Mathematical models for some types of chemical information systems

Ant species on a high evolutionary level have evolved chemical recruitment systems such as mass recruitment or quality recruitment. The recruitment process from the nest to a food source may be damped by crowding effects at the source. For four patterns of behavior (mass/quality recruitment; with/without damping) we study mathematical models for the time development of the quantity of food at the source. Each of the models can be reduced to a second order time-delayed differential equation which will be studied in the equivalent form of a first order (nonlinear) functional differential equation. We discuss the complete exploitation of a given source. In case of mass recruitment there possibly remains a threshold quantity of food not worth exploiting. However, every source will be exploited completely (in finite time) provided that the volatility of the trail pheromone is small compared with the exploitation activities of the colony and the distance from the nest to the source. In addition, for the damped models the capacity of the crowded source must be large compared with the initial quantity of food offered. The efficiency of the exploitation activities of some species allows conclusions on their evolutionary development.     

The diversity and evolution of recruitment behaviour in ants,with a discussion of the usefulness of parsimony criteria in the reconstruction of evolutionary histories

Summary Three alternative hypotheses about the evolution of recruitment behaviour in ants, based on accounts in the literature, are compared by means of a cladistic analysis. The three hypotheses are the following:Hypothesis 1. Increasingly efficient recruitment behaviours exhibited by different ant species have been shaped by or are correlated with ant phylogeny.Hypothesis 2. Increasingly efficient recruitment behaviours represent necessary evolutionary steps independently followed during the evolution of different ant clades.Hypothesis 3. Differently efficient recruitment behaviours have been selected in a convergent way among different species by similar population/environmental constraints.In a first stage of the analysis, these hypotheses have been compared in terms of parsimony (i.e. in terms of tree length = TL) of alternative cladograms based on recruitment behaviour only. The analysis gave the following results: Hypothesis 1, TL = 4; Hypothesis 2, TL = 18; Hypothesis 3, TL = 11. At least in terms of parsimony, hence, Hypothesis 1 appears to be the best. This hypothesis, however, cannot be retained for its total lack of congruence with current views on ant phylogeny. Among the remaining two hypotheses, Hypothesis 3 is again much (ca. 40%) more parsimonious than Hypothesis 2, but the retention index for recruitment behaviour on the relative cladogram is 0.2 as compared with 0.7 for Hypothesis 2. Practically, this implies biologically very implausible behavioural evolution indicated by very improbable ancestors for the species included in the analysis. In the case of recruitment evolution the biological credibility of each hypothesis is inversely proportional to its parsimony.The three hypotheses on the evolution of recruitment behaviour are compared again taking into account the morphological and behavioural correlates of recruitment. The results confirm those obtained by simple cladistic analysis of behaviour alone, namely that an obligatory (i. e. neither reversible nor random) increase in recruitment efficiency has been repeatedly selected within different ant clades. Inclusion of the recruitment correlates allows, in addition, a more precise formulation of the implications of each hypothesis and a tentative test of two other alternatives deduced from the literature. Most papers dealing with recruitment assume this behaviour to be controlled by a single gland, while at least two experimental analyses show that more than one gland is likely to be involved as behavioural releaser. A cladistic approach allowed testing of the following two adaptational hypotheses: A) Synergic behavioural control by several glands, allowing shift of the dominant role from one gland to another. B) Single gland control, making improbable the replacement of one gland by another that performs the same function. The results of the analysis appear to favour alternative A slightly, though neither alternative results in implausible evolutionary paths.It is stressed that parsimony remains the sole decisional criterion when no other criteria are available but it can by no way be preferred to the slightest trace of biological common sense.     

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.     

The allocation of foragers in red wood ants

Abstract. 1. We studied how colonies of the red wood ant, Formica polyctena , adjust the numbers of foragers allocated to different foraging trails. In a series of field experiments, foragers were marked and transferred from one nest to another, related nest, where they joined the foraging force. Transferred workers acted as a reserve of uncommitted, available foragers.
2. Previous work shows that each individual forager habitually uses one trail. We found that for an uncommitted forager, the influence of recruitment initially is stronger than that of directional fidelity. Transferred workers were likely to use trails leading to new food sources. When transferred to a new nest, foragers were not likely to use a trail in the same direction as their original trail in the donor nest.
3. After a week, transferred foragers tended to develop route fidelity. Even after bait was no longer present, they continued to use the trail that had formerly led to a bait source.
4. We examined how colonies adjust numbers on a trail by experimentally depleting some trails. Colonies usually did not compensate for depletion: foragers were not recruited to depleted trails.
5. In general, the dynamics of foraging in this species facilitate a consistent foraging effort rather than rapid adjustments of forager allocation.     

On foraging,recruitment systems and optimum number of scouts in eusocial colonies

Summary A numerical model of an eusocial colony foraging for food showed that, for each set of values of resource density, resource size and recruitment system employed, a given optimal proportion of scouts in the colony maximize the amount of resources retrieved by a colony during a fixed period. The model predicts that ants using mass recruitment systems should have larger colonies with small foragers, and should forage on large food sources. Retrieval of small food sources by small colonies is best achieved with large workers using individual foraging strategies. For mass foragers, several food sources are best retrieved using democratic decision-making systems in recruitment, whereas for very large food sources at very low mean food patch density, autocratic decision-making systems are optimal. Some of the experimental evidence available is discussed in the light of these findings, as they confirm the prediction that large colonies with small workers have mass recruitment systems, whereas workers of small colonies with large workers are generally lone foragers.     

A mathematical model for mechanotransduction at the early steps of suture formation

Growth and patterning of craniofacial sutures is subjected to the effects of mechanical stress. Mechanotransduction processes occurring at the margins of the sutures are not precisely understood. Here, we propose a simple theoretical model based on the orientation of collagen fibres within the suture in response to local stress. We demonstrate that fibre alignment generates an instability leading to the emergence of interdigitations. We confirm the appearance of this instability both analytically and numerically. To support our model, we use histology and synchrotron X-ray microtomography and reveal the fine structure of fibres within the sutural mesenchyme and their insertion into the bone. Furthermore, using a mouse model with impaired mechanotransduction, we show that the architecture of sutures is disturbed when forces are not interpreted properly. Finally, by studying the structure of sutures in the mouse, the rat, an actinopterygian (Polypterus bichir) and a placoderm (Compagopiscis croucheri), we show that bone deposition patterns during dermal bone growth are conserved within jawed vertebrates. In total, these results support the role of mechanical constraints in the growth and patterning of craniofacial sutures, a process that was probably effective at the emergence of gnathostomes, and provide new directions for the understanding of normal and pathological suture fusion.     

A mathematical description of a pump/leak/buffer model for plant nitrate uptake

A pump/leak/buffer model of nitrate in plants is presented in the form of a differential equation, based on the computer simulation model of Scaife (1989). The equation is solved, and the long-term behaviour of the rate of nitrate uptake of the plants described.     

A delay recruitment model of the cardiovascular control system

We develop a nonlinear delay-differential equation for the human cardiovascular control system, and use it to explore blood pressure and heart rate variability under short-term baroreflex control. The model incorporates an intrinsically stable heart rate in the absence of nervous control, and allows us to compare the baroreflex influence on heart rate and peripheral resistance. Analytical simplifications of the model allow a general investigation of the rôles played by gain and delay, and the effects of ageing.     

A mathematical model of iron metabolism

A mathematical model of iron metabolism is presented. It comprises the following iron pools within the body: transferrin-bound iron in the plasma, iron in circulating red cells and their bone marrow precursors, iron in mucosal, parenchymal and reticuloendothelial cells. The control exerted by a hormone, called erythropoietin, on bone marrow utilization of iron for hemoglobin synthesis is taken into account. The model so obtained consists of a system of functional differential equations of retarded type. Most model parameters can be estimated from radiotracer experiments, others can be measured and numerical values can be assigned to the remaining ones making few reasonable assumptions according to the available physiological knowledge. Iron metabolism behavior under different therapeutical treatments was simulated. Model predictions were compared to experimental data collected in clinical routine.This work has been partially supported by C.N.R. (Italy) through grants N. 80.01227.07 and N. 81.00888.07     

A model for the interaction of two chemicals

We describe a method for studying the interaction of two anesthetic agents, Morphine and Midazolam, acting simultaneously in the same individual. Representing the levels of the two chemicals by diffusion processes, we assume their interaction is governed by a linear combination of the separate components. Pharmacological data is used to estimate the model parameters and, in particular, to determine the coefficient in the linear combination. This leads to the conclusion that the two chemicals have a counteractive effect.     

Respective contributions of leader and trail during recruitment to food inTetramorium bicarinatum (Hymenoptera: Formicidae)

Summary This paper describes the food recruitment strategy of the antTetramorium bicarinatum, at both the individual and collective levels. The general organisation of recruitment used by this species during the exploitation of sucrose solutions shows similarities with group recruitment described for other species. However, our experiments demonstrate that inT. bicarinatum, the first trail laid by a recruiter during its return trip to the nest is more efficient than in other species using group recruitment, for exampleTetramorium impurum. Moreover, the efficiency of the first trail of aT. bicarinatum recruiter is comparable with that ofTapinoma erraticum, a species that uses mass recruitment. Despite the efficiency of the trail, choice experiments show that the recruited workers prefer to follow the leader rather than the first trail, suggesting the emission of a more attractive signal by the leader on its way back to the food. The function of the leader in this strategy is discussed in terms of collective decisions.     

Global asymptotic stability for a vector disease model with spatial spread

Summary We analyze the global behaviour of a vector disease model which involves spatial spread and hereditary effects. This model can be applied to investigate growth and spread of malaria. No immunization is considered. We prove that, if the recovery rate is less than or equal to a threshold value, the disease dies out, otherwise the infectious people density tends to a homogeneous distribution. Our results follow using contracting convexes techniques and agree with the results given by K. L. Cooke for the model without diffusion.Work supported by C.N.R., Grant No. 79.00696.01.     

A genetic model for disruptive selection on colony social organisation,reproduction, and ecotype distribution in wood ants inhabiting different woodland habitats

Summary Wood ants of theFormica rufa species complex are typically bound to woodland habitats and their eventual colonization of more open habitats depends on the presence of a sufficient number of woody plants. Computer simulations of the long-term development of large wood ant populations in woodland systems of differing structure were performed. A cellular, stochastic automation model simulated the alternatives of a compact, coherent woodland system and a fragmented, coarse-grained woodland system. The simulation of a compact woodland system gave evidence that disruptive selection alone may be sufficient to produce two distinct ectotypes, which are known asFormica rufa andFormica polyctena, even if there is a full fertility of the crosses. The simulation of a fragmented, coarse-grained woodland system has supported the view, that a high local frequency of nests with intermediate phenotypes might be explainable by a particular woodland pattern which favour mixed strategies. It was shown that queen dominance and intraspecific social parasitism are likely to be important factors in the dynamics of large wood ant populations.     

A mathematical model for the design of fibrin microcapsules with skin cells

The use of fibrin in tissue engineering has greatly increased over the last 10 years. The aim of this research was to develop a mathematical model to relate the microcapsule-size and cell-load to growth and oxygen depletion. Keratinocytes were isolated from rat skins and microencapsulated dropping fibrinogen and thrombin solutions. The cell growth was measured with MTT-assay and confirmed using histochemical technique. The oxygen was evaluated using a Clark sensor. It was found that Fick–Monod model explained the cell growth for the first 48 h, but overestimated the same thereafter. It was necessary to add a logistic equation to reach valid results. In relation to the preferred implant alternative, when considering large initial cell loads, the possibility to implant small loads of fast-growing cells arises from the simulations. In relation to the microcapsule size, it was found that a critical diameter could be established from which cell growth velocity is about the same.     

Observation of group recruitment to prey in a primitive ponerine ant,Amblyopone sp. (reclinata group) (Hymenoptera: Formicidae)

Summary Group recruitment during foraging was observed in a primitive ponerine ant,Amblyopone sp. (reclinata group) under laboratory condition. Workers searched for prey singly; however, if a item of prey was stung by a worker, other ants joined the attack. After the prey became immobile, one of the workers laid a trail directly toward the nest. This scout worker recruited additional workers (between 3 and 33). They formed a single file procession to the point of prey capture, and cooperatively transported the prey. A scout worker could stimulate nest workers to leave the nest without direct contact, and the recruited workers could trace the trail without guidance by the scout worker. This is the first report of recruitment behavior during foraging in the primitive antAmblyopone.     

A standard unit for monitoring recruitment of fishes to coral reef rubble

We developed a benthic standard monitoring unit for recruitment of fishes (SMURF) to sample fishes that settle in coral rubble and tested it on three fringing reefs on the west coast of Barbados, West Indies. These rubble SMURFs are inexpensive to construct and permit newly settled fishes, including cryptic, secretive and small species, to be quickly and fairly completely (> 83%) removed by divers without returning the unit to the surface. Over a 3-month period involving 9 collections, 32 units on 3 reefs yielded 948 newly settled fishes belonging to 28 taxa (3.3 ± 0.2 S.E. recruits per sample, n = 287) in addition to numerous crustaceans. Most specimens were Sparisoma spp. (Scaridae) (41.8%) and Stegastes partitus (Pomacentridae) (13.6%), with moderate numbers of Scarus spp. (Scaridae) (8.6%), Lythrypnus spp. (Gobiidae) (7.4%), and Pseudogramma gregoryi (Serranidae) (7.2%). Sampling SMURFs at 1-day rather than 11-day intervals yielded approximately twice as many fish, although some taxa were not affected by sampling frequency, indicating taxon-specific differences in post-settlement loss rates. Netting, intended to reduce predation on settlers, did not affect settlement estimates unless algae grew on it, suggesting that the SMURF design minimizes fish predation even in the absence of the netting. Over time, however, the netting resulted in substantial algal growth that was associated with an increase in abundance of Sparisoma spp. and a decrease in the abundance of S. partitus. This suggests that settler estimates are more affected by microhabitat changes from algal growth than by predation. During low settlement periods, SMURFs performed better than light-traps in assessing the daily input of new settlers of two abundant taxa. Eight SMURFs per reef were sufficient to demonstrate differences in settlement rates and temporal correlations in settlement within and among reefs. This study underscores the potential of benthic SMURFs to measure settlement of coral reef fishes at a range of spatio-temporal scales with moderate research cost and effort, and to facilitate the study of the settlement stage of several taxa.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behaviour of a system, differential equations have proved to be powerful for modelling and unravelling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge–Kutta fourth-order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

A mathematical model of heterogeneous behavior of single muscle fibres

A mathematical model of contracting muscle fibre is studied. The model is composed of an array of segments placed in series; any segment has an elastic element (PE_i) and a contractile element (CE_i) that describes the cross bridge kinetics.The corresponding system of nonlinear partial differential equations of the model is analyzed. Existence, uniqueness and continuous dependence of the solution are proven.This work has been supported by Ministero Pubblica Istruzione (fondi per la ricerca scientifica) and by Istituto di Analisi Numerica del C.N.R., Pavia     

A mathematical model for drug administration by using the phagocytosis of red blood cells

 A mathematical model for the delivery of drug directly to the macrophages by using the phagocytosis of senescent red blood cells is proposed. The model is based on the following assumption: At time t=0 a preassigned red blood cell population n(0, a)=φ(a), a>0, loaded by the drug, is injected in the blood circulation. Among the cells of that population only those with an age a≧ā (ā=120 days) will be phagocytosed by macrophages. Of course, the lifetime of the drug must be higher than ā. Within the red blood cells it cannot be metabolized, neither can it diffuse through their membranes. The emphasis of the paper is on the mathematical properties and on the formulation of the control problem. Received 15 December 1994; received in revised form 20 July 1995