We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equilibriumof the saddle-center type can have a sequence of small parameter values for which aone-round homoclinic orbit to this equilibrium exists. This contrasts with the well-knownfindings by Amick and McLeod and others that solutions of such type do not exist inanalytic Hamiltonian systems, and that the separatrices are split by the exponentiallysmall quantity. We also discuss existence of homoclinic trajectories to small periodicorbits of the Lyapunov family as well as symmetric periodic orbits near the homoclinicconnection. Our further result, illustrated by simulations, concerns the complicatedstructure of orbits related to passage through a non-smooth bifurcation of a periodicorbit.

We consider mixed and hybrid variational formulations to the linearizedelasticity system in domains with cracks. Inequality type conditions areprescribed at the crack faces which results in unilateral contact problems. Thevariational formulations are extended to the whole domain including the crackswhich yields, for each problem, a smooth domain formulation. Mixedfinite element methods such as PEERS or BDM methods are designed to avoidlocking for nearly incompressible materials in plane elasticity. We study andimplement discretizations based on such mixed finite element methods for thesmooth domain formulations to the unilateral crack problems. We obtainconvergence rates and optimal error estimates and we present some numericalexperiments in agreement with the theoretical results.

Studying the evolutive and adaptative machanisms of prokayotes is a complicated task.As these machanisms cannot be easily studied "in vivo", it is necessary to consider other methods.We have therefore developed the RAevol model, a model designed to study the evolution of bacteria and their adaptationto the environment.Our model simulates the evolution of a population of artificial bacteria in a changing environment, providing us with an insight into the strategies that digital organisms develop to adapt to new conditions.In this paper we describe the principle and architecture of the model, focusing on the mechanisms of the regulatory networks of artificial organisms.Experiments were conducted on populations of artificial bacteria under conditions of stress.We study the ways in which organisms adapt to environmental changes and examine the strategies they adopt.An analysis of these adaptation strategies is presented and a brief overview was proposed concerning the patterns and topological characteristics of the evolved regulatory networks.

An overview of recently obtained authors’ results on traveling wave solutions of someclasses of PDEs is presented. The main aim is to describe all possible travelling wavesolutions of the equations. The analysis was conducted using the methods of qualitativeand bifurcation analysis in order to study the phase-parameter space of the correspondingwave systems of ODEs. In the first part we analyze the wave dynamic modes of populationsdescribed by the “growth - taxis - diffusion" polynomial models. It is shown that“suitable" nonlinear taxis can affect the wave front sets and generate non-monotone waves,such as trains and pulses, which represent the exact solutions of the model system.Parametric critical points whose neighborhood displays the full spectrum of possible modelwave regimes are identified; the wave mode systematization is given in the form ofbifurcation diagrams. In the second part we study a modified version of theFitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assumethat this propagation is (at least, partially) caused by the cross-diffusion connectionbetween the potential and recovery variables. We show that the cross-diffusion version ofthe model, besides giving rise to the typical fast travelling wave solution exhibited inthe original “diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slowtraveling wave solution. We analyze all possible traveling wave solutions of the model andshow that there exists a threshold of the cross-diffusion coefficient (for a given speedof propagation), which bounds the area where “normal" impulse propagation is possible. Inthe third part we describe all possible wave solutions for a class of PDEs withcross-diffusion, which fall in a general class of the classical Keller-Segel modelsdescribing chemotaxis. Conditions for existence of front-impulse, impulse-front, andfront-front traveling wave solutions are formulated. In particular, we show that anon-isolated singular point in the ODE wave system implies existence of free-boundaryfronts.

In this short note, we apply the technique developed in [Math. Model. Nat. Phenom., 5 (2010), No. 4, 122-149] to study the long-time evolution for Schrödinger equation with slowly decayingpotential.

In this paper, we shall obtain the symmetries of the mathematical model describingspontaneous relaxation of eastward jets into a meandering state and use these symmetriesfor constructing the conservation laws. The basic eastward jet is a spectral parameter ofthe model, which is in geostrophic equilibrium with the basic density structure and whichguarantees the existence of nontrivial conservation laws.

Since the release of the Bio 2010 report in 2003, significant emphasis has been placed on initiating changes in the way undergraduate biology and mathematics courses are taught and on creating new educational materials to facilitate those changes. Quantitative approaches, including mathematical models, are now considered critical for the education of the next generation of biologists. In response, mathematics departments across the country have initiated changes to their introductory calculus sequence, adding models, projects, and examples from the life sciences, or offering specialized calculus courses for biology majors. However, calculus-based models and techniques from those courses have been slow to propagate into the traditional general biology courses. And although modern biology has generated exciting opportunities for applications of a broad spectrum of mathematical theories, the impact on the undergraduate mathematics courses outside of the calculus/ordinary differential equations sequence has been minimal at best. Thus, the limited interdisciplinary cross-over between the undergraduate mathematics and biology curricula has remained stubbornly stagnant despite ongoing calls for integrated approaches.

We think that this phenomenon is due primarily to a lack of appropriate non-calculus-based interdisciplinary educational materials rather than inaccessibility of the essential underlying mathematical and biology concepts. Here we present a module that uses Boolean network models of the lac operon regulatory mechanism as an introduction to the conceptual importance of mathematical models and their analysis. No mathematics background beyond high school algebra is necessary to construct the model, which makes the approach particularly appropriate for introductory biology and mathematics courses. Initially the module focuses on modeling via Boolean logic, Boolean algebra, discrete dynamical systems, and directed graphs. The analysis of the model, however, leads to interesting advanced mathematical questions involving polynomial ideals and algebraic varieties that are beyond the mathematical proficiency of most biology students but are of interest in advanced level abstract algebra courses. These questions can also be used to map a path toward further research-level problems. All computations are carried out by computational algebra systems and the advanced mathematical theory implemented by the software can be covered in detail in abstract algebra courses with mathematics students. Biology students and students in lower level mathematic courses, on the other hand, can view the implementation as a “black box” and focus on the interpretation and the implications of the output.

The module is available from the authors for classroom testing and adoption.

We give a necessary and sufficient condition for pseudo-differential operators with SGsymbols to be compact from L2(ℝn)into L2(ℝn).

We study molecular motor-induced microtubule self-organization in dilute and semi-dilutefilament solutions. In the dilute case, we use a probabilistic model of microtubuleinteraction via molecular motors to investigate microtubule bundle dynamics. Microtubulesare modeled as polar rods interacting through fully inelastic, binary collisions. Ourmodel indicates that initially disordered systems of interacting rods exhibit anorientational instability resulting in spontaneous ordering. We study the existence anddynamic interaction of microtubule bundles analytically and numerically. Our resultsreveal a long term attraction and coalescing of bundles indicating a clear coarsening inthe system; microtubule bundles concentrate into fewer orientations on a slow logarithmictime scale. In semi-dilute filament solutions, multiple motors can bind a filament toseveral others and, for a critical motor density, induce a transition to an ordered phasewith a nonzero mean orientation. Motors attach to a pair of filaments and walk along thepair bringing them into closer alignment. We develop a spatially homogenous, mean-fieldtheory that explicitly accounts for a force-dependent detachment rate of motors, which inturn affects the mean and the fluctuations of the net force acting on a filament. We showthat the transition to the oriented state can be both continuous and discontinuous whenthe force-dependent detachment of motors is important.

We study a reaction diffusion system that models the dynamics of two species that displayinter-species competition and intra-species cooperation. We find that there are betweenthree and six different equilibrium states and a variety of possible travelling wavesolutions that can connect them. After examining the travelling waves that are generatedin three different ecologically-relevant initial value problems, we construct asymptoticsolutions in the limit λ ≪ 1 (fast diffusion, slow reaction for thesecond species relative to the first).

A global feedback control of a system that exhibits a subcritical monotonic instabilityat a non-zero wavenumber (short-wave, or Turing instability) in the presence of a zeromode is investigated using a Ginzburg-Landau equation coupled to an equation for the zeromode. The method based on a variational principle is applied for the derivation of alow-dimensional evolution model. In the framework of this model the investigation of thesystem’s dynamics and the linear and nonlinear stability analysis are carried out. Theobtained results are compared with the results of direct numerical simulations of theoriginal problem.

The G-function formalism has beenwidely used in the context of evolutionary games for identifyingevolutionarily stable strategies (ESS). This formalism wasdeveloped for and applied to point processes. Here, we examine the G-functionformalism in the settings of spatial evolutionarygames and strategy dynamics, based on reaction-diffusion models. We startbyextending the point process maximum principle to reaction-diffusion modelswith homogeneous, locally stable surfaces. We then develop the strategy dynamics forsuch surfaces. When the surfaces are locally stable, but nothomogenous, the standard definitions of ESS and the maximumprinciple fall apart. Yet, we show by examples that strategy dynamics leads toconvergent stable inhomogeneous strategies that are possibly ESS, in the sensethat for many scenarios which we simulated,invaders could not coexist with the exisiting strategies.

Linear oscillators are used for modeling a diverse array of natural systems, for instance acoustics, materials science, and chemical spectroscopy. In this paper I describe simple models of structural interactions in biological molecules, known as elastic network models, as a useful topic for undergraduate biology instruction in mathematical modeling. These models use coupled linear oscillators to model the fluctuations of molecular structures around the equilibrium state. I present many learning activities associated with building and understanding these models, ranging from analytical to computational. I provide a number of web resources where students can obtain structural data, perform calculations, and suggest research directions for independent projects.