Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model

Saheb Pal; Nikhil Pal; Sudip Samanta; Joydev Chattopadhyay; Saheb Pal; Nikhil Pal; Sudip Samanta; Joydev Chattopadhyay

doi:10.3934/mbe.2019258

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5146-5179. doi: 10.3934/mbe.2019258

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Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model

1.
Department of Mathematics, Visva-Bharati, Santiniketan 731235, India
2.
Department of Mathematics, Faculty of Science and Arts-Rabigh, King Abdulaziz University, Rabigh-25732, Saudi Arabia
3.
Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India

Received: 22 December 2018 Accepted: 29 May 2019 Published: 05 June 2019

The predation strategy for predators and the avoidance strategy of prey are important topics in ecology and evolutionary biology. Both prey and predators adjust their behaviours in order to gain the maximal benefits and to increase their biomass for each. In the present paper, we consider a modified Leslie-Gower predator-prey model where predators cooperate during hunting and due to fear of predation risk, prey populations show anti-predator behaviour. We investigate step by step the impact of hunting cooperation and fear effect on the dynamics of the system. We observe that in the absence of fear effect, hunting cooperation can induce both supercritical and subcritical Hopf- bifurcations. It is also observed that fear factor can stabilize the predator-prey system by excluding the existence of periodic solutions and makes the system more robust compared to hunting cooperation. Moreover, the system shows two different types of bi-stabilities behaviour: one is between coexisting equilibrium and limit cycle oscillation, and another is between prey-free equilibrium and coexisting equilibrium. We also observe generalized Hopf-bifurcation and Bogdanov-Takens bifurcation in two parameter bifurcation analysis. We perform extensive numerical simulations for supporting evidence of our analytical findings.
- Leslie-Gower model,
- fear effect,
- cooperation,
- multiple limit cycles,
- bifurcation,
- bi-stability
Citation: Saheb Pal, Nikhil Pal, Sudip Samanta, Joydev Chattopadhyay. Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5146-5179. doi: 10.3934/mbe.2019258

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Abstract

The predation strategy for predators and the avoidance strategy of prey are important topics in ecology and evolutionary biology. Both prey and predators adjust their behaviours in order to gain the maximal benefits and to increase their biomass for each. In the present paper, we consider a modified Leslie-Gower predator-prey model where predators cooperate during hunting and due to fear of predation risk, prey populations show anti-predator behaviour. We investigate step by step the impact of hunting cooperation and fear effect on the dynamics of the system. We observe that in the absence of fear effect, hunting cooperation can induce both supercritical and subcritical Hopf- bifurcations. It is also observed that fear factor can stabilize the predator-prey system by excluding the existence of periodic solutions and makes the system more robust compared to hunting cooperation. Moreover, the system shows two different types of bi-stabilities behaviour: one is between coexisting equilibrium and limit cycle oscillation, and another is between prey-free equilibrium and coexisting equilibrium. We also observe generalized Hopf-bifurcation and Bogdanov-Takens bifurcation in two parameter bifurcation analysis. We perform extensive numerical simulations for supporting evidence of our analytical findings.

References

[1]	P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interac-tion between two species, Biometrika, 47 (1960), 219–234.
[2]	P. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213–245.
[3]	M. Aziz-Alaouiand M.D. Okiye, Boundednessand globalstability for apredator-prey modelwith modified LeslieGower and Holling-type II schemes, Appl. Math. Lett., 16 (2003), 1069–1075.
[4]	A. Nindjin, M. Aziz-Alaoui and M. Cadivel, Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with time delay, Nonlinear Anal. R. World Appl., 7 (2006), 1104–1118.
[5]	R. Gupta and P. Chandra, Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.
[6]	Y. Zhu and K. Wang, Existence and global attractivity of positive periodic solutions for a preda-torprey model with modified Leslie–Gower Holling-type II schemes, J. Math. Anal. Appl., 384 (2011), 400–408.
[7]	P. E. Stander, Cooperative hunting in lions: the role of the individual, Behav. Ecol. Sociobiol., 29 (1992), 445–454.
[8]	S. Creel and N. M. Creel, Communal hunting and pack size in African wild dogs, Lycaon pictus, Animal Behav., 50 (1995), 1325–1339.
[9]	C. Boesch, Cooperative hunting in wild chimpanzees, Animal Behav., 48 (1994), 653–667.
[10]	D. L. Mech, The Wolf, Natural History Press, New York, 1970.
[11]	D. P. Hector, Cooperative hunting and its relationship to foraging success and prey size in an avian predator, Ethology, 73 (1986), 247–257.
[12]	P. M. Kappeler and C. P. Van Schaik, Cooperation in Primates and Humans: Mechanisms and Evolution, Springer, Berlin, 2006.
[13]	P. S. Rodman, Inclusive fitness and group size with a reconsideration of group sizes in lions and wolves, Am. Nat., 118 (1981), 275–283.
[14]	J. McNutt, L. Boggs, H. Heldring, et al., Running wild: dispelling the myths of the African wild dog, Smithsonian Institution Press, Washington D.C., 1996.
[15]	I. Bailey, J. P. Myatt and A. M. Wilson, Group hunting within the carnivora: physiological, cognitive and environmental influences on strategy and cooperation, Behav. Ecol. Sociobiol., 67 (2013), 1–17.
[16]	M. Fox, Behaviour of wolves dogs and related canids, Harper and Row, New York, 1971.
[17]	D. W. Macdonald, The ecology of carnivore social behaviour, Nature, 301 (1983), 379.
[18]	J. C. Bednarz, Cooperative hunting Harris' hawks (Parabuteo unicinctus), Science, 239 (1988), 1525–1527.
[19]	T. Pitcher, A. Magurran and I. Winfield, Fish in larger shoals find food faster, Behav. Ecol. Sociobiol., 10 (1982), 149–151.
[20]	H. J. Brockmann and C. Barnard, Kleptoparasitism in birds, Animal Behav., 27 (1979), 487–514.
[21]	J. A. Vucetich, R. O. Peterson and T. A. Waite, Raven scavenging favours group foraging in wolves, Animal Behav., 67 (2004), 1117–1126.
[22]	C. Packer and L. Ruttan, The evolution of cooperative hunting, Am. Nat., 132 (1988), 159–198.
[23]	S. L. Lima, Nonlethal effects in the ecology of predator-prey interactions, Bioscience, 48 (1998), 25–34.
[24]	W. Cresswell, Predation in bird populations, J. Ornithol., 152 (2011), 251–263.
[25]	K. B. Altendorf, J. W. Laundré, C. A. López González, et al., Assessing effects of predation risk on foraging behavior of mule deer, J. Mammal., 82 (2001), 430–439.
[26]	S. Creel, D. Christianson, S. Liley, et al., Predation risk affects reproductive physiology and demography of elk, Science, 315 (2007), 960.
[27]	L. Y. Zanette, A. F. White, M. C. Allen, et al. Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398–1401.
[28]	F. Hua, K. E. Sieving, R. J. Fletcher, et al., Increased perception of predation risk to adults and offspring alters avian reproductive strategy and performance, Behav. Ecol., 25 (2014), 509–519.
[29]	J. P. Suraci, M. Clinchy, L. M. Dill, et al., Fear of large carnivores causes a trophic cascade, Nat. Commun., 7 (2016), 10698.
[30]	M. J. Peers, Y. N. Majchrzak, E. Neilson, et al., Quantifying fear effects on prey demography in nature, Ecology, 99 (2018), 1716–1723.
[31]	A. Sih, Optimal behavior: can foragers balance two conflicting demands?, Science, 210 (1980), 1041–1043.
[32]	A. Sih, Prey uncertainty and the balancing of antipredator and feeding needs, Am. Nat., 139 (1992), 1052–1069.
[33]	A. I. Houston and J. M. McNamara, Models of adaptive behaviour: an approach based on state, Cambridge University Press, Cambridge, 1999.
[34]	Z. Abramsky, M. L. Rosenzweig and A. Subach, The costs of apprehensive foraging, Ecology, 83 (2002), 1330 –1340.
[35]	M. A. Elgar, Predator vigilance and group size in mammals and birds: a critical review of the empirical evidence, Biol. Rev., 64 (1989), 13–33.
[36]	A. Sih and T. M. McCarthy, Prey responses to pulses of risk and safety: testing the risk allocation hypothesis, Animal Behav., 63 (2002), 437–443.
[37]	D. T. Blumstein and J. C. Daniel, Isolation from mammalian predators differentially affects two congeners, Behav. Ecol., 13 (2002), 657–663.
[38]	S. Creel and J. A. Winnie Jr., Responses of elk herd size to fine-scale spatial and temporal variation in the risk of predation by wolves, Animal Behav., 69 (2005), 1181–1189.
[39]	J. Duarte, C. Januário, N. Martins, et al., Chaos and crises in a model for cooperative hunting: A symbolic dynamics approach, Chaos, 19 (2009), 043102.
[40]	L. Berec, Impacts of foraging facilitation among predators on predator–prey dynamics, Bull. Math. Biol., 72 (2010), 94–121.
[41]	M. Teixeira Alves and F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13–22.
[42]	L. Pˇ ribylová and A. Peniaˇ sková, Foraging facilitation among predators and its impact on the stability of predator–prey dynamics, Ecol. Complex., 29 (2017), 30–39.
[43]	S. Pal, N. Pal and J. Chattopadhyay, Hunting cooperation in a discrete-time predator-prey system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 28 (2018), 1850083.
[44]	X. Wang, L. Y. Zanette and X. Zou, Modelling the fear effect in predator–prey interactions, J. Math. Biol, 73 (2016), 1179–1204.
[45]	X. Wang and X. Zou, Modeling the fear effect in predator-prey interactions with adaptive avoid-ance of predators, Bull. Math. Biol., 79 (2017), 1325–1359.
[46]	P. Panday, N. Pal, S. Samanta, et al., Stability and bifurcation analysis of a three-species food chain model with fear, Int. J. Bifurc. Chaos Appl. Sci. Eng., 28 (2018), 1850009.
[47]	S. K. Sasmal, Population dynamics with multiple Allee effects induced by fear factors–A mathe-matical study on prey–predator interactions, Appl. Math Model., 64 (2018), 1–14.
[48]	S. Pal, S. Majhi, S. Mandal, et al., Role of fear in a predator–prey model with Beddington-DeAngelis functional response, Z. Naturforsch. A, 74 (2019) DOI: 10.1515/zna-2018-0449.
[49]	A. Sha, S. Samanta, M. Martcheva, et al., Backward bifurcation, oscillations and chaos in an eco-epidemiological model with fear effect, J. Biol. Dyn., 13 (2019), 301–327.
[50]	P. A. Schmidt and L. D. Mech, Wolf pack size and food acquisition, Am. Nat., 150 (1997), 513–517.
[51]	N. Courbin, A. J. Loveridge, D. Macdonald, et al. Reactive responses of zebras to lion encounters shape their predator–prey space game at large scale, Oikos, 125 (2016), 829–838.
[52]	S. Periquet, M. Valeix, A. J. Loveridge, et al., Individual vigilance of African herbivores while drinking: the role of immediate predation risk and context, Animal Behav., 79 (2010), 665–671.
[53]	W. J. Ripple and E. J. Larsen, Historic aspen recruitment, elk, and wolves in northern Yellowstone National Park, USA, Biol. Conserv., 95 (2000), 361–370.
[54]	S. Creel, J. A. Winnie Jr, B. Maxwell, et al., Elk alter habitat selection as an antipredator response to wolves, Ecology, 86 (2005), 3387–3397.
[55]	J. A. Winnie Jr, D. Christianson, S. Creel, et al., Elk decision-making rules are simplified in the presence of wolves, Behav. Ecol. Sociobiol., 61 (2006), 277.
[56]	J. A. Winnie Jr, and S. Creel, Sex-specific behavioural responses of elk to spatial and temporal variation in the threat of wolf predation, Animal Behav., 73 (2007), 215–225.
[57]	D. W. Stephens and J. R. Krebs, Foraging theory, Princeton University Press, Princeton, New Jersey, 1986.
[58]	Z. Zhang, Mutualism or cooperation among competitors promotes coexistence and competitive ability, Ecol. Model., 164 (2003), 271–282.
[59]	M. J. Hamilton, O. Burger, J. P. DeLong, et al., Population stability, cooperation, and the invasi-bility of the human species, Proc. Natl. Acad. Sci. U.S.A., 106 (2009), 12255–12260.
[60]	K. Kundu, S. Pal, S. Samanta, et al., Impact of fear effect in a discrete-time predator-prey system, Bull. Calcutta Math. Soc., 110 (2018), 245–264.
[61]	H. R. Thieme, Mathematics in population biology, Princeton University Press, Princeton, New Jersey, 2003.
[62]	F. Chen, On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33–49.
[63]	S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Vol. 2, Springer, New York, 1990.
[64]	Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Vol. 112, Springer, New York, USA, 1998.
[65]	S. Pal, N. Pal, S. Samanta, et al., A predator-prey model with hunting cooperation and fear, Ecol. Complex., Under review, (M.S. no: ECOCOM-2019-21).

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