Optimal escapement levels in stochastic and deterministic harvesting models

A constant-escapement feedback policy is shown to be optimal in maximizing expected discounted net revenue from an animal resource whose dynamics are described by a stochastic stock-recruitment model, provided that unit harvesting costs satisfy certain conditions. The optimal escapement in this model is compared with that in the corresponding deterministic model and it is shown how the way in which unit harvesting costs vary with population abundance can be important in determining the relative sizes of the optimal escapements. In most cases, the optimal stochastic escapement is no less than the optimal deterministic escapement.     

Some stochastic differential equation models of an aquatic ecosystem

A set of stochastic differential equations has been used to model an aquatic ecosystem. The randomness in the system has been introduced through initial conditions of the state variables, parameters, and input variables (light and temperature). These models were analysed using Monte Carlo simulation procedures and the results were similar to those observed in the experimental and field data. They were different, however, from the results of a deterministic simulation. This approach allows us to incorporate the maximum degree of information in the model and to study the behavior of the system without arbitrarily manipulating the values of the parameters. Some possible refinements and generalizations of this approach are also discussed.     

Optimal foraging: The influence of intraspecific competition on diet selection

Summary Three-spined sticklebacks (Gasterosteus aculeatus) showed a relative preference for a familiar prey size when hunting for two sizes of Daphnia magna in high density. This result is not compatible with the apparent size hypothesis. Ten groups were investigated, each consisting of two stickle-backs tested under three consecutive experimental conditions, to establish whether the function of the preference for a familiar prey size could be avoidance of competition.First, the relative competitive ability of each fish was determined by the proportion it consumed of three series of 60 medium-sized daphnia, offered pairwise. Second, during the consumption of five series of 30 pairs, each consisting of a large and a small daphnia, it was determined how many items each fish caught of each prey size. As handling times were equal for both prey types, the larger prey size was more profitable. There was a significant correlation between relative competitive ability and mean proportion of large daphnia in the diet. In the last series the less successful competitiors caught a higher proportion of small prey than in the first series. Finally, each fish was given the choice between large and small daphnia in the absence of its competitor. The sticklebacks chose a diet similar to the one they had been allowed to select previously with competition. The previously more successful competitors concentrated on large daphnia, whereas the poorer competitors fed as generalists but not unselectively.The fish probably learned the distance from which they had recently attacked familiar prey successfully. This sure attack distance depends on the fish's competitive ability.     

Sampling behavior of starlings foraging in simple patchy environments

Summary Starlings were allowed to forage in patchy laboratory environments where patches contained either zero or a fixed number of prey. The condition of a given patch (prey or no prey) could only be determined from information gained while exploiting the patch. A starling's task was to determine to what extent to sample an apparently prey-less patch before giving it up as such, in a way which maximizes long-term energy intake rate. The simple model presented to predict the optimal sampling solutions was qualitatively but not quantitatively supported by the data. The main discrepancy was in the fact that an apparently prey-less patch should have been sampled to a fixed extent before leaving, whereas a distribution of sampling behavior was actually observed. The qualitative agreement was very good, however, as the modes of the observed sampling distributions often corresponded to the predicted optimal sampling solutions. Starlings seem to possess a patch-sampling ability which, at least for those simple situations analyzed, can lead to an efficient foraging strategy.     

Optimal foraging in bumblebees: Why is nectar left behind in flowers?

Summary Queen bumblebees (Bombus appositus) leave nectar behind in Delphinium nelsoni flowers with high-standing crops of nectar. Bumblebees deplete flowers in areas with lower-standing crops. Residual volumes predicted by an optimal feeding hypothesis agree with field measurements.     

Analysing sperm competition data: simple models for predicting mechanisms

Summary Prospective models are developed for analysing sperm competition data so as to predict the underlying mechanisms determining paternity in multiply mated females. The models require: 1) estimations of proportion of offspring sired by the last male to mate (P ₂), 2) knowledge of the number of sperm transferred by each male, and 3) knowledge of the sperm storage capacity of the female, should this be limited. They will distinguish between raffles (sperm mixing without displacement) and sperm displacement mechanisms. The sensitivity of the techniques can be increased by manipulating the number of sperm transferred by each male. Typically, this can be done by manipulating copula duration or number of ejaculations, given a knowledge of the rate of sperm transfer. Data from two contrasting insect species are fitted to the models to demonstrate the techniques. These models are prospective only, and their limitations are discussed. The principal limitation is that we assume that sperm used for fertilization mix randomly in a fertilization set immediately prior to fertilization; in reality this may be difficult to identify. When sperm mixing is very rapid, the fertilization set will often be equivalent to the sperm stores, but with slow mixing, the fertilization set may be equivalent to a much more restricted zone and may change with time. Offprint requests to: G.A. Parker     

Optimal foraging and territorial defence in the Great Tit (Parus major)

Summary We describe an experiment designed to investigate the trade-off between foraging and territorial vigilance in the Great Tit. Captive territorial male Great Tits were observed while foraging in a large indoor aviary. They obtained food from two operant patches in which the supply of food was gradually depleted during a visit. We predicted that during control sessions the birds would switch between patches in such a way as to maximise their overall feeding rate. In experimental sessions, we introduced briefly a rival male as an intruder at the start of the test. The foraging male could see the rival only when travelling between patches and not while feeding within patches. We predicted that during experimental sessions birds would switch between patches more often than in control tests, sacrificing food intake for territorial vigilance. Three of the four males tested behaved in approximately the predicted manner. We discuss the use of an inverse optimality argument to provide a calibration of feeding against the benefit resulting from territorial vigilance.     

Vicarious sampling: the use of personal and public information by starlings foraging in a simple patchy environment

Group foragers may be able to assess patch quality more efficiently by paying attention to the sampling activities of conspecifics foraging in the same patch. In a previous field experiment, we showed that starlings foraging on patches of hidden food could use the successful foraging activities of others to help them assess patch quality. In order to determine whether a starling could also use another individual’s lack of foraging success to assess and depart from empty patches more quickly, we carried out two experimental studies which compared the behaviour of captive starlings sampling artificial patches both when alone and when in pairs. Solitary starlings were first trained to assess patch quality in our experimental two-patch system, and were then tested on an empty patch both alone and with two types of partner bird. One partner sampled very few holes and thus provided a low amount of public information; the other sampled numerous holes and thus provided a high amount of public information. In experiment 1, we found no evidence of vicarious sampling. Subjects sampled a similar number of empty holes when alone as when with the low and high information partners; thus they continued to rely on their own personal information to make their patch departure decisions. In experiment 2, we modified the experimental patches, increasing the ease with which a bird could watch another’s sampling activities, and increasing the difficulty of acquiring accurate personal sampling information. This time, subjects apparently did use public information, sampling fewer empty holes before departure when with the high-information partner than when with the low-information partner, and sampling fewer holes when with the low-information partner than when alone. We suggest that the degree to which personal and public information are used is likely to depend both on a forager’s ability to remember where it has already sampled and on the type of environment in which foraging takes place. Received: 31 January 1995/Accepted after revision: 11 September 1995     

Survival analysis of stochastic single-species population models in polluted environments

This paper reports on the behaviors of single species models with and without pollution. We consider three basic models, one is deterministic, and others are stochastic. We first obtained the acute thresholds between local extinction and (stochastic) weakly persistent in the mean for population respectively. Then we study the attainability of population size 0 for the stochastic cases and show that a randomized non-autonomous logistic equation will be stochastic permanent under some conditions. Finally, we introduce some numerical simulink graphics to illustrate our main results.     

Integral projection models for finite populations in a stochastic environment

Continuous types of population structure occur when continuous variables such as body size or habitat quality affect the vital parameters of individuals. These structures can give rise to complex population dynamics and interact with environmental conditions. Here we present a model for continuously structured populations with finite size, including both demographic and environmental stochasticity in the dynamics. Using recent methods developed for discrete age-structured models we derive the demographic and environmental variance of the population growth as functions of a continuous state variable. These two parameters, together with the expected population growth rate, are used to define a one-dimensional diffusion approximation of the population dynamics. Thus, a substantial reduction in complexity is achieved as the dynamics of the complex structured model can be described by only three population parameters. We provide methods for numerical calculation of the model parameters and demonstrate the accuracy of the diffusion approximation by computer simulation of specific examples. The general modeling framework makes it possible to analyze and predict future dynamics and extinction risk of populations with various types of structure, and to explore consequences of changes in demography caused by, e.g., climate change or different management decisions. Our results are especially relevant for small populations that are often of conservation concern.     

The modified niche model: Including detritus in simple structural food web models

Food webs are constructed as structural directed graphs that describe “who eats whom,” but it is common to interpret them as energy flow diagrams where predation represents an energy transfer from the prey to the predator. It is the aim of this work to demonstrate that food webs are incomplete as energy flow diagrams if they ignore passive flows to detritus (dead organic material). While many ecologists do include detritus in conceptual and mathematical models, the detrital omission is still commonly found. Often detritus is either ignored or treated as an unlimited energy source, yet all organisms contribute to the detritus pool, which can be an energy source for other species in the system. This feedback loop is of high importance, since it increases the number of pathways available for energy flows, revealing the significance of indirect effects, and making the functional role of the top predators less clear. In this work we propose the modified niche model by adding a detritus compartment to the niche model. We demonstrate the effect of structural loops that result from feeding on detritus, by comparing empirical data sets to five different assembly models: (1) cascade, (2) constant connectance, (3) niche, (4) modified niche (original in this work), and (5) cyber-ecosystem. Of these models, only the last two explicitly include detritus. We show that when passive flows to detritus are included in the food web structure, the structure becomes more robust to the removal of individual nodes or connections. In addition, we show that food web models that include the detritus feedback loop perform better with respect to several structural network metrics.     

Optimal daily routines of singing and foraging in a bird singing to attract a mate

Summary In this paper we develop a dynamic programming model to explore the optimal organization of daily routines of singing and foraging in a small bird. While singing the bird may attract a mate but uses up energy. Most of the patterns of daily variation in singing generated have basic features very characteristic of typical passerine song output. The predictions are remarkably robust to changes in a wide range of parameters, showing which parameters are important. A peak of singing at dawn can result from variability in overnight energy expenditure in the absence of any circadian patterns in the environment.     

A complex social structure with fission–fusion properties can emerge from a simple foraging model

Precisely how ecological factors influence animal social structure is far from clear. We explore this question using an agent-based model inspired by the fission–fusion society of spider monkeys (Ateles spp). Our model introduces a realistic, complex foraging environment composed of many resource patches with size varying as an inverse power law frequency distribution with exponent β. Foragers do not interact among them and start from random initial locations. They have either a complete or a partial knowledge of the environment and maximize the ratio between the size of the next visited patch and the distance traveled to it, ignoring previously visited patches. At intermediate values of β, when large patches are neither too scarce nor too abundant, foragers form groups (coincide at the same patch) with a similar size frequency distribution as the spider monkey’s subgroups. Fission–fusion events create a network of associations that contains weak bonds among foragers that meet only rarely and strong bonds among those that repeat associations more frequently than would be expected by chance. The latter form subnetworks with the highest number of bonds and a high clustering coefficient at intermediate values of β. The weak bonds enable the whole social network to percolate. Some of our results are similar to those found in long-term field studies of spider monkeys and other fission–fusion species. We conclude that hypotheses about the ecological causes of fission–fusion and the origin of complex social structures should consider the heterogeneity and complexity of the environment in which social animals live.     

Central place foraging in wheatears (Oenanthe oenanthe L.): foraging itineraries when feeding nestlings

Summary Parent wheatears (Oenanthe oenanthe L.), foraging to meet their own needs and to provide food for a central place (CP, i.e. the nest), have to make decisions with respect to the configuration of foraging itineraries during round trips in the territory and to the directionality of their movements. These problems were studied in two pairs breeding in an agricultural area in central Swecen. All birds started a round trip by hunting from perches close to the CP and then moving to perches progressively further away in a roughly straight line until the first prey item was loaded. Loaded prey were either delivered singly (single prey loading: SPL) or with other prey (multiple prey loading: MPL). When the bird decided to return with several items to the CP, it abruptly changed the direction of its movements by making a left, right or backward turn and started to visit perches progressively nearer the CP, again following a roughly straight line. The decision to return continue loading was affected by the size of the prey as shown by the fact that prey carried singly was significantly larger than the first item in an MPL for all individuals. The distance to the CP also seemed to affect this decision as prey provisioned singly on average were collected significantly closer to the nest than the first item in an MPL by one pair. Both the size of prey loaded singly and load size of MPL increased with distance from the CP in one pair. The concentrated use of the territory in the other pair made any effect of distance difficult to detect. Great flexibility in foraging decisions was observed in cases when an individual, because of the size of the last prey captured, altered its decision to deliver an MPL and transported an SPL to the nest instead. On the basis of these results we propose a set of rules followed by predators such as wheatears when making decisions about delivering prey to a CP (Fig. 4).     

Variation in foraging effort by lactating Antarctic fur seals: response to simulated increased foraging costs

Seasonally breeding predators, which are limited in the time available for provisioning young at a central location, and by the fasting abilities of the young, are likely to maximize energy delivery to the young by maximizing the rate of energy delivery averaged over the whole period of investment. Reduction in food availability or increased foraging costs will alter the optimal behavior of individuals. This study examined the behavioral adaptations of a diving predator, the Antarctic fur seal, to increased foraging costs during lactation. One group of mothers (n=5, treatment) was fitted with additional drag to increase the cost of transport in comparison with a control group (n=8). At the scales of the individual dives, the treatment group made more shorter, shallower (< 30 m) dives. Compensation for slower swimming speeds was achieved by diving at a steeper angle. Overall, diving behavior conformed to several specific theoretical predictions but there were also departures from theory, particularly concerning swimming speed during diving. Diving behavior appears to be adjusted to maximize the proportion of time spent at the bottom of dives. At the scale of diving bouts, no difference was observed between the treatment and control groups in terms of the frequency and duration of bouts and there was also no difference between the two groups in terms of the proportion of time spent diving. At the scale of complete foraging cycles, time taken to return to the pup was significantly longer in the treatment group but there was no difference in the rate of delivery of energy (measured from pup growth rate) to the pups in each group. Since mothers in the treatment group did not use significantly more body reserves, we conclude that behavioral adjustments at the scale of individual dives allowed mothers in the treatment group to compensate for the additional foraging costs. Pup growth rate appears to be less sensitive to the foraging conditions experienced by mothers than foraging trip duration. Received: 14 June 1996 / Accepted after revision: 16 November 1996     

Using stochastic volatility models to analyse weekly ozone averages in Mexico City

In this paper we make use of some stochastic volatility models to analyse the behaviour of a weekly ozone average measurements series. The models considered here have been used previously in problems related to financial time series. Two models are considered and their parameters are estimated using a Bayesian approach based on Markov chain Monte Carlo (MCMC) methods. Both models are applied to the data provided by the monitoring network of the Metropolitan Area of Mexico City. The selection of the best model for that specific data set is performed using the Deviance Information Criterion and the Conditional Predictive Ordinate method.