Optimal foraging in bumblebees and coevolution with their plants

Summary The aims of this paper were to consider the coevolution between bumblebee movement patterns within plants and various properties of the plants such as the spatial distribution of their flowers, and to determine the extent to which the bumblebees and the plants can be considered to be maximally adaptive or optimal. Attention was restricted to plants which have flowers arranged on vertical inflorescences and to the bumblebees which visit these plants.It was found that the bumblebees tend to commence foraging at the bottom of each infloresence, that they tend to move from one flower to the closest vertically higher flower, that they miss flowers as they move upwards and that they tend to leave each inflorescence before reaching the top. It was also found for the four common plant species considered that nectar abundance per flower decreases with flower height on an inflorescence, that the flowers with receptive stigmas are restricted to the bottoms of the inflorescences while the flowers shedding pollen occur above them, and that the flowers are arranged approximately in spirals on the inflorescences.The pattern of movements of the bumblebees and the various properties of the plants appear to represent coevolved adaptations. Furthermore the bumblebees' movement patterns appear to be optimal in the sense that they result in the maximum net rate of energy gain to the bumblebees. Further studies are necessary, however, to determine whether or not the plants can be considered to be optimal.An exception to the above scheme is provided by a plant which is quite uncommon in the study area. This plant also has flowers on vertical inflorescences and appears to be pollinated by bumblebees. However, while the pattern of movements of the bumblebees on this plant species are extremely similar to those on the four common species, this plant species exhibits quite different properties from the other four. Two possible explanations for this exception are presented.     

A test of spatial memory and movement patterns of bumblebees at multiple spatial and temporal scales

Naive bumblebee foragers appear to use movement rules at smallspatial and temporal scales, but it is not clear whether theserules determine movement patterns as the scales increase. Onestrategy for efficient foraging used by bumblebees is near-farsearch, involving short flights when in good patches of flowersand longer flights when in poor patches. Bumblebees also demonstratethe use of a spatial memory strategy by returning repeatedlyto patches of flowers, and even following the same route betweenflowers, over periods of days. We attempted to determine atwhat spatial scales bumblebees use spatial memory while foragingwithin a patch and after how many flower visits spatial memoryoutweighs near-far search. Bumblebees in the laboratory foragedon a 4 x 4 array of artificial flowers with distances rangingfrom 10 to 80 cm between flowers in two simple spatial patterns.The proportion of visits to flowers containing a sucrose rewardwas monitored for either 100 or 400 flower visits in two separateexperiments, after which the locations of the rewarding andnonrewarding flowers were interchanged, producing a mirror image.A drop in accuracy after the mirror image switch would indicatethat the bees had memorized the location of rewarding flowers.Mirror image tests, and comparisons to a simulation model ofnear-far search based on actual flight distances, indicate thatnaive bumblebees used near-far search on flowers 10 cm apartbut increasingly used spatial memory as experience and spatialseparation increased. Bumblebees thus have multiple tacticsavailable to forage efficiently in different environments.     

Simple rules for complex landscapes: the case of hilltopping movements and topography

Empirical data on the signals and processes that direct animal movement during dispersal in heterogeneous landscapes are scarce. Our understanding could benefit from utilising simulation approaches and searching for simple rules across species and landscapes. This study sought to identify general movement behaviours that optimise butterfly movements during hilltopping. This widespread dispersal‐like phenomenon in butterflies, where males and virgin females ascend to mountain summits for the purpose of mating, benefits from the uniqueness of knowing the purpose (mating) and the orientation signal (topography). We used an individual‐based simulation model to search for movement rules that can optimise mating success, mating time, and the success of mated females in subsequently finding habitat patches across differently structured landscapes. We found that a strong response to topography was optimal for males and virgin females to reach summits, but slight deviation from ‘perfect’, namely a certain level of randomness in response to topography, was inherently essential to avoid the risk of being trapped on local summits. The optimal response of mated females deviated only slightly from a random movement, indicating a potential weak response to topography which could be easily overlooked by field studies. Notably, the parameter values identified by the model as optimal corresponded with the observed behaviour of butterflies in the field. Finally, the optimal movement behaviours were affected by the lifespan of butterflies and the spatial distribution of host plant patches, but less so by landscape structure. We therefore suggest that modelling approaches that build on simple biological rules can facilitate the development and parameterisation of models for understanding and potentially predicting dispersal and connectivity in complex landscapes, also in circumstances where observed patterns seem complex and empirical data are scarce.     

Remote perception of floral nectar by bumblebees

Summary On both artificial flowers in the laboratory and certain plant species in the field, bumblebees often closely approached flowers and then departed without probing for nectar. In laboratory experiments where nectar rewards were associated with subtle visual or olfactory cues, bumblebees approached and avoided non-rewarding flowers. Flowers that bees entered and probed for nectar contained rewards much more frequently than predicted by chance alone. When there were no external cues associated with nectar content, bees visited rewarding flowers by chance alone, provided rewarding flowers were not spatially clumped. In the field, bumblebees approached and rejected a large proportion of dogbane flowers and red clover inflorescences. On both species, flowers or inflorescences probed by bees contained more nectar than those rejected by bees or those that I chose at random. On fireweed and monkshood, bees rarely or never approached and rejected healthy-looking flowers. Predictions generated by an optimal foraging model were tested on data from four bumblebee species foraging on red clover. The model was highly successful in qualitatively predicting the relationship between handling time and proportion of inflorescences rejected by individual bees, and the relationship between threshold nectar content for acceptance by bees and average resource availability. Thus, bees appeared to use remotely perceived cues to maximize their rates of nectar intake.     

How simple grazing rules can lead to persistent boundaries in vegetation communities

Natural landscape boundaries between vegetation communities are dynamically influenced by the selective grazing of herbivores. Here we show how this may be an emergent property of very simple animal decisions, without the need for any sophisticated choice rules etc., using a model based on biased diffusion. Animal grazing intensity is coupled with plant competition, resulting in reaction-diffusion dynamics, from which stable boundaries spontaneously emerge. In the model, animals affect their resources by both consumption and trampling. It is assumed that forage consists of two heterogeneously distributed competing resource species, one that is preferred (grass) over the other (heather) by the animals. The solutions to the resulting system of differential equations for three cases a) optimal foraging, b) random walk foraging and c) taxis-diffusion are presented. Optimal and random foraging gave unrealistic results, but taxis-diffusion accorded well with field observations. Persistent boundaries between patches of near-monoculture vegetation were predicted, with these boundaries drifting in response to overall grazing pressure (grass advancing with increased grazing and vice versa). The reaction-taxis-diffusion model provides the first mathematical explanation for such vegetation mosaic dynamics and the parameters of the model are open to experimental testing.     

Optimal body size in bumblebees

Summary It is hypothesized that the body size of a bumblebee will be that size which maximizes its average net rate of energy intake while collecting nectar. A mathematical model is developed with the result that the net rate of energy intake of a nectar-collecting bumblebee is expressed as a function of the body size of the bumblebee. From this model the body size which maximizes the net rate of energy intake (i.e., optimal body size) is found (as the solution of an implicit equation). In this situation the advantage of large size is that larger bumblebees fly faster and hence take less flight time than smaller bumblebees. The disadvantage of larger size is greater energetic costs.The parameters of the model are estimated using data obtained from the foraging behavior of bumblebees on monkshood (Aconitum columbianum). The optimal body size is then calculated for workers of Bombus appositus which obtained almost all their nectar from monkshood. The observed and expected (i.e., optimal) body size are found to be close and not significantly different.The model also predicts that, from the bumblebee's point of view, there should be a positive correlation between the size of the bumblebee and the average amount of nectar obtained per flower. Evidence of this correlation is presented and the possible significance of the correlation from the plant's point of view is discussed. A possible extension of the model to general relationships between predator body size, prey size and prey density is discussed.     

Optimal foraging in bumblebees: Rule of movement between flowers within inflorescences

It is hypothesized that nectar-collecting bumblebees will be found to forage in ways that maximize their net rate of energy intake. Attention is focused, in this paper, on the manner in which these bumblebees move from one flower to another within inflorescences. Observations were made on workers of Bombus appositus, which were collecting nectar from Aconitum columbianum (monkshood). The rule of movement of the bumblebees was determined and compared, in terms of net rate of energy intake, with several possible alternative rules. Two of these alternatives gave equally high net rates of energy intake. The observed rule was very similar in nature to one of these and indistinguishable from both in terms of net rate of energy intake.     

Scale-free foraging by primates emerges from their interaction with a complex environment

Scale-free foraging patterns are widespread among animals. These may be the outcome of an optimal searching strategy to find scarce, randomly distributed resources, but a less explored alternative is that this behaviour may result from the interaction of foraging animals with a particular distribution of resources. We introduce a simple foraging model where individual primates follow mental maps and choose their displacements according to a maximum efficiency criterion, in a spatially disordered environment containing many trees with a heterogeneous size distribution. We show that a particular tree-size frequency distribution induces non-Gaussian movement patterns with multiple spatial scales (Lévy walks). These results are consistent with field observations of tree-size variation and spider monkey (Ateles geoffroyi) foraging patterns. We discuss the consequences that our results may have for the patterns of seed dispersal by foraging primates.     

Dynamic optimization of the sit-to-stand movement

The purpose of this study was to clarify criteria that can predict trajectories during the sit-to-stand movement. In particular, the minimum jerk and minimum torque-change models were examined. Three patterns of sit-to-stand movement from a chair, i.e., upright, natural, and leaning forward, were measured in five young participants using a 3-D motion analysis device (200 Hz). The trajectory of the center of mass and its smoothness were examined, and the optimal trajectories predicted by both models were evaluated. Trajectories of the center of mass predicted by the minimum torque-change model, rather than the minimum jerk model, resembled the measured movements in all rising movement patterns. The upright pattern required greater extension torque of the knee and ankle joints at the instant of seat-off. The leaning-forward pattern required greater extension hip torque and higher movement cost than the natural and upright patterns. These results indicate that the natural sit-to-stand movement might be a result of dynamic optimization.     

Computational simulation of fracture healing: influence of interfragmentary movement on the callus growth

Bone fractures heal through a complex process involving several cellular events. This healing process can serve to study factors that control tissue growth and differentiation from mesenchymal stem cells. The mechanical environment at the fracture site is one of the factors influencing the healing process and controls size and differentiation patterns in the newly formed tissue. Mathematical models can be useful to unravel the complex relation between mechanical environment and tissue formation. In this study, we present a mathematical model that predicts tissue growth and differentiation patterns from local mechanical signals. Our aim was to investigate whether mechanical stimuli, through their influence on stem cell proliferation and chondrocyte hypertrophy, predict characteristic features of callus size and geometry. We found that the model predicted several geometric features of fracture calluses. For instance, callus size was predicted to increase with increasing movement. Also, increases in size were predicted to occur through increase in callus diameter but not callus length. These features agree with experimental observations. In addition, spatial and temporal tissue differentiation patterns were in qualitative agreement with well-known experimental results. We therefore conclude that local mechanical signals can probably explain the shape and size of fracture calluses.     

The distribution of bumblebee colour patterns worldwide: possible significance for thermoregulation, crypsis, and warning mimicry

Bumblebee colour patterns can be highly variable within species, but are often closely similar among species. The present study takes a quantitative approach to survey bumblebee colour patterns in order to address some of the most basic questions concerning resemblances: (1) do colour‐pattern groups exist; (2) are species within colour‐pattern groups geographically clumped; and (3) are some colour‐pattern groups associated with particular kinds of habitat? The results using data for 632 worker patterns from all of the world’s bumblebee species show that: (1) there are many repeating colour patterns, forming relatively few groups of species with similar patterns; (2) colour‐pattern groups can be recognized using simple rules; and (3) species within the 24 largest colour‐pattern groups are significantly aggregated in particular areas of the world. Three principal divisions of colour‐pattern groups are associated with three likely functions: (1) the darkest bumblebees are associated primarily with the tropics, where a thermoregulatory function is suggested; (2) the palest bumblebees are associated with intermediate northern latitudes, where a cryptic function in drying grasslands is suggested; and (3) the intermediate, strongly banded bumblebees are widespread, although these patterns predominate where banding may have advantages as collective warning signals to predators (Müllerian mimicry). Further studies are needed to test these explanations. © The Natural History Museum, London. Journal compilation © 2007 The Linnean Society of London, Biological Journal of the Linnean Society, 2007, 92 , 97–118.     

Why do shallow-water predators migrate? Strategic models and empirical evidence from an estuarine mysid

We investigated factors potentially affecting tidal migrations over the littoral zone by invertebrate predators via a combination of strategic modelling, field observations and laboratory experiments using the mysid Neomysis integer. The models predict the distribution of individuals over the immersed intertidal region under the three different scenarios of diffusive movement, movement up a gradient of prey abundance, and movement towards a specific water depth. We reject the diffusive spread hypothesis since the predicted changes in spatial patterns are qualitatively inconsistent with those of density estimates over the tidal cycle in the field. The foraging hypothesis was consistent with the field samples only if there was a consistent upshore food gradient. Gut contents analysis showed meiofaunal prey were rare, but that organic detritus was the main dietary component. Sediment samples indicated some evidence of an organic matter gradient, but in the laboratory, Neomysis showed no preference for sediment with high organic content. We therefore rejected the foraging hypothesis. The depth-seeking model was not rejected, but laboratory experiments involving responses to various predators of Neomysis provided only weak evidence that depth-seeking was a result of predator avoidance. We hypothesise that the proximate mechanism is most likely to involve behavioural responses to flow conditions.     

A discrete cell model with adaptive signalling for aggregation of Dictyostelium discoideum.

Dictyostelium discoideum (Dd) is a widely studied model system from which fundamental insights into cell movement, chemotaxis, aggregation and pattern formation can be gained. In this system aggregation results from the chemotactic response by dispersed amoebae to a travelling wave of the chemoattractant cAMP. We have developed a model in which the cells are treated as discrete points in a continuum field of the chemoattractant, and transduction of the extracellular cAMP signal into the intracellular signal is based on the G protein model developed by Tang & Othmer. The model reproduces a number of experimental observations and gives further insight into the aggregation process. We investigate different rules for cell movement the factors that influence stream formation the effect on aggregation of noise in the choice of the direction of movement and when spiral waves of chemoattractant and cell density are likely to occur. Our results give new insight into the origin of spiral waves and suggest that streaming is due to a finite amplitude instability.     

Modelling dispersal of a temperate insect in a changing climate

We construct a novel individual-based random-walk model to assess how predicted global climate change might affect the dispersal rates of a temperate insect. Using a novel approach we obtained accurate field measurements of daily movements for individuals over time to parameterize our model. Males were found to move significantly further on average than females. Significant variation in movement was evident among individuals; the most dispersive individuals moved up to five (females) and seven (males) times as far on average as the least dispersive individuals. Mean relative daily movement of both males and females were exponentially related to maximum daily temperature recorded within the grass sward. Variability, both within and among individuals, in relative daily movement was incorporated into the model using gamma probability distributions. Resultant dispersal functions for seasonal movement are predicted to be highly leptokurtic, which agrees well with observations from the field. Predictions of the model suggest that for populations at the polewards edge of the current range an increase of 3-5 degrees C in daily maximum temperature may increase the proportion of long-distance dispersers (those characterized as comprising the top 0.1% of furthest dispersing individuals under local conditions experienced during the 1963-1990 period) by up to 70%.     

Analysis of a model for wolf territories

 A mechanism for territorial pattern formation in wolves is analysed using a spatially explicit mathematical model which incorporates wolf movement and scent marking. Model results reflect field observations: buffer zones where wolves are scarce arise between adjacent packs and near these buffer zones there are increased levels of scent marking. It is shown how the precise behavioral response of wolves to foreign scent-marks determines the qualitative form of the spatial territories. Realistic territories in two spatial dimensions require ‘switching’ of the movement and scent marking behavior in response to foreign scent marks. Received 20 October 1995; received in revised form 17 May 1996     

Data-oriented analyses of ciliate foraging behaviors

Optimal foraging theory states that natural selection makes foragers efficient food harvesters and maximizing a colony’s energy intake. This study presumed that the ciliates foraging trajectories follow optimal foraging theory, verified the presumption and discover specific rules and patterns hidden in the ciliate’s trajectories data using methodologies of statistical, cluster analyses, and decision tree analysis. This study examined the foraging behaviors of ciliates by video recordings and quantitative analyses of movement trajectories under four nourishment conditions (low, medium, high, and highest concentrations). Similar biological studies adopt statistical analyses to certain locomotion indices to determine the responses of plankton to various aquatic environments. In addition to statistical analyses, cluster analysis was used in this study to confirm the observations of the statistical analyses. The statistical analysis and cluster analysis results in this study revealed two distinct groups of trajectories or behaviors, which matched the optimal foraging theory. Decision tree analysis was then applied to acquire objective information regarding foraging behaviors, and further detailed the foraging behaviors with explicit classification rules using locomotion indices. The production rules can play an alternative role to assess the sustainability of an aquatic environment in terms of algae concentration.     

Mathematical modelling of host-parasitoid systems: effects of chemically mediated parasitoid foraging strategies on within- and between-generation spatio-temporal dynamics.

In this paper we develop a novel discrete, individual-based mathematical model to investigate the effect of parasitoid foraging strategies on the spatial and temporal dynamics of host-parasitoid systems. The model is used to compare na?ve or random search strategies with search strategies that depend on experience and sensitivity to semiochemicals in the environment. It focuses on simple mechanistic interactions between individual hosts, parasitoids, and an underlying field of a volatile semiochemical (emitted by the hosts during feeding) which acts as a chemoattractant for the parasitoids. The model addresses movement at different spatial scales, where scale of movement also depends on the internal state of an individual. Individual interactions between hosts and parasitoids are modelled at a discrete (micro-scale) level using probabilistic rules. The resulting within-generation dynamics produced by these interactions are then used to generate the population levels for successive generations. The model simulations examine the effect of various key parameters of the model on (i) the spatio-temporal patterns of hosts and parasitoids within generations; (ii) the population levels of the hosts and parasitoids between generations. Key results of the model simulations show that the following model parameters have an important effect on either the development of patchiness within generations or the stability/instability of the population levels between generations: (i) the rate of diffusion of the kairomones; (ii) the specific search strategy adopted by the parasitoids; (iii) the rate of host increase between successive generations. Finally, evolutionary aspects concerning competition between several parasitoid subpopulations adopting different search strategies are also examined.     

Direction change of plant root growth by the impenetrable boundary

Spatial boundary conditions must be considered when utilizing mathematical modeling of plant root growth in the container or along with the imbedding solid obstacle. Using basic root growth principles and the geometry of the boundary surface, a mathematical model can be designed to keep all root elements inside the container or outside the obstacle without passing through the boundary after the minimum deflection of growth direction, and it is based on the minimum friction between root tips and soil and energy saving principles. Such a mathematical method is used to simulate the spatial distribution of root growth and the morphological architecture of the root system near the boundary. The validity of this model is supported by experimental observations that confirm some typical characteristics predicted by the simulations. This model can be widely used in resolving boundary condition complications where water and nutrients are consumed by plants in a spatially limited or heterogeneous resource field.     

Breathing patterns in anesthetized cats and the concept of minimum respiratory effort

Theoretical studies dealing with the principle of minimal respiratory effort usually make use of sinusoidal or saw-tooth-like breathing patterns. Recent observations in anesthetized cats have shown that the driving pressure waveform for inspiration can be described by a power function of time and that most of expiration is passive. This driving pressure waveform, however, results in breathing patterns that differ from those described above. For this reason, we have reevaluated in anesthetized cats the principle of minimal respiratory effort by computing optimal duration of inspiration (TI) and optimal tidal volume (VT) for different ventilatory conditions using actual driving pressure waveforms. The results are in qualitative agreement with the experimental observations; i.e., optimal TI decreases and optimal VT increases with increasing minute ventilation. On the average, a good agreement is found between measured and computed values of TI. In some cats, however, there are substantial differences between observed and predicted values of TI, which can probably be ascribed to inaccuracies in the data used in our computations. Despite its limitations, the present model analysis is more realistic than previous ones because actual driving pressure waveforms are used together with actual values of effective inspiratory impedance.