Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation

Abstract

Abstract. In this paper we study the equationwith homogeneous boundary conditions of the Neumann type, as a model of aggregating population with a migration rate determined by φ, and total birth and mortality rates characterized by f . We will show that the aggregating mechanism induced by φ(u) allows the survival of a species in danger of extinction. Numerical simulations suggest that the solutions stabilize asymptotically in time to a not necessarily homogeneous stationary solution. This is shown to be the case for a particular version of the function φ(u).

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

“…Moreover, by using (75), similar to the proof of (46), we have J(u(•, t 0 )) < J(u 0 ), i.e. u(•, t 0 ) ∈ J J(u0) (see (17)). Then u(•, t 0 ) ∈ N J(u0) (since N J(u0) = N ∩ J J(u0) ) and then u(•, t 0 ) H 1 0 ≤ Λ J(u0) (see (19)).…”

“…The concept "pseudo-parabolic" was proposed by Showalter and Ting in 1970 in the paper [20], where the linear case was considered. Pseudoparabolic equations describe a variety of important physical processes, such as the seepage of homogeneous fluids through a fissured rock [1], the unidirectional propagation of nonlinear, dispersive, long waves [2,23], and the aggregation of populations [17].…”

Well-posedness of the solution of the fractional semilinear pseudo-parabolic equation

“…The pseudo-parabolic equation is used in diverse fields such as seepage theory of homogeneous liquid through cracked rock [3] (the coefficient of the third-order term represents the degree of cracks in the rock, and its decrease corresponds to the increase in the degree of cracking), the unidirectional propagation of nonlinear dispersive long waves [4,5] (where u is amplitude or curl), and the description of racial migration [6] (where u is the population density). Because of the wide range of applications of pseudo-parabolic equations, they attract great attention of mathematicians.…”

Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity

“…Pseudo-parabolic equations describe a variety of important physical processes, such as the seepage of homogeneous fluids through a fissured rock [2], the unidirectional propagation of nonlinear, dispersive, long waves [4,37], and the aggregation of populations [31]. The pseudo-parabolic equation…”