This paper aims to study the dynamics of a predator–prey model, where both prey and predator show herd behaviour. Due to this behaviour, predator–prey interaction occurs only at the outer edge of herds formed by the populations. Positivity and boundedness of the system are discussed. A criteria for the extinction of prey is established. A steady-state analysis has been performed. Some criteria for Hopf bifurcation are derived. The stochastic version of the model is formulated to take into account the effect of fluctuating environment. A criterion of asymptotic mean square stability of this model is derived. Numerical simulations are carried out to validate our analytical findings. Implications of our analytical and numerical findings are discussed critically.
LotkaA.Elements of physical biology. Baltimore: Williams and Wilkins, 1925.
2.
VolterraV.Variazioni e fluttuazioni del numero di individui in specie animali conviventi. Mem Accd Lincei1926; 2(1): 31–113.
3.
KotM.Elements of mathematical biology. Cambridge: Cambridge University Press, 2001.
4.
PearlRReedL.On the rate of growth of the population of the United States. Proc Nat Acad Sci1920; 6(6): 275–288.
5.
KingslandS.Modeling nature: episodes in the history of population ecology. Chicago: University of Chicago Press, 1985.
6.
CourchampFClutton-BrockTGrenfellB.Inverse dependence and the allee effect. Trends Ecol Evol1999; 14(10): 405–410.
7.
Clutton-BrockTHGaynorDMcIlrathGMet al. Predation, group size and mortality in a cooperative mongoose, Suricata suricatta. J Anim Ecol1999; 68(4): 672–683.
8.
MooringMSFitzpatrickTANishihiraTTet al. Vigilance, predation risk, and the allee effect in desert bighorn sheep. J Wildl Manage2004; 68(3): 519–532.
9.
BerecLAnguloECourchampF.Multiple allee effects and population management. Trends Ecol Evol2006; 22(4): 185–191.
10.
CourchampFBerecLGascoigneJ.Allee effects in ecology and conservation. Oxford: Oxford University Press, 2008.
11.
RinellaDJWipfliMSStrickerCAet al. Pacific salmon (Oncorhynchus sp.) runs and consumer fitness: growth and energy storage in stream-dwelling salmonids increase with salmon spawner density. Canad J Fish Aquat Sci2012; 69(1): 73–84.
12.
StephensPASutherlandWJFreckletonRP.What is the allee effect?Oikos1999; 87(1): 185–190.
AlleeW.Studies in animal aggregations: some physiological effects of aggregation on the brittle starfish, Ophioderma brevispina. J Exp Zool1927; 48(2):475–495.
15.
AlleeW.Studies in animal aggregations: mass protection against colloidal silver among goldfishes. J Exp Zool1932; 61(2): 185–207.
16.
OdumE.Fundamentals of ecology. Philadelphia: Saunders, 1953.
17.
StephensPASutherlandWJ.Consequences of the allee effect for behaviour, ecology and conservation. Trends Ecol Evol1999; 14(10): 401–405.
18.
van VoornGAKHemerikLBoerMPet al. Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong allee effect. Math Biosci2007; 209(2): 451–469.
19.
WangMHKotM.Speeds of invasion in a model with strong or weak allee effects. Math Biosci2001; 171(1): 83–97.
20.
WangJShiJWeiJ.Predator-prey system with strong allee effect in prey. J Math Biol2011; 62(3): 291–331.
21.
ClarkCW.Mathematical bioeconomics: the optimal management of renewable resources. New York: Wiley, 1990.
22.
ClarkCW.The worldwide crisis in fisheries: economic models and human behavior. Cambridge: Cambridge University Press, 2007.
23.
LiermannMHilbornR.Depensation: evidence, models and implications. Fish Fish2001; 2(1): 33–58.
24.
WangGLiangXGWangFZ.The competitive dynamics of populations subject to an allee effect. Ecol Model1999; 124(2–3): 183–192.
ConwayEDSmollerJA.Global analysis of a system of predator prey equations. SIAM J Appl Math1986; 46(4): 630–642.
27.
FloresJDMena-LorcaJGonzález-Ya nezBet al. Consequences of depensation in a smith’s bioeconomic model for open-access fishery. In: Proceedings of International Symposium on Mathematical and Computational Biology (eds MondainiRDilaoR), pp. 219–232, 2007. Rio De Janeiro: E-paper Servicos Editorials Ltda.
28.
González-OlivaresERojas-PalmaA.Multiple limit cycles in a gause type predator-prey model with holling type iii functional response and allee effect on prey. Bull Math Biol2011; 73(6): 1378–1397.
29.
AguirrePGonzález-OlivaresESáezE.Two limit cycles in a Leslie-Gower predator-prey model with additive allee effect. Nonlinear Anal RWA2009; 10(3): 1401–1416.
30.
AguirrePGonzález-OlivaresESáezE.Three limit cycles in a leslie-gower predator-prey model with additive allee effect. SIAM J Appl Math2009; 69(3): 1244–1262.
31.
MaitiASenPMannaDet al. A predator-prey system with herd behaviour and strong allee effect. Nonlinear Dyn and Syst Theory2016; 16(1): 86–101.
32.
CosnerCDeAngelisDLAultJSet al. Effects of spatial grouping on the functional response of predators. Theor Pop Biol1999; 56(1): 65–75.
33.
AjraldiVPittavinoMVenturinoE.Modelling herd behavior in population systems. Nonlinear Anal RWA2011; 12(4): 2319–2338.
BeraSPMaitiASamantaGP.Modelling herd behavior of prey: analysis of a prey-predator model. World J Model Simul2015; 11(1): 3–14.
36.
AnguloERasmussenGSAMacdonaldDWet al. Do social groups prevent allee effect related extinctions?: The case of wild dogs. Front Zool2013; 10(11): 2–13.
37.
RenshawE.Modelling biological population in space and time. Cambridge: Cambridge University Press, 1995.
38.
MayRM.Stability and complexity in model ecosystems. Princeton: Princeton University Press, 1974.
39.
NisbetRMGurneyWSC. Modelling fluctuating populations. New York: Wiley Interscience, 1982.
40.
DimentbergMF.Lotka–Volterra system in a random environmentPhys Rev E2002; 65(036204): 16.
41.
SamantaGP.Stochastic analysis of a prey-predator system. Int J Math Educ Sci Technol1994; 25(6): 797–803.
42.
SamantaGP.Influence of environmental noises on gomatam model of interacting species. Ecol Model1996; 91(1): 283–291.
43.
MaitiASamantaGP.Deterministic and stochastic analysis of a prey-dependent predator-prey system. Int J Math Educ Sci Technol2005; 36(1): 65–83.
44.
MaitiASamantaGP.Deterministic and stochastic analysis of a ratio-dependent prey-predator system. Int J Syst Sci2006; 37(12): 817–826.
45.
MaitiAJanaMMSamantaGP.Deterministic and stochastic analysis of a ratio-dependent predator-prey system with delay. Nonlinear Anal Model Control2007; 12(3): 383–398.
46.
BandyopadhyayMChakrabartiCG.Deterministic and stochastic analysis of a non-linear prey-predator system. J Biol Syst2003; 11(2): 161–172.
47.
BandyopadhyayMChattopadhyayJ.Ratio-dependent predator-prey model: effect of environmental fluctuation and stability. Nonlinearity2005; 18(2): 913–936.
48.
BeraSPMaitiASamantaGP.Stochastic analysis of a prey-predator model with herd behaviour of prey. Nonlinear Anal Model Control2016; 21(3): 345–361.
49.
MaitiASenPSamantaGP.Deterministic and stochastic analysis of a prey-predator model with herd behaviour in both. Syst Sci Control Eng2016; 4(1): 259–269.
