AM  Vol.4 No.10 B , October 2013
Managing Populations with Unimodal Dynamics
ABSTRACT

In this work, we analyzed the impact of interventions on populations which exhibit unimodal dynamics. The six landmarks that characterize the shape of the unimodal reproduction curve f ( x ) of the difference equation, X n+1 = f ( X n ) , are defined and used in order to examine and determine the behavior of dynamics of populations. By using the Li-Yorke criterion for determination of chaos we propose a qualitative intervention rule that can be applied without any explicit population equation. This proposed strategy for intervention brings out many interesting behaviors in population dynamics. A qualitative decision rule can be applied with a straight edge without any population equation and therefore offers a robust strategy for the management of populations.


Cite this paper
R. Levins, T. Awerbuch and H. Park, "Managing Populations with Unimodal Dynamics," Applied Mathematics, Vol. 4 No. 10, 2013, pp. 85-91. doi: 10.4236/am.2013.410A2009.
References
[1]   T. Awerbuch, R. Levins and M. Predescu, “Co-Dynamics of Consciousness and Populations of the Dengue Vector in a Spraying Interventions System Modeled by Difference Equations,” Far East Journal of Applied Mathematics, Vol. 37, No. 2, 2009, pp. 215-228.

[2]   M. C. Akogbeto, R. Djouaka and H. Noukpo, “Use of Agricultural Insecticides in Benin,” Bulletin de la Société de Pathologie Exotique, Vol. 98, No. 5, 2005, pp. 400405.

[3]   Food and Agriculture Organization of the United Nations, “International Code of Conduct on the Distribution and Use of Pesticides,” 2002. http://www.fao.org/waicent/faoinfo/agricult/agp/agpp/pesticid/code/download/code.pdf

[4]   D. Tilman and D. Wedin, “Oscillations and Chaos in the Dynamics of a Perennial Grass,” Nature, Vol. 353, 1991, pp. 653-655. http://dx.doi.org/10.1038/353653a0

[5]   T. Awerbuch, A. Kiszewski and R. Levins, “Surprise, Nonlinearity and Complex Behavior,” In: Martens and Mcmichael, Eds., Health Impacts of Global Environmental Change: Concepts and Methods, 2002, pp. 96-102.

[6]   R. A. Desharnais, B. Dennis, J. M. Cushing, S. M. Henson and R. F. Costantino, “Chaos and Population Control of Insect Outbreaks,” Ecology Letters, Vol. 4, No. 3, 2001, pp. 229-235. http://dx.doi.org/10.1046/j.1461-0248.2001.00223.x

[7]   E. A. Grove, G. Ladas, R. Levins and C. Puccia, “Oscillation and Stability in Models of a Perennial Grass,” Proceedings of. Dynamic Systems & Applications, Vol. 1, 1994, pp. 87-91.

[8]   R. Levins, “The Butterfly ex Machine,” In: R. S. Singh and C. B. Krimbas, Eds., Evolutionary Genetics: From Molecules to Morphology, Cambridge University Press, Cambridge, 2000, pp. 529-543.

[9]   Y. Kang, “Pre-Images of Invariant Sets of a DiscreteTime Two-Species Competition Model,” Journal of Difference Equations and Applications, Vol. 18, No. 10, 2012, pp. 1709-1733.

[10]   T. Y. Li and J. Yorke, “Period Three Implies Chaos,” The American Mathematical Monthly, Vol. 82, No. 10, 1975, pp. 985-992. http://dx.doi.org/10.2307/2318254

[11]   A. Weersink, W. Deen and S. Weaver, “Defining and Measuring Economic Threshold Levels,” Canadian Journal of Agricultural Economics, Vol. 39, No. 4, 1991, pp. 619625. http://dx.doi.org/10.1111/j.1744-7976.1991.tb03613.x

[12]   M. Krkosek, M. A. Lewis, A. Morton, L. N. Frazer and J. Volpe, “Epizootic of Wild Fish Induced by Farm Fish,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 103, No. 42, 2006, pp. 15506-15510. http://dx.doi.org/10.1073/pnas.0603525103

 
 
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