Amy Veprauskas

A juvenile–adult population model: climate change, cannibalism, reproductive synchrony, and strong Allee effects

ABSTRACT We study a discrete time, structured population dynamic model that is motivated by recent field observations concerning certain life history strategies of colonial-nesting gulls, specifically the glaucous-winged gull (Larus glaucescens). The model focuses on mechanisms hypothesized to play key roles in a populations response to degraded environment resources, namely, increased cannibalism and adjustments in reproductive timing. We explore the dynamic consequences of these mechanics using a juvenile–adult structure model. Mathematically, the model is unusual in that it involves a high co-dimension bifurcation at which, in turn, leads to a dynamic dichotomy between equilibrium states and synchronized oscillatory states. We give diagnostic criteria that determine which dynamic is stable. We also explore strong Allee effects caused by positive feedback mechanisms in the model and the possible consequence that a cannibalistic population can survive when a non-cannibalistic population cannot.

Abelian periods, partial words, and an extension of a theorem of Fine and Wilf

Recently, Constantinescu and Ilie proved a variant of the wellknown periodicity theorem of Fine and Wilf in the case of two relatively prime abelian periods and conjectured a result for the case of two nonrelatively prime abelian periods. In this paper, we answer some open problems they suggested. We show that their conjecture is false but we give bounds, that depend on the two abelian periods, such that the conjecture is true for all words having length at least those bounds and show that some of them are optimal. We also extend their study to the context of partial words, giving optimal lengths and describing an algorithm for constructing optimal words.

Analysis of lethal and sublethal impacts of environmental disasters on sperm whales using stochastic modeling

Mathematical models are essential for combining data from multiple sources to quantify population endpoints. This is especially true for species, such as marine mammals, for which data on vital rates are difficult to obtain. Since the effects of an environmental disaster are not fixed, we develop time-varying (nonautonomous) matrix population models that account for the eventual recovery of the environment to the pre-disaster state. We use these models to investigate how lethal and sublethal impacts (in the form of reductions in the survival and fecundity, respectively) affect the population’s recovery process. We explore two scenarios of the environmental recovery process and include the effect of demographic stochasticity. Our results provide insights into the relationship between the magnitude of the disaster, the duration of the disaster, and the probability that the population recovers to pre-disaster levels or a biologically relevant threshold level. To illustrate this modeling methodology, we provide an application to a sperm whale population. This application was motivated by the 2010 Deepwater Horizon oil rig explosion in the Gulf of Mexico that has impacted a wide variety of species populations including oysters, fish, corals, and whales.

The evolution of toxicant resistance in daphniids and its role on surrogate species

Prolonged exposure to a disturbance such as a toxicant has the potential to result in rapid evolution to toxicant resistance in many short-lived species such as daphniids. This evolution may allow a population to persist at higher levels of the toxicant than is possible without evolution. Here we apply evolutionary game theory to a Leslie matrix model for a daphniid population to obtain a Darwinian model that couples population dynamics with the dynamics of an evolving trait. We use the Darwinian model to consider how the evolution of resistance to the lethal or sublethal effects of a disturbance may change the population dynamics. In particular, we determine the conditions under which a daphniid population can persist by evolving toxicant resistance. We then consider the implications of this evolution in terms of the use of daphniids as surrogate species. We show for three species of daphniids that evolution of toxicant resistance means that one species may persist while another does not. These results suggest that toxicant studies that do not consider the potential of a species (or its surrogate) to develop toxicant resistance may not accurately predict the long term persistence of the species.

Examining the effect of reoccurring disturbances on population persistence with application to marine mammals

We develop a two-state Markov chain to describe the effect of reoccurring disturbances on a population that is modeled by discrete-time matrix model. The environment is described by three parameters that define the magnitude of impact of a disturbance, the average duration of impact of a disturbance, and the average time between disturbances. We derive an approximation for the stochastic growth rate in order to examine how these three parameters affect population growth. From this approximation, we calculate the sensitivity and elasticity of the growth rate with respect to the environmental parameters. We show that the average duration of impact of a disturbance and the average time between disturbances contribute equally to the stochastic growth rate. We also show that the elasticity of the stochastic growth rate is more sensitive to changes in the magnitude of impact than to changes in either the average duration of impact of a disturbance or the average time between disturbances. These conclusions hold irrespective of the population under consideration. We then provide an application of the model formulation to examine how disturbances, such as oil spills, may affect a sperm whale population. The model results suggest that, in oder to mitigate the impact of disturbances, management strategies should focus on reducing the magnitude of impact. Meanwhile, if it is more feasible to reduce either the duration of impact or the time between impacts, managers should focus on whichever is easier to obtain. In addition, when applied to a sperm whale population, our model shows that the probability of extinction can dramatically increase when disturbance frequency increases but is not greatly impacted by the assumption that all disturbances have the same magnitude.

A nonlinear continuous-time model for a semelparous species

Periodical semelparous insects such as cicadas and May beetles exhibit synchronization in age classes such that only one age class is present at any point of time. This leads to outbreaks of adults as they all reach maturity around the same time. Discrete-time models of semelparous species have shown that this type of synchronous cycling can occur as a result of greater between-class competition relative to within-class competition. However, relatively few studies have examined continuous-time models of semelparous species. Here we develop a continuous-time model for a semelparous species using a technique called the linear chain trick to convert a non-linear McKendrick partial differential equation into a finite system of ordinary differential equations. We represent semelparity by a birth function whose age distribution can be made arbitrarily narrow. We show that a Hopf bifurcation may occur in this model as a result of competition between reproducing and non-reproducing classes. This bifurcation leads to stable cycles in which the two classes are out of phase, thus providing a continuous-time counterpart to the synchronous cycles that occur in discrete-time models.

Modeling as a complementary tool to acoustic data for understanding the impact of environmental disasters on marine mammals

This study is focused on how environmental disasters, such as the Deepwater Horizon oil rig explosion in 2010, affect the dynamics of marine mammal populations, particularly sperm whales and beaked whales, in the Northern Gulf of Mexico. We briefly describe how modeling techniques are used to estimate densities of marine mammals using passive acoustic data. We then develop a matrix model to examine the possible long-term effects of a disaster. We consider cases in which the effects of a disturbance result in reductions in either survival (lethal impacts) or fecundity (sublethal impacts). This model, combined with demographic stochasticity, allows us to study the long-term recovery process following an environmental disaster. In particular, recovery probabilities and recovery times of the population are computed, and formulas are derived to compute the sensitivity of the recovery time to changes in properties of the population or the environmental disturbance. We then extend the modeling methodology to con...

A bifurcation theorem for evolutionary matrix models with multiple traits

One fundamental question in biology is population extinction and persistence, i.e., stability/instability of the extinction equilibrium and of non-extinction equilibria. In the case of nonlinear matrix models for structured populations, a bifurcation theorem answers this question when the projection matrix is primitive by showing the existence of a continuum of positive equilibria that bifurcates from the extinction equilibrium as the inherent population growth rate passes through 1. This theorem also characterizes the stability properties of the bifurcating equilibria by relating them to the direction of bifurcation, which is forward (backward) if, near the bifurcation point, the positive equilibria exist for inherent growth rates greater (less) than 1. In this paper we consider an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection. We extend the fundamental bifurcation theorem to this evolutionary model. We apply the results to an evolutionary version of a Ricker model with an added Allee component. This application illustrates the theoretical results and, in addition, several other interesting dynamic phenomena, such as backward bifurcation induced strong Allee effects.