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On the number of births and deaths during an extinction cycle, and the survival of a certain individual in a competition process


Gómez-Corral, Antonio and Lopez-García, M. (2012) On the number of births and deaths during an extinction cycle, and the survival of a certain individual in a competition process. Computers & Mathematics with Applications , 64 (3). pp. 236-259. ISSN 0898-1221

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Competition processes, as discussed by Iglehart (1964) [26] and Reuter (1961) [25], have been frequently used in biology to describe the dynamics of population models involving some kind of interaction among various species. Our interest is in the stochastic model of a competition process analyzed by Ridler-Rowe (1978) [23], which is related to an ecosystem of two species. The ecosystem is closed in the sense that no immigration or emigration is supposed to take place. Individuals compete either directly or indirectly for common resources and, consequently, births and deaths depend on the population sizes of one or both of the species. In this paper, we focus on the number of births and deaths during an extinction cycle. Specifically, we discuss an approximation method inspired from the use of the maximum size distribution, which is equally applicable to the survival of a certain individual. We analyze three models defined in terms of the way individuals within each species are selected to die. Our results are illustrated with reference to simulated data.

Item Type:Article
Uncontrolled Keywords:Bivariate birth and death process; Competition process; Extinction time; Markov chain
Subjects:Sciences > Mathematics > Mathematical statistics
ID Code:16464

P.J. Wangersky, Lotka–Volterra population models, Annual Review of Ecology and Systematics 9 (1978) 189–218.

A.J. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, 1925.

V. Volterra, Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie, Gauthier-Villars, Paris, 1931.

N.T.J. Bailey, The Elements of Stochastic Processes, John Wiley & Sons, New York, 1964.

M.S. Bartlett, Stochastic Population Models in Ecology and Epidemiology, John Wiley & Sons, New York, 1960.

L.J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Pearson Education, New Jersey, 2003.

H. Roozen, Equilibrium and extinction in stochastic population dynamics, Bulletin of Mathematical Biology 49 (1987) 671–696.

S.W. Ali, C. Cosner, Models for the effects of individual size and spatial scale on competition between species in heterogeneous environments, Mathematical Biosciences 127 (1995) 45–76.

S. Bhattacharya, M. Martcheva, Oscillations in a size-structured prey–predator model, Mathematical Biosciences 228 (2010) 31–44.

J.M. Cushing, Two species competition in a periodic environment, Journal of Mathematical Biology 10 (1980) 385–400.

S. Ellner, Convergence to stationary distributions in two-species stochastic competition models, Journal of Mathematical Biology 27 (1989) 451–462.

K. Gopalsamy, Age-specific coexistence in two-species competition, Mathematical Biosciences 61 (1982) 101–122.

S.B. Hsu, S.P. Hubbell, Two predators competing for two species: an analysis of MacArthur’s model, Mathematical Biosciences 47 (1979) 143–171.

S.B. Hsu, On a resource based ecological competition model with interference, Journal of Mathematical Biology 12 (1981) 45–52.

T. Kostova, J. Li, M. Friedman, Two models for competition between age classes, Mathematical Biosciences 157 (1999) 65–89.

B. Li, H.L. Smith, Periodic coexistence of four species competing for three essential resources, Mathematical Biosciences 184 (2003) 115–135.

B. Li, H.L. Smith, Global dynamics of microbial competition for two resources with internal storage, Journal of Mathematical Biology 55 (2007) 481–515.

J. Loman, A graphical solution to a one-predator, two-prey system with apparent competition and mutualism, Mathematical Biosciences 91 (1988) 1–16.

T. Namba, Bifurcation phenomena appearing in the Lotka–Volterra competition equations: a numerical study, Mathematical Biosciences 81 (1986) 191–212.

M.M. Ballyk, G.S.K. Wolkowicz, Classical and resource-based competition: a unifying graphical approach, Journal of Mathematical Biology 62 (2011) 81–109.

D. Tilman, Resource Competition and Community Structure, Princeton University Press, New Jersey, 1982.

A. Gómez-Corral, M. López García, Extinction times and size of the surviving species in a two-species competition process, Journal of Mathematical Biology 64 (2012) 255–289.

C.J. Ridler-Rowe, On competition between two species, Journal of Applied Probability 15 (1978) 457–465.

M.L. Zeeman, Extinction in competitive Lotka–Volterra systems, Proceedings of the American Mathematical Society 123 (1995) 87–96.

G.E.H. Reuter, Competition processes, in: J. Neyman (Ed.), Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume II: Contributions to Probability Theory, University of California Press, Berkeley, 1961, pp. 421–430.

D.L. Iglehart, Multivariate competition processes, The Annals of Mathematical Statistics 35 (1964) 350–361.

L. Billard, Competition between two species, Stochastic Processes and their Applications 2 (1974) 391–398.

M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, second ed., Dover Publications, New York, 1994.

J.R. Artalejo, A. Economou, M.J. López-Herrero, The maximum number of infected individuals in SIS epidemic models: computational techniques and quasi-stationary distributions, Journal of Computational and Applied Mathematics 233 (2010) 2563–2574.

G.J. Kemeny, J.L. Snell, A.W. Knapp, Denumerable Markov Chains, second ed., Springer-Verlag, New York, 1976.

J. Abate, W. Whitt, Numerical inversion of Laplace transforms of probability distributions, ORSA Journal on Computing 7 (1995) 36–43.

G.L. Choudhury, D.M. Lucantoni, W. Whitt, Multidimensional transform inversion with applications to the transient M/G/1 queue, The Annals of Applied Probability 4 (1994) 719–740.

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