Aggregative Movement and Front Propagation for Bi-Stable Population Models

Abstract

Front propagation for the aggregation-diffusion-reaction equation [Formula: see text] is investigated, where f is a bi-stable reaction-term and D(v) is a diffusion coefficient with changing sign, modeling aggregating-diffusing processes. We provide necessary and sufficient conditions for the existence of traveling wave solutions and classify them according to how or if they attain their equilibria at finite times. We also show that the dynamics can exhibit the phenomena of finite speed of propagation and/or finite speed of saturation.

Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves

“…Interestingly, for the strong Allee effect, shock-fronted travelling wave solutions are obtained. Following arguments presented in [39], we show that smooth travelling wave solutions cannot be obtained for certain types of nonlinear diffusivity functions and the strong Allee effect.…”

“…This approach also leads to a new kind of Allee effect, which we call the reverse Allee effect, where the growth rate is inhibited at high density. Although some of the PDEs that we consider have been investigated previously [15,[18][19][20][21][22][23][24][25][26][27][28][29][30][36][37][38][39][40][41], they have never been linked together before using a single modelling framework. The presence of Allee kinetics allows for the more realistic description of biological and ecological phenomena, as the standard reaction-diffusion model with Fisher kinetics predicts either the population tending to extinction everywhere or the spread of the population in the form of a travelling wave.…”

“…Positive-negative nonlinear diffusivity function. For the case where F A (C) has exactly one zero on the interval 0 ≤ C ≤ 1 at C = ω, Maini et al [39] examine the existence of travelling wave solutions, and provide the necessary conditions for existence,…”

“…whereF (C) = −F (1 − C), on the interval ω < C ≤ A 1 . The final necessary and sufficient condition from Maini et al [39] for the existence of travelling wave solutions is that the minimum wave speed for Equation (35), v * 1 , is greater than, or equal to, the minimum wave speed for Equation (36), v * 2 . On the interval 0 ≤ C < ω, Equation (21) has a strictly positive F (C), where F (C) ≤ D i , and strong Allee kinetics.…”

“…The sign of the wave speed indicates whether successful invasion occurs, and the magnitude of the wave speed provides an estimate of how quickly a population invades or recedes. More complicated descriptions of invasion processes with either Fisher or Allee kinetics and density-dependent nonlinear diffusion have been proposed, with the motivation of describing spatial aggregation or segregation [36][37][38][39][40][41].…”