Spatial effects in discrete generation population models

Abstract

A framework is developed for constructing a large class of discrete generation, continuous space models of evolving single species populations and finding their bifurcating patterned spatial distributions. Our models involve, in separate stages, the spatial redistribution (through movement laws) and local regulation of the population; and the fundamental properties of these events in a homogeneous environment are found. Emphasis is placed on the interaction of migrating individuals with the existing population through conspecific attraction (or repulsion), as well as on random dispersion. The nature of the competition of these two effects in a linearized scenario is clarified. The bifurcation of stationary spatially patterned population distributions is studied, with special attention given to the role played by that competition.

The Neumann problem for nonlocal nonlinear diffusion equations

“…and variations of it, have been recently widely used to model diffusion processes, see [2], [4], [9], [11], [12], [16], [17], [18], [19] and [20]. As stated in [16], if u(t, x) is thought of as the density of a single population at the point x at time t, and J(x−y) is thought of as the probability distribution of jumping from location y to location x, then the convolution (J * u)(t, x) = R N J(y − x)u(t, y) dy is the rate at which individuals are arriving to position x from all other places and −u(t, x) = − R N J(y − x)u(t, x) dy is the rate at which they are leaving location x to travel to all other sites.…”

“…This consideration, in the absence of external or internal sources, leads immediately to the fact that the density u satisfies equation (1.1). Moreover, a nonlinearity of the form J(x − y) (F (u(y)) − F (u(x))) dy may also appear in population models, see [9] and references therein. Equation (1.1) is called nonlocal diffusion equation since the diffusion of the density u at a point x and time t does not only depend on u(t, x), but on all the values of u in a neighborhood of x through the convolution term J * u.…”

Asymptotic behavior for a nonlocal diffusion equation with absorption and nonintegrable initial data. The supercritical case

“…See for instance [1,2,3,4,9,19]. In particular, nonlocal diffusions are of interest in biological and biomedical problems.…”

The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles

“…Nonlocal evolution equations of the form u t (t, x) = J * u − u(t, x), and variations of it, have been recently widely used to model diffusion processes, see [1], [2], [5], [7], [16], [17], [18], [20], [29] and [30].…”