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Influence of dispersal and strong Allee effect on a two-patch predator–prey model

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Abstract

This work aims to study a population-dispersal dynamics for predator–prey interactions in a two patch environment with strong Allee effect among prey species in both patches. It is assumed that the prey species are movable and their dispersal between patches is directed from lower fitness to the higher fitness patch (exhibiting balanced dispersal). Existence and stability criterion of the interior equilibrium point of the system is analyzed in presence as well as in absence of dispersal speed. It has been observed that Allee threshold takes an important role to destabilize the system while the prey individuals evolve in their own patches independently. Moreover dispersal cannot destabilize populations at the interior equilibrium, i.e., if a predator–prey equilibrium without dispersal is in stable state then this situation cannot be destabilized when prey species move between two patches. Numerical simulations using MATLAB validate the analytical results. The occurrence of transcritical as well as Hopf bifurcation has also been reported.

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References

  1. Hassel MP, May RM (1973) Stability in insect host-parasite models. J Anim Ecol 42:693–726

    Google Scholar 

  2. Hassel MP (1984) Parasitism in patchy environments: inverse density dependence can be stabilizing. IMA J Math Appl Med Biol 1:123–133

    MathSciNet  Google Scholar 

  3. Ives R (1992) Continuous-time models of host-parasitoid interactions. Am Nat 140:1–29

    Google Scholar 

  4. Reeve J (1988) Environmental variability, migration and persistence in host-parasitoid systems. Am Nat 132:810–836

    Google Scholar 

  5. Mchich R, Auger PM, Bravo De La Parra R, Raissi N (2002) Dynamics of a fishery on two fishing zones with fish stock dependent migrations: aggregation and control. Ecol Model 158:51–62

    Google Scholar 

  6. Gadgil M (1971) Dispersal: population consequences and evolution. Ecology 52:253–261

    Google Scholar 

  7. Hamilton WD, May RM (1977) Dispersal in stable habitats. Nature 269:578–581

    Google Scholar 

  8. Comins H, Hamilton W, May R (1980) Evolutionarily stable dispersal strategies. J Theor Biol 82:205–230

    MathSciNet  Google Scholar 

  9. Hastings A (1983) Can spatial variation alone lead to selection for dispersal? Theor Popul Biol 24:244–251

    MathSciNet  MATH  Google Scholar 

  10. Holt RD (1985) Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution. Theor Popul Biol 28:181–208

    MathSciNet  MATH  Google Scholar 

  11. McPeek MA, Holt RD (1992) The evolution of dispersal in spatially and temporally varying environments. Am Nat 140:1010–1027

    Google Scholar 

  12. Holt RD, McPeek MA (1996) Chaotic population dynamics favors the evolution of dispersal. Am Nat 148:709–718

    Google Scholar 

  13. Amarasekare P (1998) Interactions between local dynamics and dispersal: insights from single species models. Theor Popul Biol 53:44–59

    MATH  Google Scholar 

  14. Diffendorfer JE (1998) Testing models of source-sink dynamics and balanced dispersal. Oikos 81:417–433

    Google Scholar 

  15. Dieckmann U, O’Hara B, Weisser W (1999) The evolutionary ecology of dispersal. Trends Ecol Evol 14:88–90

    Google Scholar 

  16. Ferriere R, Belthoff JR, Olivieri I, Krackow S (2000) Evolving dispersal: where to go next? Trends Ecol Evol 15:5–7

    Google Scholar 

  17. Holt RD, Barfield M (2001) On the relationship between the idealfree distribution and the evolution of dispersal. In: Danchin JCE, Dhondt A, Nichols J (eds) Dispersal. Oxford University Press, New York, pp 83–95

    Google Scholar 

  18. Donahue MJ, Holyoak M, Feng C (2003) Patterns of dispersal and dynamics among habitat patches varying in quality. Am Nat 162:302–317

    Google Scholar 

  19. Padrón V, Trevisan MC (2006) Environmentally induced dispersal under heterogeneous logistic growth. Math Biosci 199:160–174

    MathSciNet  MATH  Google Scholar 

  20. Cantrell RS, Cosner C, DeAngelis DL, Padron V (2007) The ideal free distribution as an evolutionarily stable strategy. J Biol Dyn 1:249–271

    MathSciNet  MATH  Google Scholar 

  21. Kang Y, Sasmal SK, Messan K (2017) A two-patch prey–predator model with predator dispersal driven by the predation strength. Math Biosci Eng 14(4):843–880

    MathSciNet  MATH  Google Scholar 

  22. Pal D, Samanta GP (2018) Effects of dispersal speed and strong Allee effect on stability of a two-patch predator–prey model. Int J Dyn Control (2018). https://doi.org/10.1007/s40435-018-0407-1

    MathSciNet  Google Scholar 

  23. Levin SA (1974) Dispersion and population interactions. Am Nat 108:207–228

    Google Scholar 

  24. Hassell MP, Comins HN, May RM (1991) Spatial structure and chaos in insect population dynamics. Nature 353:255–258

    Google Scholar 

  25. Bascompte J, Soleé RV (1995) Spatially induced bifurcations in single-species population dynamics. J Anim Ecol 63:256–265

    Google Scholar 

  26. Ruxton GD (1996) Density-dependent migration and stability in a system of linked populations. Bull Math Biol 58:643–660

    MATH  Google Scholar 

  27. Tilman D, Kareiva P (1997) Spatial ecology: the role of space in population dynamics and interspecific interactions. Monographs in population biology, vol 30. Princeton University Press, Princeton

    Google Scholar 

  28. Hanski IA, Gilpin ME (1997) Metapopulation biology: ecology, genetics, and evolution. Academic Press, San Diego

    MATH  Google Scholar 

  29. Rohani P, Ruxton GD (1999) Dispersal and stability in metapopulations. IMA J Math Appl Med Biol 16:297–306

    MATH  Google Scholar 

  30. Hanski I (1999) Metapopulation ecology. Oxford University Press, New York

    Google Scholar 

  31. Fretwell DS, Lucas HL (1969) On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheor 19:16–32

    Google Scholar 

  32. Cressman R, Krivan V (2006) Migration dynamics for the ideal free distribution. Am Nat 168:384–397

    Google Scholar 

  33. Krivan V, Sirot E (2002) Habitats election by two competing species in a two-habitat environment. Am Nat 160:214–234

    Google Scholar 

  34. Murdoch WW, Briggs CJ, Nisbet RM (2003) Consumer-resource dynamics. Princeton University Press, Princeton

    Google Scholar 

  35. Abrams PA, Cressman R, Krivan V (2007) The role of behavioral dynamics in determining the patch distributions of interacting species. Am Nat 169:505–518

    Google Scholar 

  36. Abrams PA (2010) Implications of flexible foraging for interspecific interactions: lessons from simple models. Funct Ecol 24:7–17

    Google Scholar 

  37. Andersen V, Gubanova A, Nival P, Ruellet T (2001) Zooplankton community during the transition from spring bloom to oligotrophy in the open NW Mediterranean and effects of wind events. 2. Vertical distributions and migrations. J Plankton Res 23:243–261

    Google Scholar 

  38. Slusarczyck M, Dawidowicz P, Rygielska E (2005) Hide, rest or die: a light-mediated diapause response in Daphnia magna to the threat of fish predation. Freshw Biol 50:141–146

    Google Scholar 

  39. Courchamp F, Berec L, Gascoigne J (2008) Allee effects in ecology and conservation. Oxford University Press, Oxford

    Google Scholar 

  40. Stephens PA, Sutherland WJ, Freckleton RP (1999) What is the Allee effect? Oikos 87:185–190

    Google Scholar 

  41. Odum E (1953) Fundamentals of ecology. Saunders, Philadelphia

    Google Scholar 

  42. Hadjiavgousti D, Ichtiaroglou S (2008) Allee effect in a prey–predator system. Chaos Solitons Fractals 36:334–342

    MathSciNet  MATH  Google Scholar 

  43. Stephens PA, Sutherland WJ (1999) Consequences of the Allee effect for behaviour ecology and conservation. Trends Ecol Evol 14:401–404

    Google Scholar 

  44. Allee WC (1931) Animal aggregations. A study in general sociology. University of Chicago Press, Chicago

    Google Scholar 

  45. Kot M (2001) Elements of mathematical biology. Cambridge University Press, Cambridge

    Google Scholar 

  46. Van Voorn GAK, Hemerik L, Boer MP, Kooi BW (2007) Heteroclinic orbits indicate overexploitation in predator–prey systems with a strong Allee effect. Math Biosci 209:451–469

    MathSciNet  MATH  Google Scholar 

  47. Wang MH, Kot M (2001) Speeds of invasion in a model with strong or weak Allee effects. Math Biosci 171:83–97

    MathSciNet  MATH  Google Scholar 

  48. Wang J, Shi J, Wei J (2011) Predator–prey system with strong Allee effect in prey. J Math Biol 62:291–331

    MathSciNet  MATH  Google Scholar 

  49. Wang G, Liang XG, Wang FZ (1999) The competitive dynamics of populations subject to an Allee effect. Ecol Model 124:183–192

    Google Scholar 

  50. González-Olivares E, Rojas-Palma A (2011) Multiple limit cycles in a Gause type predator–prey model with Holling type III functional response and Allee effect on prey. Bull Math Biol 73:1378–1397

    MathSciNet  MATH  Google Scholar 

  51. Bazykin AD, Berezovskaya FS, Isaev AS, Khlebopros RG (1997) Dynamics of forest insect density: bifurcation approach. J Theor Biol 186:267–278

    Google Scholar 

  52. Conway ED, Smoller JA (1986) Global analysis of a system of predator–prey equations. SIAM J Appl Math 46:630–642

    MathSciNet  MATH  Google Scholar 

  53. Celik C, Duman O (2009) Allee effect in a discrete-time predator–prey system. Chaos Solitons Fractals 40:1956–1962

    MathSciNet  MATH  Google Scholar 

  54. Javidi M, Nyamorady N (2013) Allee effects in a predator–prey system with a saturated recovery function and harvesting. Int J Adv Math Sci 1:33–44

    Google Scholar 

  55. Sharma S, Samanta GP (2015) A ratio-dependent predatorprey model with Allee effect and disease in prey. J Appl Math Comput 47:345–364

    MathSciNet  MATH  Google Scholar 

  56. Wang W, Zhu Y, Cai Y, Wang W (2014) Dynamical complexity induced by Allee effect in a predator–prey model. Nonlinear Anal Real World Appl 16:103–119

    MathSciNet  MATH  Google Scholar 

  57. Zhou X, Liu Y, Wang G (2005) The stability of predator–prey systems subject to the Allee effects. Theor Popul Biol 67:23–31

    MATH  Google Scholar 

  58. Kent A, Doncaster CP, Sluckin T (2003) Consequences for predators of rescue and Allee effects on prey. Ecol Model 162:233–245

    Google Scholar 

  59. Carlos C, Braumann CA (2017) General population growth models with Allee effects in a random environment. Ecol Complex 30:26–33

    Google Scholar 

  60. Alvesa MT, Hilker FM (2017) Hunting cooperation and Allee effects in predators. J Theor Biol 419:13–22

    MathSciNet  MATH  Google Scholar 

  61. Berec L, Janous̃ková E, Theuer M (2017) Sexually transmitted infections and mate-finding Allee effects. Theor Popul Biol 114:59–69

    MATH  Google Scholar 

  62. Usainia S, Lloydb AL, Anguelova R, Garbaa SM (2017) Dynamical behavior of an epidemiological model with a demographic Allee effect. Math Comput Simul 133:311–325

    MathSciNet  Google Scholar 

  63. LaSalle JP (1976) The stability of dynamical systems. Society for Industrial and Applied Mathematics, Philadelphia

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous referees and Editor in Chief Prof. Jian-Qiao Sun for their careful reading, valuable comments and helpful suggestions, which have helped to improve the presentation of the work significantly. The authors are also grateful to Prof. Vlastimil Krivan, Department of Mathematics and Biomathematics at Faculty of Science, University of South Bohemia, Czech Republic for his help and encouragement. Part of this work was done by G. P. Samanta during his visit in January 2017 at the University of South Bohemia, Czech Republic. G. P. Samanta is thankful to Indian Institute of Engineering Science and Technology, Shibpur, India for financial support for this academic visit through CPDA. The first author (Sangeeta Saha) is thankful to the University Grants Commission, India for providing JRF.

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  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, 711103, India

    Sangeeta Saha & G. P. Samanta

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  1. Sangeeta Saha
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  2. G. P. Samanta
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Correspondence to Sangeeta Saha.

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Saha, S., Samanta, G.P. Influence of dispersal and strong Allee effect on a two-patch predator–prey model. Int. J. Dynam. Control 7, 1321–1349 (2019). https://doi.org/10.1007/s40435-018-0490-3

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