Integral projection models for species with complex demography

Matrix projection models occupy a central role in population and conservation biology. Matrix models divide a population into discrete classes, even if the structuring trait exhibits continuous variation ( e. g., body size). The integral projection model ( IPM) avoids discrete classes and potential artifacts from arbitrary class divisions, facilitates parsimonious modeling based on smooth relationships between individual state and demographic performance, and can be implemented with standard matrix software. Here, we extend the IPM to species with complex demographic attributes, including dormant and active life stages, cross- classification by several attributes ( e. g., size, age, and condition), and changes between discrete and continuous structure over the life cycle. We present a general model encompassing these cases, numerical methods, and theoretical results, including stable population growth and sensitivity/ elasticity analysis for density- independent models, local stability analysis in density- dependent models, and optimal/ evolutionarily stable strategy life- history analysis. Our presentation centers on an IPM for the thistle Onopordum illyricum based on a 6- year field study. Flowering and death probabilities are size and age dependent, and individuals also vary in a latent attribute affecting survival, but a predictively accurate IPM is completely parameterized by fitting a few regression equations. The online edition of the American Naturalist includes a zip archive of R scripts illustrating our suggested methods

Evolution of complex flowering strategies: an age- and size-structured integral projection model

We explore the evolution of delayed age- and size-dependent flowering in the monocarpic perennial Carlina vulgaris, by extending the recently developed integral projection approach to include demographic rates that depend on size and age. The parameterized model has excellent descriptive properties both in terms of the population size and in terms of the distributions of sizes within each age class. In Carlina the probability of flowering depends on both plant size and age. We use the parameterized model to predict this relationship, using the evolutionarily stable strategy (ESS) approach. Despite accurately predicting the mean size of flowering individuals, the model predicts a step-function relationship between the probability of flowering and plant size, which has no age component. When the variance of the flowering-threshold distribution is constrained to the observed value, the ESS flowering function contains an age component, but underpredicts the mean flowering size. An analytical approximation is used to explore the effect of variation in the flowering strategy on the ESS predictions. Elasticity analysis is used to partition the agespecific contributions to the finite rate of increase (u) of the survival-growth and fecundity components of the model. We calculate the adaptive landscape that defines the ESS and generate a fitness landscape for invading phenotypes in the presence of the observed flowering strategy. The implications of these results for the patterns of genetic diversity in the flowering strategy and for testing evolutionary models are discussed. Results proving the existence of a dominant eigenvalue and its associated eigenvectors in general size- and age-dependent integral projection models are presented

Rapid evolution with generation overlap: the double-edged effect of dormancy

In life histories with generation overlap, selection that acts differently on different life-stages can produce reservoirs of genetic variation, for example, in long-lived iteroparous adults or long-lived dormant propagules. Such reservoirs provide “migration from the past” to the current population, and depending on the trend of environmental change, they have the potential either to slow adaptive evolution or accelerate it by re-introducing genotypes not affected by recent selection (e.g., through storage effect in a fluctuating environment). That is, the effect of generation overlap is a “double-edged sword,” with each edge cutting in a different direction. Here, we use sexual (quantitative trait) and asexual (clonal) models to explore the effects of generation overlap on adaptive evolution in a fluctuating environment, either with or without a trend in the mean environment state. Our analyses show that when environmental stochasticity scaled by strength of selection is intermediate and when the trend in mean environment is slow, intermediate values of generation overlap can maximize the rate of response to selection and minimize the adaptation lag between the trait mean and the environmental trend. Otherwise, increased generation overlap results in smaller selection response and larger adaptation lag. In the former case, low generation overlap results in low heritable trait variance, while high generation overlap increases the “migration load” from the past. Therefore, to understand the importance of rapid evolution and eco-evolutionary dynamics in the wild for organisms with overlapping generations, we need to understand the interaction of generation overlap, environmental stochasticity, and strength of selection

Understanding Terrorist Organizations with a Dynamic Model

Terrorist organizations change over time because of processes such as recruitment and training as well as counter-terrorism (CT) measures, but the effects of these processes are typically studied qualitatively and in separation from each other. Seeking a more quantitative and integrated understanding, we constructed a simple dynamic model where equations describe how these processes change an organization's membership. Analysis of the model yields a number of intuitive as well as novel findings. Most importantly it becomes possible to predict whether counter-terrorism measures would be sufficient to defeat the organization. Furthermore, we can prove in general that an organization would collapse if its strength and its pool of foot soldiers decline simultaneously. In contrast, a simultaneous decline in its strength and its pool of leaders is often insufficient and short-termed. These results and other like them demonstrate the great potential of dynamic models for informing terrorism scholarship and counter-terrorism policy making.Comment: To appear as Springer Lecture Notes in Computer Science v2: vectorized 4 figures, fixed two typos, more detailed bibliograph

Ecological Invasion, Roughened Fronts, and a Competitor's Extreme Advance: Integrating Stochastic Spatial-Growth Models

Both community ecology and conservation biology seek further understanding of factors governing the advance of an invasive species. We model biological invasion as an individual-based, stochastic process on a two-dimensional landscape. An ecologically superior invader and a resident species compete for space preemptively. Our general model includes the basic contact process and a variant of the Eden model as special cases. We employ the concept of a "roughened" front to quantify effects of discreteness and stochasticity on invasion; we emphasize the probability distribution of the front-runner's relative position. That is, we analyze the location of the most advanced invader as the extreme deviation about the front's mean position. We find that a class of models with different assumptions about neighborhood interactions exhibit universal characteristics. That is, key features of the invasion dynamics span a class of models, independently of locally detailed demographic rules. Our results integrate theories of invasive spatial growth and generate novel hypotheses linking habitat or landscape size (length of the invading front) to invasion velocity, and to the relative position of the most advanced invader.Comment: The original publication is available at www.springerlink.com/content/8528v8563r7u2742

Scaling up animal movements in heterogeneous landscapes: the importance of behavior

Two major challenges of spatial ecology are understanding the effects of landscape heterogeneity on movement, and translating observations taken at small spatial and temporal scales into expected patterns at greater scales. Using a combination of computer simulations and micro-landscape experiments with Tribolium confusum beetles we found that conventional correlated random walk models with constant parameters severely underestimated spatial spread because organisms changed movement behaviors over time. However, a model incorporating behavioral heterogeneity between individuals, and within individuals over time, was able to account for observed patterns of spread. Our results suggest that the main challenge for scaling up movement patterns resides in the complexities of individual behavior rather than in the spatial structure of the landscape

Species fluctuations sustained by a cyclic succession at the edge of chaos

Although mathematical models and laboratory experiments have shown that species interactions can generate chaos, field evidence of chaos in natural ecosystems is rare. We report on a pristine rocky intertidal community located in one of the world's oldest marine reserves that has displayed a complex cyclic succession for more than 20 y. Bare rock was colonized by barnacles and crustose algae, they were overgrown by mussels, and the subsequent detachment of the mussels returned bare rock again. These processes generated irregular species fluctuations, such that the species coexisted over many generations without ever approaching a stable equilibrium state. Analysis of the species fluctuations revealed a dominant periodicity of about 2 y, a global Lyapunov exponent statistically indistinguishable from zero, and local Lyapunov exponents that alternated systematically between negative and positive values. This pattern indicates that the community moved back and forth between stabilizing and chaotic dynamics during the cyclic succession. The results are supported by a patch-occupancy model predicting similar patterns when the species interactions were exposed to seasonal variation. Our findings show that natural ecosystems can sustain continued changes in species abundances and that seasonal forcing may push these nonequilibrium dynamics to the edge of chaos.</p

Statistical modelling of annual variation for inference on stochastic population dynamics using Integral Projection Models

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