Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model.

In this paper we consider a modified spatiotemporal ecological system originating from the temporal Holling-Tanner model, by incorporating diffusion terms. The original ODE system is studied for the stability of coexisting homogeneous steady-states. The modified PDE system is investigated in detail with both numerical and analytical approaches. Both the Turing and non-Turing patterns are examined for some fixed parametric values and some interesting results have been obtained for the prey and predator populations. Numerical simulation shows that either prey or predator population do not converge to any stationary state at any future time when parameter values are taken in the Turing-Hopf domain. Prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability. With help of numerical simulations we have shown that Turing-Hopf bifurcation leads to onset of spatio-temporal chaos when predator's diffusivity is much higher compared to prey population. Our investigation reveals the fact that Hopf-bifurcation is essential for the onset of spatiotemporal chaos.

[1]  L. Ginzburg,et al.  The nature of predation: prey dependent, ratio dependent or neither? , 2000, Trends in ecology & evolution.

[2]  Ronald E. Mickens,et al.  Advances in the Applications of Nonstandard Finite Difference Schemes , 2005 .

[3]  J. Sherratt,et al.  Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models , 2008, Journal of The Royal Society Interface.

[4]  Teng Yong.  Chaos in a Three Species Food Chain , 2011 .

[5]  M. Banerjee,et al.  Self-replication of spatial patterns in a ratio-dependent predator-prey model , 2010, Math. Comput. Model..

[6]  U. Feudel,et al.  Turing instabilities and pattern formation in a benthic nutrient-microoganism system. , 2004, Mathematical biosciences and engineering : MBE.

[7]  Marten Scheffer,et al.  Should we expect strange attractors behind plankton dynamics―and if so, should we bother? , 1991 .

[8]  Yang Kuang,et al.  Global qualitative analysis of a ratio-dependent predator–prey system , 1998 .

[9]  Sergei Petrovskii,et al.  A minimal model of pattern formation in a prey-predator system , 1999 .

[10]  Stephen P. Ellner,et al.  Chaos in a Noisy World: New Methods and Evidence from Time-Series Analysis , 1995, The American Naturalist.

[11]  Erkki Korpimäki,et al.  Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos , 1993, Nature.

[12]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[13]  A. J. Lotka,et al.  Elements of Physical Biology. , 1925, Nature.

[14]  Eduardo Sáez,et al.  Dynamics of a Predator-Prey Model , 1999, SIAM J. Appl. Math..

[15]  H. I. Freedman,et al.  Persistence in predator-prey systems with ratio-dependent predator influence , 1993 .

[16]  Eckehard Schöll,et al.  Generic spatiotemporal dynamics near codimension-two Turing-Hopf bifurcations , 1997 .

[17]  A. J. Lotka.  Elements of Physical Biology. , 1925, Nature.

[18]  V. Volpert,et al.  Reaction-diffusion waves in biology. , 2009, Physics of life reviews.

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

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

[21]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[22]  Thilo Gross,et al.  Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations. , 2007, Journal of theoretical biology.

[23]  Christian Jost,et al.  About deterministic extinction in ratio-dependent predator-prey models , 1999 .

[24]  Hongwei Pan,et al.  Qualitative analysis of a ratio-dependent Holling-Tanner model , 2007 .

[25]  Joydev Chattopadhyay,et al.  Ratio-dependent predator–prey model: effect of environmental fluctuation and stability , 2005 .

[26]  Yoh Iwasa,et al.  Stripes, spots, or reversed spots in two-dimensional Turing systems. , 2003, Journal of theoretical biology.

[27]  Turing instability in a coupled predator-preymodel with different Holling type functional responses , 2010 .

[28]  J. Gower,et al.  The properties of a stochastic model for the predator-prey type of interaction between two species , 1960 .

[29]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[30]  S. Hsu,et al.  Global analysis of the Michaelis–Menten-type ratio-dependent predator-prey system , 2001, Journal of mathematical biology.

[31]  Daihai He,et al.  Chaotic oscillations and cycles in multi-trophic ecological systems. , 2007, Journal of theoretical biology.

[32]  Sergei Petrovskii,et al.  Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems , 2003, Bulletin of mathematical biology.

[33]  Bai-lian Li,et al.  Bifurcations and chaos in a predator-prey system with the Allee effect , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[34]  A. B. Peet,et al.  Complex dynamics in a three-level trophic system with intraspecies interaction. , 2005, Journal of theoretical biology.

[35]  Peter Turchin,et al.  Complex Population Dynamics , 2003 .

[36]  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.

[37]  L. Luckinbill.  The Effects of Space and Enrichment on a Predator‐Prey System , 1974 .

[38]  Sergei Petrovskii,et al.  Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system , 2011, Theoretical Ecology.

[39]  Junping Shi,et al.  CROSS-DIFFUSION INDUCED INSTABILITY AND STABILITY IN REACTION-DIFFUSION SYSTEMS ⁄ , 2011 .

[40]  Jonathan A. Sherratt,et al.  Periodic travelling waves in cyclic predator–prey systems , 2001 .

[41]  S. Levin,et al.  Diffusion and Ecological Problems: Modern Perspectives , 2013 .

[42]  R Arditi,et al.  Parametric analysis of the ratio-dependent predator–prey model , 2001, Journal of mathematical biology.

[43]  M. A. Aziz-Alaoui,et al.  Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes , 2003, Appl. Math. Lett..

[44]  R. Mickens.  Nonstandard Finite Difference Models of Differential Equations , 1993 .

[45]  Sergei Petrovskii,et al.  Spatiotemporal patterns in ecology and epidemiology , 2007 .

[46]  H. Meinhardt.  Models of biological pattern formation , 1982 .

[47]  Tapan Saha,et al.  Dynamical analysis of a delayed ratio-dependent Holling―Tanner predator―prey model , 2009 .

[48]  J. Huisman,et al.  Biodiversity of plankton by species oscillations and chaos , 1999, Nature.

[49]  Roger Arditi,et al.  Ratio-Dependent Predation: An Abstraction That Works , 1995 .

[50]  M. Hassell.  The dynamics of arthropod predator-prey systems. , 1979, Monographs in population biology.

[51]  PETER A. BRAZA,et al.  The Bifurcation Structure of the Holling--Tanner Model for Predator-Prey Interactions Using Two-Timing , 2003, SIAM J. Appl. Math..

[52]  A. J. Belsky.  Population and Community Processes in a Mosaic Grassland in the Serengeti, Tanzania , 1986 .

[53]  Pierre Couteron,et al.  Periodic spotted patterns in semi‐arid vegetation explained by a propagation‐inhibition model , 2001 .

[54]  C. S. Holling,et al.  The functional response of predators to prey density and its role in mimicry and population regulation. , 1965 .

[55]  Jesse A. Logan,et al.  Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees , 1988 .

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

[57]  Bai-lian Li,et al.  Transition to spatiotemporal chaos can resolve the paradox of enrichment , 2004 .

[58]  S. Petrovskii,et al.  Excitable Population Dynamics, Biological Control Failure, and Spatiotemporal Pattern Formation in a Model Ecosystem , 2009, Bulletin of mathematical biology.

[59]  Zhen Jin,et al.  Spatiotemporal complexity of a ratio-dependent predator-prey system. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[60]  H. I. Freedman.  Deterministic mathematical models in population ecology , 1982 .

[61]  H. Malchow.  Spatio-temporal pattern formation in nonlinear non-equilibrium plankton dynamics , 1993, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[62]  Robert M. May,et al.  Stability and Complexity in Model Ecosystems , 2019, IEEE Transactions on Systems, Man, and Cybernetics.

[63]  James T. Tanner,et al.  THE STABILITY AND THE INTRINSIC GROWTH RATES OF PREY AND PREDATOR POPULATIONS , 1975 .

[64]  L. Segel,et al.  Hypothesis for origin of planktonic patchiness , 1976, Nature.

[65]  L. P. White.  Brousse tigree patterns in southern Niger. , 1970 .

[66]  E Schöll,et al.  Spatiotemporal dynamics near a supercritical Turing-Hopf bifurcation in a two-dimensional reaction-diffusion system. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[67]  C. Klausmeier,et al.  Regular and irregular patterns in semiarid vegetation , 1999, Science.

[68]  J. L. Jackson,et al.  Dissipative structure: an explanation and an ecological example. , 1972, Journal of theoretical biology.

[69]  R. Arditi,et al.  Coupling in predator-prey dynamics: Ratio-Dependence , 1989 .

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

[71]  Dongmei Xiao,et al.  Global dynamics of a ratio-dependent predator-prey system , 2001, Journal of mathematical biology.

[72]  Andrei Korobeinikov,et al.  A Lyapunov function for Leslie-Gower predator-prey models , 2001, Appl. Math. Lett..

[73]  Mercedes Pascual,et al.  Diffusion-induced chaos in a spatial predator–prey system , 1993, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[74]  Noel Schutt,et al.  Bifurcations, and Temporal and Spatial Patterns of a Modified Lotka-volterra Model , 2008, Int. J. Bifurc. Chaos.

[75]  Osipov,et al.  Scenarios of domain pattern formation in a reaction-diffusion system. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[76]  S. Petrovskii,et al.  Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics. , 2001, Theoretical population biology.

[77]  Y. Iwasa,et al.  Labyrinthine versus straight-striped patterns generated by two-dimensional Turing systems. , 2005, Journal of theoretical biology.

[78]  Sze-Bi Hsu,et al.  Global Stability for a Class of Predator-Prey Systems , 1995, SIAM J. Appl. Math..