Bulletin of Mathematical Biology

Masutani, Keiichi

doi: 10.1007/BF02460292pmid: N/A

Abstract A simple one-dimensional model of single-species populations is studied by means of computer simulations. Although the model has a rich spectrum of dynamics including chaotic behavior, the introduction of survival thresholds makes the chaotic region so small that it can be hardly observed. Stochastic fluctuations further reduce the chaotic region because they accidentally lead populations to extinction. The model thus naturally explains the observation that the majority of natural populations do not show chaotic behavior but a monotonic return to a stable equilibrium point following a disturbance.

Rinaldi, Sergio;Muratori, Simona;Kuznetsov, Yuri

doi: 10.1007/BF02460293pmid: N/A

Abstract A classical predator-prey model is considered in this paper with reference to the case of periodically varying parameters. Six elementary seasonality mechanisms are identified and analysed in detail by means of a continuation technique producing complete bifurcation diagrams. The results show that each elementary mechanism can give rise to multiple attractors and that catastrophic transitions can occur when suitable parameters are slightly changed. Moreover, the two classical routes to chaos, namely, torus destruction and cascade of period doublings, are numerically detected. Since in the case of constant parameters the model cannot have multiple attractors, catastrophes and chaos, the results support the conjecture that seasons can very easily give rise to complex populations dynamics.

Totafurno, John;Bjerknes, Matthew;Cheng, Hazel

doi: 10.1007/BF02460294pmid: 8416118

Abstract Current understanding of the pattern of proliferation within intestinal crypts involves the notion of a cutoff region introduced by Cairnieet al. (Exp. Cell. Res. 39, 539–553, 1965b). (Cells produced above the cutoff are non-cycling, whereas cells produced below the cutoff are cycling.) They contrasted the predicted distribution of proliferation in the extreme cases of a cutoff of width 0 (a sharp cutoff) with one eight cells wide (a slow cutoff) and concluded that the data were better explained by the latter. We have shown that crypt size variation artificially broadens the apparent distribution of proliferating cells in the crypt (Totafurnoet al., Biophys. J. 54, 845–858, 1988). Here we show that the measurement and analysis of crypts of a specified height reduces this artifact. This work introduces the use of distance from the crypt base (in microns) to specify the location of cells within the crypt as an improvement over the cell position ordering traditionally used in the determination of the distribution of proliferating cells. We also show how to explicitly correct for several artifacts in the measurement of the labelling index. We conclude that cell proliferation within the crypt is more localized than previously realized; in fact, a cutoff as slow as eight cells wide is rejected.

Sendova-Franks, Ana;Franks, Nigel R.

doi: 10.1007/BF02460295pmid: N/A

Abstract We briefly review the literature on the division of labour in ant colonies with monomorphic worker populations, and show that there are anomalies in current theories and in the interpretation of existing data sets. Most ant colonies are likely to be in unstable situations and therefore we doubt if an age-based division of labour can be sufficiently flexible. We present data for a type of small ant colony in a highly seasonal environment, concentrating on individually marked older workers. We show that contrary to expectation such workers undertake a wide variety of tasks and can even retain their ability to reproduce, even whilst younger workers are actively foraging. Our analysis shows that old workers occupy four distinct spatial stations within the nest and that these are related to the tasks they perform. We suggest that correlations between age and task in many ant colonies might simply be based on ants foraging for work, i.e. actively seeking tasks to perform and remaining faithful to these as long as they are profitably employed. For this reason, employed older workers effectively displace unemployed younger workers into other tasks. In a companion paper, Tofts 1993,Bull. math. Biol. develops an algorithm that shows how foraging for work can be an efficient and flexible mechanism for the division of labour in social insects. The algorithm creates a correlation between age and task purely as a by-product of itsmodus operandi.

Lara-Aparicio, Miguel;de Medrano, Santiago López;Fuentes-Pardo, Beatriz;Moreno-Sáenz, Enrique

doi: 10.1007/BF02460296pmid: N/A

Abstract Based on experimental work on the ontogeny of the electroretinogram circadian rhythm in crayfish, we present a mathematical model simulating changes in both frequency and amplitude of the electroretinogram oscillation during several developmental stages until shortly before the adult age. Simultaneously, we propose a hypothetical oscillation in the hormonal release whose frequency is imposed on the electroretinogram oscillation. The model consists of two coupled nonlinear oscillators in which a dynamical response is obtained mainly through an Andronov-Hopf bifurcation. Through the construction of the model, a biological hypothesis about the essential elements underlying the ERG circadian rhythm and their interrelations is formulated and discussed.

Pritchard, W. G.

doi: 10.1007/BF02460297pmid: N/A

Abstract Several of the known scaling laws in the animal kingdom are based on a so-called allometric correlation in which some physical quantity is presumed to scale as some power of the mass of the animal. Such a simple correlation, when deduced purely as an empirical result, often hides the physical balances that fix the relevant scaling law. In particular, the emphasis on a simple allometric scaling has often masked the fundamental role played by time scales associated with the physical balances being struck. In this paper I have concentrated on three different attributes to which the use of dimensional analysis, scaling arguments and some judicious guesswork have led to new results and an understanding of some balances that occur in the animal kingdom. The running speed of animals is examined and a rationale deduced for the resolution of a conundrum first posed by A.V. Hill of why it is that many animals appear to have approximately the same maximum speed. A complete dimensional analysis for scaling the basal metabolic rate for a class of animals suggests that a detailed understanding of the physical balances that fix the metabolic rate could be quite subtle. However, the use of such an analysis has led to the discovery of a new correlation for mammals, relating the metabolic rate to the mass and the pulse rate of the animal. At the heart of many scaling laws for animal motion is the provision of an estimate of how the skeletal structure depends on the mass of the animal. It has been known for some time that the assumption of isometry between the builds of animals is too constrictive to describe the observed scaling laws. It is shown here how to relax the isometric assumption and deduce scaling laws in good agreement with observation. Thus, it appears that the skeletal dimensions of many animals with exoskeletons are fixed by the need to support static rather than dynamical loads. The scaling laws associated with endoskeletons are more complex, apparently, though the analysis does suggest that it is dynamical loading which is decisive for the skeletal design of land mammals.

Luo, Xincai;Taylor, Keith;Mezey, Paul G.

doi: 10.1007/BF02460298pmid: N/A

Abstract A class of deformations of polyhedra which preserve both the combinatorial type and the location of all but one vertex are examined, leading to the concepts of unconditional and conditional vertex mobility. A simple criterion for vertex mobility is given, and equimobility classes of polyhedra are introduced. The polyhedral mobility characterization is suggested for applications in dynamic molecular modeling, shape analysis of protein folding, and the study of rearrangements of atomic clusters.

Gusfield, Dan

doi: 10.1007/BF02460299pmid: 7680269

Abstract Multiple string (sequence) alignment is a difficult and important problem in computational biology, where it is central in two related tasks: finding highly conserved subregions or embedded patterns of a set of biological sequences (strings of DNA, RNA or amino acids), and inferring the evolutionary history of a set of taxa from their associated biological sequences. Several precise measures have been proposed for evaluating the goodness of a multiple alignment, but no efficient methods are known which compute the optimal alignment for any of these measures in any but small cases. In this paper, we consider two previously proposed measures, and given two computationaly efficient multiple alignment methods (one for each measure) whose deviation from the optimal value isguaranteed to be less than a factor of two. This is the novel feature of these methods, but the methods have additional virtues as well. For both methods, the guaranteed bounds are much smaller than two when the number of strings is small (1.33 for three strings of any length); for one of the methods we give a related randomized method which is much faster and which gives, with high probability, multiple alignments with fairly small error bounds; and for the other measure, the method given yields a non-obviouslower bound on the value of the optimal alignment.

Horváth, Gábor;Varjú, Dezsö

doi: 10.1007/BF02460300pmid: N/A

Abstract The optimal shape of the front profile of the thick lens in the eye of the scallop,Pecten is theoretically, geometric optically investigated as a function of the refractive index of the lens and the retina, as well as of the geometrical parameters of the eye. The shape of the theoretical front surfaces is compared with that of the real, experimentally determined front face of the lens. The degree of correction of the lens for spherical aberration of the reflecting spherical mirror in thePecten eye is examined. The optimal shape of the front profile of the lens depends strongly on a set of parameters, such that a certain fine tuning is required among them to assure a full correction for spherical aberration. The extreme variability of the eye parameters and the shape of the front face of the lens in the scallop is inconsistent with this fine tuning requirement. The degree of correction of thePecten lens for spherical aberration might not be as good as it could be, a possible biooptical reason for which is discussed.

Fuxman, Yakov L.

doi: 10.1007/BF02460301pmid: N/A

Abstract A population of cells suspended in a liquid nutrient medium is considered. The process of growth, division and death of a cell is interpreted mathematically as the Bellman-Harris stochastic process governed by random meetings between the cell and nutrient particles. Growth of a cell is considered to be a result of two processes: mass inflow into and mass outflow from the cell. It is found that, in the absence of food limitations and inhibitors, population growth is not exponential. However, the exponential increase is approached asymptotically over time. Population net growth rate is a variable rather than a constant, but tends over time to a constant value which is the rate of exponential growth. The rate of exponential growth, the probabilities of cell division and death, and the life expectancy of a cell are expressed analytically via average rate of meetings between a cell and nutrient particles. The paper presents an independent phase in calculating mathematical relations between the rate of exponential growth and the concentration of food in a substrate.