relatively small impact on the probability of survival of the species. Climate change will, however, increase the extinction probability of very small populations.

A net reproductive number for periodic matrix models

We give a definition of a net reproductive number R 0 for periodic matrix models of the type used to describe the dynamics of a structured population with periodic parameters. The definition is based on the familiar method of studying a periodic map by means of its (period-length) composite. This composite has an additive decomposition that permits a generalization of the Cushing–Zhou definition of R 0 in the autonomous case. The value of R 0 determines whether the population goes extinct (R 0<1) or persists (R 0>1). We discuss the biological interpretation of this definition and derive formulas for R 0 for two cases: scalar periodic maps of arbitrary period and periodic Leslie models of period 2. We illustrate the use of the definition by means of several examples and by applications to case studies found in the literature. We also make some comparisons of this definition of R 0 with another definition given recently by Bacaër.

Loss of foundation species increases population growth of exotic forbs in sagebrush steppe.

The invasion and spread of exotic plants following land disturbance threatens semiarid ecosystems. In sagebrush steppe, soil water is scarce and is partitioned between deep-rooted perennial shrubs and shallower-rooted native forbs and grasses. Disturbances commonly remove shrubs, leaving grass-dominated communities, and may allow for the exploitation of water resources by the many species of invasive, tap-rooted forbs that are increasingly successful in this habitat. We hypothesized that exotic forb populations would benefit from increased soil water made available by removal of sagebrush, a foundation species capable of deep-rooting, in semiarid shrub-steppe ecosystems. To test this hypothesis, we used periodic matrix models to examine effects of experimental manipulations of soil water on population growth of two exotic forb species, Tragopogon dubius and Lactuca serriola, in sagebrush steppe of southern Idaho, USA. We used elasticity analyses to examine which stages in the life cycle of T. dubius and L. serriola had the largest relative influence on population growth. We studied the demography of T. dubius and L. serriola in three treatments: (1) control, in which vegetation was not disturbed, (2) shrubs removed, or (3) shrubs removed but winter-spring recharge of deep-soil water blocked by rainout shelters. The short-term population growth rate (Lambda) of T. dubius in the shrub-removal treatment was more than double that of T. dubius in either sheltered or control treatments, both of which had limited soil water. All L. serriola individuals that emerged in undisturbed sagebrush plots died, whereas Lambda of L. serriola was high (Lambda > 2.5) in all shrub-removal plots, whether they had rainout shelters or not. Population growth of both forbs in all treatments was most responsive to flowering and seed production, which are life stages that should be particularly reliant on deep-soil water, as well as seedling establishment, which is important to most plant populations, especially during invasion. These data indicate the importance of native species, in this case the dominant shrub, in influencing soil resources and restricting population growth of exotic plants. These results argue that management of invasive plants should focus not only on removal of nonnatives, but also on reestablishment of important native species.

Optimal weed management in crop rotations: incorporating economics is crucial

Van Den Berg F, Gilligan CA, Gerdessen JC, Gregoire LAH &amp; VanDen Bosch F (2010). Optimal weed management in crop rotations: incorporating economics is crucial. Weed Research50, 413–424.SummaryAlthough the effects of crop rotation sequence and length on weed population dynamics have been studied, it is not clear whether or not the best strategy, from a weed population dynamics point of view, is also the economic optimal strategy. It is also not clear which biological and economic parameters are most important in determining this optimal strategy. We use a density‐dependent periodic matrix model, integrated with an economic model, to study the effect of rotation length and weed control on the average annual net profit (ANP). The system of the weed Persicaria maculosa in carrots (crop A) and spring wheat (crop B) is used as an example case study. The bio‐economic model shows that income is roughly constant with rotation length, apart from the shortest rotation sequence. These results are in agreement with a previous model study, which was restricted to biological dynamics. However, where the purely biological model suggests that weeding effort should be focussed on spring wheat years in the rotation, our bio‐economic model shows that this is not a viable economic strategy. In fact, the mean ANP over a rotation is mainly determined by the ANP in carrot years (where the gross margin is high) and depends on the balance between the increase in seed population density in carrot years and its decrease in wheat years, in combination with weeding costs. The model can easily be extended to incorporate other damaging organisms, making the model broadly suitable for analysing a range of weed management strategies from an economic perspective.

The stage-structured epidemic: linking disease and demography with a multi-state matrix approach model

Stage-structured epidemic models provide a way to connect the interacting processes of infection and demography. Reproduction and development can replenish the pool of susceptible hosts, and demographic structure leads to heterogeneous transmission and disease risk. Epidemics, in turn, can increase mortality or reduce fertility of the host population. Here we present a framework that integrates both demography and epidemiology in models for stage-structured epidemics. We use the vec-permutation matrix approach to classify individuals jointly by their demographic stage and infection status. We describe demographic and epidemic processes as alternating in time with a periodic matrix model. The application of matrix calculus to this framework allows for the calculation of \({\mathcal{R}_0}\) and sensitivity analysis.

Positive effects of native shrubs on Bromus tectorum demography

There is increasing recognition that overall interactions among plant species are often the net result of both positive and negative effects. However, the positive influence of other plants has rarely been examined using detailed demographic methods, which are useful for partitioning net effects at the population level into positive and/or negative effects on individual vital rates. This study examines the influence of microhabitats created by the native shrubs Artemisia tridentata and Purshia tridentata on the demography of the invasive annual grass Bromus tectorum in the Great Basin Desert, California, USA. Shrub understory environments differed significantly from intershrub space and were characterized by higher soil fertility and less extreme microclimates. There existed a strong spatial association between B. tectorum and the shrubs across four years, with more than double the density of B. tectorum in shrub microhabitats compared to intershrub space. Periodic matrix models were used to calculate population growth (lamda) and reproductive potential (RP, expected lifetime fecundity of seedlings) of B. tectorum in different microhabitats over two years. Modeled population growth was significantly increased in shrub microhabitats in the first of two years. This was primarily due to increased seedling establishment in Artemisia microhabitats, rather than effects during the growing season. In the following year, B. tectorum individuals in shrub microhabitats had a significantly greater reproductive potential than those in intershrub microhabitats, indicating shrub facilitation during the growing season. Loop analysis revealed an interacting effect of year and microhabitat on B. tectorum life history pathway elasticity values, demonstrating a fundamental influence of spatiotemporal factors on which life history pathways are important and/or possible. Life table response experiment (LTRE) analysis showed that increased survival and growth rates positively contributed to population growth in both years under Purshia, but only in the second year under Artemisia. This research provides evidence that the positive effects of native shrubs on B. tectorum can be strong enough to produce net positive effects at the population level, although positive effects were variable. In this study, a rigorous demographic approach was particularly useful in partitioning overall interactions into positive and negative components.

Flooding and canopy dynamics shape the demography of a clonal Amazon understorey herb

Summary 1. Tropical forests have highly diverse and often dense understorey vegetation. However, little is known about the demography of tropical understorey species and their response to common events like flooding and gap formation that change the prevailing environmental conditions. 2. We studied the influence of flooding and canopy gaps on the demography of the widespread clonal herb Heliconia metallica in 16 populations in the Peruvian Amazon over two wet and two dry seasons. In a factorial study design, we selected several populations in canopy gaps and under a closed canopy both at flooded and non-flooded sites. 3. Using periodic matrix models, we determined projected population growth rates during seasonal and annual time intervals and applied prospective and retrospective analyses to identify the life cycle components that contributed most to the differences in annual growth rates among the four combinations of habitat conditions. 4. All populations increased in size during the wet and decreased during the dry season. In both years, projected annual growth rates of the non-flooded populations were less than 1 both in gaps and under a closed canopy. In contrast, over the 2-year study period growth rates of the flooded populations were close to 1 both in gaps and under a closed canopy. 5. The life history of H. metallica was dominated by ramet survival and clonal growth, whereas seedling recruitment was very rare. In canopy gaps, growth and clonal expansion were rapid and reproduction was frequent, whereas under a closed canopy ramets grew little and rarely flowered. Population decline at non-flooded sites was due to higher ramet mortality during the dry season. However, the effects of flooding and canopy gaps interacted and affected life-history components differently in wet and dry seasons and in successive years. 6. Synthesis . The intricate interacting effects of flooding and canopy gaps determine the population dynamics of the understorey herb H. metallica . The clonal life form and the plasticity of the ramets enable the populations to persist under variable light conditions, but may not be able to buffer the increasingly frequent periods of drought in Amazon forests.

Demographic responses of Pinguicula ionantha to prescribed fire: a regression-design LTRE approach

This study describes the use of periodic matrix analysis and regression-design life table response experiments (LTRE) to investigate the effects of prescribed fire on demographic responses of Pinguicula ionantha, a federally listed plant endemic to the herb bog/savanna community in north Florida. Multi-state mark-recapture models with dead recoveries were used to estimate survival and transition probabilities for over 2,300 individuals in 12 populations of P. ionantha. These estimates were applied to parameterize matrix models used in further analyses. P. ionantha demographics were found to be strongly dependent on prescribed fire events. Periodic matrix models were used to evaluate season of burn (either growing or dormant season) for fire return intervals ranging from 1 to 20 years. Annual growing and biannual dormant season fires maximized population growth rates for this species. A regression design LTRE was used to evaluate the effect of number of days since last fire on population growth. Maximum population growth rates calculated using standard asymptotic analysis were realized shortly following a burn event (<2 years), and a regression design LTRE showed that short-term fire-mediated changes in vital rates translated into observed increases in population growth. The LTRE identified fecundity and individual growth as contributing most to increases in post-fire population growth. Our analyses found that the current four-year prescribed fire return intervals used at the study sites can be significantly shortened to increase the population growth rates of this rare species. Understanding the role of fire frequency and season in creating and maintaining appropriate habitat for this species may aid in the conservation of this and other rare herb bog/savanna inhabitants.

Demography and management of two clonal oaks: Quercus eduardii and Q. potosina (Fagaceae) in central México

Quercus eduardii and Q. potosina are the most abundant tree species of the temperate forests of Sierra Fría, Aguascalientes, Mexico, in an area ranging between 2200 and 2600 m above see level. During the last century these clonal oaks were intensively exploited to obtain charcoal and firewood for local use; in addition, sections of the forest were transformed into grasslands for livestock. The effects of disturbance on the population dynamics of these species are poorly known. Therefore, a demographic study was carried out in order to: (a) evaluate the effects of disturbance on the population growth rate of these species; (b) assess the effects of inter-annual environmental variability on the long term population dynamics; (c) evaluate the relative importance of sexual reproduction and clonal propagation on population growth rate; (d) simulate the impact of different levels of tree harvesting on the population dynamics of these species, and (e) recommend a harvesting intensity, based on the information obtained above. Annual, mean and periodic matrices as well as stochastic simulations were used . These size-classified population matrix models also were employed to devise a schedule that maximizes the percentage of individual plants that can be harvested without affecting their population growth rates. For this reason specific entries of the matrices were modified to simulate different harvesting intensities. The study was carried out in two disturbed and two undisturbed sites during a 4-years period. Two plots were established inside each site; one plot was excluded from livestock and the other was left intact. Annual population growth rates were above or equal to unity in all sites, species and years, whereas mean, periodic and stochastic simulation matrix models showed no significant differences between species, years, sites or from unity, suggesting that logging and grazing did not have a negative effect on the population growth rate of these species. Both species produced clonal offspring during the 4 years of the study, but reproduced sexually only once, suggesting a masting reproduction. Elasticity analyses showed that the contribution of clonal propagation is more important than fecundity to the population growth rate of both species. Mean annual and periodic matrix models showed that extractions as low as 5% cause a population decline, while with stochastic simulations extractions of up to 5% are possible for both species; however, the environmental stochasticity will drive populations to local extinctions.

Incidental and intentional catch threatens Galápagos waved albatross

As large, long-lived seabirds with delayed and slow reproduction, albatrosses have low intrinsic mortality rates and are especially vulnerable to extinction from extrinsic sources of mortality such as fishery bycatch. Leg-band recovery information for waved albatrosses revealed mortality from both incidental catch and intentional catch for human consumption. Annual adult survival in 1999–2005, estimated from capture-mark-recapture data, was lower than historical estimates. This recent increase in adult mortality probably contributed to recent and dramatic shrinkage of the breeding population; periodic matrix models confirm that population growth rate is most sensitive to changes in adult survival. Banding data and recovery information also suggest that capture by fisheries is male-biased, which should reduce fecundity in this species with obligate bi-parental care. This new documentation of bycatch, harvesting, and associated demographic consequences provides reason for serious concern about the persistence of the single breeding population of the waved albatross.