Search

We study the well-known three-component nonlinear system of equations of reaction-diffusion-type simulating the processes of competition and coexistence of different-language speakers. A simplified modification of this system is proposed. For this system, we describe the Lie symmetries, construct exact solutions in the form of traveling fronts, and establish their properties. We present the plots of traveling fronts and propose the corresponding interpretation for the description of the language shift in Ukraine during the Soviet times.

Background and Aims Experimental studies and computational modeling show increased hydration of peritoneal tissue close to peritoneal surface after intraperitoneal (ip) administration of hypertonic dialysis fluid. This overhydration - due to fluid inflow from peritoneal cavity (driven by increased intraperitoneal pressure) and from blood (due to high interstitial concentration of osmotic agent diffusing from the cavity) - may lead to tissue swelling, as observed in experiments and in disturbed physiological conditions. We estimated the degree of swelling using linear poroelastic theory with fluid and solute transport parameters obtained from clinical studies. Method The spatially distributed model of peritoneal transport was extended by equations for tissue deformation and stress derived from linear poroelastic theory. The model describes also fluid and osmotic agent flows across tissue and capillary wall. We assumed that transport and deformation occur across a layer of tissue with initial intact width L0 and deformed width L; the deformation is described as the ratio L/L0. Transport parameters are assumed as average values estimated for intact tissue by Stachowska-Pietka (2019). As tissue stiffness (Lame coefficient) for muscle is not known, we examined stiffness ranging from 110 mmHg (connective tissue; interstitium) to 700 mmHg (solid tumor). We assumed that for initial periods of peritoneal dialysis when osmotic pressure of dialysis fluid is high: 1) osmotic pressure gradient across the capillary wall prevails over the combined Starling forces, 2) spatial profile of osmotic agent concentration in tissue (interstitial fluid) can be approximated by exponential function with the penetration depth ΛS. The model yields an equation for L/L0 to be solved numerically, but an approximated closed formula also works well for typical dialysis conditions. Results The model predicts that swelling of peritoneal tissue depends on factors such as tissue stiffness, tissue width, solute penetration depth, and transport parameters for tissue and capillary wall, and on the forces that induce fluid transport: intraperitoneal pressure and the increment of osmolality of dialysis fluid over plasma osmolality. Examples of L/L0 yielded by the model - with use of glucose 1.36% dialysis fluid and for two levels of ip hydrostatic pressure (Pip) - are shown. In Figure, left panel, for L0 = 1 cm representing human abdominal muscle, and solute diffusional penetration ΛS=ΛD=0.055 cm, or lower, as due to diffusion against fluid flow, ΛS=ΛD/2=0.027 cm, is plotted versus the tissue stiffness; the dialysis fluid with glucose 1.36% is applied (osmolality increment of 60 mmol/L at the beginning of peritoneal dwell, Waniewski et al, 1996) and Pip is 15 mmHg. As stiffness of abdominal and bowel muscles may be expected around 300 mm Hg, swelling might be up to 15%; it decreases with lower ip hydrostatic and osmotic pressures. Hypothetical dialysis at Pip = 0 (isobaric with interstitial fluid) would reduce swelling by factor 2, see Figure, right panel. The depth of osmotic agent penetration into the tissue impacts tissue hydration and swelling, see Figure 1 for L/L0 with twice reduced ΛS. The model and its approximation by the closed formula provide practically the same outcomes for clinical peritoneal dialysis, see Figure 1, but some discrepancy between them may occur for thin tissue, as rat abdominal wall. The approximate formula for L/L0 works well if ΛS is much shorter than L0. Nevertheless, for high degree of swelling a nonlinear theory should be constructed. Conclusion In peritoneal dialysis, exposure of peritoneal tissue to hypertonic dialysis fluid at increased hydrostatic pressure contributes to overhydration and swelling (by 5-15% after fluid infusion) of the tissue. The extent by which this swelling may contribute to changes in peritoneal tissue structure and function warrants further studies.

A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

Background and Aims The width of the peritoneum (composed mainly of connective tissue and relatively free of vasculature) is increased in patients on peritoneal dialysis compared to healthy subjects. We investigated to what extent increased intraperitoneal (ip) hydrostatic and osmotic pressures following the infusion of dialysis fluid will change hydration status and width of the peritoneum. Method Using linear theory of poroelasticty, clinical data on transport parameters and experimental data on elastic characteristics of the interstitium, the relative change of the width of the poroelastic layer subject to the combined effect of external hydrostatic and effective osmotic pressures (that is, ideal osmotic pressure multiplied by reflection coefficient for osmotic agent) can be described as a function of effective pressure and elastic modulus of the layer: Lmod/L0 = 1/(1-deltaP*/lambda*), where L0 is initial thickness of the tissue, Lmod is modified thickness of the layer, deltaP* is change in effective combined pressure, and lambda* is the elastic modulus of the poroelastic material. The same formula describes also the change in fractional free fluid volume ratio, thetaF. The elastic modulus of the connective tissue was assumed to be 110 mmHg, as measured for the subcutaneous layer of the tip of mouse tail by Swartz et al (J Biomech, 1999), and reflection coefficient for glucose in the interstitium of 0.0035 as estimated by Stachowska-Pietka et al (NDT, 2019) from clinical data for patients on peritoneal dialysis. Results The ip hydrostatic pressure increases by 2-3 mmHg to 15 mmHg at rest depending on infused volume of dialysis fluid (and posture, body weight and location in abdominal cavity), and may increase to 100 mmHg during activities as coughing, whereas the osmotic pressure of glucose 3.86% dialysis fluid is around 400 mmHg above the osmotic pressure of plasma and interstitial fluid (in equilibrium with plasma). However, due to the low reflection coefficient of interstitium, the effective osmotic pressure of dialysis fluid minus the physiological value of interstitial osmotic pressure is only 1.4 mmHg, and is quickly decreasing with dwell time. Therefore, hydrostatic pressure is the dominant factor for interstitial hydration. Assuming ip pressure of 15 mmHg, the stretch of the peritoneum increases its equilibrium width (at 0 mmHg and isotonic interstitial fluid) by 15%. During physical activities peritoneum may transiently thicken even more. Conclusion The peritoneum becomes overhydrated after infusion of dialysis fluid, which increases interstitial hydrostatic pressure; the thickness and fractional free fluid volume of the peritoneum increase by 15% although transiently higher increases may occur following activities that increase intraperitoneal pressure. The mechanical changes in the peritoneum may contribute to the biological changes in cells present there, as fibroblasts and mesothelial cells. The swelling of the peritoneum is in agreement with the increase in the fractional free fluid volume of the intramuscular interstitium behind the peritoneum as reported by Zakaria et al (Am J Physiol Heart Circ Physiol, 1999).

A class of nonlinear reaction-diffusion-convection equations describing various processes in physics, biology, chemistry etc. is under study in the case of time and two space variables. The group of equivalence transformations is constructed, which is applied for deriving a Lie symmetry classification for the class of such equations by the well-known algorithm. It is proved that the algorithm leads to 32 reaction-diffusion-convection equations admitting nontrivial Lie symmetries. Furthermore a set of form-preserving transformations for this class is constructed in order to reduce this number of the equations and obtain a complete Lie symmetry classification. As a result, the so called canonical list of all inequivalent equations admitting nontrivial Lie symmetry (up to any point transformations) and their Lie symmetries are derived. The list consists of 22 equations and it is shown that any other reaction-diffusion-convection equation admitting a nontrivial Lie symmetry is reducible to one of these 22 equations. As a nontrivial example, the symmetries derived are applied for the reduction and finding exact solutions in the case of the porous-Fisher type equation with the Burgers term., 30 pages, 5 tables. Under review in CNSNS since July 2019

Q-conditional (non-classical) symmetries of the known three-component reaction-diffusion (RD) system [K. Aoki et al. Theor. Popul. Biol. 50, 1–17 (1996)] modelling interaction between farmers and hunter-gatherers are constructed for the first time. A wide variety of Q-conditional symmetries are found, and it is shown that these symmetries are not equivalent to the Lie symmetries. Some operators of Q-conditional (non-classical) symmetry are applied for finding exact solutions of the RD system in question. Properties of the exact solutions (in particular, their asymptotic behaviour) are identified and possible biological interpretation is discussed.

Fluid and solute transport in poroelastic media is studied. Mathematical modeling of such transport is a complicated problem because of the volume change of the specimen due to swelling or shrinking and the transport processes are nonlinearly linked. The tensorial character of the variables adds also substantial complication in both theoretical and experimental investigations. The one-dimensional version of the theory is less complex and may serve as an approximation in some problems, and therefore, a one-dimensional (in space) model of fluid and solute transport through a poroelastic medium with variable volume is developed and analyzed. In order to obtain analytical results, the Lie symmetry method is applied. It is shown that the governing equations of the model admit a non-trivial Lie symmetry, which is used for construction of exact solutions. Some examples of the solutions are discussed in detail.

The well known nonlinear model for describing the solid tumour growth (Byrne et al. 2003) is under study using an approach based on Lie symmetries. It is shown that the model in the two-dimensional (in space) approximation forms a (1+2)-dimensional boundary value problem, which admits a highly nontrivial Lie symmetry. The special case involving the power-law nonlinearities is examined in details. The symmetries derived are applied for the reduction of the nonlinear boundary value problem in question to problems of lower dimensionality. Finally, the reduced problems with correctly-specified coefficients were exactly solved and the exact solutions derived were analysed, in particular, some plots were build in order to understand the time-space behaviour of these solutions and to discuss their biological interpretation.

A new definition of conditional invariance for boundary value problems involving a wide range of boundary conditions (including initial value problems as a special case) is proposed. It is shown that other definitions worked out in order to find Lie symmetries of boundary value problems with standard boundary conditions, follow as particular cases from our definition. Simple examples of direct applicability to the nonlinear problems arising in applications are demonstrated. Moreover, the successful application of the definition for the Lie and conditional symmetry classification of a class of (1+2)-dimensional nonlinear boundary value problems governed by the nonlinear diffusion equation in a semi-infinite domain is realised. In particular, it is proved that there is a special exponent, $k=-2$, for the power diffusivity $u^k$ when the problem in question with non-vanishing flux on the boundary admits additional Lie symmetry operators compared to the case $k\not=-2$. In order to demonstrate the applicability of the symmetries derived, they are used for reducing the nonlinear problems with power diffusivity $u^k$ and a constant non-zero flux on the boundary (such problems are common in applications and describing a wide range of phenomena) to (1+1)-dimensional problems. The structure and properties of the problems obtained are briefly analysed. Finally, some results demonstrating how Lie invariance of the boundary value problem in question depends on geometry of the domain are presented., Comment: 25 pages; the main results were presented at the Conference Symmetry, Methods, Applications and Related Fields, Vancouver, Canada, May 13-16, 2014

A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete Lie symmetry classification, including a number of the cases characterised as being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications (tumour growth). Graphical representations of the solutions are provided and a biological interpretation is briefly addressed. The results are generalised on multi-dimensional case under the assumption of the radially symmetrical shape of the tumour.

This review is devoted to search for Lie and Q-conditional (nonclassical) symmetries and exact solutions of a class of reaction-diffusion-convection equations with exponential nonlinearities. A complete Lie symmetry classification of the class is derived via two different algorithms in order to show that the result depends essentially on the type of equivalence transformations used for the classification. Moreover, a complete description of Q-conditional symmetries for PDEs from the class in question is also presented. It is shown that all the well-known results for reaction-diffusion equations with exponential nonlinearities follow as particular cases from the results derived for this class of reaction-diffusion-convection equations. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. The exact solutions are compared with those found by means of different techniques. Finally, an application of the exact solutions for solving boundary-value problems arising in population dynamics is presented.

Nowadays, Lie and conditional symmetries are widely applied to study nonlinear partial differential equations (PDEs) (including multidimensional PDEs), notably for their reductions to ordinary differential equations and construction of exact solutions. [...]

A recently developed theoretical background for searching Q-conditional (nonclassical) symmetries of systems of evolution partial differential equations is presented. We generalize the standard definition of Q-conditional symmetry by introducing the notion of Q-conditional symmetry of the p-th type and show that different types of symmetry of a given system generate a hierarchy of conditional symmetry operators. It is shown that Q-conditional symmetry of the p-th type possesses some special properties, which distinguish them from the standard conditional symmetry. The general class of two-component nonlinear reaction-diffusion systems is examined in order to find the Q-conditional symmetry operators. The relevant systems of so-called determining equations are solved under additional restrictions. As a result, several reaction-diffusion systems possessing conditional symmetry are constructed. In particular, it is shown that the diffusive Lotka–Volterra system, the Belousov–Zhabotinskii system (with the correctly specified coefficients) and some of their generalizations admit Q-conditional symmetry.

Two- and three-component diffusive Lotka–Volterra systems are examined in order to find Q-conditional symmetries, to construct exact solutions and to provide their biological interpretation. An exhaustive description of Q-conditional symmetries of the first type (a special subset of nonclassical symmetries) of these nonlinear systems is derived. An essential part of this chapter is devoted to the construction of exact solutions of the systems in question using the symmetries obtained. Starting from examples of travelling fronts (finding such solutions is important from the applicability point of view), we concentrate mostly on finding exact solutions with a more complicated structure. As a result, a wide range of exact solutions are constructed for the two-component diffusive Lotka–Volterra system and some examples are presented for the three-component diffusive Lotka–Volterra system. Moreover, a realistic interpretation for two and three competing species is provided for some exact solutions.

All the main results on Q-conditional symmetry (nonclassical symmetry) of the general class of nonlinear reaction-diffusion-convection equations are summarized. Although some of them were published about 25 years ago, and the others were derived in the 2000s, it is the first attempt to present an extensive review of this matter. It is shown that several well-known equations arising in applications and their direct generalizations possess conditional symmetry. Notably, the Murray, Fitzhugh–Nagumo, and Huxley equations and their natural generalizations are identified. Moreover, several exact solutions (including travelling fronts) are constructed using the conditional symmetries obtained in order to find exact solutions with a biological interpretation.

Two classes of two-component nonlinear reaction-diffusion systems are studied in order to find Q-conditional symmetries of the first type (a special subset of nonclassical symmetries), to construct exact solutions, and to show their applicability. The first class involves systems with constant coefficient of diffusivity, while the second contains systems with variable diffusivities only. The main theoretical results are given in the form of two theorems presenting exhaustive lists (up to the given sets of point transformations) of the reaction-diffusion systems belonging to the above classes and admitting Q-conditional symmetries of the first type. The reaction-diffusion systems obtained allow one to extract specific systems occurring in real-world models. A few examples are presented, including a modification of the classical prey–predator system with diffusivity and a system modelling the gravity-driven flow of thin films of viscous fluid. Exact solutions with attractive properties are found for these nonlinear systems and their possible biological and physical interpretations are presented.

Lie symmetries of a simplified Keller–Segel system are found and applied for construction of exact solutions. The algorithms for constructing all possible traveling wave and self-similar solutions of the system in question are presented. Several families of such solutions in an explicit form are found, their properties examined and possible applicability for chemotaxis modeling is discussed.

A complete description of Q -conditional symmetries for a class of reaction–diffusion–convection equations with exponential diffusivities is derived. It is shown that all the known results for reaction–diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. The exact solutions are compared with those found earlier by means of different approaches. An application of the solutions for solving a boundary-value problem arising in population dynamics is presented.