Mathematical Model Of Competition Research Articles

A mathematical model of competition between fiber and mucin degraders in the gut provides a possible explanation for mucus thinning

The human gut microbiota relies on complex carbohydrates (glycans) for energy and growth, primarily dietary fiber and host-derived mucins. We introduce a mathematical model of a glycan generalist and a mucin specialist in a two-compartment chemostat model of the human colon. Our objective is to characterize the influence of dietary fiber and mucin supply on the abundance of mucin-degrading species within the gut ecosystem. Current mathematical gut reactor models that include the enzymatic degradation of glycans do not differentiate between glycan types and their degraders. The model we present distinguishes between a generalist that can degrade both dietary fiber and mucin, and a specialist species that can only degrade mucin. The integrity of the colonic mucus barrier is essential for overall human health and well-being, with the mucin specialist Akkermanisa muciniphila being associated with a healthy mucus layer. Competition, particularly between the specialist and generalists like Bacteroides thetaiotaomicron, may lead to mucus layer erosion, especially during periods of dietary fiber deprivation. Our model treats the colon as a gut reactor system, dividing it into two compartments that represent the lumen and the mucus of the gut, resulting in a complex system of ordinary differential equations with a large and uncertain parameter space. To understand the influence of model parameters on long-term behavior, we employ a random forest classifier, a supervised machine learning method. Additionally, a variance-based sensitivity analysis is utilized to determine the sensitivity of steady-state values to changes in model parameter inputs. By constructing this model, we can investigate the underlying mechanisms that control gut microbiota composition and function, free from confounding factors.

Numerical study of the impact of directed migration of non-indigenous species on invasion scenarios

A mathematical model of competition under conditions of biological invasion, written in the form of a system of nonlinear parabolic equations, is considered. The competition of two closely related species — resident and invader — is studied. The dynamics of populations in a heterogeneous area is determined by local interaction and diffusion. For the invader population, interspecific taxis and directed migration caused by heterogeneity of living conditions are taken into account. In computational experiments, sets of migration parameters corresponding to various invasion scenarios are determined. An analysis of the influence of initial distributions on competitive exclusion and coexistence of species is given.

Role of resource allocation and transport in emergence of cross-feeding in microbial consortia

Microbial communities that implement mutual cross-feeding are commonly observed in nature and with synthetic constructs in laboratory experiments. A mathematical model of competition in a chemostat is developed to investigate the role that resource allocation and transport of metabolites play in cooperation. The model contains four cell types that differ by whether they produce two, one, or none of two essential metabolites. Producing cell types may export these resources into the environment, and those that do not produce both metabolites must import the missing resource. The contribution to the emergence of a collaborative consortium of single resource producers from the transport rate of these metabolites and the type of transport used by the cell (active vs. passive) is studied. Multiple instances of bi-stability and tri-stability are observed, and the effect of the initial concentration of a non-cooperative cheater cell type on the final outcome of the competition is examined. When the cost of producing metabolites is introduced into the model, significant changes to the outcome of the competition are observed, including coexistence of multiple cell types.

The influence of social behaviour on competition between virulent pathogen strains

Infectious disease interventions like contact precautions and vaccination have proven effective in disease control and elimination. The priority given to interventions can depend strongly on how virulent the pathogen is, and interventions may also depend partly for their success on social processes that respond adaptively to disease dynamics. However, mathematical models of competition between pathogen strains with differing natural history profiles typically assume that human behaviour is fixed. Here, our objective is to model the influence of social behaviour on the competition between pathogen strains with differing virulence. We couple a compartmental Susceptible-Infectious-Recovered model for a resident pathogen strain and a mutant strain with higher virulence, with a differential equation of a population where individuals learn to adopt protective behaviour from others according to the prevalence of infection of the two strains and the perceived severity of the respective strains in the population. We perform invasion analysis, time series analysis and phase plane analysis to show that perceived severities of pathogen strains and the efficacy of infection control against them can greatly impact the invasion of more virulent strain. We demonstrate that adaptive social behaviour enables invasion of the mutant strain under plausible epidemiological scenarios, even when the mutant strain has a lower basic reproductive number than the resident strain. Surprisingly, in some situations, increasing the perceived severity of the resident strain can facilitate invasion of the more virulent mutant strain. Our results demonstrate that for certain applications, it may be necessary to include adaptive social behaviour in models of the emergence of virulent pathogens, so that the models can better assist public health efforts to control infectious diseases.

Using the Theory of Games to Modelling the Equipment and Prices of Car Parking

In large cities there are two serious problems connected with increasing number of cars. The first problem is the congestion of vehicles’ movement. The second one is too small of car parks, especially in centres of the cities. Authorities of cities and management of municipal streets introduce limitations in vehicles’ movement and reduce the number of car parks to minimalize streets crowd. That acting seems logical, but this is only the one point of view. From the other point of view municipal governments should aim to improve the level of the occupants’ life and assure the financial incomes, which enable to cover indispensable expenses. From this point of view, the municipal car parks are needed and bringing the profits element of municipal infrastructure. Cracow, which is one of the largest cities in Poland (about 760 thousands of occupants, and Cracovian agglomeration is about 1.4 million persons), was chosen as the object of the investigations. The zone of paid parking in Cracow, administered by the company belonging to city, has possessed 28837 parking places in 28.01.2016. In the zone there are assigned car parks or parking places near to the curbs and on pavements. The zone operates from Monday to Friday, from 10.00 to 20.00. Assuming using car parks only in 50% and fare of about 0.7 euro per hour, we receive incomes figuring out about 740000 euro/month. The purpose of the investigations was the identification of technical parameters of car parks being preferred by drivers. The investigations had been executed by method of questionnaires. Next the mathematical model of competition was made. The model was executed basing on the theory of games. Strategies of “Player 1” were prices and technical equipment of car parks and parking places lying in the zone of paid parking, administered by municipal company. Strategies of “Player 2” were prices and technical equipment of car parks belonging to private owners and two commercial centres in the city center. There were assumed that from the city’s point of view it is one rival, independently on the actual ownership of private car parks on whose strategy of acting the city do not have an influence. It is consistent with the basic foundations of the game theory. Both players compete for consumers - drivers using car parks, with such parameters like: price, distance from the destination, car parks’ equipment. The built model allows to indicate the best strategy. Knowing that strategy, one can form prices and equipment of car parks. That model may be used by municipal governments and companies which administer car parks. However, one should remember about limitations, which occur in reality, e.g. law restrictions referring to maximum prices for parking. The increasing pressure of cities’ authorities in Poland for changing those regulations, and examples of such solutions received, e.g. in USA, which concern varying prices for parking according to the demand on different car parks, and different hours, permit to have hopes, that the proposed model will be possible to use practically, soon.

Identification of biological models described by systems of nonlinear differential equations

Abstract Four simple mathematical models of pharmacokinetic, competition between immune and tumor cells, infectious disease and tuberculosis epidemic are considered. An optimization approach for identification those models based on gradient type methods is introduced. Inverse problems are formulated in the form of an operator equation and then reduced to the minimization of the corresponding misfit functionals. The adjoint problems are used for the calculation of gradients. A mathematical model of competition between immune and tumor cells is considered numerically. The results of a numerical experiment are demonstrated.

Efficiency of Small Business Management in the Communications Industry: Simulation of the Competitive Environment

Abstract The level of management in a small business is not always high. Therefore, the need has to develop evidence-based recommendations for the effective management in a highly competitive market. The aim of this study is to develop a model assessing the impact of competitive environment on the effectiveness of strategic management of a small business in the telecommunications industry. We have taken into account in this model specifics of Russian small businesses, including in the telecommunications sector. The basis of our model were known mathematical models of competition in the market – the model of Cournot oligopoly and the model of Stackelberg duopoly. According to the results of simulation, the authors have proposed the use of 4 typical control strategies. Conditions of their application have also been identified. Approbation of the author's model is made on the example of small business communication of Russia.

Competition and allelopathy with resource storage: Two resources

Allelopathy is added to a familiar mathematical model of competition between two species for two essential resources in a chemostat environment. Both species store the resources, and each produces a toxin that induces mortality in the other species. The corresponding model without toxins displays outcomes of competitive exclusion independent of initial conditions, competitive exclusion that depends on initial conditions (bistability), and globally stable coexistence, depending on tradeoffs between competitors in growth requirements and consumption of the resources. Introducing toxins that act only between, and not within species, can destabilize coexistence leading to bistability or other multiple attractors. Invasibility of the missing species into a resident׳s semitrivial equilibrium is related to competitive outcomes. Mutual invasibility is necessary and sufficient for a globally stable coexistence equilibrium, but is not necessary for coexistence at a locally stable equilibrium. Invasibility of one semitrivial equilibrium but not the other is necessary but not sufficient for competitive exclusion independent of initial conditions. Mutual non-invasibility is necessary but not sufficient for bistability. Numerical analysis suggests that when competitors display bistability in the absence of toxin production, increases in the overall magnitude of resource supply cause bistability to arise over a larger range of supply ratios between the two resources. When competitors display coexistence in the absence of toxin production, increases in overall resource supply destabilize coexistence and produce bistability or other configurations of multiple attractors over large ranges of supply ratios. The emergence of multiple attractors at high resource supplies suggests that blooms of harmful algae producing allelopathic toxins could be difficult to predict under such rich conditions.

The role of resource types in the control of chemostats using feedback linearization

In this paper we apply the differential geometric-based method of feedback linearization to mathematical models of competition in a chemostat. We first consider competition for a single resource as an illustrative example of the methodology. We then focus on competition for two resources. Here we show that by using the input nutrient concentrations as control inputs, some models for chemostats can be made equivalent to controllable linear dynamical systems, and suitably chosen control objectives can be met. The role of resource type in this process is our primary concern.

Hopf bifurcation and analysis of equilibrium for a third-order differential equation in a model of competition

In this paper, a mathematical model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor is investigated. The model is in the form of a system of nonlinear differential equations. By using qualitative methods, the conditions for the existence and local stability of the equilibria are obtained. The existence and stability of periodic solutions of the Hopf type are studied. Numerical simulations about the Hopf bifurcation value and Hopf limit cycle are also given.

Competition between three microbial populations for a single limiting resource in continuous culture

Investigation of global analysis of the behaviour of solutions of a mathematical model of competition between three microbial populations for a single limiting nutrient is given. Main results on competition are formulated.

A mathematical model of competition and colonization in a community of marine benthic algae

A mathematical model has been constructed for the algal community on the rocky shores of a Norwegian fjord. We report here on the studies of competition and colonization along a vertical transect from the upper intertidal to the sublittoral habiats. Results on species abundance and distribution (patterns of zonation) and time to reach maturity have been compared to observations both in the fjord area and in other rocky shore areas. Competition coefficients for the algae were inferred from plant morphology and shown to be in agreement with observations of algal abundance and their zone-forming ability. Competition restricts the distribution of the species, especially at the lower elevations, but does not alter their relative position. However, increasing uniform competition prolongs the time in which zone-forming can occur, and it also decreases the overall biomass which an area can sustain. Colonization by a single species may create transient stages in community development of the same order of magnitude as algae longevity, and probably also alters the zonation pattern to some degree. The simulation results indicate that the large-scale algal distribution pattern in the Hardangerfjord area results from global stability of the rocky shore community.

Competition between species: Theoretical models and experimental tests

Experimental determinations of Drosophila population dynamics cannot be explained by the Lotka-Volterra model of interspecific competition. This paper presents other possible mathematical models of competition between species, and gives the results of experiments designed to test the validity of such models. Eight of the ten new models presented contain the Lotka-Volterra model as a special case. The experiments made to test the models are of two kinds. Type 1 experiments are continuous one- or two-species populations, which permit the estimation of the carrying capacity of each species and the numbers of the two species at the point of stable equilibrium. Type 2 experiments measure the change in numbers over a short time interval in populations started with many different initial densities of the two species. Type 2 experiments give information on the dynamics of the two-species system in the phase plane whose coordinates are the number of individuals of each species. The models accounting best for the results are models five and seven (Table II). Each of these two models contains one parameter more than the Lotka-Volterra model. Model five adds a nonlinear term of self-interaction (− β i N 2 i ). Model seven has the form, dN i / dt = r i N/ K θ i i( K θ i i − N θ i i − α ij N j 1− θ i i). The exponential parameter θ removes the restriction of the logistic theory of population growth, that each individual added to the population decrease the rate of growth of the population by a constant amount. With model seven the rate of growth of a population of a single species need not have its maximum at K 2 , that is when the number of individuals is half the carrying capacity of the environment.