Spatiotemporal complexity of a predator–prey system with the effect of noise and external forcing

Abstract In this paper, we present a spatial version of the Ivlev-type predator–prey model which contains some important factors, such as noise on predator, external periodic forcing and diffusion processes on both predator and prey. From the numerical results, we know that noise or external periodic forcing can induce instability and enhance the oscillation of the species density, and the cooperation between noise and external periodic forcing inherent to the deterministic dynamics of periodically driven models gives rise to the appearance of a rich transport phenomenology. Furthermore, we demonstrate that the spatially extended system exhibits a resonant patterns and frequency-locking phenomena. Our results show that noise and external periodic forcing play a prominent role in the predator–prey model.

[1]  B. Spagnolo,et al.  Role of the noise on the transient dynamics of an ecosystem of interacting species , 2002 .

[2]  Zhujun Jing,et al.  Bifurcations and chaos in three-well Duffing system with one external forcing , 2009 .

[3]  K. Staliūnas.  Spatial and temporal spectra of noise driven stripe patterns. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Chunbiao Gan,et al.  Noise-induced chaos in a quadratically nonlinear oscillator , 2006 .

[5]  S. Hubbard,et al.  Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications. , 2006, Journal of theoretical biology.

[6]  Shigui Ruan,et al.  Global Analysis in a Predator-Prey System with Nonmonotonic Functional Response , 2001, SIAM J. Appl. Math..

[7]  C. Tessone,et al.  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 , 2000, cond-mat/0012366.

[8]  Jonathan A. Sherratt,et al.  Oscillations and chaos behind predator–prey invasion: mathematical artifact or ecological reality? , 1997 .

[9]  Jürgen Kurths,et al.  Noise-sustained and controlled synchronization of stirred excitable media by external forcing , 2005 .

[10]  Kristina Todorović,et al.  Dynamics of noisy FitzHugh–Nagumo neurons with delayed coupling , 2009 .

[11]  Jim M Cushing,et al.  Periodic Time-Dependent Predator-Prey Systems , 1977 .

[12]  Mark A. J. Chaplain,et al.  Disease induced dynamics in host–parasitoid systems: chaos and coexistence , 2006, Journal of The Royal Society Interface.

[13]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[14]  Gregoire Nicolis,et al.  Stochastic resonance , 2007, Scholarpedia.

[15]  Zhi-Xi Wu,et al.  Effects of inhomogeneous activity of players and noise on cooperation in spatial public goods games. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  R. Tian,et al.  Toward standard parameterizations in marine biological modeling , 2006 .

[17]  M. Perc.  Coherence resonance in a spatial prisoner's dilemma game , 2006 .

[18]  J. A. J. Metz,et al.  A Gentle Introduction to Structured Population Models: Three Worked Examples , 1986 .

[19]  Zhonghuai Hou,et al.  Ordering chaos by random shortcuts. , 2003, Physical review letters.

[20]  J. M. G. Vilar,et al.  Effects of Noise in Symmetric Two-Species Competition , 1998, cond-mat/9801260.

[21]  Y. Iwasa,et al.  Turing Pattern Formation with Two Kinds of Cells and a Diffusive Chemical , 2007, Bulletin of mathematical biology.

[22]  Robert E. Kooij,et al.  A predator-prey model with Ivlev's functional response , 1996 .

[23]  Hailing Wang,et al.  The dynamical complexity of a Ivlev-type prey–predator system with impulsive effect ☆ , 2008 .

[24]  Mark A. J. Chaplain,et al.  On Growth and Form: Spatio-temporal Pattern Formation in Biology , 2000 .

[25]  Lei Zhang,et al.  Pattern formation of a predator-prey system with Ivlev-type functional response , 2008, 0801.0796.

[26]  B. Ai,et al.  Noise in an insect outbreak model , 2003, physics/0306177.

[27]  J. F. Gilliam,et al.  FUNCTIONAL RESPONSES WITH PREDATOR INTERFERENCE: VIABLE ALTERNATIVES TO THE HOLLING TYPE II MODEL , 2001 .

[28]  Xin Houwen,et al.  NOISE INDUCED PATTERN TRANSITION AND SPATIOTEMPORAL STOCHASTIC RESONANCE , 1998 .

[29]  Fen-Ni Si,et al.  Propagation of travelling waves in sub-excitable systems driven by noise and periodic forcing , 2007, 0706.1917.

[30]  Lev R. Ginzburg,et al.  Assuming reproduction to be a function of consumption raises doubts about some popular predator–prey models , 1998 .

[31]  Frederic Bartumeus,et al.  MUTUAL INTERFERENCE BETWEEN PREDATORS CAN GIVE RISE TO TURING SPATIAL PATTERNS , 2002 .

[32]  Marcus R. Garvie.  Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in MATLAB , 2007, Bulletin of mathematical biology.

[33]  Alan A. Berryman,et al.  The Orgins and Evolution of Predator‐Prey Theory , 1992 .

[34]  Bruce E. Kendall,et al.  Cycles, chaos, and noise in predator–prey dynamics , 2001 .

[35]  Sergei Petrovskii,et al.  Noise and productivity dependence of spatiotemporal pattern formation in a prey-predator system , 2004 .

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

[37]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .

[38]  R. Mankin,et al.  Colored-noise-induced Hopf bifurcations in predator-prey communities. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Ioannis Andreadis,et al.  Noise perturbation of the thermostat in constant temperature molecular dynamics simulations , 2004 .

[40]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[41]  Jitsuro Sugie.  Two-Parameter Bifurcation in a Predator–Prey System of Ivlev Type , 1998 .

[42]  B. Spagnolo,et al.  Spatio-temporal patterns in population dynamics , 2002 .

[43]  M A Lewis,et al.  Ecological chaos in the wake of invasion. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[44]  B. Ai,et al.  Correlated noise in a logistic growth model. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  V. S. Ivlev,et al.  Experimental ecology of the feeding of fishes , 1962 .

[46]  J. M. Sancho,et al.  Spatiotemporal order out of noise , 2007 .

[47]  Zhen Jin,et al.  Resonance and frequency-locking phenomena in spatially extended phytoplankton–zooplankton system with additive noise and periodic forces , 2007, 0705.3724.

[48]  R. Arditi,et al.  Functional responses and heterogeneities: an experimental test with cladocerans , 1991 .

[49]  Sunita Gakkhar,et al.  Dynamics of modified Leslie–Gower-type prey–predator model with seasonally varying parameters , 2006 .