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:
[1]

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]

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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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]

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

[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.  Google Scholar

[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.  Google Scholar

[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]

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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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]

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.  Google Scholar

[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.  Google Scholar

[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]

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.  Google Scholar

[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]

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.  Google Scholar

[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]

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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[57]

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

[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.  Google Scholar

[59]

M. G. Neubert and M. Kot, The subcritical collapse of predator populations in discrete-time predator-prey models,, Mathematical Biosciences, 110 (1992), 45.   Google Scholar

[60]

M. G. Neubert, M. Kot and M. A. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model,, Theoretical Population Biology, 48 (1995), 7.  doi: 10.1006/tpbi.1995.1020.  Google Scholar

[61]

A. J. Nicholson and V. A. Bailey, The balance of animal populations.,, in, 105 (1935), 551.  doi: 10.2307/954.  Google Scholar

[62]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,", Springer, (1980).   Google Scholar

[63]

M. Rambeaud, R. Almeida, G. Pighetti and S. Oliver, Dynamics of leukocytes and cytokines during experimentally induced streptococcus uberis mastitis,, Veterinary Immunology and Immunopathology, 96 (2003), 193.  doi: 10.1016/j.vetimm.2003.08.008.  Google Scholar

[64]

L. Reichl, "A Modern Course in Statistical Physics,", Wiley, (1998).   Google Scholar

[65]

B. A. Reid, U. C. Tauber and J. C. Brunson, Reaction-controlled diffusion: Monte Carlo simulations,, Physical Review E, 68 (2003).   Google Scholar

[66]

M. L. Rosenzweig et al., Paradox of enrichment: destabilization of exploitation ecosystems in ecological time,, Science, 171 (1971), 385.   Google Scholar

[67]

A. Rozenfeld, C. Tessone,E. Albano, and H. Wio, On the influence of noise on the critical and oscillatory behavior of a predator-prey model: coherent stochastic resonance at the proper frequency of the system,, Physics Letters A, 280 (2001), 45.   Google Scholar

[68]

R. Rudnicki and K. Pichor, Influence of stochastic perturbation on prey-predator systems,, Mathematical Biosciences, 206 (2007), 108.   Google Scholar

[69]

M. Schaeffer, S.-J. Han, T. Chtanova, G. G. van Dooren, P. Herzmark, Y. Chen, B. Roysam, B. Striepen and E. A. Robey, Dynamic imaging of t cell-parasite interactions in the brains of mice chronically infected with toxoplasma gondii,, The Journal of Immunology, 182 (2009), 6379.  doi: 10.4049/jimmunol.0804307.  Google Scholar

[70]

J. A. Sherratt, Invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave,, Physica D: Nonlinear Phenomena, 117 (1998), 145.  doi: 10.1016/S0167-2789(97)00317-5.  Google Scholar

[71]

J. A. Sherratt and M. J. Smith, Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models,, Journal of the Royal Society Interface, 5 (2008), 483.  doi: 10.1098/rsif.2007.1327.  Google Scholar

[72]

N. Shnerb, E. Bettelheim, Y. Louzoun, O. Agam and S. Solomon, Adaptation of autocatalytic fluctuations to diffusive noise,, Physical Review E, 63 (2001).  doi: 10.1103/PhysRevE.63.021103.  Google Scholar

[73]

N. M. Shnerb, Y. Louzoun, E. Bettelheim and S. Solomon, The importance of being discrete: Life always wins on the surface,, Proceedings of the National Academy of Sciences, 97 (2000), 10322.  doi: 10.1073/pnas.180263697.  Google Scholar

[74]

J. Skellam, Random dispersal in theoretical populations,, Biometrika, (1951), 196.   Google Scholar

[75]

K. A. Takeuchi, M. Kuroda, H. Chate and M. Sano, Directed percolation criticality in turbulent liquid crystals,, Physical review letters, 99 (2007).  doi: 10.1103/PhysRevLett.99.234503.  Google Scholar

[76]

P.-F. Verhulst, Recherches mathematiques sur la loi daccroissement de la population,, Memoires de lAcademie Royale des Sciences et des Belles-Lettres de Bruxelles, 18 (1845), 1.   Google Scholar

[77]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi,, Mem. Acad, (1926).   Google Scholar

[78]

M. J. Washenberger, M. Mobilia and U. C. Tauber, Influence of local carrying capacity restrictions on stochastic predator-prey models,, Journal of Physics: Condensed Matter, 19 (2007).  doi: 10.1088/0953-8984/19/6/065139.  Google Scholar

[79]

G. Yaari, S. Solomon, M. Schiffer and N. M. Shnerb, Local enrichment and its nonlocal consequences for victim-exploiter metapopulations,, Physica D: Nonlinear Phenomena, 237 (2008), 2553.  doi: 10.1016/j.physd.2008.04.004.  Google Scholar

show all references

References:
[1]

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]

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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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]

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

[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.  Google Scholar

[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.  Google Scholar

[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]

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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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]

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.  Google Scholar

[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.  Google Scholar

[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]

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.  Google Scholar

[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]

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.  Google Scholar

[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]

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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[57]

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

[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.  Google Scholar

[59]

M. G. Neubert and M. Kot, The subcritical collapse of predator populations in discrete-time predator-prey models,, Mathematical Biosciences, 110 (1992), 45.   Google Scholar

[60]

M. G. Neubert, M. Kot and M. A. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model,, Theoretical Population Biology, 48 (1995), 7.  doi: 10.1006/tpbi.1995.1020.  Google Scholar

[61]

A. J. Nicholson and V. A. Bailey, The balance of animal populations.,, in, 105 (1935), 551.  doi: 10.2307/954.  Google Scholar

[62]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,", Springer, (1980).   Google Scholar

[63]

M. Rambeaud, R. Almeida, G. Pighetti and S. Oliver, Dynamics of leukocytes and cytokines during experimentally induced streptococcus uberis mastitis,, Veterinary Immunology and Immunopathology, 96 (2003), 193.  doi: 10.1016/j.vetimm.2003.08.008.  Google Scholar

[64]

L. Reichl, "A Modern Course in Statistical Physics,", Wiley, (1998).   Google Scholar

[65]

B. A. Reid, U. C. Tauber and J. C. Brunson, Reaction-controlled diffusion: Monte Carlo simulations,, Physical Review E, 68 (2003).   Google Scholar

[66]

M. L. Rosenzweig et al., Paradox of enrichment: destabilization of exploitation ecosystems in ecological time,, Science, 171 (1971), 385.   Google Scholar

[67]

A. Rozenfeld, C. Tessone,E. Albano, and H. Wio, On the influence of noise on the critical and oscillatory behavior of a predator-prey model: coherent stochastic resonance at the proper frequency of the system,, Physics Letters A, 280 (2001), 45.   Google Scholar

[68]

R. Rudnicki and K. Pichor, Influence of stochastic perturbation on prey-predator systems,, Mathematical Biosciences, 206 (2007), 108.   Google Scholar

[69]

M. Schaeffer, S.-J. Han, T. Chtanova, G. G. van Dooren, P. Herzmark, Y. Chen, B. Roysam, B. Striepen and E. A. Robey, Dynamic imaging of t cell-parasite interactions in the brains of mice chronically infected with toxoplasma gondii,, The Journal of Immunology, 182 (2009), 6379.  doi: 10.4049/jimmunol.0804307.  Google Scholar

[70]

J. A. Sherratt, Invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave,, Physica D: Nonlinear Phenomena, 117 (1998), 145.  doi: 10.1016/S0167-2789(97)00317-5.  Google Scholar

[71]

J. A. Sherratt and M. J. Smith, Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models,, Journal of the Royal Society Interface, 5 (2008), 483.  doi: 10.1098/rsif.2007.1327.  Google Scholar

[72]

N. Shnerb, E. Bettelheim, Y. Louzoun, O. Agam and S. Solomon, Adaptation of autocatalytic fluctuations to diffusive noise,, Physical Review E, 63 (2001).  doi: 10.1103/PhysRevE.63.021103.  Google Scholar

[73]

N. M. Shnerb, Y. Louzoun, E. Bettelheim and S. Solomon, The importance of being discrete: Life always wins on the surface,, Proceedings of the National Academy of Sciences, 97 (2000), 10322.  doi: 10.1073/pnas.180263697.  Google Scholar

[74]

J. Skellam, Random dispersal in theoretical populations,, Biometrika, (1951), 196.   Google Scholar

[75]

K. A. Takeuchi, M. Kuroda, H. Chate and M. Sano, Directed percolation criticality in turbulent liquid crystals,, Physical review letters, 99 (2007).  doi: 10.1103/PhysRevLett.99.234503.  Google Scholar

[76]

P.-F. Verhulst, Recherches mathematiques sur la loi daccroissement de la population,, Memoires de lAcademie Royale des Sciences et des Belles-Lettres de Bruxelles, 18 (1845), 1.   Google Scholar

[77]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi,, Mem. Acad, (1926).   Google Scholar

[78]

M. J. Washenberger, M. Mobilia and U. C. Tauber, Influence of local carrying capacity restrictions on stochastic predator-prey models,, Journal of Physics: Condensed Matter, 19 (2007).  doi: 10.1088/0953-8984/19/6/065139.  Google Scholar

[79]

G. Yaari, S. Solomon, M. Schiffer and N. M. Shnerb, Local enrichment and its nonlocal consequences for victim-exploiter metapopulations,, Physica D: Nonlinear Phenomena, 237 (2008), 2553.  doi: 10.1016/j.physd.2008.04.004.  Google Scholar

[1]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[2]

Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020396

[3]

Weinan E, Weiguo Gao. Orbital minimization with localization. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 249-264. doi: 10.3934/dcds.2009.23.249

[4]

Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021005

[5]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025

[6]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[7]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

[8]

Wenbin Lv, Qingyuan Wang. Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term. Evolution Equations & Control Theory, 2021, 10 (1) : 25-36. doi: 10.3934/eect.2020040

[9]

Yuyuan Ouyang, Trevor Squires. Some worst-case datasets of deterministic first-order methods for solving binary logistic regression. Inverse Problems & Imaging, 2021, 15 (1) : 63-77. doi: 10.3934/ipi.2020047

[10]

Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154

[11]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[12]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[13]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128

[14]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391

[15]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362

[16]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[17]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227

[18]

Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084

[19]

Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020388

[20]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (5)

[Back to Top]