Geodesic
Population models with projection matrix with some negative entries~--- a solution to the Natchez paradox
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 11 (2018) no. 3, pp. 18-28

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In this note we consider the population the model of which, derived on the basis of ethnographical accounts, includes a projection matrix with both positive and negative entries. Interpreting the eventually negative trajectories as representing the collapse of the population, we use some classical tools from convex analysis to determine a cone containing the initial conditions that give rise to the persistence of both the population and its social structure.
Keywords: population theory; Natchez civilisation; convex cone; Perron–Frobenius theory; viability cone. 
@article{VYURU_2018_11_3_a1,
     author = {J. Banasiak},
     title = {Population models with projection matrix with some negative entries~--- a solution to the {Natchez} paradox},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {18--28},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2018_11_3_a1/}
}
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J. Banasiak. Population models with projection matrix with some negative entries~--- a solution to the Natchez paradox. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 11 (2018) no. 3, pp. 18-28. http://geodesic.mathdoc.fr/item/VYURU_2018_11_3_a1/