A mathematical model for the population dynamics of army ants

A stochastic cellular automata model for the population dynamics of the army antEciton burchelli on Barro Colorado Island in Panama is set up. It is simulated on the computer and shown to give good agreement with biological data. It is analysed using two approximations akin to the mean field approximation in statistical mechanics, and good agreement with the simulations is obtained. Finally, the role of distance between successive statary phase bivouacs is discussed with regard to the rate of colony growth. There are two aspects of the biological system studied here that make it of general importance. First, the population is structured, since the size of each colony of army ants is crucial. Second, the spatial behaviour of the population, as in many others, is not diffusion-like, although it is random. This has implications for the kind of model that is chosen.     

From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves

In the vertebrate brain excitatory synaptic contacts typically occur on the tiny evaginations of neuron dendritic surface known as dendritic spines. There is clear evidence that spine heads are endowed with voltage-dependent excitable channels and that action potentials invade spines. Computational models are being increasingly used to gain insight into the functional significance of a spine with an excitable membrane. The spike-diffuse-spike (SDS) model is one such model that admits to a relatively straightforward mathematical analysis. In this paper we demonstrate that not only can the SDS model support solitary travelling pulses, already observed numerically in more detailed biophysical models, but that it has periodic travelling wave solutions. The exact mathematical treatment of periodic travelling waves in the SDS model is used, within a kinematic framework, to predict the existence of connections between two periodic spike trains of different interspike interval. The associated wave front in the sequence of interspike intervals travels with a constant velocity without degradation of shape, and might therefore be used for the robust encoding of information.     

Mathematical analysis of a basic model for epidermal wound healing

The stimuli for the increase in epidermal mitosis during wound healing are not fully known. We construct a mathematical model which suggests that biochemical regulation of mitosis is fundamental to the process, and that a single chemical with a simple regulatory effect can account for the healing of circular epidermal wounds. The numerical results of the model compare well with experimental data. We investigate the model analytically by making biologically relevant approximations. We then obtain travelling wave solutions which provide information about the accuracy of these approximations and clarify the roles of the various model parameters.     

A mathematical and experimental study of ant foraging trail dynamics

In this article, we present a mathematical model coupled to an experimental study of ant foraging trails. Our laboratory experiments on Tetramorium caespitum do not find a strong relationship between ant densities and velocities, a common assumption in traffic modeling. Rather, we find that higher order effects play a major role in observed behavior, and our model reflects this by including inertial terms in the evolution equation. A linearization of the resulting system yields left- and right-moving waves, in agreement with laboratory measurements. The linearized system depends upon Froude numbers reflecting a ratio of the energy stored in the foraging trail to the kinetic energy of the ants. The model predicts and the measurements support the existence of two distinct phase velocities.     

A spatial model for the spread of invading organisms subject to competition

 A spatially explicit integrodifference equation model is studied for the spread of an invading organism against an established competitor. Provided the invader is initially confined to a bounded region, the invasion spreads asymptotically as a travelling wave whose speed depends on the strength of the competitive interaction and on the dispersal characteristics of the invader. Even an inferior, but established, competitor can significantly reduce the invasion speed. The invasion speed is also influenced by the exact shape of the dispersal kernel (especially the thickness of the tail) as well as the mean dispersal distance for each generation. Received 10 April 1996; received in revised form 21 August 1996     

A model of survival times for predator populations: The case of the army ants

We develop a method to estimate the expected time of survival of a predator population as a function of the size of the habitat island on which it lives and the dynamic parameters of the population and its prey. The model may be thought of either as a patch occupancy model for a structured population or as a model of metapopulation type. The method is applied to a keystone predator species, the neotropical army ant Eciton burchelli. Predictions are made as to how many of the islands and habitat islands in and around Gatun Lake in the Panama Canal, most of which were formed when the canal was dug, can be expected to support such a population today, and these are compared with data.     

A cross-diffusion model of forest boundary dynamics

A simple mathematical model of mono-species forest with two age classes which takes into account seed production and dispersal is presented in the paper. This reaction — diffusion type model is then reduced by means of an asymptotic procedure to a lower dimensional reaction — cross-diffusion model. The existence of standing and travelling wave front solutions corresponding to the forest boundary is shown for the later model. On the basis of the analysis, possible changes in forest boundary dynamics caused by antropogenic impacts are discussed.     

Sociobiology of slave-making ants

Social parasitism is the coexistence in the same nest of two species of social insects, one of which is parasitically dependent on the other. Though parasitism in general is known to be of crucial importance in the evolution of host species, social parasites, though intriguing, are often considered as a phenomenon of marginal interest and are typically not taken into account in reviews on parasitism. Nevertheless, social parasites are rather common in social bees, wasps, and ants and therefore may offer unique model systems to study a number of fundamental problems in evolutionary biology. Here we review several aspects of the peculiar life history of slave-making ants, which is characterized by socially parasitic founding of colonies and the pillage of broods from neighboring host colonies during slave raids. In particular we focus on the evolution of slave-making habits (dulosis), communication mechanisms between slave makers and their hosts, sex-allocation ratios and reproductive conflict, and the effect of slave makers on host populations. Received: 2 February 2000 / Received in revised form: 21 December 2000 / Accepted: 8 January 2001     

Directional raids by army ants as an adaption to patchily distributed food: a simulation model

A typical colony of Neotropical army ants (subfamily Ecitoninae) regularly raids a large area around their bivouac by forming a narrow directional column that can reach up to one hundred meters in length. The raid is finished and then relaunched 12–17 times, each time toward different orientation. After completing all bouts the colony relocates to a new area. A hypothetical alternative to this foraging mode is raiding radially and symmetrically by expanding the search front in every direction like a circular bubble. Using an existing agent-based modeling software that simulates army ants’ behavior, we compared the two possible modes of foraging in different food distributions. Regardless of the food patch abundance, the radial raiding was superior to the directional raiding when food patches had low quality, and the directional raiding was favorable when the patches were rich. In terms of energy efficiency, the radial raiding was the better strategy in a wide range of conditions. In contrast, the directional raiding tended to yield more food per coverage area. Based on our model, we suggest that the directional raiding by army ants is an adaptation to the habitats with abundance of high-quality food patches. This conclusion fits well with the ecology of army ants.     

A nonlinear model of ionic wave propagation along microtubules

Microtubules (MTs) are important cytoskeletal polymers engaged in a number of specific cellular activities including the traffic of organelles using motor proteins, cellular architecture and motility, cell division and a possible participation in information processing within neuronal functioning. How MTs operate and process electrical information is still largely unknown. In this paper we investigate the conditions enabling MTs to act as electrical transmission lines for ion flows along their lengths. We introduce a model in which each tubulin dimer is viewed as an electric element with a capacitive, inductive and resistive characteristics arising due to polyelectrolyte nature of MTs. Based on Kirchhoff’s laws taken in the continuum limit, a nonlinear partial differential equation is derived and analyzed. We demonstrate that it can be used to describe the electrostatic potential coupled to the propagating localized ionic waves. An erratum to this article can be found at     

A diffusive SI model with Allee effect and application to FIV

A minimal reaction-diffusion model for the spatiotemporal spread of an infectious disease is considered. The model is motivated by the Feline Immunodeficiency Virus (FIV) which causes AIDS in cat populations. Because the infected period is long compared with the lifespan, the model incorporates the host population growth. Two different types are considered: logistic growth and growth with a strong Allee effect. In the model with logistic growth, the introduced disease propagates in form of a travelling infection wave with a constant asymptotic rate of spread. In the model with Allee effect the spatiotemporal dynamics are more complicated and the disease has considerable impact on the host population spread. Most importantly, there are waves of extinction, which arise when the disease is introduced in the wake of the invading host population. These waves of extinction destabilize locally stable endemic coexistence states. Moreover, spatially restricted epidemics are possible as well as travelling infection pulses that correspond either to fatal epidemics with succeeding host population extinction or to epidemics with recovery of the host population. Generally, the Allee effect induces minimum viable population sizes and critical spatial lengths of the initial distribution. The local stability analysis yields bistability and the phenomenon of transient epidemics within the regime of disease-induced extinction. Sustained oscillations do not exist.     

A one-dimensional model for the propagation of transient pressure waves through the lung

The propagation of pressure waves in the lung has been investigated by many authors concerned with respiratory physiology, ultrasound medical techniques or thoracic impact injuries. In most of the theoretical studies, the lung has been modeled as an isotropic and homogeneous medium, and by using Hooke's constitutive law (see e.g. Ganesan et al. Respir. Physiol. 110 (1997) 19; Jahed et al. J. Appl. Physiol. 66 (1989) 2675; Grimal et al. C.R. Acad. Sci., Paris 329 (IIb) (2001) 655-662), or more elaborated material laws (see, e.g. Bush and Challener (Proceedings of the International Research Council on Biokinetics Impacts (IRCOBI), Bergish-gladbach, 1988); Stuhmiller et al. J. Trauma 28 (1988) S132; Yang and Wang, Finite element modeling of the human thorax. Web page: http://wwwils.nlm.nih.gov/research/visible/vhpconf98/AUTHORS/YANG/YANG.HTM.). The hypothesis of homogeneous medium may be inappropriate for certain problems. Because of its foam-like structure, the behavior of the lung-even if the air and the soft tissue are assumed to behave like linearly elastic materials-is susceptible to be frequency dependent. In the present study, the lung is viewed as a one-dimensional stack of air and soft tissue layers; wave propagation in such a stack can be investigated in an equivalent mass-spring chain (El-Raheb (J. Acoust. Soc. Am. 94 (1993) 172; Int. J. Solids Struct. 34 (1997) 2969), where the masses and springs, respectively, represent the alveolar walls and alveolar gas. Results are presented in the time and frequency domains. The frequency dependence (cutoff frequency, variations in phase velocity) of the lung model is found to be highly dependent on the mean alveolar size. We found that short pulses induced by high velocity impacts (bullet stopped by a bulletproof jacket) can be highly distorted during the propagation. The pressure differential between two alveoli is discussed as a possible injury criterion.     

Gap junctions are required for the propagation of spreading depression

Spreading depression (SD) is a slowly propagating depression of cerebral neuronal activity and transmembrance ionic gradients, that arises in response to a variety of noxious stimuli. SD bears a strong resemblance to gap junction-mediated calcium waves among cultured astrocytes. Here, we show that gap junction-mediated intercellular diffusion is necessary for the generation of SD. Waves of SD in the isolated chicken retina were blocked by five different inhibitors of gap junctional coupling, which was assessed by the intercellular transit of Lucifer Yellow (LY). Each of these gap junction blockers inhibited both the migration of SD and the diffusion of LY in a dose-dependent manner. In contrast, glutamate-evoked calcium influx into retinal cells was not affected by these compounds. The results indicate that intercellular coupling through gap junctions is required for SD. Gap junction-mediated communication might therefore constitute an important mechanism in both normative and pathological brain function. © 1995 John Wiley & Sons, Inc.     

A Laboratory study on the visual and chemical orientation of the gastropod Nerita Fulgurans Gmelin, 1791

Behavioural responses of the gastropod Nerita fulgurans Gmelin, 1791 to flat black rectangles and intraspecific mucus trails were measured in a circular arena. Snails were tested in water either in the presence or absence of chemicals generated from a predator gastropod, Chicoreus brevifrons (Lamarck, 1822). The test hypothesis was that this snail has different behavioural responses as result of visual and chemical cue integration. Nerita fulgurans has the capacity to orient to solid targets subtending angles larger than 10° and follow its own mucus trails. In water conditioned by the predator C . brevifrons , snails exhibited an avoidance response when 10°, 20° and 45° sectors were presented, demonstrating an integration of chemical and visual information. The simultaneous presentation of two orienting cues (black sectors and mucus trails) was tested to determine the nature of the interaction. When the two cues were oriented in the same direction there was no effect. When the two cues were presented from directions 180° apart a preference for visual cues over mucus trail cues was evident when the visual angle of the visual cue subtended angles greater than 90°. This result demonstrates a hierarchical usage of the orienting references.     

A Laboratory study on the visual and chemical orientation of the gastropod Nerita Fulgurans Gmelin, 1791

Behavioural responses of the gastropod Nerita fulgurans Gmelin, 1791 to flat black rectangles and intraspecific mucus trails were measured in a circular arena. Snails were tested in water either in the presence or absence of chemicals generated from a predator gastropod, Chicoreus brevifrons (Lamarck, 1822). The test hypothesis was that this snail has different behavioural responses as result of visual and chemical cue integration. Nerita fulgurans has the capacity to orient to solid targets subtending angles larger than 10° and follow its own mucus trails. In water conditioned by the predator C . brevifrons, snails exhibited an avoidance response when 10°, 20° and 45° sectors were presented, demonstrating an integration of chemical and visual information. The simultaneous presentation of two orienting cues (black sectors and mucus trails) was tested to determine the nature of the interaction. When the two cues were oriented in the same direction there was no effect. When the two cues were presented from directions 180° apart a preference for visual cues over mucus trail cues was evident when the visual angle of the visual cue subtended angles greater than 90°. This result demonstrates a hierarchical usage of the orienting references.     

A mathematical investigation of a Clock and Wavefront model for somitogenesis

Somites are transient blocks of cells that form sequentially along the antero-posterior axis of vertebrate embryos. They give rise to the vertebrae, ribs and other associated features of the trunk. In this work we develop and analyse a mathematical formulation of a version of the Clock and Wavefront model for somite formation, where the clock controls when the boundaries of the somites form and the wavefront determines where they form. Our analysis indicates that this interaction between a segmentation clock and a wavefront can explain the periodic pattern of somites observed in normal embryos. We can also show that a simplification of the model provides a mechanism for predicting the anomalies resulting from perturbation of the wavefront.     

A field assessment of optimal foraging in ants: trail patterns and seed retrieval by the European harvester ant Messor barbarus

Summary: The ant Messor barbarus is a major seed predator on annual grasslands of the Mediterranean area. This paper is an attempt to relate the foraging ecology of this species to resource availability and to address several predictions of optimal foraging theory under natural conditions of seed harvesting.¶Spatial patterns of foraging trails tended to maximise acquisition of food resources, as trails led the ants to areas where seeds were more abundant locally. Moreover, harvesting activity concentrated on highly frequented trails, on which seeds were brought into the nest in larger numbers and more efficiently, at a higher mean rate per worker.¶The predictions of optimal foraging theory that ants should be more selective in both more resource-rich and more distant patches were tested in the native seed background. We confirm that selectivity of ants is positively related to trail length and thus to distance from the nest of foraged seeds. Conversely, we fail to find a consistent relationship between selectivity and density or species diversity of seed patches. We discuss how selectivity assessed at the colony level may depend on factors other than hitherto reported behavioural changes in seed choice by individual foragers.