Constructive effects of environmental noise in an excitable prey–predator plankton system with infected prey

Abstract An excitable model of fast phytoplankton and slow zooplankton dynamics is considered for the case of lysogenic viral infection of the phytoplankton population. The phytoplankton population is split into a susceptible ( S ) and an infected ( I ) part. Both parts grow logistically, limited by a common carrying capacity. Zooplankton ( Z ) is grazing on susceptibles and infected, following a Holling-type III functional response. The local analysis of the S – I – Z differential equations yields a number of stationary and/or oscillatory regimes and their combinations. Correspondingly interesting is the behaviour under multiplicative noise, modelled by stochastic differential equations. The external noise can enhance the survival of susceptibles and infected, respectively, that would go extinct in a deterministic environment. In the parameter range of excitability, noise can induce prey–predator oscillations and coherence resonance (CR). In the spatially extended case, synchronized global oscillations can be observed for medium noise intensities. Higher values of noise give rise to the formation of stationary spatial patterns.

[1]  Hopf bifurcations and slow-fast cycles in a model of plankton dynamics , 2002 .

[2]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[3]  Iwasa,et al.  Noise-induced regularity of spatial wave patterns in subalpine abies forests , 1998, Journal of theoretical biology.

[4]  G V Osipov,et al.  Coherence resonance in excitable and oscillatory systems: the essential role of slow and fast dynamics. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  Shigeru Shinomoto,et al.  Phase Transitions and Their Bifurcation Analysis in a Large Population of Active Rotators with Mean-Field Coupling , 1988 .

[6]  David E. Sigeti,et al.  Pseudo-regular oscillations induced by external noise , 1989 .

[7]  I. Noy-Meir,et al.  Stability of Grazing Systems: An Application of Predator-Prey Graphs , 1975 .

[8]  Alexander B. Neiman,et al.  Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments , 2003 .

[9]  E. Beltrami.  Unusual algal blooms as excitable systems: The case of “brown-tides” , 1996 .

[10]  J. Fuhrman,et al.  Bacterial viruses in coastal seawater: lytic rather than lysogenic production , 1994 .

[11]  Alexander B. Neiman,et al.  Noise-Enhanced Phase Synchronization in Excitable Media , 1999 .

[12]  T. O. Carroll,et al.  Modeling the role of viral disease in recurrent phytoplankton blooms , 1994 .

[13]  Chris Wilcox,et al.  Mobile parasitoids may restrict the spatial spread of an insect outbreak , 1997 .

[14]  Steven W. Wilhelm,et al.  Viruses as potential regulators of regional brown tide blooms caused by the alga,Aureococcus anophagefferens , 2004 .

[15]  Natalia Glushkova,et al.  A mathematical model of the ultrasonic detection of three-dimensional cracks☆ , 2002 .

[16]  G. Maruyama.  Continuous Markov processes and stochastic equations , 1955 .

[17]  A. Hastings,et al.  Unexpected spatial patterns in an insect outbreak match a predator diffusion model , 1997, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[18]  S. Habib,et al.  Stochastic PDEs: convergence to the continuum? , 2001 .

[19]  C. Suttle.  Viruses in the sea , 2005, Nature.

[20]  S. Carpenter,et al.  Catastrophic regime shifts in ecosystems: linking theory to observation , 2003 .

[21]  A. Longtin.  AUTONOMOUS STOCHASTIC RESONANCE IN BURSTING NEURONS , 1997 .

[22]  M. Cirone,et al.  Noise-induced effects in population dynamics , 2002 .

[23]  U. Feudel,et al.  Bloom dynamics in a seasonally forced phytoplankton-zooplankton model: Trigger mechanisms and timing effects , 2006 .

[24]  G. Bratbak,et al.  Flow cytometric analysis of an Emiliana huxleyi bloom terminated by viral infection , 2002 .

[25]  C. S. Holling,et al.  Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest , 1978 .

[26]  A. Nold.  Heterogeneity in disease-transmission modeling , 1980 .

[27]  Emilio Fernández,et al.  Viral activity in relation to Emiliania huxleyi blooms: a mechanism of DMSP release? , 1995 .

[28]  Frank Mathias Hilker,et al.  Spatiotemporal patterns in an excitable plankton system with lysogenic viral infection , 2005, Math. Comput. Model..

[29]  F. Hilker,et al.  Strange Periodic Attractors in a Prey-Predator System with Infected Prey , 2006 .

[30]  M. Rietkerk,et al.  Catastrophic vegetation dynamics and soil degradation in semi-arid grazing systems , 1998 .

[31]  Stig Larsson,et al.  Introduction to stochastic partial differential equations , 2008 .

[32]  J. Paul,et al.  Prophage Induction of Indigenous Marine Lysogenic Bacteria by Environmental Pollutants , 1998 .

[33]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[34]  G. Marsaglia,et al.  The Ziggurat Method for Generating Random Variables , 2000 .

[35]  L Schimansky-Geier,et al.  Regular patterns in dichotomically driven activator-inhibitor dynamics. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Joydev Chattopadhyay,et al.  Viral infection on phytoplankton–zooplankton system—a mathematical model , 2002 .

[37]  Maron,et al.  Spatial pattern formation in an insect host-parasitoid system , 1997, Science.

[38]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[39]  L. Mcdaniel,et al.  Plankton blooms: Lysogeny in marine Synechococcus , 2002, Nature.

[40]  L. Schimansky-Geier,et al.  Noise-controlled oscillations and their bifurcations in coupled phase oscillators. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  J. Kurths,et al.  Coherence Resonance in a Noise-Driven Excitable System , 1997 .

[42]  E. Ranta,et al.  Spatially autocorrelated disturbances and patterns in population synchrony , 1999, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[43]  Alexander B. Neiman,et al.  COHERENCE RESONANCE AT NOISY PRECURSORS OF BIFURCATIONS IN NONLINEAR DYNAMICAL SYSTEMS , 1997 .

[44]  M. Scheffer,et al.  Geometric Analysis of Ecological Models with Slow and Fast Processes , 2000, Ecosystems.

[45]  M. Rietkerk,et al.  Self-Organized Patchiness and Catastrophic Shifts in Ecosystems , 2004, Science.

[46]  Horst Malchow,et al.  Spatiotemporal Complexity of Plankton and Fish Dynamics , 2002, SIAM Rev..

[47]  L. Rockwood.  Introduction to population ecology , 2006 .

[48]  D. Valenti,et al.  Stochastic resonance and noise delayed extinction in a model of two competing species , 2003, cond-mat/0310582.

[49]  D. Valenti,et al.  Moment equations for a spatially extended system of two competing species , 2005, cond-mat/0510104.

[50]  D. Valenti,et al.  Noise in ecosystems: a short review. , 2004, Mathematical biosciences and engineering : MBE.

[51]  S. Carpenter,et al.  Hares and Tortoises: Interactions of Fast and Slow Variablesin Ecosystems , 2000, Ecosystems.

[52]  Sunny C. Jiang,et al.  Significance of Lysogeny in the Marine Environment: Studies with Isolates and a Model of Lysogenic Phage Production , 1998, Microbial Ecology.

[53]  H. McCallum,et al.  How should pathogen transmission be modelled? , 2001, Trends in ecology & evolution.

[54]  John Brindley,et al.  Ocean plankton populations as excitable media , 1994 .

[55]  S. Carpenter,et al.  Catastrophic shifts in ecosystems , 2001, Nature.

[56]  Curtis A. Suttle,et al.  Marine cyanophages infecting oceanic and coastal strains of Synechococcus: abundance, morphology, cross-infectivity and growth characteristics , 1993 .

[57]  L. Allen.  An introduction to stochastic processes with applications to biology , 2003 .

[58]  G. Hutchinson,et al.  An Introduction to Population Ecology , 1978 .

[59]  Hu,et al.  Phase synchronization in coupled nonidentical excitable systems and array-enhanced coherence resonance , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[60]  Lutz Schimansky-Geier,et al.  Noise-Sustained Pulsating Patterns and Global Oscillations in Subexcitable Media , 1999 .

[61]  C. Suttle,et al.  Lysogeny and Lytic Viral Production during a Bloom of the Cyanobacterium Synechococcus spp. , 2002, Microbial Ecology.

[62]  F. Hilker,et al.  Oscillations and waves in a virally infected plankton system. Part II. Transition from Lysogeny to Lysis. , 2006 .

[63]  J. Fuhrman.  Marine viruses and their biogeochemical and ecological effects , 1999, Nature.

[64]  M. E. Muller,et al.  A Note on the Generation of Random Normal Deviates , 1958 .

[65]  J. García-Ojalvo,et al.  Effects of noise in excitable systems , 2004 .

[66]  H. Haken,et al.  Stochastic resonance without external periodic force. , 1993, Physical review letters.

[67]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[68]  Akira Okubo,et al.  Oceanic diffusion diagrams , 1971 .

[69]  Sergio Rinaldi,et al.  Slow-fast limit cycles in predator-prey models , 1992 .

[70]  Werner Ebeling,et al.  Influence of Noise on Duffing‐Van der Pol Oscillators , 1986 .

[71]  Sergio Rinaldi,et al.  Limit cycles in slow-fast forest-pest models☆ , 1992 .

[72]  R. Sarkar,et al.  Dynamics of nutrient-phytoplankton interaction in the presence of viral infection. , 2003, Bio Systems.

[73]  K. Schulten,et al.  Noise-induced neural impulses , 1986, European Biophysics Journal.

[74]  Sergei Petrovskii,et al.  Oscillations and waves in a virally infected plankton system. Part I: The lysogenic stage , 2004 .

[75]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[76]  Ulrich Hillenbrand,et al.  Subthreshold dynamics of the neural membrane potential driven by stochastic synaptic input. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[77]  Rachel Kuske,et al.  Sustained oscillations via coherence resonance in SIR. , 2007, Journal of theoretical biology.