The scalar wave equation in a two-dimensional semi-infinite wave guide is considered. The recently proposed Hagstrom–Warburton (H–W) local high-order absorbing boundary conditions (ABCs), which are based on a modification of the Higdon ABCs, are presented in this context. The P-order ABC involves the free parameters 0 < aj ≤ 1, for j = 0, 1, …, P, which have to be chosen. The choice aj = 1 for all j is shown to be satisfactory, in general, although not necessarily optimal. The optimal choice of the parameters is discussed via both theoretical analysis and numerical experiments. In addition, an adaptive scheme which controls the time-varying values of P and aj is presented and tested.

A spatially explicit metapopulation model with positive density-dependent migration is analysed. We obtained conditions under which a previously stable system can be driven to instability caused by a density-dependent migration mechanism. The stability boundary depends on the rate of increase of the number of migrants on each site at local equilibrium, on the intrinsic rate of increase at local level, on the number of patches, and on topological aspects regarding the connectivity between patches. A concrete example is presented illustrating the dynamics on the dispersal-induced unstable regime.

During the past few years there has been a gro"'ing intercst in studics of population dynamics in spatial models. In a recent paper, Solé and Gamarra ( 1998) obtained a very simple condition for the stability ofthe synchroniz:ed state (ali local populations oscillate in synchrony) involving just two kcy parametcrs: the fraction of individuais that disperse to nearby patches per gcneration and thc local exponcntial rate of separation between close orbits (Lyapunov expoent) of the uncoupled population dynamics. The ma in restriction to the above result is that it was established only for an ensemble of two local populations. Thc mctapopulation siz:e (number of subpopulations) ean play a dccisive role in thc cnsemblc persistence. In fact, Commins et ai. ( 1992) have shown by means of numerical simulations in a host-parasitoid mctapopulation modcl that the probability of eX1inction of the ensemble decreases with the metapopulation siz:e Of course. this result is of great importance to conservation of species issues, since it rdatcs cxtinction likelyhood with the number ofhabitat fTagments forming the whole population Given the importance of thcse results we pro pose in this paper to CX1end Solé and Gamarra's result to a more general metapopulation We consider a spatially cxplicit mctapopulation model with interaction among the two nearest neighbors to relate, with a simple mathcmatical expression, chaos in the local, uncoupled populations, thc dcgrcc of interaction among patches, siz:e of the metapopulation and the stability of the synchronized atractor We have obtained a necessary condition for the stability of the synchronizcd state involving the same key parametcrs of Solé and Gamarra's introduci ng only thc si:z:c of metapopuJation 11. Since synchronism is strongl y correlated with extinction, our results can provide useful information on factors leading to JX)pulation extinction Refcrences Commins. 11 N., M P llasscl, and R M May (1 992) Thc spaual dynamics of ltost parasitoid systcms. J. Solé, R V and J P G Gamarra ( 1998) Chaos, d1spersal and extinction in coupled ecosystcms J. 711eor. r r E L

We present a discrete model for a metapopulation of a single species with overlapping generations. Based on the dynamical behavior of the system in absence of dispersal, we have shown that a migration mechanism which depends only on age can not stabilize a previously unstable homogeneous equilibrium, but can drive a stable uncoupled system to instability if the migration rules are strongly related to age structure.

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