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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[3]

A. Agranovich, Y. Louzoun, N. Shnerb and S. Moalem, Catalyst-induced growth with limited catalyst lifespan and competition,, Journal of Theoretical Biology, 241 (2006), 307. doi: 10.1016/j.jtbi.2005.11.031.

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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.

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

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G. Domokos and I. Scheuring, Discrete and continuous state population models in a noisy world,, Journal of Theoretical Biology, 227 (2004), 535.

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Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment,, Journal of Differential Equations, 203 (2004), 331. doi: 10.1016/j.jde.2004.05.010.

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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.

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[44]

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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.

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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.

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show all references

References:
[1]

A. Abbas and A. Lichtman, Cellular and medical immunology,, Saunders, (2003), 243.

[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.

[3]

A. Agranovich, Y. Louzoun, N. Shnerb and S. Moalem, Catalyst-induced growth with limited catalyst lifespan and competition,, Journal of Theoretical Biology, 241 (2006), 307. doi: 10.1016/j.jtbi.2005.11.031.

[4]

P. W. Anderson, Absence of diffusion in certain random lattices,, Physical Review, 109 (1958).

[5]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence,, Journal of Theoretical Biology, 139 (1989), 311. doi: 10.1016/S0022-5193(89)80211-5.

[6]

K. E. Atkinson, "An Introduction to Numerical Analysis,", John Wiley & Sons, (2008).

[7]

M. F. Bachmann, B. Odermatt, H. Hengartner and R. M. Zinkernagel, Induction of long-lived germinal centers associated with persisting antigen after viral infection,, The Journal of Experimental Medicine, 183 (1996), 2259. doi: 10.1084/jem.183.5.2259.

[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.

[9]

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

[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.

[11]

S. R. Broadbent and J. M. Hammersley, Percolation processes i. crystals and mazes,, in, 53 (1957), 629. doi: 10.1017/S0305004100032680.

[12]

D. H. Busch and E. G. Pamer, T lymphocyte dynamics during listeria monocytogenes infection,, Immunology Letters, 65 (1999), 93. doi: 10.1016/S0165-2478(98)00130-8.

[13]

G. Cai and Y. Lin, Stochastic analysis of predator-prey type ecosystems,, Ecological Complexity, 4 (2007), 242. doi: 10.1016/j.ecocom.2007.06.011.

[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.

[15]

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

[16]

U. Dieckmann, P. Marrow and R. Law, Evolutionary cycling in predator-prey interactions: Population dynamics and the red queen,, Journal of Theoretical Biology, 176 (1995), 91. doi: 10.1006/jtbi.1995.0179.

[17]

U. Dobramysl and U. C. Tauber, Spatial variability enhances species fitness in stochastic predator-prey interactions,, Physical Review Letters, 101 (2008). doi: 10.1103/PhysRevLett.101.258102.

[18]

G. Domokos and I. Scheuring, Discrete and continuous state population models in a noisy world,, Journal of Theoretical Biology, 227 (2004), 535.

[19]

Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment,, Journal of Differential Equations, 203 (2004), 331. doi: 10.1016/j.jde.2004.05.010.

[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.

[21]

H. Freedman and Y. Takeuchi, Predator survival versus extinction as a function of dispersal in a predator-prey model with patchy environment,, Applicable Analysis, 31 (1989), 247. doi: 10.1080/00036818908839829.

[22]

H. Freedman and G. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited,, Bulletin of Mathematical Biology, 48 (1986), 493. doi: 10.1016/S0092-8240(86)90004-2.

[23]

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

[24]

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

[25]

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

[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.

[27]

P. Grassberger, H. Chate and G. Rousseau, Spreading in media with long-time memory,, Physical Review E, 55 (1997). doi: 10.1103/PhysRevE.55.2488.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

[35]

I. Jensen and R. Dickman, Series analysis of the generalized contact process,, Physica A: Statistical Mechanics and its Applications, 203 (1994), 175. doi: 10.1016/0378-4371(94)90151-1.

[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.

[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.

[38]

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

[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.

[40]

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

[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.

[42]

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

[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.

[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.

[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.

[46]

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

[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).

[48]

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

[49]

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

[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).

[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.

[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.

[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.

[54]

T. R. Malthus, An essay on the principle of population, as it affects the future improvement of society: With remarks on the speculations of mr. Godwin, mr. Condorcet, and other writers,, New York: Penguin, (1798).

[55]

M. Mobilia, I. T. Georgiev and U. C. Täuber, Fluctuations and correlations in lattice models for predator-prey interaction,, Physical Review E, 73 (2006). doi: 10.1103/PhysRevE.73.040903.

[56]

M. Mobilia, I. T. Georgiev and U. C. Täuber, Spatial stochastic predator-prey models,, Stochastic models in biological sciences, (2008), 253. doi: 10.4064/bc80-0-16.

[57]

M. Murray, "Jd Mathematical Biology,", 1989., ().

[58]

D. R. Nelson and N. M. Shnerb, Non-hermitian localization and population biology,, Physical Review E, 58 (1998), 1383. doi: 10.1103/PhysRevE.58.1383.

[59]

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