Traveling wave in backward and forward parabolic equations from population dynamics

Abstract

This work is concerned with the properties of the traveling wave of the backward and forward parabolic equationwhere D(u) changes its sign once, from negative to positive value, in the interval u ∈ [0, 1] and g(u) is a mono-stable nonlinear reaction term. The existence of infinitely many traveling wave solutions is proven. These traveling waves are parameterized by their wave speed and monotonically connect the stationary states u ≡ 0 and u ≡ 1.2010 Mathematics Subject Classification. Primary: 35K57, 92D25; Secondary: 34B16.

Continuum and lattice models analysis of the aggregation diffusion cell movement

“…There are three categories of mathematical model that researchers have developed to describe this biological phenomena [23]. The first one is discrete model where space is divided into a lattice of points with the variables defined only at the points and time changes in "jump" [5], [9], [21], [22]. The second one is the continuous model where all variables are considered to be defined at every point in space and time changes continuously.…”

“…Discrete models of biophysical processes are of use when we are interested in the behavior of individual cells, as well as their interactions with other cells and the medium which surrounds them [5], [9]. Different derivations may lead to very different behavior of the movement.…”

“…For example, when the discrete population model of [9] is considered: one can show that cells can move toward the high-density region and flee from low-density. But in [5], cells will move toward low-density and flee otherwise. Usually, the cells are considered to be points which move on a lattice according to certain rules.…”

“…Usually, the cells are considered to be points which move on a lattice according to certain rules. These rules can be modified according to the states of neighboring points, such as the density of the neighboring points [5] or the volume and adhesion of the neighboring points [1], [3]. Individual based models have found useful application to many physical systems and even simple rules of interaction can give rise to remarkably complex behavior.…”

“…These are useful when the length scale over which we wish to investigate the phenomenon is much greater than the diameter of the individual elements composing it. These models have been found to be particularly useful in the study of pattern formation in nature, especially the phenomenon of "diffusion driven instability" [1], [3], [5], [9]. The application of hybrid models -where cells are models as discrete entities with their movements being influenced by continuous spatial fields -has also been found to be a useful approach [6].…”