Optimal reproductive strategies in age-structured populations of zooplankton

SUMMARY. We describe a model of zooplankton population dynamics that accounts for differences in mortality and physiology among animals of different ages or sizes. The model follows changes in numbers of individuals and changes in individual and egg biomass through time and it expresses mortality and net assimilation as functions of animal size.
We investigated the effect of egg size, age at first reproduction, and size at first reproduction on the per capita growth rates of populations growing under different conditions. In the absence of predation or when exposed to vertebrate predators that prefer large prey, populations achieve maximum growth rates when animals hatch from small eggs and reach maturity quickly at small sizes. Populations exposed to invertebrate predators that concentrate on small animals may increase r in two different ways. One way is for animals to increase juvenile survivorship by hatching from large eggs and by shortening the juvenile period. An alternative strategy is for animals to hatch from small eggs and to postpone maturity until they grow beyond the range of sizes available to their predators. Certain life history strategies maximize r if animals continue to grow after they reach maturity. By growing larger, non-primiparous females are able to hatch larger clutches and thereby increase the overall rate of population growth.
The model analysis shows how to assess age-dependent mortality rates from field data. The net rate of population increase and the age distribution of eggs together provide specific, quantitative information about mortality.     

MODELING AGE-SPECIFIC MORTALITY FOR MARINE MAMMAL POPULATIONS

A method is presented for estimating age-specific mortality based on minimal information: a model life table and an estimate of longevity. This approach uses expected patterns of mammalian survivorship to define a general model of age-specific mortality rates. One such model life table is based on data for northern fur seals (Callorhinus ursinus) using Siler's (1979) 5-parameter competing risk model. Alternative model life tables are based on historical data for human females and on a published model for Old World monkeys. Survival rates for a marine mammal species are then calculated by scaling these models by the longevity of that species. By using a realistic model (instead of assuming constant mortality), one can see more easily the real biological limits to population growth. The mortality estimation procedure is illustrated with examples of spotted dolphins (Stenella attenuata) and harbor porpoise (Phocoena phocoena).     

Ecological and mathematical considerations on self-thinning in even-aged pure stands

It is emphasized in growth analysis of self-thinning populations that relative mortality rate pertains to the difference between relative growth rates and net assimilation rates, each of which are definable on a mean plant size basis or on a biomass basis. The time trends of the ratio of relative mortality rate to relative growth rates to be expected according to Tadaki's, Shinozaki's and Hozumi's models are compared with that of the eastern white pine population, and a good agreement is exhibited. As an alternative to Hozumi's model, a new model is constructed to unite the logistic theory of plant growth and the 3/2 power law concerning self-thinning, which so far have usually been applied independently to growth analysis. To construct the model the following assumptions are made: the fundamental equation to relate mean plant weight with density in self-thinning population proposed by Shinozaki, and a special population with a specific initial density which follows thew-p trajectory of the 3/2 power law type and has an exponential decrease in its density with biological time. Properties of the model are examined from ecological and mathematical viewpoints.     

Estimating growth and mortality rates from size data

Summary A method is presented for estimating rates of individual growth and population mortality utilizing average individual size at two times during a year. The model assumes a constant rate of mortality, Brody-Bertalanffy growth, a stationary age distribution, and recruitment confined to one month each year. A hypothetical example is presented to show the interrelationships of the growth and mortality constants, size at recruitment, asymptotic size and average individual size. Three examples are presented using data from the literature: Flathead sole (Hippoglossoides elassodon), a sea urchin(Echinus esculentus), and the crown-of-thorns starfish(Acanthaster planci). The method appears to be a means of obtaining reasonable approximations of growth and mortality rates for a variety of organisms.     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

The natural variability of vital rates and associated statistics

The first concern of this work is the development of approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. The Lexis diagram is seen to make direct use of the result. It is suggested that in a variety of cases, it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits.     

Unimodal Tree Size Distributions Possibly Result from Relatively Strong Conservatism in Intermediate Size Classes

Tree size distributions have long been of interest to ecologists and foresters because they reflect fundamental demographic processes. Previous studies have assumed that size distributions are often associated with population trends or with the degree of shade tolerance. We tested these associations for 31 tree species in a 20 ha plot in a Dinghushan south subtropical forest in China. These species varied widely in growth form and shade-tolerance. We used 2005 and 2010 census data from that plot. We found that 23 species had reversed J shaped size distributions, and eight species had unimodal size distributions in 2005. On average, modal species had lower recruitment rates than reversed J species, while showing no significant difference in mortality rates, per capita population growth rates or shade-tolerance. We compared the observed size distributions with the equilibrium distributions projected from observed size-dependent growth and mortality. We found that observed distributions generally had the same shape as predicted equilibrium distributions in both unimodal and reversed J species, but there were statistically significant, important quantitative differences between observed and projected equilibrium size distributions in most species, suggesting that these populations are not at equilibrium and that this forest is changing over time. Almost all modal species had U-shaped size-dependent mortality and/or growth functions, with turning points of both mortality and growth at intermediate size classes close to the peak in the size distribution. These results show that modal size distributions do not necessarily indicate either population decline or shade-intolerance. Instead, the modal species in our study were characterized by a life history strategy of relatively strong conservatism in an intermediate size class, leading to very low growth and mortality in that size class, and thus to a peak in the size distribution at intermediate sizes.     

Estimation of growth and survival in size-structured cohort data: an application to larval striped bass (Morone saxatilis)

We present a method for estimating growth and mortality rates in size-structured population models. The methods are based on least-square fits to data using approximate models (using spline approximations) for the underlying partial differential equation population model. In a series of numerical tests, we compare our approach to an existing method (due to Hackney and Webb). As an example, we apply our techniques to experimental data from larval striped bass field studies.Research supported in part under grants at Brown University from the National Science Foundation: UINT-8521208, NSFDMS-8818530 (H.T.B., F.K. and CW.); from the Air Force Office of Scientific Research: AFOSR F49620-86-C-0111 (H.T.B., C.W.); and at University of California, Davis from the Alford P. Sloan Foundation (L.W.B.)     

Simultaneous estimation of mortality and dispersal rates of an artificially released population

Mark-recapture methods cannot estimate both mortality and dispersal rates of a wild population simultaneously. However, when an artificially cultured population is released into an area, the initial population size and the initial population distribution are usually known. If artificially cultured individuals are released with marks or distinguished from wild individuals or if no wild individual exists in the study area, we can estimate both the mortality and dispersal rates of the artificial population. The numbers of dispersed and dead individuals are estimated from the dispersal rate from the diffusion model and the total decreasing rate estimated from a mark-recapture data. We can estimate both the time-dependent and time-independent dispersal rates from the data. We choose the best fit model that has the smallest value of Akaike's Information Criteria. We also consider ‘concentric circles approximation” of spatial distribution, in which the cumulative and frequency distributions are analytically obtained.     

A model for the dynamics of an annual plant population

A model for the dynamics of a single species population of plants is proposed and its use demonstrated by the analysis of a simple example. The model incorporates the effects of microsite variation by allowing for individual differences in growth and death rates within each season. We demonstrate that an increase in the variance in individual growth rates may increase both the chances that a plant population will persist and the equilibrium size of that population. We also show that even if size-dependent death is occurring, it may not have a significant effect on the shape of the size frequency distribution. An extension of the model to multispecies communities of plants suggests an experimental procedure to determine whether competition is responsible for excluding a particular plant species from a community that appears otherwise to be suitable. A more detailed analysis of the model for a two-species community produces conditions for competitive coexistence reminiscent of those from the Lotka-Volterra competition equations. Another extension suggests that selection will favor those genotypes that maximize the product of germination probability and mass of seeds produced, if survivorship and growth are not substantially altered. Finally, an analog to r- and K-selection theory for animal populations is developed. Selection in low-density populations favors increasing growth rate, and in high-density populations favors minimizing the effect of neighbors on one's own growth rate.     

A power analysis of microsatellite-based statistics for inferring past population growth

We present results concerning the power to detect past population growth using three microsatellite-based statistics available in the current literature: (1) that based on between-locus variability, (2) that based on the shape of allele size distribution, and (3) that based on the imbalance between variance and heterozygosity at a locus. The analysis is based on the single-step stepwise mutation model. The power of the statistics is evaluated for constant, as well as variable, mutation rates across loci. The latter case is important, since it is a standard procedure to pool data collected at a number of loci, and mutation rates at microsatellite loci are known to be different. Our analysis indicates that the statistic based on the imbalance between allele size variance and heterozygosity at a locus has the highest power for detection of population growth, particularly when mutation rates vary across loci.     

Population dynamics of the ribbed mussel,Geukensia demissa: The costs and benefits of an aggregated distribution

Summary Although mussel beds are common in many intertidal habitats, the ecological significance of the aggregated distribution of mussels has not been examined. The ribbed mussel, Geukensia demissa, is found in dense aggregations on the seaward margin of many salt marshes in New England. Here, we examine the population structure of G. demissa in a New England salt marsh and investigate experimentally the costs and benefits of aggregation.Size, growth rate, and settlement rates of mussels decrease with increasing tidal height, whereas survivorship and longevity increase with increasing tidal height. Winter ice dislodges mussels from the substratum, resulting in mortality over all size classes, whereas crab predation results in the mortality of smaller mussels. The intensity of each of these mortality agents decreases with increasing tidal height. Effects of intraspecific competition on individual growth and mortality also decrease with increasing tidal height.At high densities, individual growth rates were reduced, with depression of growth rates most pronounced on smaller individuals. Mortality from sources other than intraspecific crowding, however, was reduced at high mussel densities, including mortality due to winter ice and crab predators. As a result, our data suggest that the mussel population at our study site would be reduced by 90% in only five years and no juveniles would survive through their second year without an aggregated distribution.Juveniles settle gregariously with or without adults present. The aggregated distribution of settlers and the postsettlement movement of smaller mussels to favorable microhabitats result in size and age class segregation within the population. This probably reduces intraspecific competition for food, while maintaining the survivorship advantages of an aggregated distribution.     

An assessment of the maximum sustainable yield of ivory from African elephant populations.

A general, logistic population model is used to explore the dynamics of harvested elephant populations. The model includes two features peculiar to elephant populations and the harvesting of ivory. First, because of the shape of the growth curve of tusks with age, the conversion factor that relates the number of elephants killed to the ivory yield in weight is not constant, but a function of the population size. Second, tusks from animals that die from natural causes can be retrieved and included in the total yield of ivory. The implications of the relationship between tusk size and age of an animal on the maximum sustainable yield in terms of ivory tonnage and in terms of the number of tusks are explored. The nonequilibrium implications of the tusk growth curve on the population dynamics under different harvesting strategies are also investigated. Results indicate that the maximum sustainable yield is achieved at very low harvest rates with population levels close to the pristine equilibrium. When tusks from animals that die of natural causes are included in the harvest, the maximum yield may, depending on the mortality and recruitment parameters, occur when there is no direct harvest.     

Density-dependence as a size-independent regulatory mechanism

The growth function of populations is central in biomathematics. The main dogma is the existence of density-dependence mechanisms, which can be modelled with distinct functional forms that depend on the size of the population. One important class of regulatory functions is the theta-logistic, which generalizes the logistic equation. Using this model as a motivation, this paper introduces a simple dynamical reformulation that generalizes many growth functions. The reformulation consists of two equations, one for population size, and one for the growth rate. Furthermore, the model shows that although population is density-dependent, the dynamics of the growth rate does not depend either on population size, nor on the carrying capacity. Actually, the growth equation is uncoupled from the population size equation, and the model has only two parameters, a Malthusian parameter rho and a competition coefficient theta. Distinct sign combinations of these parameters reproduce not only the family of theta-logistics, but also the van Bertalanffy, Gompertz and Potential Growth equations, among other possibilities. It is also shown that, except for two critical points, there is a general size-scaling relation that includes those appearing in the most important allometric theories, including the recently proposed Metabolic Theory of Ecology. With this model, several issues of general interest are discussed such as the growth of animal population, extinctions, cell growth and allometry, and the effect of environment over a population.     

Population Viability and Vital Rate Sensitivity of an Endangered Avian Cooperative Breeder,the White-Breasted Thrasher (Ramphocinclus brachyurus)

Social behaviors can significantly affect population viability, and some behaviors might reduce extinction risk. We used population viability analysis to evaluate effects of past and proposed habitat loss on the White-breasted Thrasher (Ramphocinclus brachyurus), a cooperatively breeding songbird with a global population size of <2000 individuals. We used an individual-based approach to build the first demographic population projection model for this endangered species, parameterizing the model with data from eight years of field study before and after habitat loss within the stronghold of the species’ distribution. The recent habitat loss resulted in an approximately 18% predicted decline in population size; this estimate was mirrored by a separate assessment using occupancy data. When mortality rates remained close to the pre-habitat loss estimate, quasi-extinction probability was low under extant habitat area, but increased with habitat loss expected after current plans for resort construction are completed. Post-habitat loss mortality rate estimates were too high for projected populations to persist. Vital rate sensitivity analyses indicated that population growth rate and population persistence were most sensitive to juvenile mortality. However, observed values for adult mortality were closest to the threshold value above which populations would crash. Adult mortality, already relatively low, may have the least capacity to change compared to other vital rates, whereas juvenile mortality may have the most capacity for improvement. Results suggest that improving mortality estimates and determining the cause(s) of juvenile mortality should be research priorities. Despite predictions that aspects of cooperative systems may result in variation in reproduction or juvenile mortality being the most sensitive vital rates, adult mortality was the most sensitive in half of the demographic models of other avian cooperative breeders. Interestingly, vital rate sensitivity differed by model type. However, studies that explicitly modeled the species’ cooperative breeding system found reproduction to be the most sensitive rate.     

Growth and population dynamic model of the reef coral Fungia granulosa Klunzinger, 1879 at Eilat, northern Red Sea

The lack of population dynamic information for most species of stony corals is due in part to their complicated life histories that may include fission, fusion and partial mortality of colonies, leading to an uncoupling of coral age and size. However, some reef-building corals may produce compact upright or free-living individuals in which the above processes rarely occur, or are clearly detectable. In some of these corals, individual age may be determined from size, and standard growth and population dynamic models may be applied to gain an accurate picture of their life history. We measured long-term growth rates (up to 2.5 years) of individuals of the free-living mushroom coral Fungia granulosa Klunzinger, 1879 at Eilat, northern Red Sea, and determined the size structure of a population on the shallow reef slope. We then applied growth and population models to the data to obtain estimates of coral age, mortality rate, and life expectancy in members of this species. In the field, few F. granulosa polyps suffered partial mortality of >10% of their tissues. Thus, the majority of polyps grew isometrically and determinately, virtually ceasing growth by about 30-40 years of age. Coral ages as revealed by skeletal growth rings were similar to those estimated from a growth curve based on field data. The frequency of individuals in each age class on the reef slope decreased exponentially with coral age, indicating high mortality rates when corals were young. The maximum coral age observed in the field population (31 years) was similar to that estimated by application of a population dynamic model (30 years). Calculated rates of growth, mortality and life expectancy for F. granulosa were within the range of those known for other stony corals. Our results reveal a young, dynamic population of this species on Eilat reefs, with high turnover rates and short lifespans. Such information is important for understanding recovery of coral reefs from disturbances, and for application to the management of commercially exploited coral populations.     

The effects of density-dependent offspring mortality on the synchrony of reproduction

Mortality rates often depend on the size of a population. Using ideal free theory to model the optimal timing of reproduction in model populations, I considered how the specific relationship between density-dependent offspring mortality and population size affects the optimal temporal distribution of reproduction. The results suggest that the specific form of the relationship between density-dependent mortality and the number of offspring produced determines the degree to which reproduction within a population is synchronous. Specifically, reproductive synchrony decreases as density-dependent mortality becomes increasingly inversely related to the number of offspring produced and is highest when density-dependent mortality is directly density-dependent. These findings support the suggestion that predation pressure selects for greater reproductive synchrony in species where mortality is directly density-dependent, but does not affect the timing of reproduction in species with density-independent rates of mortality. This revised version was published online in July 2006 with corrections to the Cover Date.     

The adequacy of aging techniques in vertebrates for rapid estimation of population mortality rates from age distributions

As a key parameter in population dynamics, mortality rates are frequently estimated using mark–recapture data, which requires extensive, long‐term data sets. As a potential rapid alternative, we can measure variables correlated to age, allowing the compilation of population age distributions, from which mortality rates can be derived. However, most studies employing such techniques have ignored their inherent inaccuracy and have thereby failed to provide reliable mortality estimates. In this study, we present a general statistical model linking birth rate, mortality rate, and population age distributions. We next assessed the reliability and data needs (i.e., sample size) for estimating mortality rate of eight different aging techniques. The results revealed that for half of the aging techniques, correlations with age varied considerably, translating into highly variable accuracies when used to estimate mortality rate from age distributions. Telomere length is generally not sufficiently correlated to age to provide reliable mortality rate estimates. DNA methylation, signal‐joint T‐cell recombination excision circle (sjTREC), and racemization are generally more promising techniques to ultimately estimate mortality rate, if a sufficiently high sample size is available. Otolith ring counts, otolithometry, and age‐length keys in fish, and skeletochronology in reptiles, mammals, and amphibians, outperformed all other aging techniques and generated relatively accurate mortality rate estimation with a sample size that can be feasibly obtained. Provided the method chosen is minimizing and estimating the error in age estimation, it is possible to accurately estimate mortality rates from age distributions. The method therewith has the potential to estimate a critical, population dynamic parameter to inform conservation efforts within a limited time frame as opposed to mark–recapture analyses.     

If a population crashes in prehistory, and there is no paleodemographer there to hear it, does it make a sound?

Catastrophic episodes (e.g., epidemics, natural disasters) strike with only limited regard for age. A large percentage of catastrophic mortality in a population can lead to a death distribution that resembles the living distribution, which includes greater numbers of older children, adolescents, and young adults than typical mortality profiles. This paper examines both the population implications of a large catastrophic mortality event, based on the Black Death as it ravaged medieval Europe, and its long-term effects on age-at-death distributions. An increased prevalence of epidemic disease is a common feature of reconstructions of the shift to agriculture and the rise of urban centers. The model begins with a hypothetical Medieval living population. This population is stable and characterized by slow growth. It has fertility and mortality rates consistent with a natural-fertility, agrarian population. The effects of catastrophic episodes are simulated by projecting the model population and subjecting it to one large (30% mortality) catastrophic episode as part of a 100-year population projection. A pair of Leslie matrices forms the basis of the projection. The catastrophic episode has important, long-term effects on both the living population and the cumulative distribution of death. The living population fails to recover from plague losses; at the end of the projection, population is still less than 75% its pre-plague level. The age-at-death distribution takes on the juvenile-young adult-heavy profile characteristic of many archaeological samples. The cumulative death profile based on the projection differs from that produced by the stable model significantly (P < 0.05) for 25-50 years after the plague episode, depending on sample size.     

Frequency distributions from birth,death, and creation processes

The time-dependent frequency distribution of groups of individuals versus group size was investigated within a continuum approximation, assuming a simplified individual growth, death and creation model. The analogy of the system to a physical fluid exhibiting both convection and diffusion was exploited in obtaining various solutions to the distribution equation. A general solution was approximated through the application of a Green's function. More specific exact solutions were also found to be useful. The solutions were continually checked against the continuum approximation through extensive simulation of the discrete system. Over limited ranges of group size, the frequency distributions were shown to closely exhibit a power-law dependence on group size, as found in many realizations of this type of system, ranging from colonies of mutated bacteria to the distribution of surnames in a given population. As an example, the modeled distributions were successfully fit to the distribution of surnames in several countries by adjusting the parameters specifying growth, death and creation rates.