Fisher Kolmogorov Petrovskii Piskunov Research Articles

Front stability of infinitely steep travelling waves in population biology

Reaction–diffusion models are often used to describe biological invasion, where populations of individuals that undergo random motility and proliferation lead to moving fronts. Many reaction–diffusion models of biological invasion are extensions of the well–known Fisher–Kolmogorov–Petrovskii–Piskunov model that describes the spatiotemporal evolution of a 1D population density, u(x,t) , as a result of linear diffusion with flux J=−∂u/∂x , and logistic growth source term, S=u(1−u) . In 2020 Fadai introduced a new reaction–diffusion model of biological invasion with a nonlinear degenerate diffusive flux, J=−u∂u/∂x , and the model was formulated as a moving boundary problem on 0<x<L(t) , with u(L(t),t)=0 and dL(t)/dt=−κu∂u/∂x at x=L(t) (Fadai and Simpson 2020 J. Phys. A: Math. Theor. 53 095601). Fadai’s model leads to travelling wave solutions with infinitely steep, well–defined fronts at the moving boundary, and the model has the mathematical advantage of being analytically tractable in certain parameter limits. In this work we consider the stability of the travelling wave solutions presented by Fadai. We provide general insight by first presenting two key extensions of Fadai’s model by considering: (i) generalised nonlinear degenerate diffusion with flux J=−um∂u/∂x for some constant m > 0; and, (ii) solutions describing both biological invasion with dL(t)/dt>0 , and biological recession with dL(t)/dt<0 . After establishing the existence of travelling wave solutions for these two extensions, our main contribution is to consider stability of the travelling wave solutions by introducing a lateral perturbation of the travelling wavefront. Full 2D time–dependent level–set numerical solutions indicate that invasive travelling waves are stable to small amplitude lateral perturbations, whereas receding travelling waves are unstable. These preliminary numerical observations are corroborated through a linear stability analysis that gives more formal insight into short time growth/decay of wavefront perturbation amplitude. Julia–based software, including level–set algorithms, is available on Github to replicate all results in this study.

Quasiparticles for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation

We construct quasiparticles-like solutions to the one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) with a nonlocal nonlinearity using the method of semiclassically concentrated states in the weak diffusion approximation. Such solutions are of use for predicting the dynamics of population patterns using analytical or semi-analytical approach. The interaction of quasiparticles stems from nonlocal competitive losses in the FKPP model. We developed the formalism of our approach relying on ideas of the Maslov method. The construction of the asymptotic expansion of a solution to the original nonlinear evolution equation is based on solutions to an auxiliary dynamical system of ODEs. The asymptotic solutions for various specific cases corresponding to various spatial profiles of the reproduction rate and nonlocal competitive losses are studied within the framework of the approach proposed.

Domain Structure Formation in Designing of the Opened Informative Measuring Systems

The opened systems possess an increasing significance and possibilities of applying in designing of measuring devices. Now an essentially nonlinear models are used for such systems. The perturbation approach is not enough for these purposes. Models of new types have solutions in a form of soliton or kink and similar objects. The equation of Fisher–Kolmogorov–Petrovskii–Piskunov is one of such equations. This equation is used for description of convection-reaction-diffusion processes. Such processes are used for studying of a self-organisation and formation of a structure in non-equilibrium opened systems. The aim of this work was to construct of a new solution for the modified equation of Fisher–Kolmogorov– Petrovskii–Piskunov in which a space inhomogeneity is accounted.To solve this problem the direct Hirota method for nonlinear partial differential equation is applied.Some modifications into this method were introduced.The new topologically non-trivial solution of the modified Fisher–Kolmogorov–Petrovskii–Piskunov equation is constructed explicitly. This solution has a kink-like form. Some arguments on the stability of such solution are considered.A possibility of domain structure formation in the systems which describe by the Fisher–Kolmogorov– Petrovskii–Piskunov equation is demonstrated.

A Chebyshev Collocation Approach to Solve Fractional Fisher–Kolmogorov–Petrovskii–Piskunov Equation with Nonlocal Condition

We provide a detailed description of a numerical approach that makes use of the shifted Chebyshev polynomials of the sixth kind to approximate the solution of some fractional order differential equations. Specifically, we choose the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) to describe this method. We write our approximate solution in the product form, which consists of unknown coefficients and shifted Chebyshev polynomials. To compute the numerical values of coefficients, we use the initial and boundary conditions and the collocation technique to create a system of equations whose number matches the unknowns. We test the applicability and accuracy of this numerical approach using two examples.

Anisotropic diffusion and traveling waves of toxic proteins in neurodegenerative diseases

Neurodegenerative diseases are closely associated with the amplification and invasion of toxic proteins. In particular Alzheimer's disease is characterized by the systematic progression of amyloid-β and τ-proteins in the brain. These two protein families are coupled and it is believed that their joint presence greatly enhances the resulting damage. Here, we examine a class of coupled chemical kinetics models of healthy and toxic proteins in two spatial dimensions. The anisotropic diffusion expected to take place within the brain along axonal pathways is factored in the models and produces a filamentary, predominantly one-dimensional transmission. Nevertheless, the potential of the anisotropic models towards generating interactions taking advantage of the two-dimensional landscape is showcased. Finally, a reduction of the models into a simpler family of generalized Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) type systems is examined. It is seen that the latter captures well the qualitative propagation features, although it may somewhat underestimate the concentrations of the toxic proteins.

Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel–Kramers–Brillouin (WKB)–Maslov semiclassical approximation is applied to the generalized nonlocal Fisher–KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.

Influence of the Environment on Pattern Formation in the One-Dimensional Nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov Model

A self-consistent model of the dynamics of a cellular population described by the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation with nonlocal competitive losses and interaction with the environment is formulated, in which the dynamics is described by the diffusion equation with allowance for the interaction of the population and the environment. With the help of computer modeling, the formation of the population pattern under the influence of the environment is considered. Possible applications of the model and its generalizations are discussed.

An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

Front propagation and quasi-stationary distributions for one-dimensional Lévy processes

We jointly investigate the existence of quasi-stationary distributions for one dimensional Lévy processes and the existence of traveling waves for the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation associated with the same motion. Using probabilistic ideas developed by S. Harris [16], we show that the existence of a monotone traveling wave for the F-KPP equation associated with a centered Lévy processes that branches at rate $r$ and travels at velocity $c$ is equivalent to the existence of a quasi-stationary distribution for a Lévy process with the same movement but drifted by $-c$ and killed at the first entry into the negative semi-axis, with mean absorption time $1/r$. This also extends the known existence conditions in both contexts. As it is discussed in [15], this is not just a coincidence but the consequence of a relation between these two phenomena.

Approximate Solutions of the One-Dimensional Fisher–Kolmogorov–Petrovskii– Piskunov Equation with Quasilocal Competitive Losses

The modified Fisher–Kolmogorov–Petrovskii–Piskunov equation with quasilocal quadratic competitive losses and variable coefficients in the small nonlocality parameter approximation is reduced to an equation with a nonlinear diffusion coefficient. Within the framework of a perturbation method, equations are obtained for the first terms of an asymptotic expansion of an approximate solution of the reduced equation. Particular solutions in separating variables are considered for the equations determining the first terms of the asymptotic series. The problem is reduced to an elliptic integral and one linear, homogeneous ordinary differential equation.

Wave propagation in the Kolmogorov–Petrovskii–Piskunov problem with delay

The problem of density wave propagation governed by a logistic equation with delay and diffusion (Fisher–Kolmogorov–Petrovskii–Piskunov equation with delay) was studied. To analyze the qualitative behavior of solutions to this equation with periodic boundary conditions in the case of the diffusion parameter tending to zero, the normal form of the problem, i.e., the Ginzburg–Landau equation was constructed near the unit equilibrium. A numerical analysis of wave propagation showed that, for sufficiently small delays, this equation has solutions close to those of the standard Kolmogorov–Petrovskii–Piskunov equation. As the delay parameter increases, a decaying oscillatory component appears in the spatial distribution of the solution and, then, undamped (in time) and slowly propagating (in space) oscillations close to solutions of the corresponding boundary value problem with periodic boundary conditions are observed near the initial perturbation segment.

Isolation effects in a system of two mutually communicating identical patches

Starting from the Fisher–Kolmogorov–Petrovskii–Piskunov equation (FKPP) we model the dynamic of a diffusive system with two mutually communicating identical patches and isolated of the remaining matrix. For this system we find the minimal size of each fragment in the explicit form and compare with the explicit results for similar problems found in the literature. From this comparison emerges an unexpected result that for a same set of the parameters, the isolated system studied in this work with size L, can be better or worst than the non isolated systems with the same size L, uniquely depending on the parameter a0 (internal conditions of the patches). Due to the fact that this result is unexpected we propose an experimental verification.

Existence of Traveling Waves for the Generalized F-KPP Equation.

Variation in genotypes may be responsible for differences in dispersal rates, directional biases, and growth rates of individuals. These traits may favor certain genotypes and enhance their spatiotemporal spreading into areas occupied by the less advantageous genotypes. We study how these factors influence the speed of spreading in the case of two competing genotypes under the assumption that spatial variation of the total population is small compared to the spatial variation of the frequencies of the genotypes in the population. In that case, the dynamics of the frequency of one of the genotypes is approximately described by a generalized Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation. This generalized F-KPP equation with (nonlinear) frequency-dependent diffusion and advection terms admits traveling wave solutions that characterize the invasion of the dominant genotype. Our existence results generalize the classical theory for traveling waves for the F-KPP with constant coefficients. Moreover, in the particular case of the quadratic (monostable) nonlinear growth-decay rate in the generalized F-KPP we study in detail the influence of the variance in diffusion and mean displacement rates of the two genotypes on the minimal wave propagation speed.

Asymptotics semiclassically concentrated on curves for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation

In this paper we construct asymptotic solutions for the nonlocal multidimensional Fisher–Kolmogorov–Petrovskii–Piskunov equation in the class of functions concentrated on a one-dimensional manifold (curve) using a semiclassical approximation technique. We show that the construction of these solutions can be reduced to solving a similar problem for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov in the class of functions concentrated at a point (zero-dimensional manifold) together with an additional operator condition. The general approach is exemplified by constructing a two-dimensional two-parametric solution, which describes quasi-steady-state patterns on a circumference.

Turbulence spreading in gyro-kinetic theory

In this letter a new operative definition for the turbulence intensity in connection with magnetized plasmas is given. In contrast to previous definitions the new definition satisfies a Fisher–Kolmogorov–Petrovskii–Piskunov type equation. Furthermore, explicit expressions for the turbulence intensity and the turbulence intensity flux, that allow for the first time direct numerical evaluation, are derived. A carefully designed numerical experiment for the case of a tokamak is performed to study the impact of turbulence spreading. The effective turbulence diffusion coefficient is measured to be smaller than the heat conduction coefficient and the turbulence spreading length is found to be of the order of the turbulence correlation length. The results show that turbulence spreading can play a role in the non-local flux gradient relation, or in the scaling of transport coefficients with the normalized Larmor radius, only over lengths scale of the order of the turbulence correlation length. A new turbulence convection mechanism, due to the drift connected with the magnetic field inhomogeneities, is described. The convective flux integrates to zero under the flux surface average unless there is an up–down asymmetry in the tubulence intensity. The latter asymmetry can be generated through a radial inhomogeneity or plasma rotation. It is shown that the turbulence convection can lead to a spreading of the order of the correlation length.

Asymptotics of the Multidimensional Nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov Equation Near a Quasistationary Solution

Asymptotic solutions of the multidimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation with an influence function that is invariant with respect to a spatial shift are constructed. The asymptotic solutions are perturbations of a spatially-homogeneous quasistationary exact solution. General expressions are illustrated by the example of a two-dimensional equation with a Gaussian initial condition.

Asymptotic Behavior of the One-Dimensional Fisher–Kolmogorov–Petrovskii–Piskunov Equation with Anomalouos Diffusion

Asymptotic solutions of the nonlocal, one-dimensional Fisher–Kolmogorov–Petrovskii–Piskunov equation with fractional derivatives in the diffusion operator are constructed. The fractional derivative is defined in accordance with the approaches of Weyl, Grunwald–Letnilkov, and Liouville. Asymptotic solutions are constructed in a class of functions that are a perturbation of the found exact quasistationary solution and tend at large times to this quasistationary solution. It is shown that the presence of fractional derivatives leads to drift of the center of mass of the initial distribution and breaks its symmetry.

Spontaneous Radiation and Quantum Dynamics of Biological Plasma

A quantum solution of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with convection and nonlinear diffusion is obtained which can provide the basis for the quantum biology and quantum microphysics equation. On this basis, quantum emission of biological systems, separate microorganisms (cells or bacteria), and dust plasma particles is investigated. The possibility arises of creating a generator of hard photons with energy higher than. Life on the Earth will probably need relic radiation, which might promote occurrence of life in the Universe.

Symmetry Operators of the Nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov Equation with a Quadratic Operator

A class of nonlinear symmetry operators has been constructed for the many-dimensional nonlocal Fisher- Kolmogorov-Petrovskii-Piskunov equation quadratic in independent variables and derivatives. The construction of each symmetry operator includes an interwining operator for the auxiliary linear equations and additional nonlinear algebraic conditions. Symmetry operators for the one-dimensional equation with a constant influence function have been constructed in explicit form and used to obtain a countable set of exact solutions. The symmetry properties of a differential equation (DE) are associated with transforms leaving invariant the set of solutions of the equation. We shall refer to these transforms as symmetry operators (SOs). Symmetry operators make it possible to generate new solutions of an equation by using them sequentially to act on a known solution. The solution generation procedure only illustrates the potentialities of symmetry operators, but does not exhaust them. The problem is how to find symmetry operators or other symmetry transforms in explicit form. The basic mathematical apparatus used in a comprehensive study of symmetries of differential equations is the Lie group theory of continuous transformations leaving an (ordinary or partial) differential equation invariant. The invariance group of an equation is also termed a symmetry group, or a group admitted by the equation. Seeking a symmetry group is reduced to solving a system of linear constitutive equations for an infinitesimal Lie group operator (generator). The Lie group of finite transformations constructed with a generator can be applied to any solution of the equation admitting the group, and thus generate parametric sets of new solutions to the equation from a known one. A Lie group of pointwise invariance transformations of a differential equation can be considered as a set of symmetry operators of the equation. A detailed description of the application of Lie groups to ordinary DEs can be found, for instance, in Refs. 1-3. For partial differential equations (PDEs), the application of Lie group methods leans upon a procedure of prolongation of the action of a Lie group on higher partial derivatives (1-3). The generator of a prolonged Lie group has special structure and is a base object of examination in a study of the PDE symmetry properties. For a Lie group admitted by a PDE, the generator is determined by a linear partial differential equation in the generator coefficients that specify the so-called symmetries of the equation. A set of symmetries possesses algebraic properties, which are used to examine the properties of equations and to find sets of their solutions (1-3). However, the problem of finding SOs of nonlinear equations in explicit form by direct methods not related to Lie groups is poorly studied. In general statement, this problem has no constructive solution due to significant

Pattern formation in terms of semiclassically limited distribution on lower dimensional manifolds for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation

We have investigated the pattern formation in systems described by the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation for the cases where the dimension of the pattern concentration domain is lower than that of the domain of independent variables. We have obtained a system of integro-differential equations which describe the dynamics of the concentration domain and the semiclassically limited density distribution for a pattern in the class of trajectory concentrated functions. Also, asymptotic large time solutions have been obtained that describe the semiclassically limited distribution for a quasi-steady-state pattern on the concentration manifold. The approach is illustrated by an example for which the analytical solution is in good agreement with the results of numerical calculations.