Showing papers in "Journal of Mathematical Biology in 1980"

Biased random walk models for chemotaxis and related diffusion approximations

Abstract: Stochastic models of biased random walk are discussed, which describe the behavior of chemosensitive cells like bacteria or leukocytes in the gradient of a chemotactic factor. In particular the turning frequency and turn angle distributions are derived from certain biological hypotheses on the background of related experimental observations. Under suitable assumptions it is shown that solutions of the underlying differential-integral equation approximately satisfy the well-known Patlak-Keller-Segel diffusion equation, whose coefficients can be expressed in terms of the microscopic parameters. By an appropriate energy functional a precise error estimation of the diffusion approximation is given within the framework of singular perturbation theory.

Spatial segregation in competitive interaction-diffusion equations

Abstract: The effect of cross-population pressure on the Volterra type dynamics for two competing species is investigated. On the basis of cross-diffusion induced instability, spatial segregation is studied. Spatially discrete models are also discussed. It is shown that this effect has a tendency to enhance the stability assuring coexistence of species. In continuous and discrete cases, time-dependent segregation processes are studied numerically.

The strong-migration limit in geographically structured populations.

Abstract: Some strong-migration limits are established for geographically structured populations. A diploid monoecious population is subdivided into a finite number of colonies, which exchange migrants. The migration pattern is fixed and ergodic, but otherwise arbitrary. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus. In all the limiting results, an effective population number Ne (less than or equal to NT) appears instead of the actual total population number NT. 1. If there is no selection, every allele mutates at rate u to types not preexisting in the population, and the (finite) subpopulation numbers Ni are very large, then the ultimate rate and pattern of convergence of the probabilities of allelic identity are approximately the same as for panmixia. If, in addition, the Ni are proportional to 1/u, as NT leads to infinity, the equilibrium probabilities of identity converge to the panmictic value. 2. With a finite number of alleles, any mutation pattern, an arbitrary selection scheme for each colony, and the mutation rates and selection of coefficients proportional to 1/NT, let Pj be the frequency of the allele Aj in the entire population, averaged with respect to the stationary distribution of the backward migration matrix M. As NT leads to infinity, the deviations of the allelic frequencies in each of the subpopulations from Pj converge to zero; the usual panmictic mutation-selection diffusion is obtained for Pj, with the selection intensities averaged with respect to the stationary distribution of M. In both models, Ne = NT and all effects of population subdivision disappear in the limit if, and only if, migration does not alter the subpopulation numbers.

Integral equation models for endemic infectious diseases.

Abstract: Endemic infectious diseases for which infection confers permanent immunity are described by a system of nonlinear Volterra integral equations of convolution type. These constant-parameter models include vital dynamics (birth and deaths), immunization and distributed infectious period. The models are shown to be well posed, the threshold criteria are determined and the asymptotic behavior is analysed. It is concluded that distributed delays do not change the thresholds and the asymptotic behaviors of the models.

Two Species Competition in a Periodic Environment

Abstract: The classical Lotka-Volterra equations for two competing species have constant coefficients. In this paper these equations are studied under the assumption that the coefficients are periodic functions of a common period. As a generalization of the existence theory for equilibria in the constant coefficient case, it is shown that there exists a branch of positive periodic solutions which connects (i.e. bifurcates from) the two nontrivial periodic solutions lying on the coordinate axes. This branch exists for a finite interval or “spectrum” of bifurcation parameter values (the bifurcation parameter being the average of the net inherent growth rate of one species). The stability of these periodic solutions is studied and is related to the theory of competitive exclusion. A specific example of independent ecological interest is examined by means of which it is shown under what circumstances two species, which could not coexist in a constant environment, can coexist in a limit cycle fashion when subjected to suitable periodic harvesting or removal rates.

A Competition Model for a Seasonally Fluctuating Nutrient

Abstract: A model of two species consuming a single, limited, periodically added resource is discussed. The model is based on chemostat-type equations, which differ from the classical models of Lotka and Volterra. The model incorporates nonlinear ‘functional response’ curves of the Holling or Michaelis-Menten type to describe the dependence of the resource-exploitation rate on the amount of resource. Coexistence of two species due to seasonal variation is indicated by numerical studies.

The existence of globally stable equilibria of ecosystems of the generalized Volterra type

Abstract: In this paper, global asymptotic stability of ecosystems of the generalized Volterra type $$dx_i /dt = x_i \left( {b_{i - } \mathop \sum \limits_{j = 1}^n a_{ij} x_j } \right),{\text{ }}i = 1,...,n,$$ is investigated. We obtain the conditions for the existence of a nonnegative and stable equilibrium point of the system by applying a result of linear complementarity theory. The results of this paper show that there exists a class of systems that do not have multiple domains of attractions. This class is defined in terms of the species interactions alone, and does not involve carrying capacities or species net birth rates.

Spatial distribution of dispersing animals

Abstract: A mathematical model for the dispersal of an animal population is presented for a system in which animals are initially released in the central region of a uniform field and migrate randomly, exerting mutually repulsive influences (population pressure) until they eventually become sedentary. The effect of the population pressure, which acts to enhance the dispersal of animals as their density becomes high, is modeled in terms of a nonlinear-diffusion equation. From this model, the density distribution of animals is obtained as a function of time and the initial number of released animals. The analysis of this function shows that the population ultimately reaches a nonzero stationary distribution which is confined to a finite region if both the sedentary effect and the population pressure are present. Our results are in good agreement with the experimental data on ant lions reported by Morisita, and we can also interpret some general features known for the spatial distribution of dispersing insects.

The effect of integral conditions in certain equations modelling epidemics and population growth.

Abstract: Models of epidemics that lead to delay differential equations often have subsidiary integral conditions that are imposed by the interpretation of these models. The neglect of these conditions may lead to solutions that behave in a radically different manner from solutions restricted to obey them. Examples are given of such behavior, including cases where periodic solutions may occur off the natural set defined by these conditions but not on it. A complete stability analysis is also given of a new model of a disease propagated by a vector where these integral conditions play an important role.

Segregation of a population in an environment

Abstract: A mathematical model for spatial patterns and the segregation of a population is presented. Individuals in a population are assumed to move at random under the influence of a given environment potential V(x). The notion of kinetic excitation K(x) and intensity excitation Q(x) of a population is introduced. Then equilibrium states of a population are defined through a macroscopic relation K(x) + Q(x) + V(x) = constant. The problem of finding out equilibrium distributions is reduced to an eigenvalue problem. It is shown that a population is segregated by the nodal surfaces of the eigenfunctions, if it is excited. Some applications of the model to biological and ecological problems are indicated.

Population models in a periodically fluctuating environment

Abstract: We investigate the behavior of population models in the presence of a periodically fluctuating environment. We consider in particular single-species models and models of interspecific competition. It is shown that the fluctuations cause constant equilibrium states to be replaced by periodic equilibrium states, with a shift in the mean value relative to the constant-environment state. It is shown also that the locations of points of exchange of stability may be changed as a result of the fluctuations.

When is one estimate of evolutionary relationships a refinement of another

Abstract: A new way to view a certain type of taxonomic character is presented and several fundamental results are rederived using this approach.

Lotka-Volterra equations: Decomposition, stability, and structure

Abstract: The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.

A mathematical model of canine granulocytopoiesis

Abstract: The granulocyte cell renewal system of the dog is represented by a mathematical model consisting of the following compartments: The pool of pluripotential stem cells, the committed stem cell pool, divided into a blood and a bone marrow compartment, the proliferation pool, the maturation pool, the reserve pool and the blood pool of functional granulocytes. This chain of compartments is described by a system of non-linear differential equations. Cell losses anyplace in the system provoke increased production in all pools containing cells capable to divide. A reduced number of granulocytes in the blood pool stimulates production of a “granulocyte releasing factor” which mobilizes a rising number of cells to transit from the marrow reserve into the blood pool. The model was simulated on a digital computer. It was found to be capable to reproduce the steady state conditions and it also fits the data of two distinct experimental perturbations of the system both equally well. These perturbations are a loss of proliferating cells as it occurs after the administration of cytostatic drugs and losses of functional cells as they are induced by leukapheresis experiments of differing leukapheresis rates.

A mechanical model for epithelial morphogenesis.

Abstract: During embryogenesiG sheets of epithelial cells bend, buckle and deform themselves in precisely defined ways. Examples are (i) the invagination of the blastula accompanying gastrulation, (ii) the folding of the neural plate accompanying neurulation, (iii) invagination and extrusion of glandular and intestinal epithelia, (iv) the infolding of the optic vesicle. These processes involve hundreds or thousands of cells, each changing its shape in coordination with the others so as to produce a coherent global pattern. How these cell shape changes are orchestrated constitutes one of the major mysteries ofmorphogenesis. In this paper we announce a model for epithelial tissue folding which differs considerably from previous models. We base our model on the experimental observation that each epithelial cell has, prior to its deformation, a subcortical band of microfilaments at its apical periphery. In our model this filamentous band generates the force driving the cell to change its shape by shortening, much like the drawing of a purse-string. The evidence for this type of mechanical action is convincing. The morphogenetic problem of coordinating the cell shape changes can be accomplished solely by mechanical interactions between the cells simply by endowing the viscoelastic response of the microfilament bundle with a particular type of constitutive relation.

Optimal strategies in immunology III. The IgM-IgG switch

Abstract: During a primary immune response generally two classes of antibody are produced, immunoglobulin M (IgM) and immunoglobulin G (IgG). It is currently thought that some lymphocytes which initially produce IgM switch to the production of IgG with the same specificity for antigen. During a secondary immune response IgG is the predominant antibody made throughout the response. In this paper we address the question of why such apparently complicated modes of response should have been adapted by evolution. We construct mathematical models of the immune response to growing antigens which incorporate complement dependent cell lysis. By comparing the times required to eliminate antigen we show that under certain conditions it is advantageous for an animal to switch some of its lymphocytes from IgM to IgG production during a primary response, but yet to secrete only IgG during a secondary response. The sensitivity of such a conclusion to parameter variations is studied and the biological basis and implications of our models are fully discussed.

Global asymptotic stability for a vector disease model with spatial spread.

Abstract: 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.

Existence and stability of periodic travelling wave solutions to Nagumo's nerve equation

Abstract: Nagumo's nerve conduction equation has a one-parameter family of spatially periodic travelling wave solutions. First, we prove the existence of these solutions by using a topological method. (There are some exceptional cases in which this method cannot be applied in showing the existence.) A periodic travelling wave solution corresponds to a closed orbit of a third-order dynamical system. The Poincare index of the closed orbit is determined as a direct consequence of the proof of the existence. Second, we prove that the periodic travelling wave solution is unstable if the Poincare index of the corresponding closed orbit is + 1. By using this result, together with the result of the author's previous paper, it is concluded that “the slow periodic travelling wave solutions” are always unstable. Third, we consider the stability of “the fast periodic travelling wave solutions”. We denote by L(c) the spatial period of the travelling wave solution with the propagation speed c. It is shown that the fast solution is unstable if its period is close to Lmin, the minimum of L(c).

The frequency of cyclic processes in biological multistate systems.

Abstract: Cyclic processes in stochastic models of macromolecular biological systems are considered. The diagram solution of the model equations (master equation) gives rise to special functions of the rate constants, called the circuit (or one-way cycle) fluxes. As Hill has shown, these functions are the fundamental theoretical components of the operational fluxes, i.e., of the rates of reaction, of transport, of energy conversion, etc. Evidence recently has been found by Monte Carlo simulations that the circuit fluxes can be interpreted as the frequencies of circuit completions. Making use of the theory of graphs, we prove that this physical interpretation of the circuit fluxes is generally valid.

Non-linear age-dependent population growth.

Abstract: A non-linear problem arising from age-dependent population dynamics is studied. Existence and uniqueness and a priori bounds for the growth of population are proved. Moreover the existence and the stability of equilibrium age distributions is investigated.

Periodic solutions of an epidemic model.

Abstract: The existence of periodic solutions of the equation $$x(t) = k\left( {P - \int_{ - \infty }^t A(t - s)x(s)ds } \right)\int_{ - \infty }^t a(t - s)x(s)ds$$ is established. This equation arises in the study of the spread of a disease which does not induce permanent immunity.

A theoretical model for studying the rate of oxygenation of blood in pulmonary capillaries.

Abstract: A mathematical analysis of the process of gas exchange in the lung is presented taking into account the transport mechanisms of molecular diffusion, convection and facilitated diffusion of the species due to haemoglobin. Since the rate at which blood gets oxygenated in the pulmonary capillaries is very fast, it is difficult to set up an experimental study to determine the effects of various parameters on equilibration rate. The proposed study is aimed at determining the effects of various physiological parameters on equilibration rate in pathological conditions. Among the significant results are that 1. dissolved oxygen takes longer to achieve equilibration across the pulmonary membrane and carbon dioxide attains equilibration faster, 2. the equilibration length increases with increase in blood velocity, haemoglobin concentration, calibre of pulmonary capillaries and fall in alveolar PO2, 3. the alveolar PCO2 and forward and backward reaction rates of haemoglobin with CO2 do not materially affect the equilibration rate or length. 4. At complete equilibration, by the end of the pulmonary capillary 92% of the total haemoglobin has combined with oxygen and 8% free pigment is left which is present as carbamino haemoglobin, met haemoglobin, carboxy haemoglobin etc. These results are of some importance for anaemic conditions, muscular exercise, meditation, altitude physiology, hypo-ventilation, hyperventilation, etc.

Frequency spectra of neutral and deleterious alleles in a finite population

Abstract: One of the major goals of population genetics is to discover the nature and amount of genetic variation in natural populations. Various measures, including the population heterozygosity at any locus and the number of alleles extant at the locus, have been used for this purpose. An important task of theoretical population genetics is thus to provide expressions for the mean values of these two quantities (when calculated from a sample of genes) for various models of selection, mutation and random drift. This aim has been achieved for the selectively neutral case, where all alleles at the locus are assumed to be selectively equivalent. It is, however, generally agreed that classes of (evolutionarily unimportant) selectively deleterious alleles exist, so that the neutral theory calculations should be extended to cover this case. This has previously been done only for extremely weak selection. In this paper we obtain, via the confluent hypergeometric function and three allied functions, concise and simple exact and approximate formulae for the means of the above measures of population variation for arbitrary selective values. These all derive from the allelic “frequency spectrum”, which is of independent interest in assessing likely models of population variation.

Phototactic Attraction in Light Trap Experiments: A Mathematical Model

Abstract: A mathematical model has been developed to evaluate the contribution of phototactic responses in light-induced accumulations. A set of differential equations describes the organism density inside and outside of the light trap as well as on its border.

The number of distinguishable alleles according to the Ohta-Kimura model of neutral mutation.

Abstract: We calculate how many alleles one can expect to distinguish in a large sample from a large population which develops according to the Ohta-Kimura model. This number tends to infinity with the sample size, but so slowly that it is bounded for all practical purposes.

Convergence to genetically uniform state in stepping stone models of population genetics

Abstract: We investigate continuous time stepping stone models. Extending the models treated in population genetics, we consider the system described by the following infinite dimensional stochastic differential equation, $$dx_k (t) = a_k (x_k )dB_k + \left\{ {\sum\limits_{j \in S} {q_{k,jXj} } } \right\}dt, k \in S$$ which contains the effects of random sampling drift and a kind of stochastic fluctuation in selection. We obtain a necessary and sufficient condition for the system to converge to a genetically uniform state.

Transition from polar to duplicate patterns

Abstract: The existence of symmetric nonuniform solutions in nonlinear reaction-diffusion systems is examined. In the first part of the paper, we establish systematically the bifurcation diagram of small amplitude solutions in the vicinity of the two first bifurcation points. It is shown that: i) The system can adopt a stable symmetric solution (basic wave number 2) if the value of the bifurcation parameter is changed or if the initial polar structure (basic wave number 1) is sufficiently perturbed. ii) This behavior is independent of the particular reaction-diffusion model proposed and of the number of intermediate components (⩾2) involved.

The extinction of slowly evolving dynamical systems

Abstract: The time evolution of slowly evolving discrete dynamical systems xi + 1= T(ri,xi), defined on an interval [0, L], where a parameter richanges slowly with respect to i is considered. For certain transformations T, once ri reaches a critical value the system faces a non-zero probability of extinction because some xj ∋ [0, L]. Recent ergodic theory results of Ruelle, Pianigiani, and Lasota and Yorke are used to derive a simple expression for the probability of survival of these systems. The extinction process is illustrated with two examples. One is the quadratic map, T(r, x) = rx(1 − x), and the second is a simple model for the growth of a cellular population. The survival statistics for chronic myelogenous leukemia patients are discussed in light of these extinction processes. Two other dynamical processes of biological importance, to which our results are applicable, are mentioned.