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Investigating Anthropogenic Mammoth Extinction with Mathematical Models Michael Frank North Carolina State University Spora: A Journal of Biomathematics Volume 1 | Issue 1 Article 3 2015 Investigating Anthropogenic Mammoth Extinction with Mathematical Models Michael Frank North Carolina State University Anneliese Slaton Mary Baldwin College Teresa Tinta University of Maryland Eastern Shore Alex Capaldi Valparaiso University, [email protected] Follow this and additional works at: https://ir.library.illinoisstate.edu/spora Recommended Citation Frank, Michael; Slaton, Anneliese; Tinta, Teresa; and Capaldi, Alex (2015) "Investigating Anthropogenic Mammoth Extinction with Mathematical Models," Spora: A Journal of Biomathematics: Vol. 1: Iss.1, . DOI: https://doi.org/10.30707/SPORA1.1Frank Available at: https://ir.library.illinoisstate.edu/spora/vol1/iss1/3 This Mathematics Research is brought to you for free and open access by ISU ReD: Research and eData. It has been accepted for inclusion in Spora: A Journal of Biomathematics by an authorized editor of ISU ReD: Research and eData. For more information, please contact [email protected]. Investigating Anthropogenic Mammoth Extinction with Mathematical Models Cover Page Footnote Research was conducted during the 2013 Valparaiso Experience in Research by Undergraduate Mathematicians, which was supported by the National Science Foundation (NSF DMS-1262852). We would also like to thank the anonymous reviewers for their helpful comments and suggestions. This mathematics research is available in Spora: A Journal of Biomathematics: https://ir.library.illinoisstate.edu/spora/vol1/iss1/3 Investigating Anthropogenic Mammoth Extinction with Mathematical Models Michael Frank1, Anneliese Slaton2, Teresa Tinta3, Alex Capaldi4,* *Correspondence: Abstract Prof. Alex Capaldi, Dept. of One extinction hypothesis of the Columbian mammoth (Mammuthus columbi), called Mathematics and Statistics, overkill, theorizes that early humans overhunted the animal. We employ two different ap- Valparaiso University, proaches to test this hypothesis mathematically: analyze the stability of the equilibria of a 1900 Chapel Dr., Valparaiso, 2D ordinary differential equations (ODE) system and develop a metapopulation differential IN 46383, USA equations model. The 2D ODE system is a modified predator-prey model that also includes [email protected] migration. The metapopulation model is a spatial expansion of the first model on a rectan- gular grid. Using this metapopulation system, we model the migration of humans into North America and the response in the mammoth population. These approaches show evidence that human-mammoth interaction would have affected the extinction of the Columbian mammoth during the late Pleistocene. Keywords: mammoth, extinction, predator-prey model, metapopulation, population dynam- ics, differential equations, mathematical modeling, ecology 1 Introduction mastodons by humans was enough to affect the extinction of other species in the ecosystem [13]. The Columbian mammoth (Mammuthus columbi), a rel- We have chosen to study the Columbian mammoth ative of modern day elephants (genera Elephas and Lox- specifically because its range was completely contained odonta), lived in North America during the Pleistocene, within North America and encompassed nearly all of the an era of time that spanned from approximately 1.8 mil- continental United States. In contrast, the woolly mam- lion to 11,700 years ago. The Columbian mammoth went moth (Mammuthus primigenius), while found in North extinct at the end of the last ice age (in the Late Pleis- America, was also found outside it, predominantly in tocene) about 13,000 years ago. At this time, there was a Siberia [15]. In this manner, we can treat the human mass extinction of megafauna on every continent except migration into North America as an invasive species. Africa and Southeast Asia [9, 10, 15]. In North America, In 2001, Alroy conducted a stochastic difference equa- 32 out of 41 large prey species went extinct during this tion simulation of the end-Pleistocene megafaunal extinc- event [1]. tion which incorporated the population dynamics of in- There are four dominant hypotheses as to why vading humans and 41 large mammalian herbivores. He the megafaunal extinction took place: disease, climate found that, using some realistic combinations of parame- change, a meteorite shower, and overkill. It is likely that ter values, the model accurately predicted as many as 32 a combination of these factors (whether in series or paral- of the fates (extinction vs survival) of the prey species. In lel) lead to the extinction [2, 17]. Our project focuses on most instances, extinction of species occurred in multiple overkill, a theory first proposed by Martin in 1973 [14]. temporal waves, and interestingly, proboscideans (includ- Martin proposed that the migration of modern humans ing Columbian mammoth) did not die off in the primary (Homo sapiens) from Asia into North America (over the wave. Alroy concludes that anthropogenic extinction was Bering land bridge) around 13,500 years ago was the lead- unavoidable for many of the megafauna [1]. We hope ing cause of megafaunal extinction. Despite the argument to strengthen this conclusion through additional evidence that the lack of kill-sites discovered in North America dis- from deterministic differential equation models. proves this theory [12], it has continued to remain a valid Previous models of prehistoric scenarios have used dif- extinction theory. Some supporters of Martin's theory ferential equations to model the interaction and competi- have even theorized that the extinction of mammoths and tion between early Homo sapien and Neanderthal (Homo 1Department of Mathematics, North Carolina State University, Raleigh, NC, 2Department of Mathematics, Mary Baldwin College, Staunton, VA, 3Department of Mathematics, University of Maryland Eastern Shore, Princess Anne, MD, 4Department of Mathematics and Statistics, Valparaiso University, Valparaiso, IN www.sporajournal.org 2015 Volume 1(1) page 8 Investigating Anthropogenic Mammoth Extinction with Mathematical Models Frank, Slaton, Tinta, Capaldi neanderthalensis) populations and were able to conclude accounts for the predators being generalists with alterna- that even a competitive advantage as small as 1% would tive food supplies [7]. lead to the demise of the Neanderthal [11]. Mathematical A migration term in each equation was included to research into population dynamics has resulted in the cre- reflect the movement of both species into and out of the ation and analysis of models of invading species as well as patch. This migration term assumes that humans will im- conditions for extinction or assimilition [4]. Other work migrate into an area with a high population of mammoths with models of interacting populations analyzed multi- (when M > NM ) and emigrate out of an area with a stability and extinction of various forms of predator-prey lower mammoth population (when M < NM ). Similarly, systems of differential equations [5, 6]. mammoths will emigrate out of an area with a large pop- In this paper, we attempt to determine if human over- ulation of humans (when H > NH ) and immigrate into hunting alone could have caused the extinction of the areas where there are few humans (when H < NH ). Columbian mammoth. To do this, we use two different Our model is given by the equations approaches to mathematically model human-mammoth interactions. First, we use an ordinary differential equa- dH H = rH H 1 − tions system to model the two populations interacting dt KH in a relatively small area, a single \patch." We analyze a M 2H M + H + c H − 1 ; (1a) the stability of the single-patch model's equilibria to de- M 2 + b2 H N termine the long-term behavior of the system. Second, M we expand the single-patch model into a metapopulation dM M M differential equations model to describe the two species = rM M 1 − − 1 interacting across a larger area. dt KM A 2 This paper is organized as follows: In Section 2 we aM M H H − 2 2 + cM M 1 − : (1b) present and analyze the single-patch differential equation M + b NH model. We expand the 2D system into a metapopulation model with migration between the patches in Section 3. Each of the twelve model parameters and the assumed Section 4 outlines our results. We conclude with a dis- ranges of their values are given in Table 1. We obtained cussion in Section 5. literature values for appropriate ranges for the first seven model parameters. Five parameters values required as- sumptions on their ranges. The predation saturation 2 Single-Patch ODE Model constant b was assumed to be between 1 and 10 mam- moths, as a small human population is likely to be easily 2.1 Model sustained for a long time by very few mammoths. The relative migration rates cH and cM were assumed to be We begin by modeling the human-mammoth interac- generally on the same order of magnitude as birth rates tion in a single sufficiently small area as a Rosenzweig- for humans and an order of magnitude higher for mam- MacArthur predator-prey ordinary differential equation moths. Finally, the external population sizes NH and NM system [21]. Thus, the model is composed of two first- had their assumed ranges chosen such that the mammoths order ODEs, one describing the changes in the human would leave the patch when humans were at least halfway population H(t) and another describing the dynamics of to their carrying capacity and humans would leave the the mammoth population M(t). patch when mammoths were at most below the Allee The growth term for each population incorporates lo- threshold A. gistic growth
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