Stage-classified Matrix Model Research Articles

Lost reproductive value reveals a high burden of juvenile road mortality in a long-lived species.

Adult mortality is often the most sensitive vital rate affecting at-risk wildlife populations. Therefore, road ecology studies often focus on adult mortality despite the possibility for roads to be hazardous to juvenile individuals during natal dispersal. Failure to quantify concurrent variation in mortality risk and population sensitivity across demographic states can mislead the efforts to understand and mitigate the effects of population threats. To compare relative population impacts from road mortality among demographic classes, we weighted mortality observations by applying reproductive value analysis to quantify expected stage-specific contributions to population growth. We demonstrate this approach for snapping turtles (Chelydra serpentina) observed on roads at two focal sites in Ontario, Canada, where we collected data for both live and dead individuals observed on roads. We estimated reproductive values using stage-classified matrix models to compare relative population-level impacts of adult and juvenile mortality. Reproductive value analysis is a tractable approach to assessing demographically variable effects for applications covering large spatial scales, nondiscrete populations, or where abundance data are lacking. For one site with long-term life-history data, we compared demographic frequency on roads to expected general population frequencies predicted by the matrix model. Our application of reproductive value is sex specific but, as juvenile snapping turtles lack external secondary sex characters, we estimated the sex ratio of road-crossing juveniles after dissecting and sexing carcasses collected on roads at five sites across central Ontario, Canada. Juveniles were more abundant on roads than expected, suggesting a substantial dispersal contribution, and the road-killed juvenile sex ratio approached 1:1. A higher proportion of juveniles were also found dead compared with adults, and cumulative juvenile mortality had similar population-level importance as adult mortality. This suggests that the impact of roads needs to be considered across all life stages, even in wildlife species with slow life histories, such as snapping turtles, that are particularly sensitive to adult mortality.

Variance in animal longevity: contributions of heterogeneity and stochasticity

Variance in longevity among individuals may arise as an effect of heterogeneity (differences in mortality rates experienced at the same age or stage) or as an effect of individual stochasticity (the outcome of random demographic events during the life cycle). Decomposing the variance into components due to heterogeneity and stochasticity is crucial for evolutionary analyses.In this study, we analyze longevity from ten studies of invertebrates in the laboratory, and use the results to partition the variance in longevity into its components. To do so, we fit finite mixtures of Weibull survival functions to each data set by maximum likelihood, using the EM algorithm. We used the Bayesian Information Criterion to select the most well supported model. The results of the mixture analysis were used to construct an age × stage-classified matrix model, with heterogeneity groups as stages, from which we calculated the variance in longevity and its components. Almost all data sets revealed evidence of some degree of heterogeneity. The median contribution of unobserved heterogeneity to the total variance was 35%, with the remaining 65% due to stochasticity. The differences among groups in mean longevity were typically on the order of 30% of the overall life expectancy. There was considerable variation among data sets in both the magnitude of heterogeneity and the proportion of variance due to heterogeneity, but no clear patterns were apparent in relation to sex, taxon, or environmental conditions.

Demographic effects of deltamethrin resistance in the Chagas disease vector Triatoma infestans.

Triatoma infestans (Heteroptera: Reduviidae) Klug is the main vector of Chagas disease in Latin America. Resistance to deltamethrin was reported in Argentina and recently associated with reproductive and longevity trade-offs. The objectives of the present study were to describe the demographic consequences of deltamethrin resistance in T. infestans and to establish possible target stages for chemical control in susceptible and resistant colonies. A stage-classified matrix model was constructed based on the average stage length for susceptible, resistant and reciprocal matings' progeny. The differences between colonies were analysed by prospective and retrospective analysis. The life table parameters indicated reduced fecundity, fertility and population growth in resistant insects. The retrospective analysis suggested the latter was associated with lower reproductive output and increased fifth-instar nymph stage length. The prospective analysis suggested that the adult stage should be the main target for insecticide control. Although, fifth-instar nymphs should also be targeted when resistance has been detected. The presented results show demographic effects of deltamethrin resistance in T. infestans. While the older stages could be the main targets for chemical control, this approach is impeded by their higher tolerance to insecticides. It is concluded that the different mode of action insecticides would be more effective than a dose increase for the control of deltamethrin-resistant T. infestans.

The stage-classified matrix models project a significant increase in biomass carbon stocks in China’s forests between 2005 and 2050

China’s forests are characterized by young age, low carbon (C) density and a large plantation area, implying a high potential for increasing C sinks in the future. Using data of provincial forest area and biomass C density from China’s forest inventories between 1994 and 2008 and the planned forest coverage of the country by 2050, we developed a stage-classified matrix model to predict biomass C stocks of China’s forests from 2005 to 2050. The results showed that total forest biomass C stock would increase from 6.43 Pg C (1 Pg = 1015 g) in 2005 to 9.97 Pg C (95% confidence interval: 8.98 ~ 11.07 Pg C) in 2050, with an overall net C gain of 78.8 Tg C yr−1 (56.7 ~ 103.3 Tg C yr−1; 1 Tg = 1012 g). Our findings suggest that China’s forests will be a large and persistent biomass C sink through 2050.

Assessment of selenium toxicity on the life cycle of Caenorhabditis elegans.

Selenium (Se) is a growing problem of global concern. Se can cause adverse effects on reproductive systems, which have been linked to declines in animal populations. The soil nematode Caenorhabditis elegans (C. elegans) is a ubiquitous soil organism that is increasingly utilized as a model organism in aquatic and soil toxicology. In the present study, the experimental data for individual body length, survival rate, brood size, and hatching rate were used to evaluate the possible effects of selenite [Se(IV)] on C. elegans. A stage-classified matrix model was applied to the experimental data to provide information on the population dynamics of C. elegans and to assess the Se(IV)-affected asymptotic population growth rate. Estimates of the survival probability showed significant decreases in survival at all stages when C. elegans was exposed to Se(IV). The growth probability of C. elegans in the L1 stage showed the most significant decline, from 0.11 h(-1) (for the control) to 0.04 h(-1) [for exposure to 3 mM Se(IV)]. These results showed that Se(IV) has a profound impact on C. elegans population dynamics. The asymptotic population growth rate (λ) was found to range from 1.00 to 0.64 h(-1) for increasing Se(IV) concentrations, implying a potential risk of population decrease for C. elegans exposure to a Se(IV)-contaminated environment. Our study shows how a mechanistic view based on the Se(IV) effects on the soil nematode C. elegans can promote a life cycle toxicity assessment. An important implication of this analysis is that mathematical models can be used to produce a population stage structure, to give clarity to the analysis of the key population-level endpoint (the asymptotic population growth rate) of population dynamics, and to evaluate the influences for the response of other species to environmental Se. These models sequentially provide candidate environmental criteria for the evaluation of the population impact of Se.

Effect of localized oil spills on Atlantic loggerhead population dynamics

The purpose of this study is to analyze the population dynamics of loggerhead sea turtles (Caretta caretta) affected by localized oil spills. Methods include development of a spatial, stage-classified matrix model parameterized for the following primary loggerhead populations: Gulf of Mexico, western North Atlantic Ocean, and Florida peninsula. Oil spills are simulated deterministically in each population's nesting region, with 1) oil-induced mortality ranging from 25% to 100% and 2) stage classes affected either proportionally or equally. A transient sensitivity analysis was performed to determine the parameters most influential to the population growth rate. Results suggest that increased protection and understanding of young sea turtles found in the Sargasso Sea is essential to the survival of the species. In addition, findings provide insights into the population dynamics of the At- lantic loggerhead turtles and identify conservation measures appropriate in each oil spill case.

Population dynamics of the endangered fan mussel Pinna nobilis in a marine lake: a metapopulation matrix modeling approach

A metapopulation, time-invariant, stage-classified matrix model was developed to assess the dynamics of an important Pinna nobilis population in the marine Lake Vouliagmeni (Korinthiakos Gulf, Greece). The main aim of the study was to provide insight on the life cycle of the fan mussel and reveal influential factors for its population dynamics, with a special focus on the effect of poaching. The size of the fan mussel shell was selected as a state variable, and the model consisted of five size classes. The lake was divided in two regions, a shallow region of high (illegal) fishing mortality and high recruitment (region 1) and a deeper region of low mortality and low recruitment (region 2). The estimation of the transition matrix (stage-specific growth and mortality probabilities) was based on a tagging survey between 2005 and 2006, while independent annual surveys for abundance estimation using distance sampling techniques were utilized for the estimation of recruitment and stage-specific fertilities. The population was found to be increasing with an intrinsic rate of increase r = 0.038; however, r was not statistically different from zero. The life expectancy and expected lifetime offspring production of individuals in region 1 was markedly lower than that of individuals in region 2. Due to poaching, the life expectancy of a yearling fan mussel was less than 2.5 years in region 1, while it was almost 12 years in region 2. The highest expected annual natural mortality of fan mussels occurred on their first year of life after settlement (~43%) and greatly declined at greater sizes. Perturbation analysis revealed that the population growth rate was most sensitive to the vital rates of the larger size classes in region 2 and to fertilities corresponding to offspring that settled in the same region. The spatial distribution and abundance of the species was greatly dependent on the extent of poaching, which caused a size segregation of individuals, with small and young individuals being abundant in region 1, and larger and older individuals being restricted in region 2. If poaching ceased, the fan mussel population would be increasing with a significantly higher intrinsic rate of increase (r = 0.186), while if region 2 was also illegally exploited at the same intensity as region 1, the fan mussel population would be decreasing with r = −0.364 and would eventually collapse. The existence of refuge areas, where fan mussels may grow and reproduce, providing adjacent areas with offspring, seems crucial for the viability of local populations. Transplantation of fan mussels from high mortality areas to low mortality refuges might prove to be an effective measure to protect local populations of the species.

Stage‐structured matrix models for organisms with non‐geometric development times

Matrix models have been used to model population growth of organisms for many decades. They are popular because of both their conceptual simplicity and their computational efficiency. For some types of organisms they are relatively accurate in predicting population growth; however, for others the matrix approach does not adequately model growth rate. One of the reasons for the lack of accuracy is that most matrix-based models implicitly assume a specific degree of variability in development times for the organism. Because the variability is implicit, the implied variances are often not verified with experimental data. In this paper, we shall present extensions to the stage-classified matrix models so that organisms with arbitrary means and standard deviations of development times can be modeled.

Demography and life history of two rattan species, Eremospatha macrocarpa and Laccosperma secundiflorum, in Côte d'Ivoire

Abstract:Two rattans species were studied in Côte d'Ivoire using a stage-classified matrix model to compare their demography and life histories. Respectively 854 and 1009 genets ofEremospatha macrocarpa(pleonanthic) andLaccosperma secundiflorum(hapaxanthic) were censused every 6 mo over 18 mo. The population growth rates ofE. macrocarpa(λ = 0.979) andL. secundiflorum(λ = 0.959) were not significantly different from 1. This indicates that the populations were close to equilibrium. However, the difference between the stable stage distributions and the observed distributions indicated temporal variation in vital rates. Elasticity analysis showed that growth and fecundity had lower contributions to λ than the survival rates for the two species. A Life Table Response Experiment revealed that the survival of the first juvenile stage (all stems < 6 m in length), fecundity, growth of the second juvenile stage (at least one stem > 6 m in length) and adult (reproductive) survival highly contribute to the differences between the demography of the two species. Reproduction is postponed longer forL. secundiflorumthan forE. macrocarpaand the mean remaining life span for adult genets is shorter forL. secundiflorumthan forE. macrocarpa. Finally, our results suggest the existence of two trade-offs within reproduction which suggest that, although the two species have different demographic features, the λ of their populations are not significantly different from 1.

Demography of Machaerium cuspidatum, a shade-tolerant neotropical liana

The population dynamics of the liana Machaerium cuspidatum (Fabaceae) were studied in an Amazonian forest in Ecuador using a stage-classified matrix model. Growth, mortality and number of new seedlings were recorded over two 1.5-y periods in eight 20 × 250-m transects. The population growth rate (λ) was 1.032, which was significantly higher than 1. The dynamics of the species were most influenced by survival of large plants, which is typical for slow-growing woody species. λ varied little among years or transects in spite of the high seedling production in one period. The importance of canopy openness and crown height for population growth was studied using bootstraps on λ for different subsets of the whole population. The results suggest that the population growth rate was lower in tall forest (height > 10 m) than in the population in general. The population growth rates in areas with shaded understorey and in low forest (≤ 20 m) were the same as for the whole population. A life table response experiment (LTRE) revealed that the lower λ in tall forest resulted from lower growth and survival of large plants in tall forest. These results indicate that the species is shade-tolerant but that it is sensitive to variations in forest dynamics. The dominance of the species and the increasing population size suggest that the forest has had a low disturbance rate for a long time.

Population dynamics of four understorey shrub species in beech forest

Abstract Question: In the population dynamics of four understorey shrub species (Hp, Hydrangea paniculata Sieb. et Zucc.; Lu, Lindera umbellata Thunb. var. membranacea (Maxim.) Momiyama; Ms, Magnolia salicifolia (Sieb. et Zucc.) Maxim.; Vf, Viburnum furcatum Blume ex Maxim.), (1) What is the relative importance of seedling regeneration versus vegetative growth? (2) Can these shrubs persist stably for a long time in the understorey? (3) What kind of variation in demographic features is observed among these shrubs? Location: 780m a.s.l., north-eastern Japan. Methods: Population dynamics were analyzed by using stage-classified matrix models. Models were mainly constructed from five years stem-census data, including current-year seedlings and sprouts. Results: Current-year sprouts emerged every year in every species. Current-year seedlings emerged every year in Lu and Vf, but densities were very low. In every species, population growth rate (λ) was close to the equilibrium value 1.0 and no statistical differe...

Survival, growth, and escape from herbivory are determined by habitat and herbivore species for three Australian woodland plants.

To understand how plant communities are structured by herbivory it is essential to investigate the roles of different herbivores and the responses of a variety of plant species in different habitats. We examined the effects of mammalian herbivores on survival and growth of transplanted seedlings of two native trees (Eucalyptus albensand Callitris glaucophylla), and one native grass (Themeda australis) in white box ( E. albens) woodlands in eastern Australia over 3 years. Herbivores were manipulated using four fencing treatments that successively excluded livestock, macropods, and rabbits from woodland and grassland (cleared pasture). Survival was highest in the absence of mammalian herbivores and in woodlands, and patterns differed among plant species. Survival of T. australis was low, especially in grasslands, and mortality by overgrowth was common in ungrazed treatments. All plant species were taller in fenced plots, and differences between treatments were greater in grassland. Rabbits and livestock had the greatest influence on C. glaucophylla, while T. australis and E. albens were most affected by livestock and macropods. We used field data to parameterize stage-classified matrix models to predict escape from herbivory (escape height >100 cm) for tree species. Reduced herbivory increased the proportion of individuals reaching escape height after 15 years. Rate of escape was greater in grassland, and this faster growth appeared to counteract much of the negative impact of herbivores. While T. australis was unable to escape herbivory, larger, ungrazed individuals were more likely to flower and therefore contribute to the maintenance of the population. Our results show that habitat and herbivore species strongly influence the effect of herbivory on vegetation.

Simple Methods for Calculating Age‐Based Life History Parameters for Stage‐Structured Populations

Stage—classified matrix models are important analytical and theoretical tools for the study of population dynamics; in particular, these models may be appropriate for populations in which survivorship and fecundity are dependent on size or developmental stage, populations in which the age of individuals is difficult to determine, and populations in which there are multiple types of newborns. Nevertheless, methods for analyzing the implications of a population's stage—transition matrix have been limited in comparison to methods available for age—structured models (life tables or Leslie matrices). In this paper we show that all of the standard age—based measures of life history traits can be derived from a stage—transition model. By decomposing the transition matrix into separate birth, survival, and fission matrices we derive simple, direct formulas for age—based life history traits such as the discrete survivorship function, lx, maternity function, fx, mean age at maturity, and net reproductive rate, Ro, and also population parameters, including the stable age distribution, age—specific reproductive value, and generation time. These provide a common set of parameters for comparing age—structured and stage—structured populations or comparing populations with differently structured life cycles. In addition, we define four measures of age and life—span that summarize the relationship between stage and age in a stage—structured population: age distribution and mean age of residence for each stage class, expected remaining life—span for individuals in each stage class, and total life—span conditional on reaching a given stage class. We illustrate the use of our methods to address specific ecological questions by applying them to several previously published demographic data sets. These questions include: (1) what are the demographic effects of crowding on the tropical palm Astrocaryum mexicanum?; (2) how important is the initial rosette size in determining life history of teasel, Dipsacus sylvestris?; and (3) how old are reproducing adults in a stage—classified population of pink lady's—slipper, Cypripedium acaule? Our results may also be useful for evaluating the adequacy of a given stage—transition model.