%0 Book Section %1 MR2223351 %A Gourley, S. A. %A Wu, J. %B Nonlinear dynamics and evolution equations %C Providence, RI %D 2006 %I Amer. Math. Soc. %K epidemic invasions nonlocal_dispersal survey reaction-diffusion travelling_wave %P 137--200 %T Delayed non-local diffusive systems in biological invasion and disease spread %V 48 %X A couple of years ago the authors surveyed results on the biological modeling, dynamics analysis and numerical simulation of nonlocal spatial effects induced by time delays in reaction-diffusion models for single species population (published in both Russian and English see S. A. Gourley, J. W.-H. So and J. H. Wu, Sovrem. Mat. Fundam. Napravl. 1 (2003), 84--120 (electronic); MR2129130 (2006b:35182)). Due to the rapid development in the field since then, the authors present a new survey to update the progress towards modeling and analysis of long time behaviors of related biological and epidemiological systems where the non-locality arises exclusively due to the interaction of time lag of feedback and the spatial movement of biological species. It starts with the derivation of a reaction-diffusion equation with nonlocal delayed nonlinearity. Variations of the model are then discussed, such as systems of ordinary delay differential equations and lattice delay differential equations with global interaction (when space is discrete), and systems of neutral functional differential equations and hyperbolic-parabolic equations with non-local delayed reaction (when time lags for spatial movement are considered). More complicated models are described along with the results for the asymptotic behaviors of solutions to the model equations. All these show how interweaving diffusion and delay results in various interesting and challenging mathematical problems. Some unsolved problems are addressed along with issues for further research. Over one hundred references are cited in this survey article.

@incollection{MR2223351, abstract = {A couple of years ago the authors surveyed results on the biological modeling, dynamics analysis and numerical simulation of nonlocal spatial effects induced by time delays in reaction-diffusion models for single species population (published in both Russian and English [see S. A. Gourley, J. W.-H. So and J. H. Wu, Sovrem. Mat. Fundam. Napravl. 1 (2003), 84--120 (electronic); MR2129130 (2006b:35182)]). Due to the rapid development in the field since then, the authors present a new survey to update the progress towards modeling and analysis of long time behaviors of related biological and epidemiological systems where the non-locality arises exclusively due to the interaction of time lag of feedback and the spatial movement of biological species. It starts with the derivation of a reaction-diffusion equation with nonlocal delayed nonlinearity. Variations of the model are then discussed, such as systems of ordinary delay differential equations and lattice delay differential equations with global interaction (when space is discrete), and systems of neutral functional differential equations and hyperbolic-parabolic equations with non-local delayed reaction (when time lags for spatial movement are considered). More complicated models are described along with the results for the asymptotic behaviors of solutions to the model equations. All these show how interweaving diffusion and delay results in various interesting and challenging mathematical problems. Some unsolved problems are addressed along with issues for further research. Over one hundred references are cited in this survey article. }, added-at = {2009-11-12T04:19:12.000+0100}, address = {Providence, RI}, author = {Gourley, S. A. and Wu, J.}, biburl = {https://www.bibsonomy.org/bibtex/21887983e32134003642cc3638fa1c451/peter.ralph}, booktitle = {Nonlinear dynamics and evolution equations}, description = {MR: Selected Matches for: Items authored by or related to Gourley, Stephen A.}, interhash = {698b351253c8235a0ff16fba7f61796e}, intrahash = {1887983e32134003642cc3638fa1c451}, keywords = {epidemic invasions nonlocal_dispersal survey reaction-diffusion travelling_wave}, mrclass = {35K57 (34K60 35-02 35B40 35K55 92D25 92D30)}, mrnumber = {MR2223351 (2007a:35079)}, mrreviewer = {Si Ning Zheng}, pages = {137--200}, publisher = {Amer. Math. Soc.}, series = {Fields Inst. Commun.}, timestamp = {2013-09-12T22:23:01.000+0200}, title = {Delayed non-local diffusive systems in biological invasion and disease spread}, volume = 48, year = 2006 }