Biological semelparity is a life history adaptation in which an individual organism reproduces once and then, or shortly thereafter, dies. This reproductive strategy is found throughout the plant and animal kingdoms. The trade-offs between reproduction and survival and the distinctions between semelparous and iteroparous life cycles have long been recognized as key issues involved in the study of life history strategies. Major topics of interest are the population dynamic consequences and the evolutionary advantages (or disadvantages) of semelparity versus iteroparity. Recent developments in the mathematical modeling of semelparity, using methods of nonlinear dynamics and bifurcation theory, have established a fundamental dynamic dichotomy that is of both biological and mathematical interest. From a mathematical point of view, models for the dynamics of semelparous species lie outside the standard theory of general structured population dynamics. Specifically, the fundamental bifurcation theorem that deals with the passage from population extinction to persistence (as the expected lifetime number of newborns produced by a newborn increases through the critical value of one) fails to hold. The challenge of determining the dynamic consequences of this fact have been met only in low dimensional cases (i.e., short maturation periods) and even then not thoroughly. These studies have established that, in lower dimensional cases, semelparous models exhibit a dynamic dichotomy that consists, roughly speaking, of an alternative between equilibration with overlapping generations and oscillations with non-overlapping generations. The oscillations in the later case can be strictly periodic, but also might be aperiodic. (They result from an invariant loop whose structure is a heteroclinic cycle.) Which of the two dynamics results (i.e., which is mathematically stable) depends on the magnitude of inter-stage competition present (relative to intra-stage competition). The first part of this project addresses the conjecture that this dynamic dichotomy is also present in semelparous models of higher dimension, to quantify the amount of inter-stage competition that results in an oscillatory dynamic, and to clarify the nature of these oscillations. The methods involve stability analysis, bifurcation methods, perturbation expansions, monotone semi-flow theory, the use average Lyapunov functions, persistence theory, and numerical simulations. The second part of the project addresses questions about the evolution of semelparity and the possibility of its being an evolutionary stable strategy (ESS). The method to be used is based on evolutionary game theory (and is called Darwinian dynamics), a methodology that extends a population dynamic model to include the dynamics of an evolving (mean phenotypic) trait, which in turn affects the population dynamics (through its influence on vital birth, growth, and death rates). The approach is primarily by means of bifurcation theory and will depend on the dynamic studies in the first part of the project. Indeed, part 2 will obtain (among other things) generalizations of the results in part 1 to an evolutionary setting. The theoretical results and methodology developed in part 2 will then be used in applications that address specific evolutionary questions. Using biologically reasonable trade-offs to build sub-models for fecundity and survivorships as functions of an evolving trait, we will study the circumstances under which semelparity is evolutionarily favored and when it is not. The Darwinian dynamics approach allows the methods of nonlinear dynamics and bifurcation theory to be applied to these evolutionary questions.

Investigations of many problems in biological sciences are based fundamentally on an understanding of population dynamics. This includes problems concerning the effects of climate change on ecosystems, the spread and control of diseases and pests, the protection of endangered species, the invasion of non-native species, the management of agricultural systems, the operation of fisheries, the design of wildlife refuges, and many others. Mathematical models derived to study problems such as these must, if one hopes to obtain accurate descriptions and predictions, be based on accurate dynamic models of the populations involved. For example, there is currently a great deal of research being carried out to "downscale" global climate data, i.e., to resolve the data to smaller scales, so that it can be used in population (ecosystem) dynamic models, the ultimate goal being an ability to predict the effects of future climate change on specific species of plants and animals. Accurate models of population dynamics must take into account, to some level of resolution, details concerning the life history strategy of species, i.e., the growth and reproduction schedule by means of they optimize fitness. Species with one type of life history will likely be quite differently affected by climate change (or by an invasive species or diseases or management decisions, etc.) than will be a species with a different life history. Biologists recognize two broad types of life histories: one in which individuals reproduce and then, or shortly thereafter, die (referred to as semelparity) and individuals who have repeated reproductive events throughout their life (referred to as iteroparity). There are numerous species throughout the plant and annual kingdoms that are semelparous (annual plants, a great many insects, some species of salmon, etc.). Models of semelparous population dynamics have not received the attention, with regard to many of their important aspects, as have those for iteroparous populations. Recent preliminary studies have shown that semelparous populations exhibit dynamic features that are, in several fundamental ways, very different from those typical iteroparous populations. These features, among others, have to do with the propensity of semelparous populations to exhibit periodic crashes and booms (as, for example, seen in the notorious cicada cycles or disastrous outbreaks of forest insect pests). The main goals of this research project are: (1) to develop a broad based theory of semelparous population dynamics and understand the properties of population oscillations (periodic outbreaks) and ascertain the conditions under which they do and do not occur; (2) extend the population dynamic theory to an evolutionary context so as to provide an understanding of how semelparous populations adapt and evolve; (3) to apply the findings to carefully selected and derived models of specific, important types of life histories studied in both theoretical and applied ecology. The mathematical models to be used in this research are of a type that is particularly accessible to those with limited backgrounds in the mathematics of dynamical systems. Because of the quick learning curve associated with these kinds of models, the project provides abundant research opportunities for students (both undergraduate and graduate) that, on the one hand, introduces them in an accessible context to sophisticated concepts and methods in the mathematical theory of dynamical systems and, on the other hand, permits them to carry out interesting applications that make solid contributions to biological problems.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Mary Ann Horn
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University of Arizona
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