2012, 9(3): 461-485. doi: 10.3934/mbe.2012.9.461

Stochastic models for competing species with a shared pathogen

1. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States

2. 

Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, United States

Received  December 2011 Revised  March 2012 Published  July 2012

The presence of a pathogen among multiple competing species has important ecological implications. For example, a pathogen may change the competitive outcome, resulting in replacement of a native species by a non-native species. Alternately, if a pathogen becomes established, there may be a drastic reduction in species numbers. Stochastic variability in the birth, death and pathogen transmission processes plays an important role in determining the success of species or pathogen invasion. We investigate these phenomena while studying the dynamics of deterministic and stochastic models for $n$ competing species with a shared pathogen. The deterministic model is a system of ordinary differential equations for $n$ competing species in which a single shared pathogen is transmitted among the $n$ species. There is no immunity from infection, individuals either die or recover and become immediately susceptible, an SIS disease model. Analytical results about pathogen persistence or extinction are summarized for the deterministic model for two and three species and new results about stability of the infection-free state and invasion by one species of a system of $n-1$ species are obtained. New stochastic models are derived in the form of continuous-time Markov chains and stochastic differential equations. Branching process theory is applied to the continuous-time Markov chain model to estimate probabilities for pathogen extinction or species invasion. Finally, numerical simulations are conducted to explore the effect of disease on two-species competition, to illustrate some of the analytical results and to highlight some of the differences in the stochastic and deterministic models.
Citation: Linda J. S. Allen, Vrushali A. Bokil. Stochastic models for competing species with a shared pathogen. Mathematical Biosciences & Engineering, 2012, 9 (3) : 461-485. doi: 10.3934/mbe.2012.9.461
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show all references

References:
[1]

Mathematical Modelling: Theory and Applications, 22, Springer, Dordrecht, The Netherlands, 2007.  Google Scholar

[2]

Stoch. Anal. Appl., 26 (2008), 274-297.  Google Scholar

[3]

Prentice Hall, Upper Saddle River, NJ, 2007. Google Scholar

[4]

2nd edition, CRC Press, Boca Raton, FL, 2011.  Google Scholar

[5]

Int. J. Numer. Anal. Modeling, 2 (2005), 329-344.  Google Scholar

[6]

Phil. Trans. R. Soc. Lond. B, 314 (1986), 533-570. doi: 10.1098/rstb.1986.0072.  Google Scholar

[7]

Reprint of the 1964 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990.  Google Scholar

[8]

Appl. Statist., 13 (1965), 2-8.  Google Scholar

[9]

Am. Nat., 139 (1992), 1131-1150. doi: 10.1086/285379.  Google Scholar

[10]

Technical Report ORST-MATH-11-02, Oregon State Univ., 2011. Citation URL: http://ir.library.oregonstate.edu/xmlui/handle/1957/13738/. Google Scholar

[11]

Proc. Natl. Acad. Sci. U. S. A., 104 (2007), 5473-5478. Google Scholar

[12]

J. Theor. Biol., 187 (1997), 95-109. doi: 10.1006/jtbi.1997.0418.  Google Scholar

[13]

Oxford Univ. Press, Oxford, 2006. Google Scholar

[14]

Am. Nat., 164 (2004), S64-S78. doi: 10.1086/424681.  Google Scholar

[15]

Mem. Am. Math. Soc., 156 (2002), viii+118 pp.  Google Scholar

[16]

Ann. Appl. Probab., 19 (2009), 477-496. doi: 10.1214/08-AAP590.  Google Scholar

[17]

Ann. Appl. Probab., 7 (1997), 10-45.  Google Scholar

[18]

J. Anim. Ecol., 70 (2001), 1053-1061. doi: 10.1046/j.0021-8790.2001.00558.x.  Google Scholar

[19]

Academic Press, Inc., Boston, MA, 1992.  Google Scholar

[20]

J. Theor. Biol., 185 (1997), 345-356. doi: 10.1006/jtbi.1996.0309.  Google Scholar

[21]

Math. Biosci., 231 (2011), 144-158. doi: 10.1016/j.mbs.2011.02.011.  Google Scholar

[22]

Ph.D thesis, Oregon State Univ., Corvallis, OR, 2009. Google Scholar

[23]

Math. Comput. Model., 37 (2003), 87-108. doi: 10.1016/S0895-7177(03)80008-0.  Google Scholar

[24]

Nonlinear Anal. Real World Appl., 10 (2009), 723-744. doi: 10.1016/j.nonrwa.2007.11.005.  Google Scholar

[25]

Die Grundlehren der Mathematischen Wissenschaften, Bd. 119, Springer-Verlag, Berlin, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1963.  Google Scholar

[26]

Ecol. Lett., 9 (2006), 1253-1271. doi: 10.1111/j.1461-0248.2006.00964.x.  Google Scholar

[27]

SIAM Rev., 50 (2008), 347-368. doi: 10.1137/060666457.  Google Scholar

[28]

Oxford Univ. Press, Oxford, (2006), 2-27. Google Scholar

[29]

Am. Nat., 126 (1985), 196-211. doi: 10.1086/284409.  Google Scholar

[30]

Trends Ecol. Evol., 13 (1998), 387-390. Google Scholar

[31]

Wiley Series in Probability and Mathematical Statistics Applied Probability and Statistics, Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1975.  Google Scholar

[32]

2nd edition, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[33]

Ecology, 80 (1999), 2442-2448. Google Scholar

[34]

Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992.  Google Scholar

[35]

Ann. Appl. Probab., 16 (2006), 1385-1410. doi: 10.1214/105051606000000394.  Google Scholar

[36]

Ann. Appl. Probab., 16 (2006), 448-474. doi: 10.1214/105051605000000782.  Google Scholar

[37]

Ph.D thesis, Oregon State Univ., Corvallis, OR, 2012.  Google Scholar

[38]

Special issue on mathematics and the social and biological sciences, SIAM J. Appl. Math., 29 (1975), 243-253. doi: 10.1137/0129022.  Google Scholar

[39]

Trends Ecol. Evol., 16 (2001), 295-300. doi: 10.1016/S0169-5347(01)02144-9.  Google Scholar

[40]

Ph.D thesis, Texas Tech Univ., Lubbock, TX, 2006. Google Scholar

[41]

in "Differential & Difference Eqns. Appl.," Hindawi Publ. Corp., New York, (2006), 775-785.  Google Scholar

[42]

Math. Med. Biol., 24 (2007), 17-34. doi: 10.1093/imammb/dql021.  Google Scholar

[43]

Modern Analytic and Computational Methods in Science and Mathematics, No. 34 , American Elsevier Publishing Co., Inc., New York, 1971.  Google Scholar

[44]

Bull. Math. Biol., 73 (2011), 2707-2730.  Google Scholar

[45]

Ann. Appl. Probab., 9 (1999), 1226-1259.  Google Scholar

[46]

J. Theor. Biol., 200 (1999), 111-118. doi: 10.1006/jtbi.1999.0982.  Google Scholar

[47]

The University Series in Mathematics, Plenum Press, New York, 1987.  Google Scholar

[48]

Ph.D thesis, Institut für Mathematik der Unversität Potsdam, Germany, 2010. Google Scholar

[49]

Math. Biosci. Eng., 3 (2006), 219-235.  Google Scholar

[50]

Ecol. Lett., 3 (2000), 10-14. Google Scholar

[51]

Ecol. Lett., 6 (2003), 189-196. Google Scholar

[52]

Math. Biosci., 180 (2002), 29-48.  Google Scholar

[53]

SIAM J. Appl. Dyn. Sys., 3 (2004), 601-619.  Google Scholar

[54]

Math. Biosci., 174 (2001), 111-131. doi: 10.1016/S0025-5564(01)00081-5.  Google Scholar

[55]

Biometrika, 42 (1955), 116-122. doi: 10.2307/2333427.  Google Scholar

[56]

Trans. Am. Math. Soc., 355 (2003), 713-734. doi: 10.1090/S0002-9947-02-03103-3.  Google Scholar

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