Trinity University

Food web dynamics have long-since been a topic of interest in the ecological world. Various food web configurations can be used to depict the interactions between species of different trophic levels. However, it is important to note that interactions do exist between species of the same trophic level, whether it is a predatorprey relationship or a competitive relationship. Competition and predation have repeatedly been recognized as important factors in community ecology, but it was not until recently that they were considered together. Ecologists began to acknowledge an interaction between the two where potentially competing species are also involved in a predator-prey relationship. Thus, intraguild predation (IGP) has commonly been defined as this mixture of competition and predation. Here, we look at the intraguild relationship between three species: gypsy moth, white-footed mouse, and oak. Gypsy moths are a primary defoliator of oak trees, while whitefooted mice are significant predators of the gypsy moth. Acorns, produced by oak trees, are also a primary resource for the white-footed mouse. Populations typically remain stable, but gypsy moths continue to experience intermittent outbreaks. Previous studies have recognized acorn abundance as the main cause for these outbreaks. We develop a semi-discrete model based on theoretical analyses and specific assumptions in order to isolate principal parameters leading to gypsy moth outbreaks.

Abstract-Biological tissue can be treated as porous media as it is composed of dispersed cell separated by connective voids which allow flow of nutrients, minerals etc. to reach all cell within tissue. In the present study, a mathematical model has been developed to study the phase change heat transfer phenomena during freezing and thawing process in biological tissues considered as porous media. Effective heat capacity formulation is used for phase change problem. Numerical simulation is used to study the effect of porosity, on the motion of freezing and thawing front and transient temperature distribution in biological tissue. It is observed that porosity has significant effect on transient temperature profile and phase change interfaces, further decrease in freezing rate has been found with increased value of porosity.

We study the influence of a donor-controlled resource subsidy on predator-prey interactions. The prey increases logistically, the subsidy appears arithmetically, and the predator experiences satiation. In one model, the prey and subsidy are found together, and in a second they are spatially separated. Criteria for feasibility and stability of the different equilibrium states are discussed. Implications for a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy) are considered.

The Iron Hypothesis, formulated by oceanographer John Martin, and first tested in 1993, suggested that fertilizing the oceans with iron would draw down atmospheric CO2.  Unfortunately, the overwhelming majority of the scientific evidence available to date indicates that fertilization results in a short term increase in phytoplankton biomass that is then consumed and respired by heterotrophs, returning the CO2 immediately to the atmosphere. In this paper we use mathematical tools to investigate the validity of the Iron Hypothesis. The model which we study recognizes the role played by the phyotoplankton aggregates which are viewed as individual building blocks labeled by their size. The evolution of the system is captured by nonlinear integro-differential equation containing terms responsible for the coalescence, fragmentation, growth and removal of the aggregates. Based on the outcomes of the analysis, we reformulate the Iron Hypothesis and demonstrate that the level of atmospheric CO2 could be dramatically reduced. In particular, the prescribed iron fertilization method happens to accelerate fish production.  

We consider and analyze a mathematical model for the marine bacteriophage infection with latency period (L). In recent time many researchers work on marine virus and they show that the marine bacteriophage infection induces bacterial lysis by which viruses release into the marine environment on average rate. In addition with this we assume that within the constant latency period L, the released virus newly infect susceptible bacteria at the rate ψ for which the growth of virus is at the rate The dynamical behavior of the modified system is studied by showing bounds of solution, stability analysis, hopf bifurcation and persistence criterion. A threshold b* exists and the disease free equilibrium Ef is locally and globally stable for 1 < b < b* and for b > b* the positive equilibrium E+ bifurcates from Ef . A concluding discussion using numerical simulation is then presented.

Pathogens may use different routes of transmission to maximize their spread among host populations. Horizontal (i.e., through host contacts) and vertical (i.e., from mother to offspring) transmission modes are generally associated with different parasite strategies. Highly virulent pathogens are more efficiently transmitted horizontally because they usually produce larger parasite loads. On the other hand, because the efficiency of vertical transmission is dependent on host fitness, less virulent pathogens causing less harm to their hosts seem be associated with this transmission route. Among directly-transmitted diseases, more virulent pathogens also tend to be favored in large host populations in which impairing host fitness does not compromise disease transmission. In vector-borne diseases, pathogen transmission rate not only depends on the total population size but on the relative proportion of vectors and hosts. How this could affect the evolution of different transmission modes has been largely overlooked. The parasite Trypanosoma cruzi which is transmitted by Triatomine bugs to several Vertebrate hosts is responsible of Chagas disease in Latin America. T. cruzi is also widespread in the Southeastern U.S. where it infects sylvatic hosts such as raccoons, opossums or rodents. Besides classical transmission involving vectors feeding on hosts, other alternative transmission routes such as vertical and oral transmission (via the consumption of vectors by hosts) have been reported in the sylvatic T. cruzi cycles. The two major T. cruzi strains occurring in the U.S. show different efficiencies at vertical and horizontal transmission, which suggests that a trade-off occurs between these two transmission modes. We were able to show mathematically with an epidemiological two-strain model that the outcome of the competition between the two parasites is affected by the ratio of the number of vectors per host. Our model does not involve parasite virulence (typically, T. cruzi does not harm sylvatic hosts) and the result holds providing that oral transmission also occurs and that oral and horizontal transmission rates saturate for different vector / host ratios. This result opens new perspectives for the understanding of the competition and specialization of the T. cruzi strains occurring in the U.S. but also provides new insights into the evolution of transmission modes in vector-borne diseases.

Cell migration plays a vital role throughout the life span of organisms. In many cases, this movement is governed by the presense of certain chemicals which attract or repell the cells. We use a previously developed cellular automata model that describes the movement of cells which are attracted to a chemicalt in the environment. We assume that the concentration of the chemical varies in space and the cells move toward greater chemical concentrations with greater probability.

In addition, we allow cell to remain in their current place with certain probability depending on the concentration of the chemical in the current site. Next, we upscale the dicrete mathematical model to a continuous partial differential equation model using transition probabilities dependent on space and time. Other works have done the upscaling of discrete models were cell are constantly moving and the direction of the movemnt is governed by a constant concentration of chemoattractants, while in our CA, we allow cells to stay in their current site and the chemoatractant varies in space and time.

Species abundance distributions may reflect the dynamic processes that influence fitness in a community, but the question of how they do so remains open in ecology. Recent studies focus on the relative importance of two types of mechanisms: niche and neutral dynamics. Neutral dynamics are based on demographic stochasticity and immigration and niche dynamics are generated by trait differences that affect the fitnesses of competing species. One recent study showed that abundance patterns produced under niche and neutral dynamics are very similar, especially when diversity is relatively high compared with the number of niches. In this study, species in separate niches are essentially non-interacting. To investigate these issues, we compare the patterns resulting from a stochastic Lotka-Volterra competition model in which all species are interacting with one another. This model generates neutral communities when competition does not depend on traits, and communities with niches when competition declines with trait distance. We find more substantial differences in species abundance distributions of these two types of communities than have been previously shown, suggesting that mechanisms that produce niches also influence abundance patterns. In particular we show that interactions between niches increases mean species abundance, and that fitness differences within a niche largely influence the differences we see species abundance distributions between the two types of communities.

In this paper, a non-iterative numerical integration method is developed on a uniform mesh for a class of singularly perturbed two-point boundary value problems exhibiting internal and twin boundary layers.

This method is non-iterative on a small deviating argument which converts the original second order boundary value problem to the first order differential equation with the deviating argument. By applying numerical integration method on first order differential equation, tridiagonal scheme is obtained and is solved efficiently. This method is noniterative and very easy to implement. Relative errors with L2-norm are presented to illustrate the proposed method.

 

Healthcare associated infections caused by medical-facility-borne antibiotic-resistant bacteria have become a costly problem that compromises medical care in hospitals worldwide. We developed a compartmental model that focuses on the evolution of two bacterial strains (drug-resistant and non-drug resistant) residing within the patient population and healthcare workers in a hospital. Reformulating the model as an optimal control problem, the model predicts that the non-resistant bacteria will decrease and eventually reach a very low level after the control is applied. Moreover, in contrast to the differential equation model, the drug-resistant bacteria will persist, possibly because of mutation, but it will take about five times longer to reach the maximum levels.

Heterogeneity is a fundamental issue in mathematical epidemiology. We expect many factors influencing disease transmission to vary across populations and different spatial scales. Many results exist for the effect of heterogeneity on the spread of disease for SIR type models, where transmission occurs as a result of direct contact with infected individuals. Waterborne disease, such as cholera, may be spread through contact with a contaminated water source as well as through direct person-person transmission. We investigate the effect of heterogeneity in both transmission pathways on the value of the basic reproductive number R0 in multi-patch SIWR models, specifically a system of N patches sharing a common water source.