A theory of stochastic harvesting in stochastic environments

We investigate how model populations respond to stochastic harvesting in a stochastic environment. In particular, we show that the effects of variable harvesting on the variance in population density and yield depend critically on the autocorrelation of environmental noise and on whether the endogenous dynamics of the population display over- or undercompensation to density. These factors interact in complicated ways; harvesting shifts the slope of the renewal function, and the net effect of this shift will depend on the sign and magnitude of the other influences. For example, when environmental noise exhibits a positive autocorrelation, the relative importance of a variable harvest to the variance in density increases with overcompensation but decreases with undercompensation. For a fixed harvesting level, an increasing level of autocorrelation in environmental noise will decrease the relative variation in population density when overcompensation would otherwise occur. These and other intricate interactions have important ramifications for the interpretation of time series data when no prior knowledge of demographic or environmental details exists. These effects are important whenever the harvesting rate is sufficiently high or variable, conditions likely to occur in many systems, whether the harvesting is caused by commercial exploitation or by any other strong agent of density-independent mortality.     

Individual adaptations in stochastic environments

Summary Many natural populations undergo radical and unpredictable fluctuations, associated with stochastic environmental conditions. Under such circumstances, fitness of a genotype (or strategy) is defined as the geometric mean of the intergenerational genotypic population growth ratel(t). Unfortunately, this population-level criterion has proved difficult to apply at the level of individual organisms.After developing a formula for the variance ofl as the sum of developmental and environmental variance, we discuss several models of individual adaptations, involving clutch size, progeny size and number, and foraging behaviour under risk of predation, based on the geometric-mean fitness concept. We then show how the method of dynamic programming can be extended to deal with facultative behaviour in stochastic environments. Finally we discuss the concept of an evolutionarily stable strategy in a stochastic environment.Our analysis suggests several novel interpretations of field and laboratory observations. Under the geometric mean criterion behaviour may be determined primarily by the worst likely environment; behaviour may appear suboptimal if observed only under normal or average conditions. For example,except under extreme environmental conditions, avian clutches larger than those that are observed might result in increased fecundity, with little if any cost of reproduction in terms of parental survival; however, in unusually bad years such large clutches might be disastrous, in terms of parental survival. This consideration may help explain some recently reported experimental clutch-size manipulation results. Similarly, our analysis indicates that the known phenomenon of seasonal reduction in seed size may constitute a double bet-hedging strategy, determined by parental mortality risk and future seed survival probability. We also discuss circumstances in which phenotypic polymorphism is an adaptation to environmental uncertainty. Thus almost any individual life history or behavioural adaptation may be affected by environmental stochasticity.     

Foraging behavior in stochastic environments

How do temporally stochastic environments affect risk sensitivity in foraging behavior? We build a simple model of foraging under predation risks in stochastic environments, where the environments change over generations. We analyze the effects of stochastic environments on risk sensitivity of foraging animals by means of the difference between the geometric mean fitness and the arithmetic mean fitness. We assume that foraging is associated with predation risks whereas resting in the nest is safe because it is free of predators. In each generation, two different environments with given food amounts and predation risks occur with a certain probability. The geometric mean optimum is independent of food amounts. In most cases of stochastic environments, risk-averse tendency is increased, but in some limited conditions, more risk-prone behavior is favored. Specifically, risk-prone tendency is increased when the variation in food amount increases. Our results imply that the optimal behavior depends on the probability distribution of environmental effects under all selection regimes.     

Evolution and population dynamics in stochastic environments

Inter-generational temporal variability of the environment is important in the evolution and adaptation of phenotypic traits. We discuss a population-dynamic approach which plays a central role in the analysis of evolutionary processes. The basic principle is that the phenotypes with the greatest long-term average growth rate will dominate the entire population. The calculation of longterm average growth rates for populations under temporal stochasticity can be highly cumbersome. However, for a discrete non-overlapping population, it is identical to the geometric mean of the growth rates (geometric mean fitness), which is usually different from the simple arithmetic mean of growth rates. Evolutionary outcomes based on geometric mean fitness are often very different from the predictions based on the usual arithmetic mean fitness. In this paper we illustrate the concept of geometric mean fitness in a few simple models. We discuss its implications for the adaptive evolution of phenotypes, e.g. foraging under predation risks and clutch size. Next, we present an application: the risk-spreading egg-laying behaviour of the cabbage white butterfly, and develop a two-patch population dynamic model to show how the optimal solution diverges from the ssual arithmetic mean approach. The dynamics of these stochastic models cannot be predicted from the dynamics of simple deterministic models. Thus the inclusion of stochastic factors in the analyses of populations is essential to the understanding of not only population dynamics, but also their evolutionary dynamics.     

From stochastic environments to life histories and back

Environmental stochasticity is known to play an important role in life-history evolution, but most general theory assumes a constant environment. In this paper, we examine life-history evolution in a variable environment, by decomposing average individual fitness (measured by the long-run stochastic growth rate) into contributions from average vital rates and their temporal variation. We examine how generation time, demographic dispersion (measured by the dispersion of reproductive events across the lifespan), demographic resilience (measured by damping time), within-year variances in vital rates, within-year correlations between vital rates and between-year correlations in vital rates combine to determine average individual fitness of stylized life histories. In a fluctuating environment, we show that there is often a range of cohort generation times at which the fitness is at a maximum. Thus, we expect ‘optimal’ phenotypes in fluctuating environments to differ from optimal phenotypes in constant environments. We show that stochastic growth rates are strongly affected by demographic dispersion, even when deterministic growth rates are not, and that demographic dispersion also determines the response of life-history-specific average fitness to within- and between-year correlations. Serial correlations can have a strong effect on fitness, and, depending on the structure of the life history, may act to increase or decrease fitness. The approach we outline takes a useful first step in developing general life-history theory for non-constant environments.     

Spatial network structure and amphibian persistence in stochastic environments

In the past few years, the framework of complex networks has provided new insight into the organization and function of biological systems. However, in spite of its potential, spatial ecology has not yet fully incorporated tools and concepts from network theory. In the present study, we identify a large spatial network of temporary ponds, which are used as breeding sites by several amphibian species. We investigate how the structural properties of the spatial network change as a function of the amphibian dispersal distance and the hydric conditions. Our measures of network topology suggest that the observed spatial structure of ponds is robust to drought (compared with similar random structures), allowing the movement of amphibians to and between flooded ponds, and hence, increasing the probability of reproduction even in dry seasons.     

The spikes trains probability distributions: A stochastic calculus approach

We discuss the statistics of spikes trains for different types of integrate-and-fire neurons and different types of synaptic noise models. In contrast with the usual approaches in neuroscience, mainly based on statistical physics methods such as the Fokker-Planck equation or the mean-field theory, we chose the point of the view of the stochastic calculus theory to characterize neurons in noisy environments. We present four stochastic calculus techniques that can be used to find the probability distributions attached to the spikes trains. We illustrate the power of these techniques for four types of widely used neuron models. Despite the fact that these techniques are mathematically intricate we believe that they can be useful for answering questions in neuroscience that naturally arise from the variability of neuronal activity. For each technique we indicate its range of applicability and its limitations.     

The evolution of parental care in stochastic environments

Parental care is of fundamental importance to understanding reproductive strategies and allocation decisions. Here, we explore how parental care strategies evolve in variable environments. Using a set of life-history trait trade-offs, we explore the relative costs and benefits of parental care in stochastic environments. Specifically, we consider the cases in which environmental variability results in varying adult death rates, egg death rates, reproductive rate and carrying capacity. Using a measure of fitness appropriate for stochastic environments, we find that parental care has the potential to evolve over a wide range of life-history characteristics when the environment is variable. A variable environment that affects adult or egg death rates can either increase or decrease the fitness of care relative to that in a constant environment, depending on the specific costs of care. Variability that affects carrying capacity or adult reproductive rate has negligible effects on the fitness associated with care. Increasing parental care across different life-history stages can increase fitness gains in variable environments. Costly investment in care is expected to affect the overall fitness benefits, the fitness optimum and rate of evolution of parental care. In general, we find that environmental variability, the life-history traits affected by such variability and the specific costs of care interact to determine whether care will be favoured in a variable environment and what levels of care will be selected.     

The evolution of patch selection in stochastic environments

A null model for habitat patch selection in spatially heterogeneous environments is the ideal free distribution (IFD), which assumes individuals have complete knowledge about the environment and can freely disperse. Under equilibrium conditions, the IFD predicts that local population growth rates are zero in all occupied patches, sink patches are unoccupied, and the fraction of the population selecting a patch is proportional to the patch's carrying capacity. Individuals, however, often experience stochastic fluctuations in environmental conditions and cannot respond to these fluctuations instantaneously. An evolutionary stability analysis for fixed patch-selection strategies reveals that environmental uncertainty disrupts the classical IFD predictions: individuals playing the evolutionarily stable strategy may occupy sink patches, local growth rates are negative and typically unequal in all patches, and individuals prefer higher-quality patches less than predicted by their carrying capacities. Spatial correlations in environmental fluctuations can enhance or marginalize these trends. The analysis predicts that continually increasing environmental variation first selects for range expansion, then selects for persisting coupled sink populations, and ultimately leads to regional extinction. In contrast, continually increasing habitat degradation first selects for range contraction and may select for persisting coupled sink populations before regional extinction. These results highlight the combined roles of spatial and temporal heterogeneity on the evolution of habitat selection.     

ESS emergence pattern of male butterflies in stochastic environments

Summary The evolutionarily stable (or ESS) emergence schedule for males of univoltine butterflies is analysed in an environment in which the female emergence schedule fluctuates stochastically between years. The ESS emergence curve, computed using the mutant invadability criterion, is shown to be the one that maximizes mean logarithmic lifetime mating success in the population in which it dominates. If males have accurate information about the female emergence schedule within each year, their emergence curve would evolve to the one predicted by a deterministic game model. The male emergence curve would then shift between years, closely following year to year changes in the female emergence pattern. If, instead, males have uncertainty about the female emergence schedule, the ESS male emergence curve becomes broader than the one predicted by the deterministic game model and will not track the between-year fluctuation of female emergence well. In a special case, we show how the between-year variation of mean emergence date, the variance of emergence date, the sexual difference in mean emergence dates (protandry) and the between-year correlation of mean emergence dates of both sexes should change with the degree of accuracy of information available to males.     

Extinction of populations by inbreeding depression under stochastic environments

Inbreeding depression may induce rapid extinction due to positive feedbacks between inbreeding depression and reduction of population size, which is often referred to as extinction vortex by inbreeding depression. The present analysis has demonstrated that the extinction vortex is likely to happen with realistic parameter values of genomic mutation rate of lethals or semilethals, equilibrium population size, intrinsic rate of natural increase, and rate of population decline caused by nongenetic extrinsic factors. Simulation models incorporating stochastic fluctuations of population size further indicated that extinction by inbreeding depression is facilitated by environmental fluctuations in population size. The results suggest that there is a positive interaction between genetic stochasticity and environmental stochasticity for extinction of populations by inbreeding depression. Received: May 10, 1999 / Accepted: November 5, 1999     

Population persistence time under intermittent control in stochastic environments

If there exists a critical population size above which environmental degradation becomes serious, the population should be suppressed or reduced upon reaching that level. Since population size control is accompanied by costs, a reduction in control frequency may be preferable from an economic viewpoint. Although this can be realized by decreasing the population size drastically in each control, such management may result in increased population extinction probability according to environmental stochasticity. The effects of population management on both mean population persistence time and management cost were analyzed theoretically using a diffusion process. The model showed the functional forms of both mean persistence time and control frequency explicitly; these decreased with an increasing number of individuals removed from the population in each control operation. Based on the analysis, indices representing management costs are proposed. Mean persistence time is generally an increasing function of the cost indices. Nevertheless, if the cost of each control increases with the number of individuals removed, even the most conservative management practice (continuous control) may not be overly expensive.     

Evolution of alternative insect life histories in stochastic seasonal environments

Deterministic seasonality can explain the evolution of alternative life history phenotypes (i.e., life history polyphenism) expressed in different generations emerging within the same year. However, the influence of stochastic variation on the expression of such life history polyphenisms in seasonal environments is insufficiently understood. Here, we use insects as a model and explore (1) the effects of stochastic variation in seasonality and (2) the life cycle on the degree of life history differentiation among the alternative developmental pathways of direct development and diapause (overwintering), and (3) the evolution of phenology. With numerical simulation, we determine the values of development (growth) time, growth rate, body size, reproductive effort, adult life span, and fecundity in both the overwintering and directly developing generations that maximize geometric mean fitness. The results suggest that natural selection favors the expression of alternative life histories in the alternative developmental pathways even when there is stochastic variation in seasonality, but that trait differentiation is affected by the developmental stage that overwinters. Increasing environmental unpredictability induced a switch to a bet‐hedging type of life history strategy, which is consistent with general life history theory. Bet‐hedging appeared in our study system as reduced expression of the direct development phenotype, with associated changes in life history phenotypes, because the fitness value of direct development is highly variable in uncertain environments. Our main result is that seasonality itself is a key factor promoting the evolution of seasonally polyphenic life histories but that environmental stochasticity may modulate the expression of life history phenotypes.     

Three-dimensional stochastic off-lattice model of binding chemistry in crowded environments

Molecular crowding is one of the characteristic features of the intracellular environment, defined by a dense mixture of varying kinds of proteins and other molecules. Interaction with these molecules significantly alters the rates and equilibria of chemical reactions in the crowded environment. Numerous fundamental activities of a living cell are strongly influenced by the crowding effect, such as protein folding, protein assembly and disassembly, enzyme activity, and signal transduction. Quantitatively predicting how crowding will affect any particular process is, however, a very challenging problem because many physical and chemical parameters act synergistically in ways that defy easy analysis. To build a more realistic model for this problem, we extend a prior stochastic off-lattice model from two-dimensional (2D) to three-dimensional (3D) space and examine how the 3D results compare to those found in 2D. We show that both models exhibit qualitatively similar crowding effects and similar parameter dependence, particularly with respect to a set of parameters previously shown to act linearly on total reaction equilibrium. There are quantitative differences between 2D and 3D models, although with a generally gradual nonlinear interpolation as a system is extended from 2D to 3D. However, the additional freedom of movement allowed to particles as thickness of the simulation box increases can produce significant quantitative change as a system moves from 2D to 3D. Simulation results over broader parameter ranges further show that the impact of molecular crowding is highly dependent on the specific reaction system examined.     

Demography and dispersal: invasion speeds and sensitivity analysis in periodic and stochastic environments

Invasion speeds can be calculated from matrix integrodifference equation models that incorporate stage-specific demography and dispersal. These models also permit the calculation of the sensitivity and elasticity of invasion speed to changes in demographic and dispersal parameters. Such calculations have been used to understand the factors determining invasion speed and to explore possible tactics to manage invasive species. In this paper, we extend these calculations to temporally varying environments. We present formulas for the invasion speed and its sensitivity and elasticity in both periodic and stochastic environments. Periodic models can describe seasonal variation within a year, or can be used to study the frequency of occurrence of events (e.g., floods, fires) on interannual time scales. Stochastic models can incorporate variances, covariances, and temporal autocorrelation of parameters. We show that the invasion speed is calculated from a growth rate which is in turn calculated from a periodic or stochastic product of moment-generating function matrices. We present a new formulation of sensitivity analysis, using matrix calculus, that applies equally to constant, periodic, and stochastic environments.