William M. Schaffer

Selection for Optimal Life Histories: The Effects of Age Structure

The theory of optimal reproductive strategies has traditionally been studied in two ways: formal analysis of simple models that neglect the effects of age structure, and computer studies of complex life histories. Each of these approaches has disadvantages. The consequences of simple models sometimes dependmore on the nature of the simplifying assumptions than on the biological issues in question. On the other hand, computer simulations are only as general as the examples considered. The present study seeks to extend the formal analysis of optimal life histories to complex cases. I show that an optimal life history maximizes for each age class the expected fecundity at that age plus the sum of all future expected parameters. This result enables us to determine, at least inthe case of a three—stage life history, the manner in which the optimal reproductive effort at each age depends on the efforts made at the other ages, and thus, the coevolution of the various age—specific efforts. Three cases are distinguished: (1) If fertility and post—breeding survival and growth are concave functions of reproductive effort (i.e., have second derivatives that areeverywhere negative) there is a single set of age—specific reproductive rates towhich the system evolves regardless of initial conditions. This set of reproductive rates corresponds to an iteroparous life history (repeated breeding at different ages). (2) On the other hand, if fertility and subsequent growth and survival are convex functions of effort (positive second derivatives), semelparity (a single, herculean reproductive effort, followed by death) will most often evolve. However, an alternative, iteroparous life history sometimes exists, although stability considerations suggest that it may be transitory. (3)More realistic fertility and growth survival functions can generate alternative reproductive strategies that are stable since each represents a local maximum infitness. Often one of these alternatives corresponds to semelparity, the second to repeated reproduction. In such cases, the evolutionary outcome depends on initial conditions. This suggests that related species, with similar ecologies, may have very different life histories, the differences resulting from historical accidents that have trapped each on a different adaptive peak. The Salmonid genera, Salmo and Oncorhynchus, are suggested as possible examples.

Optimal Reproductive Effort in Fluctuating Environments

When the life history functions B (E) and P (E)-fecundity and postbreeding survival-are subject to environmentally induced fluctuations, one of two patterns is selected for: If the functions are concave (iteroparity in constant environments), the optimal population is monomorphic. Variation in B (E) selects for reduced effort in all individuals; variation in P (E) for increased breeding. If functions are convex (semelparity in constant environments) and fecundity is the parameter at issue, the optimal population can be polymorphic, with only a fraction of the population reproducing annually. Increasing the severity of fluctuations reduces the optimal value of this proportion, even if the average rate of reproductive success is not changed.

The Adaptive Significance of Variations in Life History among Local Populations of Atlantic Salmon in North America

In this paper variations in life history data among local populations of Atlantic salmon have been examined. The following patterns emerge: (1) The mean age of first spawning increases with the difficulty of upstream migration as estimated by the distance the fish ascend into freshwater. Other indices of river harshness yield similar results. (2) The effect of commercial fishing has been to eliminate larger and older fish from the run. Available evidence suggests that this has selected for an early age of first return on the Miramichi River and that the frequencies of genes coding for different ages of first spawning have been altered. (3) The mean age of first spawning is positively correlated with marine growth rates after the grilse stage. Rapid growth at sea subsequent to the grilse stage is associated with delayed reproduction; slower growth with an earlier age of first breeding. This result suggests different paths of high—seas migration. (4) The variability about the mean age of first spawning first increases and then decreases as one moves north over the salmons range from Maine to Ungava. In addition, we have observed that these results are in accord with predictions made by recent theoretical analyses of the optimal reproductive response to differing environmental conditions. We, therefore, conclude that the observed patterns of variation in life history are adaptive. We further take the agreement between theory and nature as a validation of the hypothesis that populations will in general differ from each other in the manner of their respective optima.

Order and Chaos in Ecological Systems

Systems of differential equations exhibiting complex periodic or chaotic behavior can sometimes be associated with one-dimensional mappings (difference equations) which encapsulate the properties of the attractor governing the full n-dimensional system. Recent advances make it possible to reconstruct the underlying attractor, and hence the one-dimensional map, from time-series data for a single state variable (species). The present paper illustrates this technique with reference to a hypothetical system consisting of a single predator species and two species of victims. Trapping records for Canadian lynx suggest that the method may have application to real-world populations.

Discrete-time growth-dispersal models

Abstract Integrodifference equations are discrete-time models that share many of the attributes of scalar reaction-diffusion equations. At the same time, they readily exhibit period doubling and chaos. We examine the properties of some simple integrodifference equations.

Nearly one dimensional dynamics in an epidemic

The incidence of measles in New York City and Baltimore was studied using recently developed techniques in nonlinear dynamics. The data, monthly case reports for the years 1928-1963, suggest almost two dimensional, chaotic flows whose essential attributes are captured by one dimensional, unimodal maps. The effects of noise, inevitable in ecological and epidemiological systems are discussed.

The Application of Optimal Control Theory to the General Life History Problem

The application of optimal control theory to life history evolution in species with discrete breeding seasons and overlapping generations is discussed. For each age class, the objective functional maximized consists of an integral (total reproduction for that age class) plus a final function (residual reproductive value). A simple example, for which monocarpy is the optimal strategy, is given. The present results complement previous studies (e.g., Schaffer 1979) of life history evolution as a problem in static optimization. The works of Leon (1976), who applied control theory to the case of species with continuous reproduction, and Mirmirani and Oster (1978), who did the same organisms with annual life histories, are thus extended.

Oscillations and chaos in epidemics: a nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark.

Using traditional spectral analysis and recently developed non-linear methods, we analyze the incidence of six childhood diseases in Copenhagen, Denmark. In three cases, measles, mumps, rubella, the dynamics suggest low dimensional chaos. Outbreaks of chicken pox, on the other hand, conform to an annual cycle with noise superimposed. The remaining diseases, pertussis and scarlet fever, remain problematic. The real epidemics are compared with the output of a Monte Carlo analog of the SEIR model for childhood infections. For measles, mumps, rubella, and chicken pox, we find substantial agreement between the model simulations and the data.

The Adaptive Significance of Variations in Reproductive Habit in the Agavaceae II: Pollinator Foraging Behavior and Selection for Increased Reproductive Expenditure

A coevolutionary hypothesis accounting for the evolution of Big Bang reproduction in yuccas and agaves is proposed. Particular attention is paid to the selective consequences of pollinator foraging behavior and reciprocally to the effects of plant reproductive expenditures on pollinator foraging strategies. Several lines of evidence, both observational and experimental, supporting the theory are presented. This evidence incidentally supports the notion that insect pollinators visit flowers in qualitative accord with the dictates of optimal diet theory. Although most nonannual plants and animals breed more than once in their lives, a few species exhibit an all or nothing pattern of reproductive investment. In

Stretching and folding in lynx fur returns: evidence for a strange attractor in nature?

A three-dimensional phase portrait for fur returns of Canadian lynx is constructed from a time series for a single variable. The resulting orbit is essentially sheetlike and exhibits successive stretching and folding, a hallmark of low-dimensional, strange attractors. Analysis of the associated one-dimensional map suggests that the observed fluctuations, and hence the stretching and folding, most likely represent an orbit which was at least twice periodic with noise superimposed.