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1991 | Book

The Problem of Relevant Detail Read chapter Read first chapter

Differential Equations Models in Biology, Epidemiology and Ecology

Proceedings of a Conference held in Claremont California, January 13–16, 1990

Editors: Stavros Busenberg, Mario Martelli

Publisher: Springer Berlin Heidelberg

Book Series : Lecture Notes in Biomathematics

Included in: Professional Book Archive

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About this book

The past forty years have been the stage for the maturation of mathematical biolo~ as a scientific field. The foundations laid by the pioneers of the field during the first half of this century have been combined with advances in ap­ plied mathematics and the computational sciences to create a vibrant area of scientific research with established research journals, professional societies, deep subspecialty areas, and graduate education programs. Mathematical biology is by its very nature cross-disciplinary, and research papers appear in mathemat­ ics, biology and other scientific journals, as well as in the specialty journals devoted to mathematical and theoretical biology. Multiple author papers are common, and so are collaborations between individuals who have academic bases in different traditional departments. Those who seek to keep abreast of current trends and problems need to interact with research workers from a much broader spectrum of fields than is common in the traditional mono-culture disciplines. Consequently, it is beneficial to have occasions which bring together significant numbers of workers in this field in a forum that encourages the exchange of ideas and which leads to a timely publication of the work that is presented. Such an occasion occurred during January 13 to 16, 1990 when almost two hun­ dred research workers participated in an international conference on Differential Equations and Applications to Biology and Population Dynamics which was held in Claremont.

Table of Contents

Frontmatter

Mathematical Biology

Frontmatter
The Problem of Relevant Detail
Abstract
In recent years, welcome attention has been directed to population structure in mathematical models, improving substantially upon the overly aggregated characterizations of classical interest. This conference, in honor of Kenneth Cooke and his contributions to theories involving age structure, social structure, and delays, is testament to the modern trend, and to the recognition that simplistic theories that ignore population structure often are seriously flawed in their predictions.
Simon Levin
Lifespans in Population Models: Using Time Delays
Abstract
Population models with a compartment structure have to incorporate the duration of stay in each class. Using a two-compartment model of the regulation of blood cell production, we investigate two elimination mechanisms: a “random”, constant rate destruction process, and a lifespan of finite duration. The limiting cases of each death mechanism alone are considered in turn, and the dynamics are compared to the full model, from the point of view of the equilibrium solutions, and their (local) asymptotic stability. The crucial quantity not to be neglected is seen to be the maturation time in the precursors’ compartment.
Jacques Bélair
Convergence to Equilibria in General Models of Unilingual-Bilingual Interactions
Abstract
We derive general models for the interaction of unilingual and bilingual components of a population, and classify those models for which the dynamics is trivial by obtaining criteria for the nonexistence of periodic solutions.
H. I. Freedman, I. Baggs
The Sherman-Rinzel-Keizer Model for Bursting Electrical Activity in the Pancreatic β-Cell
Abstract
Pancreatic β-cells exhibit periodic bursting electrical activity (BEA) consisting of active and silent phases. The Sherman-Rinzel-Keizer (SRK) model of this phenomenon consists of three coupled first-order nonlinear differential equations which describe the dynamics of the membrane potential, the activation parameter for the voltage-gated potassium channel, and the intracellular calcium concentration. These equations are nondimensionalized and transformed into a Liénard differential equation coupled to a single first-order differential equation for the slowly changing nondimensional calcium concentration. Leading-order perturbation problems are derived for the silent and active phases of the BEA on slow and fast time scales. Numerical solutions of these leading-order problems are compared with those for the exact equation in their respective regions. The leading-order solution in the active phase has a limit cycle behavior with a slowly varying frequency. It is observed that the “damping term” in the Liénard equation is small numerically.
Mark Pernarowski, Robert M. Miura, J. Kevorkian

Epidemiology

Frontmatter
Models for the Spread of Universally Fatal Diseases II
Abstract
We consider a simple model for a universally fatal disease with an infective period long enough to allow natural deaths during the infective period. The analysis of this model is considerably more complicated than the analysis of a model with an infective period short enough that the population dynamics are confined to the susceptible class. However, the basic result that in some circumstances the stability of an endemic equilibrium may depend on the distribution of infective periods is shared by both models.
Fred Brauer
Nonexistence of Periodic Solutions for a Class of Epidemiological Models
Abstract
Disease transmission models are formulated under assumptions that the size of the population varies and the force of infection is of the proportionate mixing type. Conditions are given that rule out the possibility of periodic solutions for such models. Examples are considered and sharp thresholds identified.
Stavros Busenberg, P. van den Driessche
On the Solution of the Two-Sex Mixing Problem
Abstract
In this paper we describe an axiomatic framework that allows for the general incorporation of sexual structure into two-sex pair-formation models for sexuallytransmitted diseases. A representation theorem describing all solutions to this mixing framework as perturbations of particular solutions is proved. Two-sex age structured demographic and age-structured epidemiological models that make use of our framework, and are therefore capable of describing the dynamics of individuals and/or pairs of individuals, are formulated.
C. Castillo-Chavez, S. Busenberg
Modelling the Effects of Screening in HIV Transmission Dynamics
Abstract
In this paper, we study the effect of screening on the transmission dynamics of HIV virus in a male homosexual population using a simple mathematical model along the lines of those proposed by Anderson, et al. (1986) and others. Analytical results will be given on threshold for a successful screening program which will prevent an epidemic for the simple case when infectivity is assumed to be constant throughout the incubation period. For the model with variable infectivity, we will use numerical simulation and phase plane analysis to explore the possible effects (positive and negative) of random screening as a control policy for AIDS. We also give a formulation of a heterogeneous population model with proportionate mixing and discuss two random screening schemes for the model.
Ying-Hen Hsieh
An S→E→I Epidemic Model with Varying Population Size
Abstract
An S → E → I epidemic model with a general shape of density-dependent mortality and incidence rate is studied analytically and numerically.
The combined effect of a latent period and of varying population size can produce oscillations in this ODE model. When fertility of exposed individuals is the same as that of susceptibles, there is a clear threshold. On the contrary, when both exposed and infectives do not contribute to birth rate, there may exist multiple endemic states also below the threshold.
When the contact rate is independent of population size, the global behaviour is established: all trajectories converge to an equilibrium.
Andrea Pugliese
Stability Change of the Endemic Equilibrium in Age-Structured Models for the Spread of S—I—R Type Infectious Diseases
Abstract
Age-structured endemic models of S → I → R type can exhibit a stability change for the endemic equilibrium if the rate of a susceptible individual to be infected by an infective individual does not depend of the age of the susceptible individual, but is highly concentrated in a particular age class of the infectives.
Horst R. Thieme

Ecology and Population Dynamics

Frontmatter
Mathematical Model for the Dynamics of a Phytoplankton Population
Abstract
The problem of modelling the vertical structure of a phytoplankton population in sea water has been treated in many papers (see e.g. A. Wörz-Busekros [9], N. Shigesada and A. Okubo [7], H. Ishii and I. Takagi [4]). In [9] the coupled dynamics of phytoplankton and nutrient is considered with a sinking term for the phytoplankton and both the algae and the nutrient diffusing with the same diffusion coefficient. A unique solution of the corresponding initial-boundary value problem was shown to exist. In [ 7] the self-shading effect of the biomass in a nutrient-saturated environment was taken into account (neglecting light absorption by water in order to have an autonomous system at equilibrium). The global stability of the unique positive stationary solution for an improved model was proved in [4].
E. Beretta, A. Fasano
Some Delay Models for Juvenile vs. Adult Competition
Abstract
Models of competition have played a central role in theoretical population dynamics and ecology. The vast majority of mathematical models of competitve interactions that have been formulated have been done so with regard to highly aggregate state variables at the total population level and have ignored differences between individual organisms, in effect treating all individuals of a species as identical. Biological populations generally consist, however, of individuals with diverse physiological characteristics, such as age, body size or weight, life cycle stages, etc., (with intra-species variances that in fact can exceed inter-specific variances amongst competing species) and it has become widely recognized that this diversity can have a significant influence upon population level dynamics (Werner and Gilliam (1984), Ebenman and Persson (1988), Metz and Diekmann (1986)). Models that ignore individual level physiological variances cannot, except in the simplest of cases, adequately account for the mechanisms that result in competition between individual organisms for limited resources. In particular, intra-specific competition can be accounted for in such models only in highly qualitative ways at best.
J. M. Cushing
McKendrick Von Foerster Models for Patch Dynamics
Abstract
For many years, Lotka-Volterra equations served as the basic paradigm in ecological modelling. These equations, and models based on them, were responsible for the embodiment of ideas such as the competitive exclusion principle, and even quite recently, many ideas concerning the dynamics of food webs. However, as has long been recognized, these models represent a vast simplification of ecological reality. In particular, they ignore the consequences of structure within the populations modelled. This can be spatial structure, age structure, physiological structure, genetic or phenotypic structure, or possibly other kinds of structure. My goal in this paper will be to describe how models based on systems of first order partial differential equations, known as the McKendrick or Von Foerster equation, can be used to describe interacting spatially structured populations.
Alan Hastings
Generic Failure of Persistence and Equilibrium Coexistence in a Model of m-species Competition in an n-vessel Gradostat when m > n
Abstract
A mathematical model of competition between m species for a single limiting resource in an n vessel gradostat is studied. It is shown that the system has no steady state coexistence in general if m > n, and there is always a saturated equilibrium which must be on the boundary of the state space. Thus the system does not exhibit any form of persistence.
Willi Jäger, Hal Smith, Betty Tang
Boundedness of Solutions in Neutral Delay Predator-Prey and Competition Systems
Abstract
In this paper, we establish conditions under which solutions of
$$ \left\{ \begin{gathered} \dot{x}\left( t \right) = rx\left( t \right)\left[ {1 - \int_{0}^{{{{\tau }_{1}}}} {x\left( {t - s} \right)d\mu \left( s \right) - \rho \dot{x}} \left( {t - {{\tau }_{2}}} \right) - y\left( {t - {{\tau }_{3}}} \right)g\left( {x\left( t \right)} \right)} \right], \hfill \\ y\left( t \right) = y\left( t \right)\left[ {a + bx\left( {t - {{\tau }_{4}}} \right)g\left( {x\left( {t - {{\tau }_{4}}} \right)} \right) - cy\left( {t - {{\tau }_{5}}} \right)} \right], \hfill \\ \end{gathered} \right. $$
(2.1)
will be bounded. This partially answers the open questions proposed by this author in his recent works on neutral predator-prey and competition systems.
Y. Kuang
Some Examples of Nonstationary Populations of Constant Size
Abstract
Examples of nonstationary populations of constant size are explicitly constructed. Necessary and sufficient conditions for a constant birth function are established under mild restrictions on the death rate and the initial distribution and examples are provided.
Fabio A. Milner, Tanya Kostova
Coexistence in Competition-Diffusion Systems
Abstract
It is a central problem in population ecology to understand the mechanism of spatial patterning of ecological communities. In this paper, we will be concerned with regionally segregation of competing species in a homogeneous environment from a theoretical aspect. Suppose the situation where n species are competing with each other and moving by diffusion. Let u i (t,x) be the population density of the i-th species at time t and position x for i = 1,2,..., n. Then the dynamics of u i (t,x) are described by
$$ \frac{\partial }{{\partial t}}{{u}_{i}} = {{d}_{i}}\Delta {{u}_{i}} + \left( {{{r}_{i}} - \sum\limits_{{j = 1}}^{n} {{{a}_{{ij}}}{{u}_{j}}} } \right){{u}_{i}}, \left( {t,x} \right) \in \left( {0,\infty } \right) \times \Omega \left( {i = 1,2 \ldots ,n} \right), $$
(1.1)
where Δ is the Laplace operator in R N ,rj is the intrinsic growth rate, a ii and a ii (i ≠ j) are respectively the coefficients of intra- and inter-specific competition and d i is the diffusion coefficient (i = 1,2...,n). We assume that a habitat Ω is bounded in R N . First define a basically homogeneous environment for competing species by the following assumptions
(A)
r i , a ij and di} (i = 1,2, ...,n) are positive constants;
 
(B)
The boundary condition at the boundary ∂Ω is of zero flux, i.e.
 
Masayasu Mimura
Population Interactions with Growth Rates Dependent on Weighted Densities
Abstract
Differential and difference equation models of interacting populations are presented and analyzed. The per capita growth rates are functions of linear combinations of individual population densities. The asymptotic behavior of these systems is discussed, including the occurrence of strange attractors. For the two dimensional differential equation, a general formula for the stability coefficient of a Hopf bifurcation is derived.
James F. Selgrade, Gene Namkoong
Global Stability in a Population Model with Dispersal and Stage Structure
Abstract
Recently there has been much interest in modeling population growth where the population disperses among patches in a patchy environment [3], [4], [5], [6], [7], [8], [9], [10], [11], [17], [22], [23]. A second group of papers have dealt with models of populations where the life history of the particular species involves two or more stages [1], [2], [13], [18], [24], [25]. In [12], we have combined these two concepts and analyzed a model incorporating stage structure and dispersal.
J.-H. Wu, H. I. Freedman

Erratum

Erratum
Stavros Busenberg, Mario Martelli
Backmatter
Metadata
Title
Differential Equations Models in Biology, Epidemiology and Ecology
Editors
Stavros Busenberg
Mario Martelli
Copyright Year
1991
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-45692-3
Print ISBN
978-3-540-54283-4
DOI
https://doi.org/10.1007/978-3-642-45692-3