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Exact travelling wave solutions of threespecies competitiondiffusion systems
Steady states in hierarchical structured populations with distributed states at birth
1.  Department of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA, United Kingdom 
2.  Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 532010413 
References:
[1] 
A. S. Ackleh, K. Deng and S. Hu, A quasilinear hierarchical sizestructured model: Wellposedness andapproximation,, Appl. Math. Optim., 51 (2005), 35. Google Scholar 
[2] 
A. S. Ackleh and K. Ito, Measurevalued solutions for a hierarchically sizestructured population,, J. Differential Equations, 217 (2005), 431. Google Scholar 
[3] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2^{nd} edition,, Pure and Applied Mathematics (Amsterdam), 140 (2003). Google Scholar 
[4] 
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,, SIAM Rev., 18 (1976), 620. Google Scholar 
[5] 
W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, "OneParameter Semigroups of Positive Operators,", Lecture Notes in Mathematics, 1184 (1986). Google Scholar 
[6] 
À. Calsina and J. Saldań, Asymptotic behavior of a model ofhierarchically structured population dynamics,, J. Math. Biol., 35 (1997), 967. Google Scholar 
[7] 
À. Calsina and J. Saldañ, Basic theory for a class of models ofhierarchically structured population dynamics with distributed states in the recruitment,, Math. Models Methods Appl. Sci., 16 (2006), 1695. Google Scholar 
[8] 
J. M. Cushing, The dynamics of hierarchical agestructured populations,, J. Math. Biol., 32 (1994), 705. Google Scholar 
[9] 
J. M. Cushing, "An Introduction to Structured Population Dynamics,", CBMSNSF Regional Conference Series in Applied Mathematics, 71 (1998). Google Scholar 
[10] 
K. Deimling, "Nonlinear Functional Analysis,", SpringerVerlag, (1985). Google Scholar 
[11] 
K.J. Engel and R. Nagel, "OneParameter Semigroups for Linear Evolution Equations,", Graduate Texts in Mathematics, 194 (2000). Google Scholar 
[12] 
J. Z. Farkas, D. M. Green and P. Hinow, Semigroup analysis ofstructured parasite populations,, Math. Model. Nat. Phenom., 5 (2010), 94. Google Scholar 
[13] 
J. Z. Farkas and T. Hagen, Stability and regularity results for asizestructured population model,, J. Math. Anal. Appl., 328 (2007), 119. Google Scholar 
[14] 
J. Z. Farkas and T. Hagen, Asymptotic analysis of a sizestructured cannibalism model with infinite dimensional environmental feedback,, Commun. Pure Appl. Anal., 8 (2009), 1825. Google Scholar 
[15] 
J. Z. Farkas and T. Hagen, Hierarchical sizestructured populations: The linearized semigroup approach,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 639. Google Scholar 
[16] 
J. Z. Farkas and P. Hinow, On a sizestructured twophase population model with infinite statesatbirth,, Positivity, 14 (2010), 501. Google Scholar 
[17] 
A. Grabosch and H. J. A. M. Heijmans, Cauchy problems with statedependent time evolution,, Japan J. Appl. Math., 7 (1990), 433. Google Scholar 
[18] 
M. E. Gurtin and R. C. MacCamy, Nonlinear agedependent population dynamics,, Arch. Rational Mech. Anal., 54 (1974), 281. Google Scholar 
[19] 
S. M. Henson and J. M. Cushing, Hierarchical models of intraspecific competition: Scramble versus contest,, J. Math. Biol., 34 (1996), 755. Google Scholar 
[20] 
S. R.J. Jang and J. M. Cushing, A discrete hierarchical model ofintraspecific competition,, J. Math. Anal. Appl., 280 (2003), 102. Google Scholar 
[21] 
S. R.J. Jang and J. M. Cushing, Dynamics of hierarchical models in discretetime,, J. Difference Equ. Appl., 11 (2005), 95. Google Scholar 
[22] 
N. Kato, A principle of linearized stability for nonlinear evolution equations,, Trans. Amer. Math. Soc., 347 (1995), 2851. Google Scholar 
[23] 
J. A. J. Metz and O. Diekmann, Age dependence,, in, 68 (1983). Google Scholar 
[24] 
J. Prüss, On the qualitative behavior of populations with agespecific interactions,, Comput. Math. Appl., 9 (1983), 327. Google Scholar 
[25] 
S. L. Tucker and S. O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables,, SIAM J. Appl. Math., 48 (1988), 549. Google Scholar 
[26] 
Ch. Walker, Global bifurcation of positive equilibria in nonlinear population models,, J. Diff. Eq., 248 (2010), 1756. Google Scholar 
[27] 
G. F. Webb, "Theory of Nonlinear AgeDependent Population Dynamics,", Monographs and Textbooks in Pure and Applied Mathematics, 89 (1985). Google Scholar 
[28] 
K. Yosida, "Functional Analysis,", Reprint of the sixth (1980) edition, (1980). Google Scholar 
show all references
References:
[1] 
A. S. Ackleh, K. Deng and S. Hu, A quasilinear hierarchical sizestructured model: Wellposedness andapproximation,, Appl. Math. Optim., 51 (2005), 35. Google Scholar 
[2] 
A. S. Ackleh and K. Ito, Measurevalued solutions for a hierarchically sizestructured population,, J. Differential Equations, 217 (2005), 431. Google Scholar 
[3] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2^{nd} edition,, Pure and Applied Mathematics (Amsterdam), 140 (2003). Google Scholar 
[4] 
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,, SIAM Rev., 18 (1976), 620. Google Scholar 
[5] 
W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, "OneParameter Semigroups of Positive Operators,", Lecture Notes in Mathematics, 1184 (1986). Google Scholar 
[6] 
À. Calsina and J. Saldań, Asymptotic behavior of a model ofhierarchically structured population dynamics,, J. Math. Biol., 35 (1997), 967. Google Scholar 
[7] 
À. Calsina and J. Saldañ, Basic theory for a class of models ofhierarchically structured population dynamics with distributed states in the recruitment,, Math. Models Methods Appl. Sci., 16 (2006), 1695. Google Scholar 
[8] 
J. M. Cushing, The dynamics of hierarchical agestructured populations,, J. Math. Biol., 32 (1994), 705. Google Scholar 
[9] 
J. M. Cushing, "An Introduction to Structured Population Dynamics,", CBMSNSF Regional Conference Series in Applied Mathematics, 71 (1998). Google Scholar 
[10] 
K. Deimling, "Nonlinear Functional Analysis,", SpringerVerlag, (1985). Google Scholar 
[11] 
K.J. Engel and R. Nagel, "OneParameter Semigroups for Linear Evolution Equations,", Graduate Texts in Mathematics, 194 (2000). Google Scholar 
[12] 
J. Z. Farkas, D. M. Green and P. Hinow, Semigroup analysis ofstructured parasite populations,, Math. Model. Nat. Phenom., 5 (2010), 94. Google Scholar 
[13] 
J. Z. Farkas and T. Hagen, Stability and regularity results for asizestructured population model,, J. Math. Anal. Appl., 328 (2007), 119. Google Scholar 
[14] 
J. Z. Farkas and T. Hagen, Asymptotic analysis of a sizestructured cannibalism model with infinite dimensional environmental feedback,, Commun. Pure Appl. Anal., 8 (2009), 1825. Google Scholar 
[15] 
J. Z. Farkas and T. Hagen, Hierarchical sizestructured populations: The linearized semigroup approach,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 639. Google Scholar 
[16] 
J. Z. Farkas and P. Hinow, On a sizestructured twophase population model with infinite statesatbirth,, Positivity, 14 (2010), 501. Google Scholar 
[17] 
A. Grabosch and H. J. A. M. Heijmans, Cauchy problems with statedependent time evolution,, Japan J. Appl. Math., 7 (1990), 433. Google Scholar 
[18] 
M. E. Gurtin and R. C. MacCamy, Nonlinear agedependent population dynamics,, Arch. Rational Mech. Anal., 54 (1974), 281. Google Scholar 
[19] 
S. M. Henson and J. M. Cushing, Hierarchical models of intraspecific competition: Scramble versus contest,, J. Math. Biol., 34 (1996), 755. Google Scholar 
[20] 
S. R.J. Jang and J. M. Cushing, A discrete hierarchical model ofintraspecific competition,, J. Math. Anal. Appl., 280 (2003), 102. Google Scholar 
[21] 
S. R.J. Jang and J. M. Cushing, Dynamics of hierarchical models in discretetime,, J. Difference Equ. Appl., 11 (2005), 95. Google Scholar 
[22] 
N. Kato, A principle of linearized stability for nonlinear evolution equations,, Trans. Amer. Math. Soc., 347 (1995), 2851. Google Scholar 
[23] 
J. A. J. Metz and O. Diekmann, Age dependence,, in, 68 (1983). Google Scholar 
[24] 
J. Prüss, On the qualitative behavior of populations with agespecific interactions,, Comput. Math. Appl., 9 (1983), 327. Google Scholar 
[25] 
S. L. Tucker and S. O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables,, SIAM J. Appl. Math., 48 (1988), 549. Google Scholar 
[26] 
Ch. Walker, Global bifurcation of positive equilibria in nonlinear population models,, J. Diff. Eq., 248 (2010), 1756. Google Scholar 
[27] 
G. F. Webb, "Theory of Nonlinear AgeDependent Population Dynamics,", Monographs and Textbooks in Pure and Applied Mathematics, 89 (1985). Google Scholar 
[28] 
K. Yosida, "Functional Analysis,", Reprint of the sixth (1980) edition, (1980). Google Scholar 
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