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Impact of generalist type sexually reproductive top predator interference on the dynamics of a food chain model

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Abstract

Food uptake ability of higher trophic level species are more complicated and interesting due to their choice and availability of food and consequently their growth. In the present study, the effect of top predator interference on the dynamics of a tritrophic food chain model involving an intermediate and a generalist type top predator are considered. It is assumed that the interaction between the logistically growing prey and intermediate predator follows the Volterra scheme, while that between the top predator and its favorite food i.e. the intermediate predator follows Crowley–Martin type functional response. Also, it is considered that the growth of top predator is by sexual reproduction. The boundedness of the system, existence of an attracting set, local and global stability of non-negative equilibrium points and uniform persistence are established by very complicated and highly nonlinear biological considerations. With the help of empirical results from different field and laboratory experiment, it is clarified that Crowley–Martin type functional response gives better result than other predator dependent functional responses for the issue of interference among own population. Also it is clarified from literature survey that high trophic species are mostly generalist type i.e. they can survive rather than their favorite food depending upon the present environmental situation and their growth rate depends upon the mating success. Moreover, increasing the top predator interference stabilizes the system, while increasing the normalization of the residual reduction in the top predator population destabilizes the system. Different numerical examples support these ecological phenomenon.

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References

  1. Lotka AJ (1925) Elements of physical biology. Williams and Wilkins, Baltimore

    MATH  Google Scholar 

  2. Berryman AA (1992) The origins and evolutions of predator–prey theory. Ecology 73:1530–1535

    Article  Google Scholar 

  3. Holling CS (1959a) The components of predation as revealed by a study of small mammal predation of the European pine sawfly. Can Entomol 91:293–320

    Article  Google Scholar 

  4. Holling CS (1959b) Some characteristics of simple types of predation and parasitism. Can Entomol 91:385–398

    Article  Google Scholar 

  5. Alstad D (2001) Basic populas models of ecology. Prentice Hall Inc, Upper Saddle River

    Google Scholar 

  6. Anderson O (1984) Optimal foraging by largemouth bass in structured environments. Ecology 65:851–861

    Article  Google Scholar 

  7. Anderson TW (2001) Predator responses, prey refuges and density-dependent mortality of a marine fish. Ecology 82(1):245–257

    Article  Google Scholar 

  8. Johnson WD (2006) Predation, habitat complexity and variation in density dependent mortality of temperate reef fishes. Ecology 87(5):1179–1188

    Article  Google Scholar 

  9. Cosner C, DeAngelis DL, Ault JS, Olson DB (1999) Effects of spatial grouping on the functional response of predators. Theor Popul Biol 56:65–75

    Article  MATH  Google Scholar 

  10. Beddington JR (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J Anim Ecol 44:331–340

    Article  Google Scholar 

  11. DeAngelis DL, Goldstein RA, O’Neill RV (1975) A model for trophic interaction. Ecology 56:881–892

    Article  Google Scholar 

  12. Collazo JA, Gilliam JF, Castro LM (2010) Functional response models to estimate feeding rates of wading birds. Waterbirds 33(1):33–40

    Article  Google Scholar 

  13. Freedman HI, Waltman P (1984) Persistence in models of three interacting predator–prey populations. Math Biosci 68:213–231

    Article  MATH  MathSciNet  Google Scholar 

  14. Jost C, Ellner SP (2000) Testing for predator dependence in predator–prey dynamics: a non-parametric approach. Proc R Soc Lond B 267:1611–1620

    Article  Google Scholar 

  15. Skalski GT, Gilliam JF (2001) Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology 82(11):3083–3092

    Article  Google Scholar 

  16. Villemereuila PBD, Sepulcrea AL (2010) Consumer functional responses under intra and inter-specific interference competition. Ecol Modell 222(3):419–426

    Article  Google Scholar 

  17. Crowley PH, Martin EK (1989) Functional responses and interference within and between year classes of a dragonfly population. J N Am Benthol Soc 8:211–221

    Article  Google Scholar 

  18. Zimmermann B, Sand HK, Wabakken P, Liberg O, Andreassen HP (2015) Predator-dependent functional response in wolves: from food limitation to surplus killing. J Anim Ecol 84:102–112

    Article  Google Scholar 

  19. Hongying L, Weiguo W (2011) Dynamics of a delayed discrete semi-ratio dependent predator–prey system with Holling type IV functional response. Adv Differ Equ 7:2–19

    MATH  MathSciNet  Google Scholar 

  20. Lu C, Zhang L (2010) Permanence and global attractivity of a discrete semi-ratio dependent predator–prey system with Holling II type functional response. J Appl Math Comput 33(1):125–135

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang Q, Fan M, Wang K (2001) Dynamics of a class of nonautonomous semi-ratio-dependent predator–prey systems with functional responses. Math Anal Appl 278:443–471

    Article  MATH  MathSciNet  Google Scholar 

  22. Braza PA (2003) The bifurcation structure of the Holling–Tanner model for predator–prey interactions using two-timing. SIAM J Appl Math 63(3):889–904

    Article  MATH  MathSciNet  Google Scholar 

  23. Hsu SB, Hwang TW (1999) Hopf bifurcation analysis for a predator–prey system of Holling and Leslie type. Taiwan J Math 3(1):35–53

    Article  MATH  MathSciNet  Google Scholar 

  24. Jana D (2014) Stabilizing effect of prey refuge and predator’s interference on the dynamics of prey with delayed growth and generalist predator with delayed gestation. Int J Ecol. doi:10.1155/2014/429086

    Google Scholar 

  25. May RM (1974) Stability and complexity in model ecosystems. Princeton University Press, Princeton

    Google Scholar 

  26. Murray JD (1989) Mathematical biology. Springer, berlin

    Book  MATH  Google Scholar 

  27. Chen S, Shi J, Wei J (2012) Global stability and Hopf bifurcation in a delayed diffusive Leslie–Gower predator–prey system. Int J Bifurc Chaos 22(3):1250061

    Article  MATH  MathSciNet  Google Scholar 

  28. Du Y, Peng R, Wang M (2009) Effect of a protection zone in the diffusive Leslie predator–prey model. J Differ Equ 246(10):3932–3956

    Article  MATH  MathSciNet  Google Scholar 

  29. Huang J, Ruan S, Song J (2014) Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response. J Differ Equ 257:1721–1752

    Article  MATH  MathSciNet  Google Scholar 

  30. Ma Y (2012) Global Hopf bifurcation in the Leslie–Gower predator–prey model with two delays. Nonlinear Anal Real World Appl 13(1):370–375

    Article  MATH  MathSciNet  Google Scholar 

  31. Tsai HC, Ho CP (2004) Global stability for the lesliegower predator–prey system with time-delay and holling’s type functional response. Tunghai Sci 6:43–72

    Google Scholar 

  32. Aziz-Alaoui MA, Okiye MD (2003) Boundedness and global stability for a predator–prey model with modified Leslie–Gower and Holling-type II schemes. Appl Math Lett 16(7):1069–1075

    Article  MATH  MathSciNet  Google Scholar 

  33. Jana D, Agrawal R, Upadhyay RK (2015) Dynamics of generalist predator in a stochastic environment: effect of delayed growth and prey refuge. Appl Math Comput 268:1072–1094

    MathSciNet  Google Scholar 

  34. Nindjin AF, Aziz-Alaoui MA, Cadivel M (2006) Analysis of a predator–prey model with modified Leslie-Gower and Holling-type II schemes with time delay. Nonlinear Anal Real World Appl 7:1104–1118

    Article  MATH  MathSciNet  Google Scholar 

  35. Priyadarshi A, Gakkhar S (2013) Dynamics of Leslie–Gower type generalist predator in a tri-trophic food web system. Commun Nonlinear Sci Numer Simul 18(11):3202–3218

    Article  MATH  MathSciNet  Google Scholar 

  36. Yang W (2013) Global asymptotical stability and persistent property for a diffusive predator–prey system with modified Leslie–Gower functional response. Nonlinear Anal Real World Appl 14(3):1323–1330

    Article  MATH  MathSciNet  Google Scholar 

  37. Fischer BM, Meyer E, Maraun M (2014) Positive correlation of trophic level and proportion of sexual taxa of oribatid mites (Acari: Oribatida) in alpine soil systems. Exp Appl Acarol 63(4):465–479

    Article  Google Scholar 

  38. Parshad RD, Bhowmick S, Quansah E, Basheer A, Upadhyay RK (2016) Predator interference effects on biological control: the ”paradox” of the generalist predator revisited. Commun Nonlinear Sci Numer Simul 39:169–184

    Article  MathSciNet  Google Scholar 

  39. Upadhyay RK, Chattopadhyay J (2005) Chaos to order: role of toxin producing phytoplankton in aquatic systems. Nonlinear Anal Modell Control 10(4):383–396

    MATH  MathSciNet  Google Scholar 

  40. Upadhyay RK, Iyengar SRK, Rai V (1998) Chaos: an ecological reality? Int J Bifurc Chaos 8:1325–1333

    Article  MATH  MathSciNet  Google Scholar 

  41. Upadhyay RK, Rai V, Iyengar SRK (2001) Species extinction problem: genetic vsecological factors. Appl Math Modell 25:937–951

    Article  MATH  Google Scholar 

  42. Aziz-Alaoui MA (2002) Study of a Leslie–Gower-type tritrophic population model. Chaos Solitons Fract 14:1275–1293

    Article  MATH  MathSciNet  Google Scholar 

  43. Tripathi JP, Abbas S, Thakur M (2015) Dynamical analysis of a prey-predator model with Beddington–DeAngelis type function response incorporating a prey refuge. Nonlinear Dyn 80:177–196

    Article  MATH  MathSciNet  Google Scholar 

  44. Brikhoff G, Rota GC (1982) Ordinary differential equations. Ginn, Boston

    Google Scholar 

  45. Tripathi JP, Abbas S, Thakur M (2014) Local and global stability analysis of two prey one predator model with help. Commun Nonlinear Sci Simul 19:3284–3297

    Article  MathSciNet  Google Scholar 

  46. Butler G, Freedman HI, Waltman P (1986) Uniformly persistent systems. Proc Am Math Soc 96(3):425–430

    Article  MATH  MathSciNet  Google Scholar 

  47. Brauer F, Chavez CC (2001) Mathematical models in population biology and epidemiology. Springer, New York

    Book  MATH  Google Scholar 

  48. Freedman HI, So J (1985) Global stability and persistence of simple food chains. Math Bios 76:69–86

    Article  MATH  MathSciNet  Google Scholar 

  49. Gaurd TC, Hallam TG (1979) Persistence in food web-I. Lotka–Voltera chains. Bull Math Biol 41:877–891

    MathSciNet  Google Scholar 

  50. Mauritzen M, Belikov SE, Boltunov AN, Derocher AE, Hansen E, Ims RA, Wiig Ø, Yoccoz N (2003) Functional responses in polar bear habitat selection. OIKOS 100:112–124

    Article  Google Scholar 

  51. Mysterud A, Ims RA (1998) Functional responses in habitat use: availability influences relative use in trade-off situations. Ecology 79:1435–1441

    Article  Google Scholar 

  52. Stirling I, Andriashek D, Calvert W (1993) Habitat preferences of polar bears in the western Canadian Arctic in late winter and spring. Polar Res 29:13–24

    Article  Google Scholar 

Download references

Acknowledgments

First author Dr. D. Jana is thankful to University Grand Commission, Government of India for giving him Dr. D. S. Kothari Postdoctoral Fellowship (India; No.F.4-2/2006(BSR)/13-1004/2013(BSR)) to pursue his postdoctoral research work. The research work of second author Dr. J. P. Tripathi is fully supported by Central University of Rajasthan.

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Authors and Affiliations

  1. Ecological Modelling Laboratory, Department of Zoology, Visva-Bharati University, Santiniketan, 731 235, India

    Debaldev Jana

  2. Department of Mathematics, Central University of Rajasthan, NH-8, Bandarsindri, Kishangarh, District Ajmer, Rajasthan, 305817, India

    Jai Prakash Tripathi

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  1. Debaldev Jana
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Correspondence to Debaldev Jana.

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Jana, D., Tripathi, J.P. Impact of generalist type sexually reproductive top predator interference on the dynamics of a food chain model. Int. J. Dynam. Control 5, 999–1009 (2017). https://doi.org/10.1007/s40435-016-0255-9

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