Evolution of flowering strategies in Oenothera glazioviana: an integral projection model approach

The timing of reproduction is a key determinant of fitness. Here, we develop parameterized integral projection models of size-related flowering for the monocarpic perennial Oenothera glazioviana and use these to predict the evolutionarily stable strategy (ESS) for flowering. For the most part there is excellent agreement between the model predictions and the results of quantitative field studies. However, the model predicts a much steeper relationship between plant size and the probability of flowering than observed in the field, indicating selection for a 'threshold size' flowering function. Elasticity and sensitivity analysis of population growth rate lambda and net reproductive rate R(0) are used to identify the critical traits that determine fitness and control the ESS for flowering. Using the fitted model we calculate the fitness landscape for invading genotypes and show that this is characterized by a ridge of approximately equal fitness. The implications of these results for the maintenance of genetic variation are discussed.     

Population dynamics of the cactus Mammillaria gaumeri: an integral projection model approach

The Cactaceae family in Mexico is particularly important because members of this family exhibit a high degree of endemism. Unfortunately, many species of the Cactaceae are threatened or endangered. We employed an integral projection model for studies of the population dynamics of Mammillaria gaumeri, an endemic cactus of the Yucatán characterized by a small population size. The integral projection model provides estimates of the asymptotic growth rate, stable size distribution, reproductive values, and sensitivities and elasticities of the growth rate to changes in vital rates. Nine locations of this species were studied along the Yucatan coast over a 9-year period. Individuals were classified by plant volume. Most population growth rate (λ) values were below unity. The highest elasticity values corresponded to the survival of intermediate size individuals. The percentage of germination in the field was low, and consequently, fecundity values were also low. Reproductive values were observed to increase with plant volume. The stable size distribution of M. gaumeri was skewed toward small individuals. For all years, the kernel showed that individual survival determined the population growth rate.     

Evolution of gametophytic apomixis in flowering plants: an alternative model from Maloid Rosaceae

Gametophytic apomixis, asexual reproduction involving megagametophytes, occurs in many flowering-plant families and as several variant mechanisms. Developmental destabilization of sexual reproduction as a result of hybridization and/or polyploidy appears to be a general trigger for its evolution, but the evidence is complicated by ploidy-level changes and hybridization occurring with facultative apomixis. The repeated origins of polyploid apomictic complexes in the palaeopolyploid Maloid Rosaceae suggest a new model of evolutionary transitions that may have wider applicability. Two conjectures are fundamental to this model: (1) that as previously suggested by Rutishauser, like many sexual flowering plants the polyploid apomicts require maternal–paternal balance in the second fertilization event that gives rise to the endosperm, and (2) that the observed variation in endosperm ploidy levels relates less to flexibility late in development than to the known variation in developmental origin of the megagametophyte between mechanisms loosely categorized as diplospory and apospory. The model suggests explanations for the relative frequencies of apospory and diplospory, and for the wide but incomplete associations of apospory with a pollination requirement (pseudogamy) and of diplospory with autonomous development of the endosperm. It is suggested that pollination from other taxa may provide some adaptive advantage to pseudogamous apospory. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

Evolution of life history strategies for an asexual annual plant model

At any given moment in time a plant is partitioning total growth mass into its various component parts such as leaves, roots, reproductive material, etc. The view is taken that the plant has evolved a life history strategy to control this partitioning process. This paper illustrates the utility of optimal control theory for use in determining life history strategies which maximize fitness for a given asexual plant model. The optimal control methods are first used on a model previously analyzed by Professor Dan Cohen, who used a different method. His results of a change from 100% vegetable growth to 100% reproductive growth at a fixed switching time is again obtained. This 100% switching result is shown to be more generally applicable by using a qualitatively described model. However the switching time in general is shown to be a function of both leaf mass and time remaining to the end of the growing season. The allocation to toxin production is also considered. It is shown that under this model an inequality between system parameters must be satisfied before the plant should allocate growth to toxin production. Although the particular model explored here may rarely be realistic in nature, these same methods of optimal control theory can be applied in a similar fashion to many other proposed models of plant resource allocation.     

A stochastic flowering model describing an asynchronically flowering set of trees

A general stochastic model is presented that simulates the time course of flowering of individual trees and populations, integrating the synchronization of flowering both between and within trees. Making some hypotheses, a simplified expression of the model, called the 'shoot' model, is proposed, in which the synchronization of flowering both between and within trees is characterized by specific parameters. Two derived models, the 'tree' model and the 'population' model, are presented. They neglect the asynchrony of flowering, respectively, within trees, and between and within trees. Models were fitted and tested using data on flowering of Psidium cattleianum observed at study sites at elevations of 200, 520 and 890 m in Reunion Island. The 'shoot' model fitted the data best and reproduced the strong irregularities in flowering shown by empirical data. The asynchrony of flowering in P. cattleianum was more pronounced within than between trees. Simulations showed that various flowering patterns can be reproduced by the 'shoot' model. The use of different levels of organization of the general model is discussed.     

A size-structured model of bacterial growth and reproduction

We consider a size-structured bacterial population model in which the rate of cell growth is both size- and time-dependent and the average per capita reproduction rate is specified as a model parameter. It is shown that the model admits classical solutions. The population-level and distribution-level behaviours of these solutions are then determined in terms of the model parameters. The distribution-level behaviour is found to be different from that found in similar models of bacterial population dynamics. Rather than convergence to a stable size distribution, we find that size distributions repeat in cycles. This phenomenon is observed in similar models only under special assumptions on the functional form of the size-dependent growth rate factor. Our main results are illustrated with examples, and we also provide an introductory study of the bacterial growth in a chemostat within the framework of our model.     

Incorporating environmental change over succession in an integral projection model of population dynamics of a forest herb

Despite seemingly obvious effects of environmental drivers, mechanisms behind long‐term changes in plant population sizes over time are often poorly known. We investigated how soil potassium concentration and seed predation are likely to change over time as a result of succession from deciduous forest to spruce forest, and how this affects population trajectories of Actaea spicata. Observations and addition experiments showed that high soil potassium concentration increased individual growth rates. Among‐site comparisons showed that soil potassium concentration was lower where proportion spruce was higher. Incorporation of a gradual increase in spruce over time in an integral projection model where individual growth depended on potassium suggested a net decrease in A. spicata population sizes over forest succession. This result suggests that small changes in factors with small effects on individual performance can influence patterns of species occupancy along successional gradients. We incorporated also density independent and density dependent effects of pre‐dispersal seed predation over succession into the same model. Seed predation influenced the tree composition at which A. spicata population growth was positive. However, significant effects of A. spicata population size on seed predation intensity did not translate into important feedback effects on population growth trajectories over succession. Our results illustrate how demographic models can be used to gain understanding of the mechanisms behind effects of environmental change on species abundances and distributions by the simultaneous inclusion of changing abiotic and biotic factors.     

Effects of size‐ and sex‐selective harvesting: An integral projection model approach

Harvesting is often size‐selective, and in species with sexual size dimorphism, it may also be sex‐selective. A powerful approach to investigate potential consequences of size‐ and/or sex‐selective harvesting is to simulate it in a demographic population model. We developed a population‐based integral projection model for a size‐ and sex‐structured species, the commonly exploited pike (Esox lucius). The model allows reproductive success to be proportional to body size and potentially limited by both sexes. We ran all harvest simulations with both lower size limits and slot limits, and to quantify the effects of selective harvesting, we calculated sex ratios and the long‐term population growth rate (λ). In addition, we quantified to what degree purely size‐selective harvesting was sex‐selective, and determined when λ shifted from being female to male limited under size‐ and sex‐selective harvesting. We found that purely size‐selective harvest can be sex‐selective, and that it depends on the harvest limits and the size distributions of the sexes. For the size‐ and sex‐selective harvest simulations, λ increased with harvest intensity up to a threshold as females limited reproduction. Beyond this threshold, males became the limiting sex, and λ decreased as more males were harvested. The peak in λ, and the corresponding sex ratio in harvest, varied with both the selectivity and the intensity of the harvest simulation. Our model represents a useful extension of size‐structured population models as it includes both sexes, relaxes the assumption of female dominance, and accounts for size‐dependent fecundity. The consequences of selective harvesting presented here are especially relevant for size‐ and sex‐structured exploited species, such as commercial fisheries. Thus, our model provides a useful contribution toward the development of more sustainable harvesting regimes.     

On periodic cohort solutions of a size-structured population model

 We consider a size-structured population model with discontinuous reproduction and feedback through the environmental variable ‘substrate’. The model admits solutions with finitely many cohorts and in that case the problem is described by a system of ODEs involving a bifurcation parameter β. Existence of nontrivial periodic n-cohort solutions is investigated. Moreover, we discuss the question whether n cohorts (n≧2) with small size differences will tend to a periodic one-cohort solution as t→∞. Received 16 March 1995; received in revised form 7 January 1997     

Evolution of body size: an optimization model

The growth of individuals is often described by bioenergetic equations which partition assimilated energy into maintenance, growth, reproduction, and so on. Such energy flows vary with body size. The bioenergetic equations in this paper, under the assumptions that organisms will adopt behavior that channels a maximum amount of energy into the production of surviving progeny, produce a model optimizing growth and reproduction. Using the Weierstrass theorem, we show that a solution of the optimization problem exists. The problem is further solved analytically using the Pontryagin maximum principle. The main conclusion of the paper is that in a given environment an optimal body size exists, one which maximizes the energy channeled to the production of progeny. This body size depends on the mortality rate, the maximum life span, and the derivative of the growth equation with respect to body size. The biological results predicted from the model are compared with ecological data for zooplankton and vertebrate species, which support the conclusions obtained.     

Competitive exclusion and coexistence for a quasilinear size-structured population model

We present a quasilinear size-structured model which describes the dynamics of a population with n competing ecotypes. We assume that the vital rates of each subpopulation depend on the total population due to competition. We provide conditions on the individual rates which guarantee competitive exclusion in the case of closed reproduction (offspring always belongs to the same ecotype as the parent). In particular, our results suggest that the ratio of the reproduction and mortality rates is a good measure to determine the winning ecotype. Meanwhile, we show that in the case of open reproduction all ecotypes coexist.     

Mating strategies in flowering plants: the outcrossing-selfing paradigm and beyond

Comparisons of the causes and consequences of cross- and self-fertilization have dominated research on plant mating since Darwin's seminal work on plant reproduction. Here, I provide examples of these accomplishments, but also illustrate new approaches that emphasize the role of floral design and display in pollen dispersal and fitness gain through male function. Wide variation in outcrossing rate characterizes animal-pollinated plants. In species with large floral displays, part of the selfing component of mixed mating can arise from geitonogamy and be maladaptive because of strong inbreeding depression and pollen discounting. Floral strategies that separate the benefits of floral display from the mating costs associated with geitonogamy can resolve these conflicts by reducing lost mating opportunities through male function. The results from experiments with marker genes and floral manipulations provide evidence for the function of herkogamy and dichogamy in reducing self-pollination and promoting pollen dispersal. Evidence is also presented indicating that increased selfing resulting from changes to floral design, or geitonogamy in large clones, can act as a stimulus for the evolution of dioecy. The scope of future research on mating strategies needs to be broadened to include investigations of functional links among flowers, inflorescences and plant architecture within the framework of life-history evolution.     

Stochastic stable population growth in integral projection models: theory and application

Stochastic matrix projection models are widely used to model age- or stage-structured populations with vital rates that fluctuate randomly over time. Practical applications of these models rest on qualitative properties such as the existence of a long term population growth rate, asymptotic log-normality of total population size, and weak ergodicity of population structure. We show here that these properties are shared by a general stochastic integral projection model, by using results in (Eveson in D. Phil. Thesis, University of Sussex, 1991, Eveson in Proc. Lond. Math. Soc. 70, 411-440, 1993) to extend the approach in (Lange and Holmes in J. Appl. Prob. 18, 325-344, 1981). Integral projection models allow individuals to be cross-classified by multiple attributes, either discrete or continuous, and allow the classification to change during the life cycle. These features are present in plant populations with size and age as important predictors of individual fate, populations with a persistent bank of dormant seeds or eggs, and animal species with complex life cycles. We also present a case-study based on a 6-year field study of the Illyrian thistle, Onopordum illyricum, to demonstrate how easily a stochastic integral model can be parameterized from field data and then applied using familiar matrix software and methods. Thistle demography is affected by multiple traits (size, age and a latent "quality" variable), which would be difficult to accommodate in a classical matrix model. We use the model to explore the evolution of size- and age-dependent flowering using an evolutionarily stable strategy (ESS) approach. We find close agreement between the observed flowering behavior and the predicted ESS from the stochastic model, whereas the ESS predicted from a deterministic version of the model is very different from observed flowering behavior. These results strongly suggest that the flowering strategy in O. illyricum is an adaptation to random between-year variation in vital rates.     

Felling the giants: integral projection models indicate adult management to control an exotic invasive palm

Plant Ecology - Population models are helpful for understanding demographic trends in invasive plants and crucial in defining effective management actions. Here, we assessed the dynamics of three...