Density-dependent Diffusion Research Articles

Speed and Shape of Population Fronts with Density-Dependent Diffusion

There is growing empirical evidence that animal movement patterns depend on population density. We investigate travelling wave solutions in reaction-diffusion models of animal range expansion in the case that population diffusion is density-dependent. We find that the speed of the selected wave depends critically on the strength of diffusion at low density. For sufficiently large low-density diffusion, the wave propagates at a speed predicted by a simple linear analysis. For small or zero low-density diffusion, the linear analysis is not sufficient, but a variational approach yields exact or approximate expressions for the speed and shape of population fronts.

Asymptotic stability in a predator-prey system with density-dependent diffusion and indirect pursuit-evasion interaction in 2D

This paper deals with a predator-prey system with density-dependent diffusion and indirect pursuit-evasion interaction under homogeneous Neumann boundary conditions. By providing appropriate conditions, the asymptotic stability of solution is investigated.

Existence of traveling wave solutions for density-dependent diffusion competitive systems

In this paper we are concerned with the existence of traveling wave solutions for two species competitive systems with density-dependent diffusion. Since the density-dependent diffusion is a kind of nonlinear diffusion and degenerates at the origin, the methods for proving the existence of traveling wave solutions for competitive systems with linear diffusion are invalid. To overcome the degeneracy of diffusion, we construct a nonlinear invariant region Ω near the origin. Then by using the method of phase plane analysis, we prove the existence of traveling wave solutions connecting the origin and the unique coexistence state, when the speed c is large than some positive value. In addition, when one species is density-dependent diffusive while the other is linear diffusive, via the change of variables and the central manifold theorem, we prove the existence of the minimal speed c∗ . And for c⩾c∗ , traveling wave solutions connecting the origin and the unique coexistence state still exist. In particular, when c=c∗ , we find that one component of the traveling wave solution is sharp type while the other component is smooth, which is a different phenomenon from linear diffusive systems and scalar equations.

Effect of density-dependent diffusion on a diffusive predator–prey model in spatially heterogeneous environment

The paper presents a class of predator–prey model with density-dependent diffusion in the spatially heterogeneous environment. We first provide the global existence and boundedness of the solution for the model. Then, by taking a variable transformation, the difficulty brought by the cross-diffusion can be overcome, and the existence, stability and local bifurcation of semi-trivial steady-state solutions for the equivalent system are further studied. Finally, the existence of positive solutions of the system is also given by using the Leray–Schauder degree theory and the method of principle eigenvalue, especially for the limit cases when the diffusion coefficient tends to zero or infinite.

Travelling waves in a minimal go-or-grow model of cell invasion

We consider a minimal go-or-grow model of cell invasion, whereby cells can either proliferate, following logistic growth, or move, via linear diffusion, and phenotypic switching between these two states is density-dependent. Formal analysis in the fast switching regime shows that the total cell density in the two-population go-or-grow model can be described in terms of a single reaction–diffusion equation with density-dependent diffusion and proliferation. Using the connection to single-population models, we study travelling wave solutions, showing that the wave speed in the go-or-grow model is always bounded by the wave speed corresponding to the well-known Fisher–KPP equation.

Boundedness of Solutions to a Fully Parabolic Indirect Pursuit–Evasion Predator–Prey System with Density-Dependent Diffusion in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{R}}}^2$$\\end{document}

This paper deals with a fully parabolic indirect pursuit–evasion predator–prey system with density-dependent diffusion ut=Δ(ψ1(w)u)+u(λ-u+αv),vt=Δ(ψ2(z)v)+v(μ-v-βu),wt=Δw-w+v,zt=Δz-z+u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{t}=\\Delta (\\psi _1(w)u)+u(\\lambda -u+\\alpha v), v_{t}=\\Delta (\\psi _2(z) v)+v(\\mu -v-\\beta u), w_{t}=\\Delta w -w+v, z_{t}=\\Delta z-z+u$$\\end{document} under a smooth bounded domain Ω⊂R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset {\\mathbb{R}}^2$$\\end{document} with homogeneous Neumann boundary conditions, where the parameters λ,μ,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda , \\mu , \\alpha$$\\end{document} and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} are assumed to be positive. Through the establishment of appropriate conditions for the density-dependent diffusion functions ψ1(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _1(w)$$\\end{document} and ψ2(z),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _2(z),$$\\end{document} it is revealed that a unique classical solution exists for the corresponding initial-boundary problem, which remains uniformly bounded over time.

Invasive-Invaded Interaction Incorporating a Bramson Model with Density-Dependent Diffusion and a Non-Lipschitz Reaction

The primary objective of the presented study is to investigate the pairwise interaction dynamics between invasive and invaded species, considering a model characterized by a non-regular, non-Lipschitz type reaction, as well as non-homogeneous diffusion. To achieve this, we begin with the foundational model proposed by Bramson in 1988 and tailor it to account for density-dependent diffusion and the non-Lipschitz type reaction, rendering it more applicable to our specific ecological scenario. Subsequently, our newly developed model is subjected to different analyses to ascertain the existence and uniqueness of positive weak solutions. It is noteworthy that density-dependent diffusive operators exhibit a property known as “finite propagation”, which manifests as the existence of a propagating front in the ecological system. Furthermore, we delve into the problem domain by employing the concept of traveling waves to identify specific solutions. A key outcome of our investigation is as follows: When both species propagate at significantly different speeds in the context of traveling waves, the interaction between them is deemed unstable, resulting in oscillations in the concentration of the invaded species. Conversely, when both species propagate within a similar range of speeds, the dynamics of the system are predominantly governed by the invasive species.AMS Subject Classification: 35K55, 35K57, 35K59, 35K65

Analysis of an optimal control problem for a spatio‐temporal SIR model with nonlinear density dependent diffusion terms

SummaryThis paper focuses on the study of an optimal control problem for a new spatio‐temporal SIR epidemic model with nonlinear density dependent diffusion terms and a class of nonlinear incidence functions. We consider two types of control variables, vaccination for the susceptible and treatment for the infected. For fixed controls, by means of Schauder fixed point theorem, we prove that the proposed model admits a weak biologically feasible solution, the uniqueness of the latter is also investigated. Furthermore, using the state and adjoint problems, first order necessary optimal conditions are obtained. Finally, numerical simulations are carried out for particular diffusion terms incorporating the heard mentality of individuals, when it comes to the spatial movement, and for particular incidence functions, as well as by varying the parameters of the objective functional, to illustrate the possible optimal control strategies and their effect on the studied population.

Modelling the interaction of invasive-invaded species based on the general Bramson dynamics and with a density dependant diffusion and advection.

The main goal of the presented study is to introduce a model of a pairwise invasion interaction with a nonlinear diffusion and advection. The new equation is based on the further general works introduced by Bramson (1988) to describe the invasive-invaded dynamics. This type of model is made particular with a density dependent diffusion along with an advection term. The new resulting model is then analyzed to explore the regularity, existence and uniqueness of solutions. It is well known that density dependent diffusion operators induce a propagating front with finite speed for compactly supported functions. Based on this, we introduce an analytical approach to determine the evolution of such a propagating front in the invasion dynamics. Afterward, we study the problem with travelling wave profiles and a numerical assessment. As a main finding to remark: When both species propagate with significantly different travelling wave speeds, the interaction becomes unstable, while when the species propagate with similar low speeds, the interaction stabilizes.

Existence of a traveling wave solution to glioblastoma multiforme model with Gompertz growth function

A mathematical model of glioblastoma multiforme with the Gompertz growth function is analyzed. The Fisher-Kolmogorov type model contains a density-dependent diffusion term, taxis and growth function and considers the properties of glioblastoma multiforme. An analytical solution by the traveling wave method is investigated and the condition for its existence is found. Numerical solutions in Matlab to find the optimal values of the model parameters are performed using the simplex Nelder-Mead algorithm.

Global stability and Hopf bifurcation of a delayed cooperative species model with density-dependent diffusion

In this article, we have proposed a newly delayed cooperative species model with density-dependent diffusion. Firstly, we prove the existence and uniqueness of positive equilibrium of this model through mathematical analysis method. Then, for this model, we investigate the persistence properties in the case of self-diffusion and global stability of positive equilibrium by constructing Lyapunov function. Further, we discuss the existence problem of Hopf bifurcation deduced by delay. Finally, the theoretical results in this article are verified by carrying out some numerical simulations. The research results show that density dependent diffusion does not affect the stability of the positive equilibrium of the model, but delay does.

Traveling waves for generalized Fisher–Kolmogorov equation with discontinuous density dependent diffusion

We are concerned with the existence and qualitative properties of traveling wave solutions for a quasilinear reaction‐diffusion equation on the real line. We consider a non‐Lipschitz reaction term of Fisher–KPP type and a discontinuous diffusion coefficient that allows for degenerations and singularities at equilibrium points. We investigate the joint influence of the reaction and diffusion terms on the existence and nonexistence of traveling waves, and assuming these terms are of power‐type near equilibria, we provide classification of solutions based on their asymptotic properties. Our approach provides a broad theoretical background for the mathematical treatment of rather general models not only in population dynamics but also in other applied sciences and engineering.

Novel exact solutions, bifurcation of nonlinear and supernonlinear traveling waves for M-fractional generalized reaction Duffing model and the density dependent M-fractional diffusion reaction equation

In the present work, we study the dynamical behavior of nonlinear traveling waves for the M-fractional generalized reaction Duffing model and density dependent M-fractional diffusion reaction equation. Novel exact solutions in the form of rational, hyperbolic and trigonometric functions are obtained using the new extended direct algebraic method. The obtained solutions are also verified for the aforesaid equations with the help of symbolic soft computations. Furthermore, some produced results are graphically illustrated using 3D surface graphs and 2D line plots, which provide useful information about the dynamical behavior. Moreover, we assurance that all the obtained solutions are accurate, efficient and an excellent contribution in the literature of solitary wave theory. Bifurcation behavior of some nonlinear traveling waves of the proposed equations is also studied with the help of bifurcation theory of planar dynamical systems through numerical simulation. It is observed that proposed equations supports nonlinear solitary waves, periodic waves and most important supernonlinear periodic waves.

Density-dependent diffusion and refuge in a spatial Rosenzweig-MacArthur model: Stability results

In this note we present a study of the solutions associated to a particular spatial extension of the Rosenzweig-MacArthur model for predator and prey. The analysis presented here shows that positive steady state solutions emerge via a transcritical bifurcation mechanism, in accordance with the insight obtained from previous numerical and analytical results. In the model under discussion, prey is assumed to move avoiding crowds via a density-dependent diffusion and also incorporates the existence of a refuge zone, where predators cannot consume prey. Saturation in prey consumption is also included through a Holling type II functional response.

Construction of multiple new analytical soliton solutions and various dynamical behaviors to the nonlinear convection-diffusion-reaction equation with power-law nonlinearity and density-dependent diffusion via Lie symmetry approach together with a couple of integration approaches

By taking advantage of three different computational analytical methods: the Lie symmetry analysis, the generalized Riccati equation mapping approach, and the modified Kudryashov approach, we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation (NCDR) with power-law nonlinearity and density-dependent diffusion. Lie symmetry analysis is one of the powerful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables. By the Lie group technique, we obtain one-parameter invariant transformations, determining equations and corresponding vectors for the considered convection-diffusion-reaction equation. By treating the parameters of the governing equation as constants, the applied methods yield a variety of new closed-form solutions, including inverse function solutions, periodic solutions, exponential function solutions, dark solitons, singular solitons, combo bright-singular solitons, and the combine of bright-dark solitons and dark-bright solitons. Moreover, using the Bäcklund transformation of the generalized Riccati equation and modified Kudryashov method, we can construct multiple solitons and other solutions of the considered equation. The obtained new solutions of this work demonstrate that the used approaches are powerful and effective in dealing with nonlinear equations, and that these solutions are required to explain many biological and physical phenomena. Comparing our obtained solutions of this paper with the ones obtained in the literature, we see that our solutions are new and not reported elsewhere. These newly formed soliton solutions will be more beneficial in the various disciplines of ocean engineering, plasma physics, and nonlinear sciences.

On a two-species competitive predator-prey system with density-dependent diffusion.

This paper deals with a two-species competitive predator-prey system with density-dependent diffusion, i.e., \begin{eqnarray*}\label{1a} \left\{ \begin{split}{}&u_t=\Delta (d_{1}(w)u)+\gamma_{1}uF_{1}(w)-uh_{1}(u)-\beta_{1}uv,&(x,t)\in \Omega\times (0,\infty),\\&v_t=\Delta (d_{2}(w)v)+\gamma_{2}vF_{2}(w)-vh_{2}(v)-\beta_{2}uv,&(x,t)\in \Omega\times (0,\infty),\\&w_t=D\Delta w-uF_{1}(w)-vF_{2}(w)+f(w),&(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{2}$, with the nonnegative initial data $\left( {u_{0}, v_{0}, w_{0}} \right) \in (W^{1,p}(\Omega))^{3}$ with $p>2$, where the parameters $D,\gamma_{1},\gamma_{2},\beta_{1},\beta_{2}>0$, $d_{1}(w)$ and $d_{2}(w)$ are density-dependent diffusion functions, $F_{1}(w)$ and $F_{2}(w)$ are commonly called the functional response functions accounting for the intake rate of predators as the functions of prey density, $h_{1}(u)$ and $h_{2}(v)$ represent the mortality rates of predators, and $f(w)$ stands for the growth function of the prey. First, we rigorously prove the global boundedness of classical solutions for the above general model provided that the parameters satisfy some suitable conditions by means of $L^{p}$-estimate techniques. Moreover, in some particular cases, we establish the asymptotic stabilization and precise convergence rates of globally bounded solutions under different conditions on the parameters by constructing some appropriate Lyapunov functionals. Our results not only extend the previous ones, but also involve some new conclusions.

Stability properties of the steady state for the isentropic compressible Navier–Stokes equations with density dependent viscosity in bounded intervals

We prove existence and asymptotic stability of the stationary solution for the compressible Navier-Stokes equations for isentropic gas dynamics with a density dependent diffusion in a bounded interval. We present the necessary conditions to be imposed on the boundary data which ensure existence and uniqueness of the steady state, and we subsequent investigate its stability properties by means of the construction of a suitable Lyapunov functional for the system. The Saint-Venant system, modeling the dynamics of a shallow compressible fluid, fits into this general framework.

Integrable nonlinear reaction-diffusion population models for fisheries

Sustainable natural resource management strategies are required to maintain a fishery. One common device is to designate a No-Take Area (NTA) or a less restrictive Marine Protected Area (MPA) within a fishery. In this work, exact time-dependent solutions to 2-D nonlinear reaction-diffusion equations of Fisher type are constructed to model the fish population density of a single species within an NTA. The spillover rate from an NTA, important for harvesters, has previously been calculated without allowing for spatial variation within the NTA. Here, that spatial dependence is incorporated in the calculation. Spatially-dependent and density-dependent diffusivity and birth rates are considered for linear, rectangular and circular geometries, the latter being the most efficient with minimal critical domain area. The effect of the spatial dependence of diffusivity and birth rates, common in fish populations where reproduction and protection are favoured towards the centre, on the minimum size of an exclusion zone that ensures species survival, is studied. It is shown that when any one of these factors is taken into account, the size of the critical domain decreases by a small amount.

Traveling waves for unbalanced bistable equations with density dependent diffusion

We study the existence and qualitative properties of traveling wave solutions for the unbalanced bistable reaction-diffusion equation with a rather general density dependent diffusion coefficient. In particular, it allows for singularities and/or degenerations as well as discontinuities of the first kind at a finite number of points. The reaction term vanishes at equilibria and it is a continuous, possibly non-Lipschitz function. We prove the existence of a unique speed of propagation and a unique traveling wave profile (up to translation) which is a non-smooth function in general. In the case of the power-type behavior of the diffusion and reaction near equilibria we provide detailed asymptotic analysis of the profile. For more information see https://ejde.math.txstate.edu/Volumes/2021/76/abstr.html

Traveling wave solutions to the density-suppressed motility model

To understand the “self-trapping” mechanism inducing spatio-temporal pattern formations observed in the experiment of [21] for bacterial motion, the following density-suppressed motility model{ut=Δ(γ(v)u)+u(a−bu),vt=Δv+u−v, was proposed in [6,21], where u(x,t) and v(x,t) represent the densities of bacteria and the chemical emitted by the bacteria, respectively; γ(v) is called the motility function satisfying γ′(v)<0 and a,b>0 are positive constants accounting for the growth and death rates of bacterial cells. The analysis of the above system is highly non-trivial due to the cross-diffusion and possible degeneracy resulting from the nonlinear motility function γ(v) and mathematical progresses on the global well-posedness and asymptotics of solutions were just made recently. Among other things, the purpose of this paper is to consider the above system with motility function γ(v)=1(1+v)m(m>0) and investigate the traveling wave solutions which are genuine patterns observed in the experiment of [21]. By introducing an auxiliary parabolic problem to which the comparison principle applies and constructing relaxed super- and sub-solutions with spatially inhomogeneous decay rates, we show that there exist two constants b⁎(m,a) and K(m,a), for b>b⁎(m,a) and K(m,a)<1, the above density-suppressed motility model admits traveling wave solutions (u,v)(x,t)=:(U,V)(x⋅ξ−ct) in RN along the direction ξ∈SN−1 for all wave speed c≥2a connecting the equilibrium (a/b,a/b) to (0,0), while positive traveling wave solutions will not exist if c<2a. As m→0, we have b⁎(m,a)→0 and K(m,a)→0, our results are well consistent with the relevant results for the well-known Fisher-KPP equation (i.e. the first equation of the above system with γ(v)=1). The main novel idea in the analysis of this paper is the construction of super- and sub-solutions with spatially inhomogeneous (i.e. non-constant) decay rates, in contrast to the constant decay rates used in the literature for reaction-diffusion equations, which was developed to cope with the difficulty caused by the density-dependent nonlinear diffusion in the system. We further discuss the selection of wave patterns and wave speeds for given initial value and use numerical simulations to illustrate that both monotone and non-monotone traveling wavefronts exist depending on whether the motility function γ(v) changes its convexity at v=a/b. Two-dimensional simulations demonstrate that the system can generate outward expanding ring (strip) pattern as observed in the experiment.