Population Models with Projection Matrix with Some Negative Entries - a Solution to the Natchez Paradox

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.

References

1. Cushing J.M. An Introduction to Structured Population Dynamics. Philadelphia, SIAM, 1998.
2. Luenberger D.G. Introduction to Dynamic Systems: Theory, Models, and Applications. N.Y., Wiley, 1979.
3. Noutsos D. On Perron-Frobenius Property of Matrices Having Some Negative Entries. Linear Algebra and its Applications, 2006, vol. 412, no. 2, pp. 132-153.
4. Schneider H., Vidyasagar M. Cross-Positive Matrices. SIAM Journal on Numerical Analysis, 1970, vol. 7, no. 4, pp. 508-519.
5. Swanton J.R. Indian Tribes of the Lower Mississippi Valley and Adjacent Coast of the Gulf of Mexico. N.Y., Dover Publications, 2013.
6. Hart C.A Reconstruction of the Natchez Social Structure. American Anthropologist, 1943, vol. 45, pp. 374-386.
7. Fischer J.L. Solutions for the Natchez Paradox. Ethnology, 1964, vol. 3, no. 1, pp. 53-65.
8. White D.R., Murdock G.P., Scaglion R. Natchez Class and Rank Reconsidered. Ethnology, 1971, vol. 10, no. 4, pp. 369-388.
9. Gritzmann P., Klee V., Tam. B.-S. Cross-Positive Matrices Revisited. Linear Algebra and Its Applications, 1995, vol. 223, pp. 285-305.
10. Fenchel W., Blackett D.W. Convex Сones, Sets, and Functions. Princeton University, Department of Mathematics, Logistics Research Project, 1953.
11. Rockafellar R.T. Convex Analysis. Princeton, Princeton University Press, 2015.