Advertisement

Bulletin of Mathematical Biology

, Volume 81, Issue 4, pp 995–1030 | Cite as

Preytaxis and Travelling Waves in an Eco-epidemiological Model

  • Andrew M. BateEmail author
  • Frank M. Hilker
Article
  • 147 Downloads

Abstract

Preytaxis is the attraction (or repulsion) of predators along prey density gradients and a potentially important mechanism for predator movement. However, the impact preytaxis has on the spatial spread of a predator invasion or of an epidemic within the prey has not been investigated. We investigate the effects preytaxis has on the wavespeed of several different invasion scenarios in an eco-epidemiological system. In general, preytaxis cannot slow down predator or disease invasions and there are scenarios where preytaxis speeds up predator or disease invasions. For example, in the absence of disease, attractive preytaxis results in an increased wavespeed of predators invading prey, whereas repulsive preytaxis has no effect on the wavespeed, but the wavefront is shallower. On top of this, repulsive preytaxis can induce spatiotemporal oscillations and/or chaos behind the invasion front, phenomena normally only seen when the (non-spatial) coexistence steady state is unstable. In the presence of disease, the predator wave can have a different response to attractive susceptible and attractive infected prey. In particular, we found a case where attractive infected prey increases the predators’ wavespeed by a disproportionately large amount compared to attractive susceptible prey since a predator invasion has a larger impact on the infected population. When we consider a disease invading a predator–prey steady state, we found some counter-intuitive results. For example, the epidemic has an increased wavespeed when infected prey attract predators. Likewise, repulsive susceptible prey can also increase the infection wave’s wavespeed. These results suggest that preytaxis can have a major effect on the interactions of predators, prey and diseases.

Keywords

Biological invasions Wavespeed Spatiotemporal oscillations Pattern formation 

Notes

References

  1. Ainseba BE, Bendahmane M, Noussair A (2008) A reaction–diffusion system modeling predator–prey with prey-taxis. Nonlinear Anal Real World Appl 9:2086–2105MathSciNetCrossRefzbMATHGoogle Scholar
  2. Arditi R, Tyutyunov Y, Morgulis A, Govorukhin V, Senina I (2001) Directed movement of predators and the emergence of density-dependence in predator–prey models. Theor Popul Biol 59:207–221CrossRefzbMATHGoogle Scholar
  3. Armstrong RA, McGehee R (1980) Competitive exclusion. Am Nat 115:151–170MathSciNetCrossRefGoogle Scholar
  4. Aronson DG, Weinberger HF (1975) Multidimensional nonlinear diffusion arising in population genetics. In: Goldstein EA (ed) Partial differential equations and related topics, vol 446. Lecture notes in mathematics. Springer, Berlin, pp 5–49CrossRefGoogle Scholar
  5. Aronson DG, Weinberger HF (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv Math 30:38–76MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bate AM, Hilker FM (2013a) Complex dynamics in an eco-epidemiological model. Bull Math Biol 75:2059–2078MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bate AM, Hilker FM (2013b) Predator–prey oscillations can shift when dieases become endemic. J Theor Biol 316:1–8CrossRefzbMATHGoogle Scholar
  8. Bate AM, Hilker FM (2014) Disease in group-defending prey can benefit predators. Theor Ecol 7:87–100CrossRefGoogle Scholar
  9. Begon M, Townsend CR, Harper JL (2002) Ecology, 4th edn. Blackwell Publishing, OxfordGoogle Scholar
  10. Bell SS, White A, Sherratt JA, Boots M (2009) Invading with biological weapons: the role of shared disease in ecological invasion. Theor Ecol 2:53–66CrossRefGoogle Scholar
  11. Berdoy M, Webster JP, Macdonald DW (2000) Fatal attraction in rates infected with toxoplasma gondii. Proc R Soc Lond B 267:1591–1594CrossRefGoogle Scholar
  12. Berleman JE, Scott J, Chumley T, Kirby JR (2008) Predataxis behavior in Myxococcus xanthus. Proc Natl Acad Sci 105:17127–17132CrossRefGoogle Scholar
  13. Britton NF (2003) Essential mathematical biology. Springer, LondonCrossRefzbMATHGoogle Scholar
  14. Chakraborty A, Singh M, Lucy D, Ridland P (2007) Predator–prey model with prey-taxis and diffusion. Math Comput Model 46:482–498MathSciNetCrossRefzbMATHGoogle Scholar
  15. Coyner DF, Schaack SR, Spalding MG, Forrester DJ (2001) Altered predation susceptibility of mosquitofish infected with eustrongylides ignotus. J Wildl Dis 37:556–560CrossRefGoogle Scholar
  16. Curio E (1976) The ethology of predation, zoophysiology and ecology, vol 7. Spring, BerlinCrossRefGoogle Scholar
  17. Dagbovie AS, Sherratt JS (2014) Absolute stability and dynamical stabilisation in predator–prey systems. J Math Biol 68:1403–1421MathSciNetCrossRefzbMATHGoogle Scholar
  18. Dickman CR (1996) Impact of exotic generalist predators on the native fauna of Australia. Wildl Biol 2:185–195CrossRefGoogle Scholar
  19. Dobson AP (1988) The population biology of parasite-induced changes in host behavior. Q Rev Biol 63:139–165CrossRefGoogle Scholar
  20. Edelstein-Keshet L (1988) Mathematical models in biology. Random House, New YorkzbMATHGoogle Scholar
  21. Errington PL (1946) Predation and vertebrate populations. Q Rev Biol 21:145–177Google Scholar
  22. Ferreri L, Venturino E (2013) Cellular automata for contact ecoepidemic processes in predator–prey systems. Ecol Complex 13:8–20CrossRefGoogle Scholar
  23. Freedman HI, Wolkowicz GSK (1986) Predator–prey systems with groups defence: the paradox of enrichment revisited. Bull Math Biol 48:493–508MathSciNetCrossRefzbMATHGoogle Scholar
  24. Grünbaum D (1998) Using spatially explicit models to characterize foraging performance in heterogeneous landscapes. Am Nat 151:97–115CrossRefGoogle Scholar
  25. Hardin G (1960) The competitive exclusion principle. Science 131:1292–1297CrossRefGoogle Scholar
  26. Hastings A (1996) Models of spatial spread: a synthesis. Biol Conserv 78:143–148CrossRefGoogle Scholar
  27. Hilker FM, Schmitz K (2008) Disease-induced stabilization of predator–prey oscillations. J Theor Biol 255:299–306CrossRefzbMATHGoogle Scholar
  28. Hilker FM, Malchow H, Langlais M, Petrovskii SV (2006) Oscillations and waves in a virally infected plankton system. Part II: transition from lysogeny to lysis. Ecol Complex 3:200–208CrossRefGoogle Scholar
  29. Hosono HG (1998) The minimal speed of traveling fronts for a diffusive Lotka–Volterra competition model. Bull Math Biol 60:435–448CrossRefzbMATHGoogle Scholar
  30. Hudson PJ, Dobson AP, Newborn D (1992) Do parasites make prey vulnerable to predation? Red grouse and parasites. J Anim Ecol 61:681–692CrossRefGoogle Scholar
  31. Johnson CG (1967) International dispersal of insects and insect-borne viruses. Neth J Plant Pathol 1:21–43CrossRefGoogle Scholar
  32. Kareiva P, Odell G (1987) Swarms of predators exhibit preytaxis if individual predators use area-restricted search. Am Nat 130:233–270CrossRefGoogle Scholar
  33. Keller EF, Segel LA (1971) Traveling band of chemotactic bacteria: a theoretical analysis. J Theor Biol 130:235–248CrossRefzbMATHGoogle Scholar
  34. Krause J, Ruxton GD (2002) Living in groups. Oxford University Press, OxfordGoogle Scholar
  35. Kubanek J (2009) Chemical defense in invertebrates. In: Hardege JD (ed) Chemical ecology. EOLSS, OxfordGoogle Scholar
  36. Lafferty KD (1992) Foraging on prey that are modified by parasites. Am Nat 140:854–867CrossRefGoogle Scholar
  37. Langer WL (1964) The black death. Sci Am 210:114–121CrossRefGoogle Scholar
  38. Lee JM, Hillen T, Lewis MA (2008) Continuous traveling waves for prey-taxis. Bull Math Biol 70:654–676MathSciNetCrossRefzbMATHGoogle Scholar
  39. Lee JM, Hillen T, Lewis MA (2009) Pattern formation in prey-taxis systems. J Biol Dyn 3:551–573MathSciNetCrossRefzbMATHGoogle Scholar
  40. LeVeque RJ (1992) Numerical methods for conservation laws. Birkhäuser, BaselCrossRefzbMATHGoogle Scholar
  41. Lewis MA, Li B, Weinberger HF (2002) Spreading speed and linear determinacy for two-species competition models. J Math Biol 45:219–233MathSciNetCrossRefzbMATHGoogle Scholar
  42. Lloyd HG (1983) Past and present distribution of red and grey squirrels. Mamm Rev 13:69–80CrossRefGoogle Scholar
  43. Malchow H, Hilker FM, Petrovskii SV, Brauer K (2004) Oscillations and waves in a virally infected plankton system. Part I: the lysogenic stage. Ecol Complex 1:211–223CrossRefGoogle Scholar
  44. Malchow H, Hilker FM, Sarkar RR, Brauer K (2005) Spatiotemporal patterns in an excitable plankton system with lysogenic viral infection. Math Comput Model 42:1035–1048MathSciNetCrossRefzbMATHGoogle Scholar
  45. Malchow H, Petrovskii SV, Venturino E (2008) Spatiotemporal patterns in ecology and epidemiology: theory, models, and simulation. Chapman and Hall/CRC, New YorkzbMATHGoogle Scholar
  46. Middleton AD (1930) Ecology of the American gray squirrel in the British Isles. Proc Zool Soc Lond 2:809–843Google Scholar
  47. Moore J (2002) Parasites and the behavior of animals. Oxford series in ecology and evolution. Oxford University Press, OxfordGoogle Scholar
  48. Morton KW, Mayers DF (2002) Numerical solution of partial differential equations. Cambridge University Press, CambridgezbMATHGoogle Scholar
  49. Murray JD (2002) Mathematical biology I: an introduction, 3rd edn. Springer, New YorkzbMATHGoogle Scholar
  50. Murray JD (2003) Mathematical biology II: spatial models and biomedical applications, 3rd edn. Springer, New YorkzbMATHGoogle Scholar
  51. Packer C, Holy RD, Hudson PJ, Lafferty KD, Dobson AP (2003) Keeping the herds healthy and alert: implications of predator control for infectious disease. Ecol Lett 6:797–802CrossRefGoogle Scholar
  52. Petrovskii SV, Li BL (2006) Exactly solvable models of biological invasion. Chapman and Hall/CRC, Boca RatonzbMATHGoogle Scholar
  53. Petrovskii SV, Malchow H (2000) Critical phenomena in plankton communities: KISS model revisited. Nonlinear Anal Real World Appl 1:37–51MathSciNetCrossRefzbMATHGoogle Scholar
  54. Roy P, Upadhyay RK (2015) Conserving Iberian Lynx in Europe: issues and challenges. Ecol Complex 22:16–31CrossRefGoogle Scholar
  55. Sapoukhina N, Tyutyunov Y, Arditi R (2003) The role of prey taxis in biological control: a spatial theoretical model. Am Nat 162:61–76CrossRefGoogle Scholar
  56. Sherratt JS, Dagbovie AS, Hilker FM (2014) A mathematical biologist’s guide to absolute and convective instability. Bull Math Biol 76:1–26MathSciNetCrossRefzbMATHGoogle Scholar
  57. Shigesada N, Kawasaki K (1997) Biological invasions: theory and practice. Oxford University Press, OxfordGoogle Scholar
  58. Sieber M, Hilker FM (2011) Prey, predators, parasites: intraguild predation or simpler community models in disguise? J Anim Ecol 80:414–421CrossRefGoogle Scholar
  59. Sieber M, Malchow H, Schimansky-Geier L (2007) Constructive effects of environmental noise in an excitable prey–predator plankton system with infected prey. Ecol Complex 4:223–233CrossRefGoogle Scholar
  60. Siekmann I, Malchow H, Venturino E (2008) Predation may defeat spatial spread of infection. J Biol Dyn 2:40–54MathSciNetCrossRefzbMATHGoogle Scholar
  61. Skellam JG (1951) Random dispersal in theoretical populations. Biometrika 38:196–218MathSciNetCrossRefzbMATHGoogle Scholar
  62. Slobodkin LB (1968) How to be a predator. Am Zool 8:43–51CrossRefGoogle Scholar
  63. Su M, Hui C (2011) The effect of predation on the prevalence and aggregation of pathogens in prey. BioSystems 105:300–306CrossRefGoogle Scholar
  64. Su M, Hui C, Zhang YY, Li Z (2008) Spatiotemporal dynamics of the epidemic transmission in a predator–prey system. Bull Math Biol 70:2195–2210MathSciNetCrossRefzbMATHGoogle Scholar
  65. Su M, Hui C, Zhang Y, Li Z (2009) How does the spatial structure of habitat loss affect the eco-epidemic dynamics? Ecol Model 220:51–59CrossRefGoogle Scholar
  66. Tompkins DM, White AR, Boots M (2003) Ecological replacement of native red squirrels by invasive greys driven by disease. Ecol Lett 6:189–196CrossRefGoogle Scholar
  67. Tyson R, Lubkin SR, Murray JD (1999) Model and analysis of chemotactic bacterial patterns in a liquid medium. J Math Biol 38:359–375MathSciNetCrossRefzbMATHGoogle Scholar
  68. Tyson R, Stern LG, LeVeque RJ (2000) Fractional step methods applied to a chemotaxis model. J Math Biol 41:455–475MathSciNetCrossRefzbMATHGoogle Scholar
  69. Tyutyunov YV, Titova LI, Senina IN (2017) Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system. Ecol Complex 31:170–180CrossRefGoogle Scholar
  70. Upadhyay RK, Roy P, Venkataraman C, Madzvamuse A (2016) Wave of chaos in a spatial eco-epidemiological system: generating realistic patterns of patchiness in rabbit-lynx dynamics. Math Biosci 281:98–119MathSciNetCrossRefzbMATHGoogle Scholar
  71. Vyas A, Kim S, Glacomini N, Boothroyd JC, Sapolsky RM (2007) Behavioral changes induced by toxoplasma infection of rodents are highly specific to aversion of cat odors. Proc Natl Acad Sci 104:6442–6447CrossRefGoogle Scholar
  72. Wang Q, Yang S (2017) Nonconstant positive steady states and pattern formation of 1D prey-taxis systems. J Nonlinear Sci 27:71–91MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Centre for Mathematical Biology, Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Environment DepartmentUniversity of YorkYorkUK
  3. 3.Institute of Environmental Systems Research, School of Mathematics/Computer ScienceOsnabrück UniversityOsnabrückGermany

Personalised recommendations