Banasiak, Jacek2019-05-092019-05-092018-08Banasiak, J. 2018, 'Population models with projection matrix with some negative entries a solution to the natchez paradox', Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, vol. 11, no. 3, pp. 18-28.2071-0216 (print)2308-0256 (online)10.14529/mmp180302http://hdl.handle.net/2263/69077In 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.en© South Ural State University. All rights reserved.Population theoryNatchez civilisationConvex conePerron-Frobenius theoryViability coneArticle