Internatonal Mathematcal Forum, Vol. 6,, no. 7, 8 Soltary and Travelng Wave Solutons to a Model of Long Range ffuson Involvng Flux wth Stablty Analyss Manar A. Al-Qudah Math epartment, Rabgh Faculty of Scence & Art Kng Abdul Azz Unversty, P. O. Box Rabgh 9, Saud Araba manarqudah@ahoo.com Marwan S. Abualrub Math epartment,the Unversty of Jordan P. O. Box 9, Amman-Jordan abualrubms@yahoo.com Abstract A model for nsect dspersal has been consdered, an equlbrum and stablty analyss has been done and the behavor to the soltary and travelng wave solutons of the model are obtaned. Mathematcs Subject Classfcaton: 9B99 Keywords: Soltary wave, travelng wave, stablty analyss. Introducton The dynamcs of populaton has been descrbed usng mathematcal models whch have been very successful n gvng good effect n the study of anmal and human populatons. Ffe [5], consdered reacton and dffuson systems whch are dstrbuted n -dmentonal spaces. Abualrub [], studed dffuson n two dmensonal spaces for whch dffuson s more realstc and applcable n lfe. Also he taled about long range dffuson wth populaton pressure n Planton-
M. A. Al-Qudah and M. S. Abualrub Herbvore populatons. In [], we ncluded long range dffuson nvolvng flux for nsect populaton and taled about the exstence and unqueness of solutons for the consdered model n the space. And we found the requred p and q n smlar approach used n []. In ths paper we study soltary wave soluton usng the generalzed Tanh functon method as n [7]. Also we fnd a travelng wave soluton then, we dscuss stablty of solutons to our model.. Long Range ffuson Involvng Flux Here we consder long range dffuson nvolvng flux n two dmensons whch s gven by: u u u u u u Δ Δ t x u x, f x ; where u u x, t s the nsect populaton densty. Here Δ represents the Laplacan operator and Δ x x., j j u t s the rate of change of the nsect populaton densty, Δ u s the long range dffuson term, where s a small constant, and, are postve constants. u s the nteracton between the males and females of the nsect populaton, and u x s the nstantaneous flux n the x drecton due to molecular dffuson. Here Δ u s the regular dffuson of the nsect populaton. For smplcty tae the one dmensonal case thus, equatons and become u t u xxxx u u u x u x uu xx u x, f x ; where x R 5. Soltary Wave Solutons Frst, we want to fnd an exact soltary wave soluton to equaton usng the generalzed Tanh functon method whch s based on the Rccat equaton whch s gven by y A By 6 dy ; where y, and A, B are constants. dz The man dea of ths method s mentoned n [7]. Lettng u x, t u ξ ; where ξ x ct 7 substtutng equaton 7 nto equaton we obtan c u u u u u uu 8
Soltary and travelng wave solutons 5 whch s a fourth order nonlnear OE. Now, by ntroducng the ndependent varable Y tanhξ the soluton of equaton 8 can be wrtten n the followng form n j u x, t u ξ a j Y 9 j ; where Y tanhξ ;n and a, a, K, an can be determned as n the descrpton of the tanh method whch s mentoned earler. In equaton 9 balancng the term u u wth the term u gves n ; that s the soluton has the form u ξ a ay ay. Substtutng 6 and nto 9 to get the followng dffcult algebrac equaton c a A a BY a AY a BY [ ] a a [ a ay ay ] [ a ay ay ] [ a A aby a AY aby ] [ a a Y a Y ][ a A a ABY 8a ABY a B Y 6a B Y ] 6a A B 6a A 6 ab Y B Y 6a A B Now, collectng the coeffcents to get a system of seven nonlnear algebrac equatons. Solvng the resultng system for a, a, a, c usng mathematca software, we obtan the followng set of solutons AB a a 6B a c We can choose one of the set of solutons for the Rccat equaton, see[] ; namely the followng: u ξ a a tanhξ sechξ Y AB Y AB Y tanhξ sechξ a B Y Concluson. If we assume that,,, then the soluton can be graphed usng mathematca software to be an ellpse; ths means that our soluton s stable snce the orgn s a center pont, see []. 5
6 M. A. Al-Qudah and M. S. Abualrub. Travelng Wave Soluton Now want to see an exact travelng wave soluton. If such soluton exsts t can be wrtten n the followng form: u x, t z, z x ct ;where c s the wave speed. Substtutng equaton n equaton to obtan: c ; where the dfferentaton n equaton s wth respect to z. Snce we are loong for a travelng wave soluton, we have to mpose the followng condtons on : and 5 Remar. The reason for mposng the boundary condtons 5 s because we see a nonnegatve soluton of equaton, for whch at one end, say as z, s at one steady state and as z t s at the other. As done n Murray [6] and from the frst term n the asymptotc wave front soluton to Fsher's equaton we expect that the soluton of equaton together wth the condtons n 5 mght tae the form: z z ae 6 ; where, as n Murray [6], we must assume that and ths wll gve a. Remar. The condtons on gven n equaton 6 and the condton means that the number of nsects n the begnnng of the experment was one unt that s larger than ts number n the mddle of the experment because half of the nsects ded out and at the end of the experment they wll become all dead. 5. Stablty of Solutons Here we study the stablty of solutons to a model of long range dffuson nvolvng flux. Consder the equaton whch can be convert nto a system of OE's as follows: [ c [ ] ] 7
Soltary and travelng wave solutons 7 Then system 7 can be wrtten n the matrx form as follows: [ ] c 8 equaton 8 can be wrtten n the followng form g A r r r 9 the last system s almost lnear system snce x g r has contnuous st, nd, rd, and th partal dervatves, and,,, s the only equlbrum pont of the system. Now, we need to study the type of the equlbrum pont,,, and dscuss ts stablty, to do ths we fnd det c I A, to get the followng equaton:. c Solvng equaton usng mathematca software to get very complcated roots, see []. Remar 5. We can use our result n fndng the soltary wave soluton by generalzed tanh functon method. Then, equaton can be wrtten as: Solvng equaton and assumng that both and are postve, we obtan the followng roots: / / / /
8 M. A. Al-Qudah and M. S. Abualrub Concluson 5. We may assume that, as n the soltary wave soluton. It s obvous that the equlbrum pont wll be unstable spral pont because and have postve real parts. References [] M. S. Abualrub, Long Range ffuson-reacton Model on Populaton ynamcs, ocumenta Mathematca, 998, -. [] M. S. Abualrub, Exstence and Unqueness of Solutons to a ffusve Predator-Prey Model, Journal of Appled Functonal Analyss,. Copyrght 9 Eudoxus press, LLC, 7-. [] M. A. Al- Qudah, and M. S. Abualrub, Exstence of Solutons to a Model of Long Range ffuson Involvng Flux, Journal of Appled Functonal Analyss., 5. Copyrght Eudoxus press, LLC, 7-76. [] M. A. Al- Qudah, Stablty of Soluton for Non-lnear Partal fferental Equatons, Ph. d thess, The Unversty of Jordan, Amman, Jun.. [5] P. C. Ffe, Statonary Patterns for Reacton-ffuson Equatons, Ptman Research Notes n Mathematcs, eds., W. E. Ftzgbbon, III and H. W. Waler. 977, 8-. [6] J.. Murray, Mathematcal Bology, Bomathematcs Texts, Sprnger- Verlage, New Yor, 989. [7] L. Wazzan, More Solton Solutons to the KV and the KV-Burgers Equatons Proc. Pastan Acad. Sc. 7, 7-. Receved: October,