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Modelling the fear effect in a two-species predator–prey system under the influence of toxic substances

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Abstract

Effect of anti-predator defense due to fear of predator felt by prey and effect of toxic substance released by external sources on prey–predator system is a serious mater of concern in mathematical biology. In the proposed model we have discussed a prey–predator system in which both the species are infected by environmental toxicant. In our consideration prey species is directly infected by environmental toxicant and predator gets infected by consuming infected prey. Prey’s growth rate is assumed to be affected by fear of predator. In this work the proposed predator–prey model is analyzed in presence of environmental fluctuation, i.e., stochastic analysis of this model is discussed. Using Itô formula: positivity, boundedness, uniform continuity criterion and global attractivity of solutions of this system have been established. Conditions for which the prey as well as the predator goes extinct have been derived. Conditions for persistence of the system have also been discussed. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different levels of toxicant and different levels of fear have been demonstrated by depicting figures in numerical simulation using MATLAB.

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References

  1. Altendorf, K.B., Laundre, J.W., Gonzalez, C.A.L., Brown, J.S.: Assessing effects of predation risk on foraging behavior of mule deer. J. Mammal. 82(2), 430–439 (2011)

    Article  Google Scholar 

  2. Boonstra, R., Hik, D., Singleton, G.R., Tinnikov, A.: The impact of predator-induced stress on the snowshoe hare cycle. Ecol. Monogr. 68, 371–394 (1998). https://doi.org/10.2307/2657244

    Article  Google Scholar 

  3. Creel, S., Christianson, D., Liley, S., Winnie Jr., J.A.: Predation risk affects reproductive physiology and demography of elk. Science 315(5814), 960 (2007). https://doi.org/10.1126/science.1135918

    Article  Google Scholar 

  4. Creel, S., Christianson, D.: Relationships between direct predation and risk effects. Trends Ecol. Evol. 23, 194–201 (2008)

    Article  Google Scholar 

  5. Cresswell, W.: Predation in bird populations. J. Ornithol. 152, 251–263 (2011)

    Article  Google Scholar 

  6. Das, A., Samanta, G.P.: Stochastic prey–predator model with additional food for predator. Phys. A Stat. Mech. Appl. 512, 121–141 (2018). https://doi.org/10.1016/j.physa.2018.08.138

    Article  MathSciNet  Google Scholar 

  7. Das, A., Samanta, G.P.: Modelling the fear effect on a stochastic prey–predator system with additional food for predator. J. Phys. A Math. Theor. 51, 465601 (2018). https://doi.org/10.1088/1751-8121/aae4c6

    Article  Google Scholar 

  8. Das, A., Samanta, G.P.: A prey–predator model with refuge for prey and additional food for predator in a fluctuating environment. Phys. A Stat. Mech. Appl. 538, 122844 (2020). https://doi.org/10.1016/j.physa.2019.122844

    Article  MathSciNet  Google Scholar 

  9. Das, M., Samanta, G.P.: A delayed fractional order food chain model with fear effect and prey refuge. Math. Comput. Simul. 178, 218–245 (2020a)

    Article  MathSciNet  Google Scholar 

  10. Das, M., Samanta, G.P.: A prey–predator fractional order model with fear effect and group defense. Int. J. Dyn. Control (2020b). https://doi.org/10.1007/s40435-020-00626-x

    Article  Google Scholar 

  11. Dubey, B., Hussain, J.: A model for the allelopathic effect on two competing species. Ecol. Modell. 129, 195–207 (2000)

    Article  Google Scholar 

  12. Eggers, S., Griesser, M., Ekman, J.: Predator-induced plasticity in nest visitation rates in the Siberian jay (Perisoreus infaustus). Behav. Ecol. 16(1), 309–315 (2005). https://doi.org/10.1093/beheco/arh163

    Article  Google Scholar 

  13. Ghalambor, C.K., Peluc, S.I., Marti, T.E.: Plasticity of parental care under the risk of predation: how much should parents reduce care? Biol. Lett. 9(4), 20130154 (2013). https://doi.org/10.1098/rsbl.2013.0154

    Article  Google Scholar 

  14. Ghosh, J., Sahoo, B., Poria, S.: Prey–predator dynamics with prey refuge providing additional food to predator. Chaos Solitons Fract. 96, 110–119 (2017)

    Article  MathSciNet  Google Scholar 

  15. Hallam, T.G., Clark, C.W.: Non-autonomous logistic equations as models of populations in a deteriorating environment. J. Theor. Biol. 93, 303–311 (1982)

    Article  Google Scholar 

  16. Hallam, T.G., Luna, T.J.: Effects of toxicants on populations: a qualitative approach III, environmental and food chain pathways. Theor. Biol. 109, 411–429 (1984)

    Article  Google Scholar 

  17. Holling, C.S.: The functional response of predator to prey density and its role in mimicry and population regulation. Mem. Ent. Soc. Can. 45, 5–60 (1965)

    Article  Google Scholar 

  18. Hua, F., Fletcher, R.J., Sieving, K.E., Dorazio, R.M.: Too risky to settle: avian community structure changes in response to perceived predation risk on adults and offspring. Proc. R. Soc. B Biol. Sci. 280(1764), 20130762 (2013). https://doi.org/10.1098/rspb.2013.0762

    Article  Google Scholar 

  19. Hua, F., Sieving, K.E., Fletcher, R.J., Wright, C.A.: Increased perception of predation risk to adults and offspring alters avian reproductive strategy and performance. Behav. Ecol. 25(3), 509–519 (2014). https://doi.org/10.1093/beheco/aru017

    Article  Google Scholar 

  20. Ji, C., Jiang, D.: Dynamics of a stochastic density dependent predator–prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 381, 441–453 (2011)

    Article  MathSciNet  Google Scholar 

  21. Kar, T.K., Chaudhuri, K.S.: On non-selective harvesting of two competing fish species in the presence of toxicity. Ecol. Modell. 161, 125–137 (2003)

    Article  Google Scholar 

  22. Krapivsky, P.L., Redner, S.: Kinetics of a diffusive capture process: lamb besieged by a pride of lions. J. Phys. A Math. Gen. 26(17). http://stacks.iop.org/0305-4470/29/i=17/a=011 (1996)

  23. Lande, R.: Risks of population extinction from demographic and environmental stochasticity and random catastrophes. Am. Nat. 142, 911–927 (1993)

    Article  Google Scholar 

  24. Mao, X.: Stochastic Differential Equations and Applications. Horwood, New York (1997)

    MATH  Google Scholar 

  25. Mao, X.: Stochastic Differential Equations and Applications. Woodhead Publishing, Oxford (2011)

    Google Scholar 

  26. May, R.M.: Stability in randomly fluctuating deterministic environments. Am. Nat. 107, 621–650 (1973)

    Article  Google Scholar 

  27. Mondal, S., Maiti, A., Samanta, G.P.: Effects of fear and additional food in a delayed predator–prey model. Biophys. Rev. Lett. 13(4), 157–177 (2018)

    Article  Google Scholar 

  28. Mondal, S., Samanta, G.P.: Dynamics of a delayed predator–prey interaction incorporating nonlinear prey refuge under the influence of fear effect and additional food. J. Phys. A Math. Theor. 53, 295601 (2020)

    Article  MathSciNet  Google Scholar 

  29. Mondal, S., Samanta, G.P.: Time-delayed predator–prey interaction with the benefit of antipredation response in presence of refuge. Z. Naturforschung A (2020). https://doi.org/10.1515/zna-2020-0195

    Article  Google Scholar 

  30. Pangle, K.L., Peacor, S.D., Johannsson, O.E.: Large nonlethal effects of an invasive invertebrate predator on zooplankton population growth rate. Ecology 88, 402–412 (2007)

    Article  Google Scholar 

  31. Peacor, S.D., Peckarsky, B.L., Trussell, G.C., Vonesh, J.R.: Costs of predator-induced phenotypic plasticity: a graphical model for predicting the contribution of nonconsumptive and consumptive effects of predators on prey. Oecologia 171(1), 1–10 (2013). https://doi.org/10.1007/s00442-012-2394-9

    Article  Google Scholar 

  32. Peckarsky, B.L., Cowan, C.A., Penton, M.A., Anderson, C.: Sublethal consequences of streamdwelling predatory stoneflies on mayfly growth and fecundity. Ecology 74, 1836–1846 (1993). https://doi.org/10.2307/1939941

    Article  Google Scholar 

  33. Pettorelli, N., Coulson, T., Durant, S.M., Gaillard, J.M.: Predation, individual variability and vertebrate population dynamics. Oecologia 167(2), 305–314 (2011). https://doi.org/10.1007/s00442-011-2069-y

    Article  Google Scholar 

  34. Preisser, E.L., Bolnick, D.I.: The many faces of fear: comparing the pathways and impacts of nonconsumptive predator effects on prey populations. PlOS One 3(6), e2465 (2008). https://doi.org/10.1371/journal.pone.0002465

    Article  Google Scholar 

  35. Samanta, G.P.: Influence of environmental noises on the Gomatam model of interacting species. Ecol. Modell. 91(1–3), 283–291 (1996)

    Article  Google Scholar 

  36. Samanta, G.P.: A two-species competitive system under the influence of toxic substances. Appl. Math. Comput. 216(1), 291–299 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Samanta, G.P., Maiti, A.: Stochastic Gomatam model of interacting species: non-equilibrium fluctuation and stability. Syst. Anal. Modell. Simul. 43, 683–692 (2003)

    Article  MathSciNet  Google Scholar 

  38. Schwarzl, M., Godec, A., Oshanin, G., Metzler, R.: A single predator charging a herd of prey: effects of self volume and predator–prey decision-making. J. Phys. A Math. Theor. 49, 225601 (2016)

    Article  MathSciNet  Google Scholar 

  39. Smith, M.J.: Models in Ecology, p. 146. Cambridge University Press, Cambridge (1974)

    Google Scholar 

  40. Suraci, J.P., Clinchy, M., Dill, L.M., Roberts, C., Zanette, L.Y.: Fear of large carnivores causes a trophic cascade. Nat. Commun. (2016). https://doi.org/10.1038/ncomms10698

    Article  Google Scholar 

  41. Wang, X., Zanette, L.Y., Zou, X.: Modelling the fear effect in predator–prey interactions. J. Math. Biol. 73(5), 1179–1204 (2016). https://doi.org/10.1007/s00285-016-0989-1

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, X., Zou, X.: Modeling the fear effect in predator–prey interactions with adaptive avoidance of predators. Bull. Math. Biol. 79, 1325–1359 (2017). https://doi.org/10.1007/s11538-017-0287-0

    Article  MathSciNet  MATH  Google Scholar 

  43. Wirsing, A.J., Heithaus, M.R., Dill, L.M.: Living on the edge: dugongs prefer to forage in microhabitats that allow escape from rather than avoidance of predators. Anim. Behav. 74(1), 93–101 (2007). https://doi.org/10.1016/j.anbehav.2006.11.016

    Article  Google Scholar 

  44. Wirsing, A.J., Ripple, W.J.: A comparison of shark and wolf research reveals similar behavioural responses by prey. Front. Ecol. Environ. 9(6), 335–341 (2011). https://doi.org/10.1890/090226

    Article  Google Scholar 

  45. Zanette, L.Y., White, A.F., Allen, M.C., Clinchy, M.: Perceived predation risk reduces the number of offspring songbirds produce per year. Science 334(6061), 1398–1401 (2011). https://doi.org/10.1126/science.1210908

    Article  Google Scholar 

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Acknowledgements

The authors are grateful to the anonymous referees, Dr. Pasquale Vetro, Editor-in-Chief, for their careful reading, valuable comments and helpful suggestions, which have helped them to improve the presentation of this work significantly.

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

    Amartya Das & G. P. Samanta

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Das, A., Samanta, G.P. Modelling the fear effect in a two-species predator–prey system under the influence of toxic substances. Rend. Circ. Mat. Palermo, II. Ser 70, 1501–1526 (2021). https://doi.org/10.1007/s12215-020-00570-x

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