Showing papers in "Journal of Mathematical Biology in 1989"
TL;DR: A predator-prey model is developed and it is shown that in the case where the uninfected predator cannot survive only on unin infected prey, the parasitization could lead to persistence of the predator provided a certain threshold of transmission is surpassed.
Abstract: A predator-prey model, where both species are subjected to parasitism, is developed and analyzed. For the case where there is coexistence of the predator with the uninfected prey, an epidemic threshold theorem is proved. It is shown that in the case where the uninfected predator cannot survive only on uninfected prey, the parasitization could lead to persistence of the predator provided a certain threshold of transmission is surpassed.
390 citations
TL;DR: Methods are presented here that allow individual phenotypes, population means, and patterns of variance and covariance to be quantified for infinite-dimensional characters.
Abstract: Infinite-dimensional characters are those in which the phenotype of an individual is described by a function, rather than by a finite set of measurements. Examples include growth trajectories, morphological shapes, and norms of reaction. Methods are presented here that allow individual phenotypes, population means, and patterns of variance and covariance to be quantified for infinite-dimensional characters. A quantitative-genetic model is developed, and the recursion equation for the evolution of the population mean phenotype of an infinite-dimensional character is derived. The infinite-dimensional method offers three advantages over conventional finite-dimensional methods when applied to this kind of trait: (1) it describes the trait at all points rather than at a finite number of landmarks, (2) it eliminates errors in predicting the evolutionary response to selection made by conventional methods because they neglect the effects of selection on some parts of the trait, and (3) it estimates parameters of interest more efficiently.
388 citations
TL;DR: For a modified Anderson and May model of host parasite dynamics it is shown that infections of different levels of virulence die out asymptotically except those that optimize the basic reproductive rate of the causative parasite.
Abstract: For a modified Anderson and May model of host parasite dynamics it is shown that infections of different levels of virulence die out asymptotically except those that optimize the basic reproductive rate of the causative parasite. The result holds under the assumption that infection with one strain of parasite precludes additional infections with other strains. Technically, the model includes an environmental carrying capacity for the host. A threshold condition is derived which decides whether or not the parasites persist in the host population.
374 citations
TL;DR: Simulation and analysis of models with cross-immunity indicate that sustained oscillations cannot be maintained by age-specific infection activity level rates when the mortality rate is constant; but are possible if mortalities are age- specific, even if activity levels are independent of age.
Abstract: Infection by one strain of influenza type A provides some protection (cross-immunity) against infection by a related strain. It is important to determine how this influences the observed co-circulation of comparatively minor variants of the H1N1 and H3N2 subtypes. To this end, we formulate discrete and continuous time models with two viral strains, cross-immunity, age structure, and infectious disease dynamics. Simulation and analysis of models with cross-immunity indicate that sustained oscillations cannot be maintained by age-specific infection activity level rates when the mortality rate is constant; but are possible if mortalities are age-specific, even if activity levels are independent of age. Sustained oscillations do not seem possible for a single-strain model, even in the presence of age-specific mortalities; and thus it is suggested that the interplay between cross-immunity and age-specific mortalities may underlie observed oscillations.
328 citations
TL;DR: In this article, the authors examined two models of single-species dynamics which incorporate non-local effects, and the emphasis is on the ability of these models to generate stable patterns.
Abstract: In this work we examine two models of single-species dynamics which incorporate non-local effects. The emphasis is on the ability of these models to generate stable patterns. Global behavior of the bifurcating branches is also investigated.
265 citations
TL;DR: In this paper, necessary and sufficient conditions for stochastically bounded coexistence in a class of models for two species competing in a randomly varying environment were given, where coexistence is implied by mutual invasibility.
Abstract: We give necessary and sufficient conditions for stochastically bounded coexistence in a class of models for two species competing in a randomly varying environment. Coexistence is implied by mutual invasibility, as conjectured by Turelli. In the absence of invasibility, a species converges to extinction with large probability if its initial population is small, and extinction of one species must occur with probability one regardless of the initial population sizes. These results are applied to a general symmetric competition model to find conditions under which environmental fluctuations imply coexistence or competitive exclusion.
150 citations
TL;DR: This study investigates systematically the role played by the reproductive number (the number of secondary infections generated by an infectious individual in a population of susceptibles) on single group populations models of the spread of HIV/AIDS, and concludes that single population models of this type are robust and hence are good building blocks for the construction of multiple group models.
Abstract: In this study, we investigate systematically the role played by the reproductive number (the number of secondary infections generated by an infectious individual in a population of susceptibles) on single group populations models of the spread of HIV/AIDS. Our results for a single group model show that if R ⩽ 1, the disease will die out, and strongly suggest that if R > 1 the disease will persist regardless of initial conditions. Our extensive (but incomplete) mathematical analysis and the numerical simulations of various research groups support the conclusion that the reproductive number R is a global bifurcation parameter. The bifurcation that takes place as R is varied is a transcritical bifurcation; in other words, when R crosses 1 there is a global transfer of stability from the infection-free state to the endemic equilibrium, and vice versa. These results do not depend on the distribution of times spent in the infectious categories (the survivorship functions). Furthermore, by keeping all the key statistics fixed, we can compare two extremes: exponential survivorship versus piecewise constant survivorship (individuals remain infectious for a fixed length of time). By choosing some realistic parameters we can see (at least in these cases) that the reproductive numbers corresponding to these two extreme cases do not differ significantly whenever the two distributions have the same mean. At any rate a formula is provided that allows us to estimate the role played by the survivorship function (and hence the incubation period) in the global dynamics of HIV. These results support the conclusion that single population models of this type are robust and hence are good building blocks for the construction of multiple group models. Our understanding of the dynamics of HIV in the context of mathematical models for multiple groups is critical to our understanding of the dynamics of HIV in a highly heterogeneous population.
147 citations
TL;DR: An epidemiological model with both a time delay in the removed class and a nonlinear incidence rate is analysed to determine the equilibria and their stability for diseases where individuals are first susceptible, then infected, then removed with temporary immunity and then susceptible again when they lose their immunity.
Abstract: An epidemiological model with both a time delay in the removed class and a nonlinear incidence rate is analysed to determine the equilibria and their stability. This model is for diseases where individuals are first susceptible, then infected, then removed with temporary immunity and then susceptible again when they lose their immunity. There are multiple equilibria for some parameter values, and, for certain of these, periodic solutions arise by Hopf bifurcation from the large nontrivial equilibrium state.
102 citations
TL;DR: Conditions that produce population persistence can be distinguished from conditions that cause extinction, and the difference between any pair of solutions eventually converges to zero.
Abstract: With x = population size, the nonautonomous equation x = xf(t,x) provides a very general description of population growth in which any of the many factors that influence the growth rate may vary through time. If there is some fixed length of time (usually long) such that during any interval of this length the population experiences environmental variability representative of the variation that occurs in all time, then definite conclusions about the population's long-term behavior apply. Specifically, conditions that produce population persistence can be distinguished from conditions that cause extinction, and the difference between any pair of solutions eventually converges to zero. These attributes resemble corresponding features of the related autonomous population growth model x = xf(x).
72 citations
TL;DR: This paper considers the distribution of the genealogical tree of a sample of genes in the infinitely-many-site model where the relative age ordering of the mutations is known and a computer implementation of a recursion for the probability of such trees is discussed when the nodes are age-labeled, or not.
Abstract: This paper considers the distribution of the genealogical tree of a sample of genes in the infinitely-many-site model where the relative age ordering of the mutations (nodes in the tree) is known. A computer implementation of a recursion for the probability of such trees is discussed when the nodes are age-labeled, or not.
58 citations
TL;DR: Two sets of sufficient conditions are given for convergence to stationary distributions, for some general models of two species competing in a randomly varying environment and Chesson's “lottery model with vacant space” to illustrate how the assumptions can be checked in specific models.
Abstract: Two sets of sufficient conditions are given for convergence to stationary distributions, for some general models of two species competing in a randomly varying environment. The models are nonlinear stochastic difference equations which define Markov chains. One set of sufficient conditions involves strong continuity and φ-irreducibility of the transition probability for the chain. The second set has a much weaker irreducibility condition, but is only applicable to monotonic models. The results are applied to a stochastic two-species Ricker model, and to Chesson's “lottery model with vacant space”, to illustrate how the assumptions can be checked in specific models.
TL;DR: A cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension is investigated and the existence of spatially hetero-geneous steady states is shown.
Abstract: We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.
TL;DR: The stability of the travelling wave previously conjectured, is shown both for one- and two-dimensional perturbations.
Abstract: Possible constitutive models are examined for the formation of a herd, under the assumption that a herd forms a travelling wave while grazing. Under quite general conditions, it is found that the only possibility for a travelling wave is a balance between food seeking and natural dispersion, such as in chemotaxis. The stability of the travelling wave previously conjectured, is shown both for one- and two-dimensional perturbations.
TL;DR: This well-known phenomenon is analysed for some low dimensional examples of Lotka-Volterra type, with special attention paid to the occurrence of heteroclinic cycles.
Abstract: Ecological communities can lose their permanence if a predator or a competitor is removed: the remaining species no longer coexist. This well-known phenomenon is analysed for some low dimensional examples of Lotka-Volterra type, with special attention paid to the occurrence of heteroclinic cycles.
TL;DR: Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population.
Abstract: Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population. All four combinations of weak or strong selection and tight or loose linkage are treated, though the proof in the case of strong selection and loose linkage is incomplete. Under certain conditions, explicit formulas are obtained for the stationary distributions of the two diffusions with loose linkage.
TL;DR: In this article, it is shown that Hopf bifurcation is supercritical in Lotka-Volterra systems and that the transition to chaotic dynamics occurs via period-doubling cascades from the limit cycle.
Abstract: It is shown that in a suitable class of Lotka-Volterra systems it is possible to characterize the centre-critical case of the Hopf bifurcation of the multipopulation equilibrium. Moreover, for three populations, it is shown that, in the non-critical case, Hopf bifurcation is supercritical. Numerical evidence of transition to chaotic dynamics, via period-doubling cascades, from the limit cycle is reported.
TL;DR: In this paper, the authors considered birth-and-death processes under the influence of disasters that arrive independently of the present population size and derived an integral equation involving a shifted and rescaled argument.
Abstract: Markov branching processes and in particular birth-and-death processes are considered under the influence of disasters that arrive independently of the present population size. For these processes we derive an integral equation involving a shifted and rescaled argument. The main emphasis, however, is on the (random) probability of extinction. Its distribution density satisfies an equation which can be solved numerically at least up to a multiplicative constant. In an example it is also found by simulation.
TL;DR: A model is derived for the description of parasitic diseases on host populations with age structure and existence and uniqueness of solutions are proved.
Abstract: A model is derived for the description of parasitic diseases on host populations with age structure. The parasite population develops according to a linear birth-death-process. The parasites influence mortality and fertility of the hosts and are acquired with a rate depending on the mean parasite load of the host population. The model consists of a system of partial differential equations with initial and boundary conditions. From the boundary condition a renewal equation for the host population is derived. The model is then generalized to describe a multitype process. Existence and uniqueness of solutions are proved. Results concerning persistent solutions are indicated.
TL;DR: The outcome of competition turns out to be surprisingly sensitive to variation of the communication rate, so several coexistence regimes in parameter space are identified and a method for obtaining “operating diagrams” for given pairs of competing microorganisms is described.
Abstract: In this paper we study a mathematical model of competition between two species of microorganisms for a single limiting nutrient in a laboratory device called a gradostat. A gradostat consists of several (we consider only two) chemostats (CSTR's) connected together so that material can flow between the vessels in such a way that a nutrient gradient is established. Our model is a slightly modified version of one considered recently by Jager et al. [3], in that the rate of exchange of material between the two vessels (the communication rate) is allowed to differ from the dilution rate. The outcome of competition turns out to be surprisingly sensitive to variation of the communication rate. We identify several coexistence regimes in parameter space and describe a method for obtaining “operating diagrams” for given pairs of competing microorganisms.
TL;DR: A model of cell cycle kinetics is proposed, which includes unequal division of cells, and a nonlinear dependence of the fraction of cells re-entering proliferation on the total number of cells in the cycle, which is analyzed using the operator semigroup theory combined with classical differential equations approach.
Abstract: A model of cell cycle kinetics is proposed, which includes unequal division of cells, and a nonlinear dependence of the fraction of cells re-entering proliferation on the total number of cells in the cycle. The model is described by a nonlinear functional-integral equation. It is analyzed using the operator semigroup theory combined with classical differential equations approach. A complete description of the asymptotic behavior of the model is provided for a relatively broad class of nonlinearities. The nonnegative solutions either tend to a stable steady state, or to zero. The simplicity of the model makes it an interesting step in the analysis of dynamics of nonlinear structure populations.
TL;DR: A model is discussed for the description of parasitic diseases on host populations with age structure and existence and stability of persistent solutions as well as the distribution of parasites on the host population are discussed.
Abstract: A model is discussed for the description of parasitic diseases on host populations with age structure. The parasite population develops according to a linear multitype birth-death-process. The parasites influence mortality and fertility of the hosts and are acquired with a rate depending on the mean parasite load of the host population. The model consists of a system of partial differential equations with initial and boundary conditions. Existence and stability of persistent solutions as well as the distribution of parasites on the host population are discussed.
TL;DR: A selection-migration model in population genetics, involving two alleles A1 and A2 such that A1 is at an advantage over A2 in certain subregions and at a disadvantage in others, is discussed.
Abstract: We discuss a selection-migration model in population genetics, involving two alleles A
1 and A
2 such that A
1 is at an advantage over A
2 in certain subregions and at a disadvantage in others. It is shown that if A
1 is at an overall disadvantage to A
2 and the rate of gene flow is sufficiently large than A
1 must die out; on the other hand, if the two alleles are in some sense equally advantaged overall, then A
1 and A
2 can coexist no matter how great the rate of gene flow.
TL;DR: In this article, the effects of competition for space in a heterogeneous environment, making use of nonlinear interaction-diffusion equations, were considered, and the main conclusion is that two populations can coexist if the interspecific repulsive forces are weaker than the intraspecific ones.
Abstract: We consider effects of competition for space in a heterogeneous environment, making use of nonlinear interaction-diffusion equations. Competition for space is assumed to mean mutual repulsive interactions that force other individuals to disperse from a crowded region. In other words, we are concerned with density-dependent dispersal forced by population pressures. Spatial heterogeneity is incorporated in the growth rates, and the environment is assumed to have a favorable habitat for two populations surrounded by largely hostile regions. Space-independent migration rates are assumed. We ignore the usual density-dependence in the growth rates to focus our attention on density-dependence in the migration rates. Our main conclusion is that two populations can coexist if the interspecific repulsive forces are weaker than the intraspecific ones. It is also emphasized that density-dependent dispersal in a heterogeneous environment is not always a stabilizing agent, and that either of two populations may become extinct by competition for space. Finally, the resemblance of our results to those from Lotka-Volterra competition equations is suggested.
TL;DR: A classification scheme for those population models which allow variation in development rates is proposed, based on two ways of modifying standard age-structured models, which can provide guidance in developing appropriate insect population models for specific modelling situations.
Abstract: A classification scheme for those population models which allow variation in development rates is proposed, based on two ways of modifying standard age-structured models. The resulting classes of models are termed development index models and sojourn time models. General formulations for the two classes of models are developed from two basic balance equations, and numerous specific models from the literature are shown to fit into the scheme. Concepts from competing risks theory are shown to be important in understanding the interplay between mortality and maturation. Relationships among the classes are investigated both for the most general forms of the models and for the simpler forms often used. The scheme can provide guidance in developing appropriate insect population models for specific modelling situations.
TL;DR: A stochastic model is proposed to study the problem of inherent resistance by cell populations when chemotherapeutic agents are used to control tumor growth and satisfactory demonstration runs indicate that it could represent a useful tool in verifying the results of experimental and clinical chemotherapy courses and planning treatment strategies.
Abstract: A stochastic model is proposed to study the problem of inherent resistance by cell populations when chemotherapeutic agents are used to control tumor growth. Stochastic differential equations are introduced and numerically integrated to simulate expected response to the chemotherapeutic strategies as a function of different parameters. Satisfactory demonstration runs of the model indicate that it could represent a useful tool in verifying the results of experimental and clinical chemotherapy courses and planning treatment strategies. Some types of behaviour are illustrated graphically.
TL;DR: In this article, the authors studied the descent of the trajectories of homogeneous hypercycles, i.e. their behavior as t → -∞, and showed that this backward behavior is not as uniform as the forward behavior.
Abstract: For homogeneous hypercycles with 2, 3 or 4 substances the future behavior of its trajectories is easily understood, in fact any trajectory converges to an equilibrium point as t → +∞. In this paper we study the descent of the trajectories, i.e. their behavior as t → -∞. It turns out that this backward behavior is not as uniform as the forward behavior. In fact, depending on the initial points some α-limit sets are singletons while others consist of certain edges of the state simplex.
TL;DR: This method allows one to calculate the expected number of segregating sites in a random sample of n genes from a population whose evolution is described by a model which is not selectively neutral.
Abstract: A numerical method is developed for solving a nonstandard singular system of second-order differential equations arising from a problem in population genetics concerning the coalescent process for a sample from a population undergoing selection. The nonstandard feature of the system is that there are terms in the equations that approach infinity as one approaches the boundary. The numerical recipe is patterned after the LU decomposition for tridiagonal matrices. Although there is no analytic proof that this method leads to the correct solution, various examples are presented that suggest that the method works. This method allows one to calculate the expected number of segregating sites in a random sample of n genes from a population whose evolution is described by a model which is not selectively neutral.
TL;DR: The existence of compensated networks is proven, the convergence and stability of simulations are investigated, and implications for cognitive systems are discussed.
Abstract: Plasticity of synaptic connections plays an important role in the temporal development of neural networks which are the basis of memory and behavior. The conditions for successful functional performance of these nerve nets have to be either guaranteed genetically or developed during ontogenesis. In the latter case, a general law of this development may be the successive compensation of disturbances. A compensation type algorithm is analyzed here that changes the connectivity of a given network such that deviations from each neuron's equilibrium state are reduced. The existence of compensated networks is proven, the convergence and stability of simulations are investigated, and implications for cognitive systems are discussed.
TL;DR: Computer simulations of some particular cases of this model support the notion that both ecological and immunological factors determine the observed convex patterns of age-prevalence and age-intensity curves of human helminth infections.
Abstract: The general solution of the mathematical model of herd immunity to human helminth infections recently proposed by Anderson and May [3] is obtained. The numerical solution of a more accurate biological model is indistinguishable from the corresponding exact solution of a more tractable mathematical model. Computer simulations of some particular cases of this model support the notion that both ecological and immunological factors determine the observed convex patterns of age-prevalence and age-intensity curves of human helminth infections.
TL;DR: It is mathematically proved that, in/on a long and narrow cylinder-like domain, any solution of reaction-diffusion system asymptotically loses its spatial dependence in the transectional/circumferential direction.
Abstract: It is often reported that an animal with spotty coat markings on its body has a tail with stripe-shaped pattern. In other various biological and chemical phenomena in/on cylinder-like domains, longitudinally periodic band patterns are observed much more often than the other non-uniform patterns. This paper mathematically explains these observations by proving that, in/on a long and narrow cylinder-like domain, any solution of reaction-diffusion system asymptotically loses its spatial dependence in the transectional/circumferential direction.