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Original Articles

Pattern formation in prey-taxis systems

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Pages 551-573 | Received 03 Jul 2007, Published online: 09 Oct 2009

References

  • Allen , M. B. and Eli , M. B. 1998 . Numerical Analysis for Applied Science , New York : Wiley-Interscience .
  • Alonso , D. , Bartemeus , F. and Catalan , J. 2002 . Mutual interference between predators can give rise to Turing spatial patterns . Ecology , 83 : 28 – 34 .
  • Arditi , R. , Tyutyunov , Y. , Morgulis , A. and Govorukhin , V. 2001 . Directed movement of predators and the emergence of density-dependence in predator-prey models . Theoret. Population Biol. , 59 : 207 – 221 .
  • Biktashev , V. N. , Brindley , J. , Holden , A. V. and Tsyganov , M. A. 2004 . Pursuit-evasion predator-prey waves in two spatial dimensions . Chaos , 91 : 218102
  • Britton , N. F. 1986 . Reaction-Diffusion Equations and Their Applications to Biology , London : Academic Press .
  • Chakraborty , A. , Singh , M. , Lucy , D. and Ridland , P. 2007 . Predator-prey model with prey-taxis and diffusion . Math. Comput. Model. , 46 : 482 – 498 .
  • Dolak , Y. and Hillen , T. 2003 . Cattaneo models for chemosensitive movement numerical solution and pattern formation . J. Math. Biol. , 46 : 153 – 170 .
  • Garvey , J. E. , Stein , R. A. and Thomas , H. M. 1994 . Assessing how fish predation and interspecific prey competition influence a crayfish assemblage . Ecology , 75 ( 2 ) : 532 – 547 .
  • Grunbaum , D. 1998 . Using spatially explicit models to characterize foraging performance in heterogeneous landscapes . Am. Nat. , 151 : 97 – 115 .
  • Hamilton , W. D. 1971 . Geometry for the selfish herd . J. Theor. Biol. , 31 : 295 – 311 .
  • Hill , A. M. and Lodge , D. M. 1999 . Replacement of resident crayfishes by an exotic crayfish: the roles of competition and predation . Ecol. Appl. , 9 ( 2 ) : 678 – 690 .
  • Hillen , T. and Painter , K. 2001 . Global existence for a parabolic chemotaxis model with prevention of overcrowding . Adv. Appl. Math. , 26 ( 4 ) : 280 – 301 .
  • Jorné , J. 1974 . The effects of ionic migration on oscillations and pattern formation in chemical systems . J. Theor. Biol. , 43 : 375 – 380 .
  • Kareiva , P. and Odell , G. 1987 . Swarms of predators exhibit ‘preytaxis’ if individual predators use arearestricted search . Am. Nat. , 130 : 233 – 270 .
  • Kolokolnikov , T. , Erneux , T. and Wei , J. 2006 . Mesa-type patterns in the one-dimensional brusselator and their stability . Phys. D , 214 : 63 – 77 .
  • Kurganov , A. and Tadmor , E. 2000 . New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations . J. Comput. Phys. , 160 : 214 – 282 .
  • Lee , J. 2006 . Prey-taxis and its applications , Edmonton, , AB, Canada : University of Alberta . Ph.D. thesis
  • Lee , J. , Hillen , T. and Lewis , M. A. 2007 . Prey-taxis equations: continuous travelling waves for pest control . in preparation
  • LeVeque , R. J. 2002 . Finite Volume Methods for Hyperbolic Problems , Cambridge : Cambridge University Press .
  • Lewis , M. A. 1994 . Spatial coupling of plant and herbivore dynamics: the contribution of herbivore dispersal to transient and persistent ‘waves’ of damage . Theor. Population Biol. , 45 : 277 – 312 .
  • Losey , J. E. and Denno , R. F. 1998c . The escape response of pen aphids to foliar-foraging predators: factors affecting dropping behaviour . Eco. Entom. , 23 : 53 – 61 .
  • Malchow , H. 1995 . Flow-and locomotion-induced pattern formation in nonlinear population dynamics . Ecol. Model. , 82 : 257 – 264 .
  • Malchow , H. 2000 . Motional instability in prey-predator system . J. Theor. Biol. , 204 : 639 – 647 .
  • Morton , T. L. , Haefner , J. , Nugala , V. , Decino , R. D. and Mendes , L. 1994 . The selfish herd revisited: do simple movement rules reduce relative predation risk . J. Theor. Biol. , 167 : 73 – 79 .
  • Murray , J. D. 1989 . Mathematical Biology , New York : Springer Verlag .
  • Okubo , A. and Levin , S. A. 2000 . Diffusion and Ecological Problems: New Perspectives , 2 , New York : Springer Verlag .
  • Owen , M. R. and Lewis , M. A. 2001 . How predation can slow, stop or reverse a prey invasion . Bull. Math. Biol. , 63 : 655 – 684 .
  • Rovinsky , A. B. and Menzinger , M. 1992 . Chemical instability induced by a differential flow . Phys. Rev. Lett. , 69 : 1193 – 1196 .
  • Segel , L. A. 1980 . Mathematical Models in Molecular and Cellular Biology , Cambridge : Cambridge University Press .
  • Segel , L. A. and Jackson , J. 1972 . Dissipative structure: an explanation and an ecological example . J. Theor. Biol. , 37 : 545 – 559 .
  • Shi , W. and Zusman , D. R. 1993 . Fatal attraction . Nat. Sci. , 366 : 414 – 415 . Correspondence
  • Strikwerda , J. C. 1989 . Finite Difference Schemes and Partial Differential Equations , CA : Wadsworth, Inc .
  • Tsyganov , M. A. , Brindley , J. , Holden , A. V. and Biktashev , V. N. 2003 . Quasisoliton interaction of pursuitevasion waves in a predator-prey system . Phys. Rev. Lett. , 91 : 218102
  • Turchin , P. 2003 . Complex Population Dynamics , Princeton and Oxford : Princeton University Press .
  • Tyson , R. , Lubkin , S. R. and Murray , J. D. 1999 . Model and analysis of chemotactic bacterial patterns in a liquid medium . J. Math. Biol. , 38 : 359 – 375 .
  • Tyson , R. , Stern , L. G. and LeVeque , R. J. 2000 . Fractional step methods applied to a chemotaxis model . J. Math. Biol. , 41 : 455 – 475 .
  • Viscido , S. V. and Wethey , D. S. 2002 . Quantitative analysis of fiddler crab flock movement: evidence for ‘selfish herd’ behaviour . Anim. Behav. , 63 : 735 – 741 .
  • Viscido , S. V. , Miller , M. and Wethey , D. S. 2001 . The response of a selfish herd to an attack from outside the group perimeter . J. Theor. Biol. , 208 : 315 – 328 .

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