Dynamics of Nonlinear Parabolic Equations with Cosymmetry (original) (raw)
Computation of a Family of Non-cosymmetrical Equilibria in a System of Nonlinear Parabolic Equations
Computing Supplementa, 2003
We study a system of two nonlinear parabolic equations with cosymmetry and find a continuous family of equilibria with nonconstant spectrum. Basing on the finite-difference approach, we develop a numerical method for solving the partial differential equations and calculating the continuous family of non-cosymmetrical equilibria. An application of the proposed system to population kinetics is demonstrated. Numerically computed families are presented. The dependence on model parameters is discussed.
A quasilinear parabolic model for population evolution
Differential Equations & Applications, 2012
A quasilinear parabolic problem is investigated. It models the evolution of a single population species with a nonlinear diffusion and a logistic reaction function. We present a new treatment combining standard theory of monotone operators in L 2 (Ω) with some orderpreserving properties of the evolutionary equation. The advantage of our approach is that we are able to obtain the existence and long-time asymptotic behavior of a weak solution almost simultaneously. We do not employ any uniqueness results; we rely on the uniqueness of the minimal and maximal solutions instead. At last, we answer the question of (long-time) survival of the population in terms of a critical value of a spectral parameter.
Degenerate parabolic Problems in population dynamics
Japan Journal of Applied Mathematics, 1985
We investigate the coexistente of prey-predator or competing species, subject to density dependent diffusion in an inhomogeneous habitat. It is proven that coexistence arises in suitable domains, where favourable conditions are satisfied. Support properties and attractivity of the resulting stationary solutions are investigated.
Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat
Discrete & Continuous Dynamical Systems - B, 2019
We explore an approach based on the theory of cosymmetry to model interaction of predators and prey in a two-dimensional habitat. The model under consideration is formulated as a system of nonlinear parabolic equations with spatial heterogeneity of resources and species. Firstly, we analytically determine system parameters, for which the problem has a nontrivial cosymmetry. To this end, we formulate cosymmetry relations. Next, we employ numerical computations to reveal that under said cosymmetry relations, a one-parameter family of steady states is formed, which may be characterized by different proportions of predators and prey. The numerical analysis is based on the finite difference method (FDM) and staggered grids. It allows to follow the transformation of spatial patterns with time. Eventually, the destruction of the continuous family of equilibria due to mistuned parameters is analyzed. To this end, we derive the so-called cosymmetric selective equation. Investigation of the selective equation gives an insight into scenarios of local competition and coexistence of species, together with their connection to the cosymmetry relations. When the cosymmetry relation is only slightly violated, an effect we call 'memory on the lost family' may be observed. Indeed, in this case, a slow evolution takes place in the vicinity of the lost states of equilibrium.
Proceedings of the American Mathematical Society, 2000
We establish existence and uniqueness of solutions for a general class of nonlocal nonlinear evolution equations. An application of this theory to a class of nonlinear reaction-diffusion problems that arise in population dynamics is presented. Furthermore, conditions on the initial population density for this class of problems that result in finite time extinction or persistence of the population is discussed. Numerical evidence corroborating our theoretical results is given.
Nonlinear parametric excitation of an evolutionary dynamical system
2012
Abstract Nonlinear parametric excitation refers to the nonlinear analysis of a system of ordinary differential equations with periodic coefficients. In contrast to linear parametric excitation, which offers determinations of the stability of equilibria, nonlinear parametric excitation has as its goal the structure of the phase space, as given by a portrait of the Poincare map.
Analysis and simulation of a coupled hyperbolic/parabolic model problem
Journal of Numerical Mathematics, 2005
We investigate a periodic, one-dimensional, linear, and degenerate advection-diffusion equation. The problem is hyperbolic in a subinterval and parabolic in the complement, and the boundary conditions only impose the periodicity of the advective-diffusive flux to ensure mass conservation. Following Gastaldi and Quarteroni (1989), a condition is added to select the "physically acceptable" solution as the limit of vanishing viscosity solutions, namely the continuity of the solution at the parabolic to hyperbolic interface. Using this condition, we establish the well-posedness of the Cauchy problem in the framework of the evolution linear semi-groups theory. We also discuss the regularity of the solution when the initial condition is too rough to be in the domain of the evolution operator. Then, we present reference solutions obtained using the additional interface condition. These solutions can be used to test the robustness of numerical schemes. Finally, we discuss results obtained with an upwind scheme, a finite volume box-scheme, and a local discontinuous Galerkin method. The three schemes (which do not enforce explicitly the additional interface condition) select automatically the physically acceptable solution, the two latter schemes being more accurate.