Internal null stabilization for some diffusive models of population dynamics

We investigate the large-time behavior of the solutions to some Fisher-type models with nonlocal terms describing the dynamics of biological populations with diffusion, logistic term and migration. Two types of logistic terms are taken into account. A necessary condition and a sufficient condition for the internal null stabilizability of the solution to a Fisher model with nonlocal term are provided. In case of null stabilizability (with state constraints) a feedback stabilizing control of harvesting type is proposed. The rate of stabilization corresponding to the feedback stabilizing control is dictated by the principal eigenvalue to a certain linear but not selfadjoint operator. A large principal eigenvalue leads to a fast stabilization to zero. Another goal is to approximate this principal eigenvalue using a method suggested by the theoretical result concerning the large time behavior of the solution to a certain Fisher model with a special logistic term. An iterative method to improve the position (by translations) of the support of the feedback stabilizing control in order to get a larger principal eigenvalue, and consequently a faster stabilization to zero is derived. Numerical tests illustrating the effectiveness of the theoretical results are given. 2013 Elsevier Inc. All rights reserved. 1. Setting of the problems After the pioneering work of Fisher [18] the mathematical modeling of spatially structured populations has been carefully analyzed, given rise to a flourishing literature on the diffusive biological models (see [14,27,28]). The local/nonlocal intra or interspecific interactions of one or several interacting populations species were taken into account by several authors (see e.g. [7,15,19,20]). The following Fisher-type model describes the dynamics of a single biological population species which is free to move in an isolated habitat X: @tyðx; tÞ dDyðx; tÞ 1⁄4 aðxÞyðx; tÞ kðxÞyðx; tÞ; x 2 X; t > 0: Here X R (N 2 f2;3g) is a bounded domain, with a smooth enough boundary @X; yðx; tÞ is the population density at position x and moment t; d > 0 is the diffusion coefficient, aðxÞ is the natural growth rate at position x, and kðxÞy2 is a logistic term. The logistic term describes a local intraspecific competition for resources. If a migration phenomenon occurs then the population dynamics is described by 0096-3003/$ see front matter 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.03.125 ⇑ Corresponding author at: Faculty of Mathematics, ‘‘Alexandru Ioan Cuza’’ University of Ias i, 700506 Ias i, Romania. E-mail addresses: [email protected] (L.-I. Anit a), [email protected] (S. Anit a), [email protected] (V. Arnăutu). Applied Mathematics and Computation 219 (2013) 10231–10244