Competitive exclusion and coexistence for a quasilinear size-structured population model

We present a quasilinear size-structured model which describes the dynamics of a population with n competing ecotypes. We assume that the vital rates of each subpopulation depend on the total population due to competition. We provide conditions on the individual rates which guarantee competitive exclusion in the case of closed reproduction (offspring always belongs to the same ecotype as the parent). In particular, our results suggest that the ratio of the reproduction and mortality rates is a good measure to determine the winning ecotype. Meanwhile, we show that in the case of open reproduction all ecotypes coexist.     

Competitive exclusion and coexistence of pathogens in a homosexually-transmitted disease model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373-390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable.     

An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.Research Supported in part by NSERC grant A-8965 and the University of Victoria Committee on Faculty Research & Travel     

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

An susceptible-infective-removed epidemic model incorporating media coverage with time delay is proposed. The stability of the disease-free equilibrium and endemic equilibrium is studied. And then, the conditions which guarantee the existence of local Hopf bifurcation are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity. However, the time delay affects the stability of the endemic equilibrium and produces limit cycle oscillations while the basic reproduction number is greater than unity. Finally, some examples for numerical simulations are included to support the theoretical prediction.     

Global stability of an SIR epidemic model with time delays

An SIR disease transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a disease free equilibrium point is globally stable if no endemic equilibrium point exists. Further the endemic point (if it exists) is globally stable with respect to the whole state space except the neighborhood of the disease free state.Research partly supported by the Ministry of Education, Science and Culture, Japan, Grant 05640256     

Competitive exclusion,coexistence and community structure

Studies of coexistence are based ultimately on the assumption that competitive exclusion is a general and accredited phenomenon in nature. However, the ecological and evolutionary impact of interspecific competition is of questionable significance. Review of three reputed examples of competitive exclusion in the field (Aphytis wasps, red and grey squirrels, and triclads) demonstrates that the widely-accepted competition-based interpretations are unlikely, that alternative explanations are overlooked, and that all other reported cases need critical reinvestigation. Although interspecific competition does undoubtedly occur, the evidence suggests it is usually too weak and intermittent a force to achieve competitive exclusion or any other ecologically-significant result, except perhaps rarely and on a minor scale. Coexistence and community structure are therefore not ecological conditions to be especially accounted for, especially since their inherently comparative methods generate information that is largely superficial. Alternative questions that relate directly to species, the units of diversity, are discussed and they emphasize the primary habitat requirements of species, which are fixed during speciation. Neither the comparative approach of coexistence studies, nor the investigation of pattern is likely to be profitable, since the acquisition of species-specific characteristics during speciation requires historical research. At least one alternative theory of ecology, based on the close adaptation of organisms to the environment, is available.     

Competitive exclusion and limiting similarity: a unified theory

Robustness of coexistence against changes of parameters is investigated in a model-independent manner by analyzing the feedback loop of population regulation. We define coexistence as a fixed point of the community dynamics with no population having zero size. It is demonstrated that the parameter range allowing coexistence shrinks and disappears when the Jacobian of the dynamics decreases to zero. A general notion of regulating factors/variables is introduced. For each population, its impact and sensitivity niches are defined as the differential impact on, and the differential sensitivity towards, the regulating variables, respectively. Either the similarity of the impact niches or the similarity of the sensitivity niches results in a small Jacobian and in a reduced likelihood of coexistence. For the case of a resource continuum, this result reduces to the usual "limited niche overlap" picture for both kinds of niche. As an extension of these ideas to the coexistence of infinitely many species, we demonstrate that Roughgarden's example for coexistence of a continuum of populations is structurally unstable.     

On the time to extinction for a two-type version of Bartlett's epidemic model

We are interested in how the addition of type heterogeneities affects the long time behaviour of models for endemic diseases. We do this by analysing a two-type version of a model introduced by Bartlett under the restriction of proportionate mixing. This model is used to describe diseases for which individuals switch states according to susceptible-->infectious-->recovered and immune, where the immunity is life-long. We describe an approximation of the distribution of the time to extinction given that the process is started in the quasi-stationary distribution, and we analyse how the variance and the coefficient of variation of the number of infectious individuals depends on the degree of heterogeneity between the two types of individuals. These are then used to derive an approximation of the time to extinction. From this approximation we conclude that if we increase the difference in infectivity between the two types the expected time to extinction decreases, and if we instead increase the difference in susceptibility the effect on the expected time to extinction depends on which part of the parameter space we are in, and we can also obtain non-monotonic behaviour. These results are supported by simulations.     

Founder control and coexistence in a simple model of asymmetric competition for light

Size asymmetry in plant light acquisition complicates predictions of competitive outcomes in light-limited communities. We present a mathematically tractable model of asymmetric competition for light and discuss its implications for predicting outcomes of competition during establishment in two-, three-, and many-species communities. In contrast to the resource-reduction model of symmetric competition for a single resource, the model we present predicts that outcomes of asymmetric competition for light will sometimes depend on the timing of establishment and the consequent hierarchy among species in canopy position. Competitive outcomes in the model depend on the minimum light requirements (L(c)) and self-shading of species lower in the canopy compared to the light available (L(out)(*)) beneath species higher in the canopy. Succession progresses towards species with decreasing values for L(c), but arrested successions occur when initial dominants have relatively high values for L(c) but low values for L(out)(*), leading to founder control. A theoretically limitless number of species may coexist in competition for light when dominance is founder controlled. These model predictions have implications for an array of applied ecological questions, including methods to control invasive species in light-limited restored ecosystems.     

Plant competition and exclusion with optimizing individuals

Most models of plant competition represent competition as taking place between species when realistically competition takes place between individuals. We model individual plants as optimally choosing biomass in order to maximize net energy that is directed into reproduction. Competition is for access to light and a plant that grows more biomass adds to the leaf area index, creating negative feedback in the form of more self shading and shading of its neighbors. In each period and for given species densities, simultaneous maximization by all plants yields an equilibrium characterized by optimum biomasses. Between periods the net energies plants obtain are used to update the densities, and if densities change the equilibrium changes in the subsequent period. A steady state is attained when all plants have net energies that just allow for replacement. Four main predictions of the resource-ratio theory of competition are obtained, providing behavioral underpinnings for species level models. However, if individual plant parameters are not identical across species, then the predictions do not follow. The optimization framework yields many other predictions, including how specific leaf areas and resource stress impact biomass and leaf area indices.     

A model for dengue disease with variable human population

 A model for the transmission of dengue fever with variable human population size is analyzed. We find three threshold parameters which govern the existence of the endemic proportion equilibrium, the increase of the human population size, and the behaviour of the total number of human infectives. We prove the global asymptotic stability of the equilibrium points using the theory of competitive systems, compound matrices, and the center manifold theorem. Received: 3 November 1997 / Revised version: 3 July 1998     

Dynamics of an SIQS epidemic model with transport-related infection and exit-entry screenings

Population dispersal, as a common phenomenon in human society, may cause the spreading of many diseases such as influenza, SARS, etc. which are easily transmitted from one region to other regions. Exit and entry screenings at the border are considered as effective ways for controlling the spread of disease. In this paper, the dynamics of an SIQS model are analyzed and the combined effects of transport-related infection enhancing and exit-entry screenings suppressing on disease spread are discussed. The basic reproduction number is computed and proved to be a threshold for disease control. If it is not greater than the unity, the disease free equilibrium is globally asymptotically stable. And there exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is greater than unity. It is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists. Exit screening and entry screening are shown to be helpful for disease eradication since they can always have the possibility to eradicate the disease endemic led by transport-related infection and furthermore have the possibility to eradicate disease even when the isolated cites are disease endemic.     

Money-in-the-bank: a model for coral reef fish coexistence

Synopsis An alternative to the reef fish lottery model is proposed for explaining instances of coexistence of reef fishes without apparent spatial resource partitioning. This model is termed money-in-the-bank because of a financial analogy used to explain it. It stresses the importance of habitats that can support only one of two or more closely related species that coexist elsewhere. Populations living in such monospecific habitats could, according to the model, produce enough larvae to repopulate these habitats plus an excess that may settle in the multispecific habitats. Possible examples among cardinalfishes are given.This paper forms part of the proceedings of a mini-symposium convened at Cornell University, Ithaca, N.Y., 18–19 May 1976, entitled Patterns of Community Structure in Fishes (G. S. Helfman, ed.).     

The identification of the periodic behaviour in an epidemic model

In this paper we study a method for the identification of the unknown parameter of the periodic function and also the first component of the state vector, in a mathematical model which describes the evolution of some diseases with an oro-fecal transmission.To solve the identification problem we use a numerical method to integrate the differential equations system, which reproduces the stability properties of the above mentioned continuous system.The numerical methods which we propose can be applied also to a spatial semi discretization of the reaction-diffusion model which is a diffusive generalization of the system that we consider in this paper.Finally, through an analysis on both the continuous and the discrete system we also obtain a necessary condition on the experimental data in order that a periodic trajectory of the system exists.Work supported by: Progetto Finalizzato Controllo Malattie da Infezione-CNR and by Ministero Pubblica Istruzione     

Evolve II: A computer model of an evolving ecosystem

The Evolve II program is a model of an ecosystem in which organisms are allowed to evolve. Organisms are subject to a changeable environment and competition from other organisms for a limited food supply. The gene structure may change through mutation. A feature of Evolve II is that the magnitude of phenotypic change resulting from mutation is itself a property of the gene. The system was studied under a number of environmental variation schemes. We report three significant findings. Two species (lineages with distinctly different survival strategies) evolved and coexisted in the same environmental conditions. Organisms developed a resistance to phenotypic change in response to mutation in slowly varying environments. However, traits which favor survival of the individual at the expense of reproduction could in some cases undergo phenotypic change in response to mutation despite the fact that this did not favor the survival of the offspring. This demonstrates that gene structures can evolve which are advantageous from the standpoint of the lineage, but not advantageous from the standpoint of individual offspring.     

Global dynamics of an SEIR epidemic model with saturating contact rate

Heesterbeek and Metz [J. Math. Biol. 31 (1993) 529] derived an expression for the saturating contact rate of individual contacts in an epidemiological model. In this paper, the SEIR model with this saturating contact rate is studied. The basic reproduction number R0 is proved to be a sharp threshold which completely determines the global dynamics and the outcome of the disease. If R0 < or =1, the disease-free equilibrium is globally stable and the disease always dies out. If R0 > 1, there exists a unique endemic equilibrium which is globally stable and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the saturating contact rate to the basic reproduction number and the level of the endemic equilibrium is also analyzed.     

Global dynamics of an SIRS epidemic model with saturation incidence

In this paper, the dynamical behavior of an SIRS epidemic model with birth pulse, pulse vaccination, and saturation incidence is studied. By using a discrete map, the existence and stability of the infection-free periodic solution and the endemic periodic solution are investigated. The conditions required for the existence of supercritical bifurcation are derived. A threshold for a disease to be extinct or endemic is established. The Poincaré map and center manifold theorem are used to discuss flip bifurcation of the endemic periodic solution. Moreover, numerical simulations for bifurcation diagrams, phase portraits and periodic solutions, which are illustrated with an example, are in good agreement with the theoretical analysis.     

An SEIS epidemic model with transport-related infection

In this paper, an SEIS epidemic model is proposed to study the effect of transport-related infection on the spread and control of infectious disease. New result implies that traveling of the exposed (means exposed but not yet infectious) individuals can bring disease from one region to other regions even if the infectious individuals are inhibited from traveling among regions. It is shown that transportation among regions will change the disease dynamics and break infection out even if infectious diseases will go to extinction in each isolated region without transport-related infection. In addition, our analysis shows that transport-related infection intensifies the disease spread if infectious diseases break out to cause an endemic situation in each region, in the sense of that both the absolute and relative size of patients increase. This suggests that it is very essential to strengthen restrictions of passengers once we know infectious diseases appeared.