2013, 10(3): 523-550. doi: 10.3934/mbe.2013.10.523

Diffusion rate determines balance between extinction and proliferation in birth-death processes

1. 

Department of Mathematics, Bar Ilan University, Ramat Gan, Israel, Israel, Israel

Received  July 2012 Revised  September 2012 Published  April 2013

We here study spatially extended catalyst induced growth processes. This type of process exists in multiple domains of biology, ranging from ecology (nutrients and growth), through immunology (antigens and lymphocytes) to molecular biology (signaling molecules initiating signaling cascades). Such systems often exhibit an extinction-proliferation transition, where varying some parameters can lead to either extinction or survival of the reactants.
    When the stochasticity of the reactions, the presence of discrete reactants and their spatial distribution is incorporated into the analysis, a non-uniform reactant distribution emerges, even when all parameters are uniform in space.
    Using a combination of Monte Carlo simulation and percolation theory based estimations; the asymptotic behavior of such systems is studied. In all studied cases, it turns out that the overall survival of the reactant population in the long run is based on the size and shape of the reactant aggregates, their distribution in space and the reactant diffusion rate. We here show that for a large class of models, the reactant density is maximal at intermediate diffusion rates and low or zero at either very high or very low diffusion rates. We give multiple examples of such system and provide a generic explanation for this behavior. The set of models presented here provides a new insight on the population dynamics in chemical, biological and ecological systems.
Citation: Hilla Behar, Alexandra Agranovich, Yoram Louzoun. Diffusion rate determines balance between extinction and proliferation in birth-death processes. Mathematical Biosciences & Engineering, 2013, 10 (3) : 523-550. doi: 10.3934/mbe.2013.10.523
References:
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show all references

References:
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A. Abbas and A. Lichtman, Cellular and medical immunology,, Saunders, (2003), 243. Google Scholar

[2]

A. Agranovich and Y. Louzoun, Predator-prey dynamics in a uniform medium lead to directed percolation and wave-train propagation,, Physical Review E, 85 (2012). doi: 10.1103/PhysRevE.85.031911. Google Scholar

[3]

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[4]

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[5]

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[6]

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[8]

H. Behar, N. Shnerb and Y. Louzoun, Balance between absorbing and positive fixed points in resource consumption models,, Physical Review E, 86 (2012). doi: 10.1103/PhysRevE.86.031146. Google Scholar

[9]

A. A. Berryman, The orgins and evolution of predator-prey theory,, Ecology, (1992), 1530. doi: 10.2307/1940005. Google Scholar

[10]

C. J. Briggs and M. F. Hoopes, Stabilizing effects in spatial parasitoid-host and predator-prey models: a review,, Theoretical Population Biology, 65 (2004), 299. doi: 10.1016/j.tpb.2003.11.001. Google Scholar

[11]

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[12]

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[13]

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[14]

A. M. de Roos, E. McCauley and W. G. Wilson, Pattern formation and the spatial scale of interaction between predators and their prey,, Theoretical Population Biology, 53 (1998), 108. Google Scholar

[15]

U. Dieckmann and R. Law, The dynamical theory of coevolution: a derivation from stochastic ecological processes,, Journal of Mathematical Biology, 34 (1996), 579. Google Scholar

[16]

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[17]

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[18]

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[19]

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[20]

R. A. Fisher, The wave of advance of advantageous genes,, Annals of Human Genetics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[21]

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[22]

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[23]

G. Gardiner, "Handbook of Stochastic Processes for Physics,", 2002., (). Google Scholar

[24]

G. F. Gause et al., Experimental analysis of vito volterras mathematical theory of the struggle for existence,, Science, 79 (1934). Google Scholar

[25]

P. Grassberger, On phase transitions in schlogls second model,, Zeitschrift fur Physik B Condensed Matter, 47 (1982), 365. doi: 10.1007/BF01313803. Google Scholar

[26]

P. Grassberger, Directed percolation in 2+ 1 dimensions,, Journal of Physics A: Mathematical and General, 22 (1989), 3673. doi: 10.1088/0305-4470/22/17/032. Google Scholar

[27]

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[28]

P. Grassberger, F. Krause and T. von der Twer, A new type of kinetic critical phenomenon,, Journal of Physics A: Mathematical and General, 17 (1999). doi: 10.1088/0305-4470/17/3/003. Google Scholar

[29]

A. Hastings, Global stability of two species systems,, Journal of Mathematical Biology, 5 (1977), 399. doi: 10.1007/BF00276109. Google Scholar

[30]

U. Hershberg, Y. Louzoun, H. Atlan and S. Solomon, Hiv time hierarchy: winning the war while, loosing all the battles,, Physica A: Statistical Mechanics and its Applications, 289 (2001), 178. doi: 10.1016/S0378-4371(00)00466-0. Google Scholar

[31]

H. Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states,, Advances In Physics, 49 (2000), 815. doi: 10.1080/00018730050198152. Google Scholar

[32]

A. R. Ives, B. J. Cardinale and W. E. Snyder, A synthesis of subdisciplines: Predator-prey interactions, and biodiversity and ecosystem functioning,, Ecology Letters, 8 (2004), 102. doi: 10.1111/j.1461-0248.2004.00698.x. Google Scholar

[33]

C. Janeway and P. Travers, "Immunobiology: The Immune System in Health and Disease,", Garland Publ., (1997). Google Scholar

[34]

H.-K. Janssen, On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state,, Zeitschrift für Physik B Condensed Matter, 42 (1981), 151. doi: 10.1007/BF01319549. Google Scholar

[35]

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[36]

Y. Kan-On, Fisher wave fronts for the lotka-volterra competition model with diffusion,, Nonlinear Analysis: Theory, 28 (1997), 145. doi: 10.1016/0362-546X(95)00142-I. Google Scholar

[37]

C. KELLY, D. CARVALHO and T. TOME, Self-organized patterns of coexistence out of a predator-prey cellular automaton,, International Journal of Modern Physics C, 17 (2006), 1647. doi: 10.1142/S0129183106010005. Google Scholar

[38]

M. Kenneth, P. Travers and M. Walport, Janeways immunobiology,, Open ISBN, (2007). Google Scholar

[39]

W. Kermack and A. McKendrick, Contributions to the mathematical theory of epidemics,, Bulletin of Mathematical Biology, 53 (1991), 33. doi: 10.1098/rspa.1927.0118. Google Scholar

[40]

H. Kesten and V. Sidoravicius, Branching random walk with catalysts,, Electron. J. Probab., 8 (2003), 1. doi: 10.1214/EJP.v8-127. Google Scholar

[41]

A. Kolmogorov, I. Petrovsky and N. Piskunov, Etude de lquation de la diffusion avec croissance de la quantit de matiere et son applicationa un probleme biologique,, Mosc. Univ. Bull. Math, 1 (1937), 1. Google Scholar

[42]

M. Kot, "Elements of Mathematical Ecology,", Cambridge University Press, (2001). doi: 10.1017/CBO9780511608520. Google Scholar

[43]

R. Law, M. J. Plank, A. James and J. L. Blanchard, Size-spectra dynamics from stochastic predation and growth of individuals,, Ecology, 90 (2009), 802. doi: 10.1890/07-1900.1. Google Scholar

[44]

P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species,, Biometrika, (1960), 219. Google Scholar

[45]

A. L. Lin, B. A. Mann, G. Torres-Oviedo, B. Lincoln, J. Käs and H. L. Swinney, Localization and extinction of bacterial populations under inhomogeneous growth conditions,, Biophysical Journal, 87 (2004), 75. doi: 10.1529/biophysj.103.034041. Google Scholar

[46]

A. J. Lotka, Undamped oscillations derived from the law of mass action,, Journal of the American Chemical Society, 42 (1920), 1595. Google Scholar

[47]

A. J. Lotka, Contribution to the energetics of evolution,, Proceedings of the National Academy of Sciences of the United States of America, 8 (1922). Google Scholar

[48]

A. J. Lotka, "Elements of Physical Biology,", Williams & Wilkins Baltimore, (1925). Google Scholar

[49]

Y. Louzoun, S. Solomon, H. Atlan and I. Cohen, The emergence of spatial complexity in the immune system,, Physica A, 297 (2001), 242. Google Scholar

[50]

Y. Louzoun, S. Solomon, H. Atlan, I. Cohen, et al., Microscopic discrete proliferating components cause the self-organized emergence of macroscopic adaptive features in biological systems,, preprint, (2000). Google Scholar

[51]

Y. Louzoun, S. Solomon, H. Atlan and I. R. Cohen, Modeling complexity in biology,, Physica A: Statistical Mechanics and its Applications, 297 (2001), 242. doi: 10.1016/S0378-4371(01)00201-1. Google Scholar

[52]

Y. Louzoun, S. Solomon, H. Atlan and I. R. Cohen, Proliferation and competition in discrete biological systems,, Bulletin of Mathematical Biology, 65 (2003), 375. Google Scholar

[53]

Y. Louzoun, S. Solomon, J. Goldenberg and D. Mazursky, World-size global markets lead to economic instability,, Artificial Life, 9 (2003), 357. doi: 10.1162/106454603322694816. Google Scholar

[54]

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