Bounds on the critical times for the Fisher-KPP equation

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

BOUNDS ON THE CRITICAL TIMES FOR THE GENERAL FISHER–KPP EQUATION

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented.

Dynamic transitions and bifurcations of 1D reaction-diffusion equations: The Self-adjoint case

This paper deals with the classification of transition phenomena in the most basic dissipative system possible, namely the 1D reaction diffusion equation. The emphasis is on the relation between the linear and nonlinear terms and the effect of the boundaries which influence the first transitions. We consider the cases where the linear part is self-adjoint with 2nd order and 4th order derivatives which is the case which most often arises in applications. We assume that the nonlinear term depends on the function and its first derivative which is basically the semilinear case for the second order reaction-diffusion system. As for the boundary conditions, we consider the typical Dirichlet, Neumann and periodic boundary settings. In all the cases, the equations admit a trivial steady state which loses stability at a critical parameter. We aim to classify all possible transitions and bifurcations that take place. Our analysis shows that these systems display all three types of transitions: continuous, jump and mixed and display transcritical, supercritical bifurcations with bifurcated states such as finite equilibria, circle of equilibria, and slowly rotating limit cycle. Many applications found in the literature are basically corollaries of our main results. We apply our results to classify the first transitions of the Chaffee-Infante equation, the Fisher-KPP equation, the Kuramoto Sivashinsky equation and the Swift-Hohenberg equation.

$m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection

In this paper, we investigate a $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection. Considering the influence of advection term and initial conditions on the long time behavior of solutions, we obtain spreading-vanishing dichotomy, spreading-transition-vanishing trichotomy, and vanishing happens with the coefficient of advection term in small amplitude, medium-sized amplitude and large amplitude, respectively. Then, the appropriate parameters are selected in the simulation to intuitively show the corresponding theoretical results. Moreover, the wave-spreading and wave-vanishing cases of the solutions are observed in our study.