We present a mathematical model of a fishery on several sites with a variable price. Themodel takes into account the evolution during the time of the resource, fishes and boatsmovements between the different sites, fishing effort and price that varies with respectto supply and demand. We suppose that boats and fishes movements as well as pricesvariations occur at a fast time scale. We use methods of aggregation of variables in orderto reduce the number of variables and we derive a reduced model governing two globalvariables, respectively the biomass of the resource and the fishing effort of the wholefishery. We look for the existence of equilibria of the aggregated model. We show that theaggregated model can have 1, 2 or 3 non trivial equilibria. We show that a variation ofthe total number of sites can induce a switch from over-exploitation to sustainablefisheries.

We consider two atomic transitions excited by two variable laser fields in a three-level system. We study the soliton-pair propagation out of resonance and under thermal bath effect. We present general analytical implicit expression of the soliton-pair shape. Furthermore, we show that when the coupling to the environment exceeds a critical value, the soliton-pair propagation through three-level atomic system will be prohibited.

The contact between two membranes can be described by a system of variationalinequalities, where the unknowns are the displacements of the membranes and theaction of a membrane on the other one. We first perform the analysis of thissystem. We then propose a discretization, where the displacements areapproximated by standard finite elements and the action by alocal postprocessing. Such a discretization admits an equivalent mixedreformulation. We prove the well-posedness of the discrete problem and establishoptimal a priori error estimates.

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransversal heteroclinic cycles. We show that bifurcations under consideration lead to the birth of Lorenz-like attractors. They can be viewed as attractors in the Poincare map for periodically perturbed classical Lorenz attractors and hence they can allow for the existence of homoclinic tangencies and wild hyperbolic sets.

Skeletal patterning in the vertebrate limb,i.e., the spatiotemporal regulation of cartilage differentiation(chondrogenesis) during embryogenesis and regeneration, is oneof the best studied examples of a multicellular developmental process.Recently [Alber et al., The morphostatic limit for a model ofskeletal pattern formation in the vertebrate limb, Bulletin ofMathematical Biology, 2008, v70, pp. 460-483], a simplified two-equationreaction-diffusion system was developed to describe the interaction of two ofthe key morphogens: the activator and an activator-dependent inhibitor ofprecartilage condensation formation. A discontinuous Galerkin (DG)finite element method was applied to solve this nonlinear system on complexdomains to study the effects of domain geometry on the pattern generated [Zhu etal., Application of Discontinuous Galerkin Methods for reaction-diffusionsystems in developmental biology, Journal of Scientific Computing, 2009, v40,pp. 391-418]. In this paper, we extend these previous results and develop a DGfinite element model in a moving and deforming domain for skeletal patternformation in the vertebrate limb. Simulations reflect the actual dynamics oflimb development and indicate the important role played by the geometryof the undifferentiated apical zone.

Blood clotting system (BCS) modelling is an important issue with a plenty of applications in medicine and biophysics. The BCS main function is to form a localized clot at the site of injury preventing blood loss. Mutual influence of fibrin clot consisting mainly of fibrin polymer gel and blood flow is an important factor for BCS to function properly. The process of fibrin polymer mesh formation has not adequately been described by current mathematical models. That is why it is not possible to define the borders of growing clot and model its interaction with a blood flow. This paper main goal is to propose physically well-founded mathematical model of fibrin polymerization and gelation. The proposed model defines the total length of fibrin polymer fibers in the unit volume, determines a position of the border between gel and liquid and allows to evaluate the permeability of growing gel. Without significant structural changes the proposed model could be modified to include the blood shear rate influence on the fibrin polymerization and gelation.

Evolution of cell populations can be described with dissipative particle dynamics, whereeach cell moves according to the balance of forces acting on it, or with partialdifferential equations, where cell population is considered as a continuous medium. Wecompare these two approaches for some model examples

Flow cytometric analysis using intracellular dyes such as CFSE is a powerful experimentaltool which can be used in conjunction with mathematical modeling to quantify the dynamicbehavior of a population of lymphocytes. In this survey we begin by providing an overviewof the mathematically relevant aspects of the data collection procedure. We then presentan overview of the large body of mathematical models, along with their assumptions anduses, which have been proposed to describe the dynamics of proliferating cell populations.While much of this body of work has been aimed at modeling the generation structure (cellsper generation) of the proliferating population, several recent models have considered themore fundamental task of modeling CFSE histogram data directly. Such models are analyzedand recent results are discussed. Finally, directions for future research aresuggested.

A two dimensional two-delays differential system modeling the dynamics of stem-like cellsand white-blood cells in Chronic Myelogenous Leukemia is considered. All three types ofstem cell division (asymmetric division, symmetric renewal and symmetric differentiation)are present in the model. Stability of equilibria is investigated and emergence ofperiodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventuallyshown. The stability of these limit cycles is studied using the first Lyapunovcoefficient.

Water flow in plant tissues takes place in two different physical domains separated bysemipermeable membranes: cell insides and cell walls. The assembly of all cell insides andcell walls are termed symplast and apoplast,respectively. Water transport is pressure driven in both, where osmosis plays an essentialrole in membrane crossing. In this paper, a microscopic model of water flow and transportof an osmotically active solute in a plant tissue is considered. The model is posed on thescale of a single cell and the tissue is assumed to be composed of periodicallydistributed cells. The flow in the symplast can be regarded as a viscous Stokes flow,while Darcy’s law applies in the porous apoplast. Transmission conditions at the interface(semipermeable membrane) are obtained by balancing the mass fluxes through the interfaceand by describing the protein mediated transport as a surface reaction. Applyinghomogenization techniques, macroscopic equations for water and solute transport in a planttissue are derived. The macroscopic problem is given by a Darcy law with a force termproportional to the difference in concentrations of the osmotically active solute in thesymplast and apoplast; i.e. the flow is also driven by the local concentration differenceand its direction can be different than the one prescribed by the pressure gradient.

Theoretical framework for linear stability of an anomalous sub-diffusive activator-inhibitor system is set. Generalized Turing instability conditions are found to depend on anomaly exponents of various species. In addition to monotonous instability, known from normal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly exponents for both species the type of unstable modes is determined by the ratio of the reactants' diffusion coefficients. When the ratio exceeds its normal critical value, the monotonous modes become stable, whereas oscillatory instability persists until the anomalous critical value is also exceeded. An exact formula for the anomalous critical value is obtained. It is shown that in the short wave limit the growth rate is a power law of the wave number. When the anomaly exponents differ, disturbance modes are governed by power laws of the distinct exponents. If the difference between the diffusion anomaly exponents is small, the splitting of the power law exponents is absent at the leading order and emerges only as a next-order effect. In the short wave limit the onset of instability is governed by the anomaly exponents, whereas the ratio of diffusion coefficients influences the complex growth rates. When the exponent of the inhibitor is greater than that of the activator, the system is always unstable due to the inhibitor enhanced diffusion relatively to the activator. If the exponent of the activator is greater, the system is always stable. Existence of oscillatory unstable modes is also possible for waves of moderate length.

We study controllability for a nonhomogeneous string and ring under an axial stretchingtension that varies with time. We consider the boundary control for a string anddistributed control for a ring. For a string, we are looking for a controlf(t) ∈ L 2(0,T) that drives the state solution to rest. We show that for a ring, two forcesare required to achieve controllability. The controllability problem is reduced to amoment problem for the control. We describe the set of initial data which may be driven torest by the control. The proof is based on an auxiliary basis property result.

We present and characterize a multi-host epidemic model of Rift Valley fever (RVF) virusin East Africa with geographic spread on a network, rule-based mitigation measures, andmosquito infection and population dynamics. Susceptible populations are depleted bydisease and vaccination and are replenished with the birth of new animals. We observe thatthe severity of the epidemics is strongly correlated with the duration of the rainy seasonand that even severe epidemics are abruptly terminated when the rain stops. Becausenaturally acquired herd immunity is established, total mortality across 25 years isrelatively insensitive to many mitigation approaches. Strong reductions in cattlemortality are expected, however, with sufficient reduction in population densities ofeither vectors or susceptible (ie. unvaccinated) hosts. A better understanding of RVFepidemiology would result from serology surveys to quantify the importance of herdimmunity in epidemic control, and sequencing of virus from representative animals toquantify the realative importance of transportation and local reservoirs in nucleatingyearly epidemics. Our results suggest that an effective multi-layered mitigation strategywould include vector control, movement control, and vaccination of young animals yearly,even in the absence of expected rainfall.

We explicitly give all stationary solutions to the focusing cubic NLS on the line, in thepresence of a defect of the type Dirac’s delta or delta prime. The models proveinteresting for two features: first, they are exactly solvable and all quantities can beexpressed in terms of elementary functions. Second, the associated dynamics is far frombeing trivial. In particular, the NLS with a delta prime potential shows two symmetrybreaking bifurcations: the first concerns the ground state and was already known. Thesecond emerges on the first excited state, and up to now had not been revealed. Wehighlight such bifurcations by computing the nonlinear and the no-defect limits of thestationary solutions.

The dynamic model of plant growth GreenLab describes plant architecture and functionalgrowth at the level of individual organs. Structural development is controlled by formalgrammars and empirical equations compute the amount of biomass produced by the plant, andits partitioning among the growing organs, such as leaves, stems and fruits. The number oforgans initiated at each time step depends on the trophic state of the plant, which isevaluated by the ratio of biomass available in plant to the demand of all the organs. Thecontrol of the plant organogenesis by this variable induces oscillations in the simulatedplant behaviour. The mathematical framework of the GreenLab model allows to compute theconditions for the generation of oscillations and the value of the period according to theset of parameters. Two case-studies are presented, corresponding to emergence ofoscillations associated to fructification and branching.

Similar alternating patterns arecommonly reported by botanists. In this article, two examples were selected: alternatepatterns of fruits in cucumber plants and alternate appearances of branches inCecropia trees. The model was calibrated from experimental datacollected on these plants. It shows that a simple feedback hypothesis of trophic controlon plant structure allows the emergence of cyclic patterns corresponding to the observedones.

We study the influence of natural convection on stability of reaction fronts in porous media. The model consists of the heat equation, of the equation for the depth of conversion and of the equations of motion under the Darcy law. Linear stability analysis of the problem is fulfilled, the stability boundary is found. Direct numerical simulations are performed and compared with the linear stability analysis.

We review the basic pathology of cyclical neutropenia in both humansand the grey collie, and examine the role that mathematical modelingof hematopoietic cell production has played in our understanding ofthe origins of this fascinating dynamical disease.

In this paper we propose a mathematical model to describe the evolution of leukemiain the bone marrow. The model is based on a reaction-diffusion system of equations in a porousmedium. We show the existence of two stationary solutions, one of them corresponds to the normalcase and another one to the pathological case. The leukemic state appears as a result of a bifurcationwhen the normal state loses its stability. The critical conditions of leukemia developmentare determined by the proliferation rate of leukemic cells and by their capacity to diffuse. Theanalytical results are confirmed and illustrated by numerical simulations.

Recently the effect of a quiescent phase (or dormant/resting phase in applications) onthe dynamics of a system of differential equations has been investigated, in particular with respectto stability properties of stationary points. It has been shown that there is a general phenomenonof stabilization against oscillations which can be cast in rigorous form. Here we investigate, forhomogeneous systems, the effect of a quiescent phase, and more generally, a phase with slowerdynamics. We show that each exponential solution of the original system produces two exponentialsolutions of the extended system whereby the stability properties can be controlled.

Two main approaches have been considered for modelling the dynamics of the SIS model oncomplex metapopulations, i.e, networks of populations connected by migratory flows whoseconfigurations are described in terms of the connectivity distribution of nodes (patches)and the conditional probabilities of connections among classes of nodes sharing the samedegree. In the first approach migration and transmission/recovery process alternatesequentially, and, in the second one, both processes occur simultaneously. Here we followthe second approach and give a necessary and sufficient condition for the instability ofthe disease-free equilibrium in generic networks under the assumption of limited (orfrequency-dependent) transmission. Moreover, for uncorrelated networks and under theassumption of non-limited (or density-dependent) transmission, we give a bounding intervalfor the dominant eigenvalue of the Jacobian matrix of the model equations around thedisease-free equilibrium. Finally, for this latter case, we study numerically theprevalence of the infection across the metapopulation as a function of the patchconnectivity.