Hristo V. Kojouharov

Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response

The dynamics of a predator-prey model with a Beddington-DeAngelis functional response and linear intrinsic growth rate of the prey population is fully analyzed. Conditions on local and global stability of the interior equilibrium are established. The equilibria type are determined. All possible global asymptotic behaviors of the system are considered, including the determination of the extinction conditions and existence of periodic orbits. It is shown that mutual interference between predators can alone stabilize predator-prey interactions even when only a linear intrinsic growth rate of the prey population is considered in the mathematical model. Additional biological implications and a set of numerical simulations supporting the analysis are also presented.

Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems

Abstract General two-dimensional autonomous dynamical systems and their standard numerical discretizations are considered. Nonstandard stability-preserving finite-difference schemes based on the explicit and implicit Euler and the second-order Runge–Kutta methods are designed and analyzed. Their elementary stability is established theoretically and is also supported by a numerical example.

Nonstandard finite-difference methods for predator-prey models with general functional response

Predator-prey systems with linear and logistic intrinsic growth rate of the prey are analyzed. The models incorporate the mutual interference between predators into the functional response which stabilizes predator-prey interactions in the system. Positive and elementary stable nonstandard (PESN) finite-difference methods, having the same qualitative features as the corresponding continuous predator-prey models, are formulated and analyzed. The proposed numerical techniques are based on a nonlocal modeling of the growth-rate function and a nonstandard discretization of the time derivative. This discretization approach leads to significant qualitative improvements in the behavior of the numerical solution. In addition, it allows for the use of an essentially implicit method for the cost of an explicit method. Applications of the PESN methods to specific predator-prey systems are also presented.

An unconditionally positivity preserving scheme for advection-diffusion reaction equations

Abstract Parabolic equations with advection, diffusion and reaction terms are used to model many physical and biological systems. In many applications the unknowns represent quantities that cannot be negative such as concentrations of chemical compounds or population sizes. Widely used schemes such as finite differences may produce negative solutions because of truncation errors and may then become unstable. We propose a new scheme that guarantees the positivity of the solutions for arbitrary step sizes. It works for reaction terms that consist of the sum of a positive function and a negative function. We develop it for one advection–diffusion reaction equation in one spatial dimension with constant velocity and diffusion and state how to generalize it. The method is applicable to both advection and diffusion dominated problems. We give some examples from different applications.

Mathematical Analysis of an HIV/AIDS Model: Impact of Educational Programs and Abstinence in Sub-Saharan Africa

We formulate a deterministic HIV/AIDS model to theoretically investigate how counselling and testing coupled with the resulting decrease in sexual activity could affect the HIV epidemic in resource-limited communities. The threshold quantities are determined and stabilities analyzed. Theoretical analysis and numerical simulations support the idea that increase in the number of sexually inactive HIV positive individuals who voluntarily abstain from sex has a positive impact on HIV/AIDS control. Results from this theoretical study suggest that effective counselling and testing have a great potential to partially control the epidemic (especially when HIV positive individuals either willingly withdraw from risky sexual activities or disclose their status beforehand) even in the absence of antiretroviral therapy (ART). Therefore, more needs to be done in resource-limited settings, such as sub-Saharan Africa, as far as the HIV/AIDS epidemic is concerned and a formalized information, education, and communication strategy should be given prominence in educational campaigns.

Bacterial Wall Attachment in a Flow Reactor

A mathematical model of microbial growth for limiting nutrients in a fully three dimensional flow reactor which accounts for the colonization of the reactor wall surface by the microbes is studied analytically. It can be viewed as a model of the large intestine or of the fouling of a commercial bioreactor or pipe flow. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid by the microbes. Only one steady state is stable for any particular set of parameter values. Sharp and explicit conditions are given for the stability of each. The effects of adding an antimicrobial agent to the reactor are examined with and without wall growth.

Analysis and numerical simulation of phytoplankton-nutrient systems with nutrient loss

The dynamics of a mathematical model of a layer of single phytoplankton species growing over a pool of nutrients, proposed by [A.H. Taylor, J.R.W. Harris, J. Aiken, The interaction of physical and biological process in a model of the vertical distribution of phytoplankton under stratification, Mar. Int. Ecohyrd., J.C. Nihoul (Ed.) 42 (1986) 313-330] is analyzed. Both cases of presence and absence of a phytoplankton in the water below the layer of interest are considered. Positive and elementary stable nonstandard (PESN) methods, having the same qualitative features as the corresponding continuous models, are formulated and analyzed. Biological implications and a set of numerical simulations supporting the mathematical and numerical analysis are also presented.

Numerical simulation of dual-species biofilms in porous media

There are bacteria that can form strong biofilms in porous media. The biofilms can be used as biobarriers to restrict the flow of pollutants. If a second species of bacteria that can actually react with the contaminants is added to the biobarrier, the result is a much more effective way of controlling the pollutants. We propose a mathematical model for the formation of these biobarriers and numerically solve the resulting equations for the flow, transport and reactions. Qualitative comparisons with some experimental results are also given.

Dynamically consistent numerical methods for general productive–destructive systems

The dynamic consistency of a class of non-standard finite-difference schemes is analysed for general 2D and 3D productive–destructive systems (PDS). Based on those results a methodology for construction of positive and elementary stable non-standard numerical methods is developed. The numerical techniques are based on a non-local modelling of the right-hand side function and a non-standard treatment of the time derivative. This discretization approach leads to significant qualitative improvements in the behaviour of the numerical solutions. The explicit form of the proposed new schemes makes them a computationally effective tool in simulations of the dynamics of systems of biological, chemical and physical interactions that are naturally modelled by PDS. Applications to several specific biological systems are presented.

Combined nonstandard numerical methods for ODEs with polynomial right-hand sides

In this paper we formulate conditions that guarantee second-order accuracy of the one-step nonstandard finite difference methods for general first order autonomous ordinary differential equations. We also develop a new class of nonstandard finite-difference methods for differential equations of the form dx/dt = polynomial(x), based on earlier results obtained in reference [H.V. Kojouharov, B.M. Chen, Nonstandard Eulerian-Lagrangian methods for multi-dimensional reactive transport problems, Appl. Numer. Math. 49 (2) (2004) 225-243]. Error analysis and numerical examples that demonstrate the performance of the proposed new method are also provided.