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On the slow-fast dynamics of a tri-trophic food chain model with fear and Allee effects. (English) Zbl 07872013

MSC:

34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
37B25 Stability of topological dynamical systems
37C83 Dynamical systems with singularities (billiards, etc.)
Full Text: DOI

References:

[1] N.Shigesada, K.Kawasaki, and E.Teramoto, Spatial segregation of interacting species, J. Theor. Biol.79 (1979), 83-99.
[2] X.Meng, R.Liu, and T.Zhang, Adaptive dynamics for a non‐autonomous Lotka‐Volterra model with size‐selective disturbance, Nonlinear Anal. Real World Appl.16 (2014), 202-213. · Zbl 1325.92074
[3] A. A.Berryman, The origins and evolution of predator‐prey theory, Ecology73 (1992), no. 5, 1530-1535.
[4] T. R.Malthus, An essay on the principle of population, as it affects the future improvement of society, with remarks on the speculations of Mr. Godwin, M. Condorcet, and other writers, The Lawbook Exchange, Ltd., London, 1798.
[5] P. F.Verhulst, Notice sur la loi que la population suit dans son accroissement. Correspondance mathematique et physique publiee par A, Quetelet10 (1838), 113-121.
[6] A.Lotka, Elements of physical biology, Williams and Wilkins, Baltimore, 1925. · JFM 51.0416.06
[7] V.Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem. Accd. Linc.2 (1926), 31-113. · JFM 52.0450.06
[8] C. S.Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Canada97, no. S45, 1-60.
[9] C. S.Elton, Animal ecology, Sidgwick and Jackson, London, 1927.
[10] R. L.Lindeman, The trophic‐dynamic aspect of ecology, Ecology23 (1942), 399-417.
[11] D.Manna, A.Maiti, and G. P.Samanta, A Michaelis-Menten type food chain model with strong Allee effect on the prey, Appl. Math. Comput.311 (2017), 390-409. · Zbl 1426.92055
[12] P. W.Prince, C. E.Bouton, P.Gross, B. A.Mcpheropn, J. N.Thompson, and A. E.Weis, Interactions among three trophic levels: influence of plants on interactions between insect herbivores and natural enemies, Annu. Rev. Ecol. Syst.11 (1980), 41-65.
[13] A.Hastings and T.Powell, Chaos in a three‐species food chain, Ecology72 (1991), 896-903.
[14] Y. A.Kuznetsov and S.Rinaldi, Remarks on food chain dynamics, Math. Biosci.134 (1996), 1-33. · Zbl 0844.92025
[15] K.McCann and P.Yodzis, Biological conditions for chaos in a three‐species food chain, Ecology75 (1994), 561-564.
[16] A.Klebanoff and A.Hastings, Chaos in three species food chains, J. Math. Biol.32 (1994), 427-51. · Zbl 0823.92030
[17] J. N.Eisenberg and D. R.Maszle, The structural stability of a three‐species food chain model, J. Theor. Biol.176 (1995), no. 4, 501-510.
[18] H. I.Freedman, Mathematical analysis of some three‐species food‐chain models, Math. Biosci.33 (1977), no. 3-4, 257-276. · Zbl 0363.92022
[19] H. I.Freedman and P.Waltman, Persistence in models of three interacting predator‐prey populations, Math. Biosci.68 (1983), no. 2, 213-231. · Zbl 0534.92026
[20] S.Lv and M.Zhao, The dynamic complexity of a three species food chain model, Chaos Soliton Fract.37 (2008), 1469-80. · Zbl 1142.92342
[21] Z.Song, B.Zhen, and J.Xu, Species coexistence and chaotic behavior induced by multiple delays in a food chain system, Ecol. Complex.19 (2014), 9-17.
[22] D.Sen, S.Ghorai, and M.Banerjee, Complex dynamics of a three species prey‐predator model with intraguild predation, Ecol. Complex.34 (2018), 9-22.
[23] B.Ghosha, D.Pal, T.Legovic, and T. K.Kar, Harvesting induced stability and instability in a tri‐trophic food chain, Math. Biosci.304 (2018), 89-99. · Zbl 1409.92263
[24] Y.Mu and W. C.Lo, Dynamics of the food‐chain population in a polluted environment with impulsive input of toxicant, Discrete Contin. Dyn. Syst. Ser. B.26 (2021), no. 8, 4173-4190. · Zbl 1466.92229
[25] B.Mondal, U.Ghosh, M. d.Rahman, P.Saha, and S.Sarkar, Sadikur, Studies of different types of bifurcations analyses of an imprecise two species food chain model with fear effect and non‐linear harvesting, Math. Comput. Simul.192 (2022), 111-135. · Zbl 1540.34006
[26] W.Allee, Animal aggregations, Q. Rev. Biol.2 (1927), 367-398.
[27] W.Allee, Studies in animal aggregations: some physiological effects of aggregation on the brittle starfish, Ophioderma brevispina, J. Exp. Zool.48 (1927), 475-495.
[28] W.Allee, Studies in animal aggregations: mass protection against colloidal silver among goldfishes, J. Exp. Zool.61 (1932), 185-207.
[29] F.Courchamp, L.Berec, and J.Gascoigne, Allee effects in ecology and conservation, Oxford University Press, Oxford, 2008.
[30] A. J.Terry, Predator-prey models with component Allee effect for predator reproduction, J. Math. Biol.71 (2015), 1325-1352, DOI 10.1007/s00285‐015‐0856‐5. · Zbl 1330.34078
[31] E.Gonzales‐Oliviars, J.Mena‐Lorca, A. R.Palma, and J. D.Flores, Dynamical complexities in the Leslie‐Gower predator‐prey model as consequences of the Allee effect on prey, Appl. Math. Model.35 (2011), no. 1, 366-381. · Zbl 1202.34079
[32] E.Gonzalez‐Olivares and A. R.Palma, 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 (2011), 1378-1397, DOI 10.1007/s11538‐010‐9577‐5. · Zbl 1215.92061
[33] J.Bhattacharyya and S.Pal, Algae-herbivore interactions with Allee effect and chemical defense, Ecol. Complex.27 (2016), 48-62.
[34] P.Mandal and A.Lahrouz, Study of dynamical properties in a bi‐dimensional model describing the prey‐predator dynamics with strong Allee effect in prey, Electron. J. Qual. Theory Differ. Equ.2017 (2017), no. 93, 1-21. · Zbl 1413.92028
[35] A. R.Palma and E.Gonzalez‐Olivares, Optimal harvesting in a predator‐prey model with Allee effect and Sigmoid functional response, Appl. Math. Model.36, no. 5, 1864-1874. · Zbl 1243.49046
[36] J.Wang, J.Shi, and J.Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol.62, no. 3, 291-331. · Zbl 1232.92076
[37] F.Capone, M. F.Carfora, R.De Luca, and I.Torcicollo, Nonlinear stability and numerical simulations for a reaction-diffusion system modelling Allee effect on predators, Int. J. Nonlinear Sci. Numer. Simul.23 (2022), no. 5, 751-760, DOI 10.1515/ijnsns‐2020‐0015. · Zbl 1525.35166
[38] S. R.Zhou, Y.Liu, and G.Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Popul. Biol.67 (2005), 23-31. · Zbl 1072.92060
[39] X.Lai, S.Liu, and R.Lin, Rich dynamical behaviours for predator-prey model with weak Allee effect, Appl. Anal.89 (2010), 1271-1292. · Zbl 1193.92083
[40] R.Lin, S.Liu, and X.Lai, Bifurcations of a predator‐prey system with weak Allee effects, J. Korean Math. Soc.50 (2013), no. 4, 695-713, DOI 10.4134/JKMS.2013.50.4.695. · Zbl 1301.34065
[41] S.Debnath, U.Ghosh, and S.Sarkar, Global dynamics of a tritrophic food chain model subject to the Allee effects in the prey population with sexually reproductive generalized‐type top predator, Comp. Math. Methods.2 (2020), no. 2, e1079, DOI 10.1002/cmm4.1079.
[42] X.Wang, L.Zanette, and X.Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol.73 (2016), no. 5, 1179-1204, DOI 10.1007/s00285‐016‐0989‐1. · Zbl 1358.34058
[43] O. J.Schmitz, A. P.Beckerman, and K. M.Brien, Behaviorally mediated trophic cascades: effects of predation risk on food web interactions, Ecology78 (1997), 1388-1399.
[44] E.Preisser, D. I.Bolnick, and M. F.Benard, Scared to death? The effects of intimidation and consumption in predator‐prey interactions, Ecology86 (2005), 501-509.
[45] L. Y.Zanette, A. F.White, M. C.Allen, and M.Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science334 (2011), 1398-1401.
[46] E. L.Preisser and D. I.Bolnick, The many faces of fear: comparing the pathways and impacts of nonconsumptive predator effects on prey populations, PloS One3 (2008), no. 6, e2465.
[47] S. D.Peacor, B. L.Peckarsky, G. C.Trussell, and J. R.Vonesh, Costs of predator‐induced phenotypic plasticity: a graphical model for predicting the contribution of nonconsumptive and consumptive effects of predators on prey, Oecologia171 (2013), no. 1, 1-10.
[48] N.Pettorelli, T.Coulson, S. M.Durant, and J. M.Gaillard, Predation, individual variability and vertebrate population dynamics, Oecologia167 (2011), no. 2, 305-314.
[49] J.Roy, D.Barman, and S.Alam, Role of fear in a predator‐prey system with ratio‐dependent functional response in deterministic and stochastic environment, Biosystems197, no. 2020, 104176.
[50] P.Panday, N.Pal, S.Samanta, and J.Chattopadhyay, Stability and bifurcation analysis of a three‐species food chain model with fear, Int. J. Bif. Chaos28 (2018), no. 1, 1850009. · Zbl 1386.34093
[51] J. P.Suraci, M.Clinchy, L. M.Dill, D.Roberts, and L. Y.Zanette, Fear of large carnivores causes a trophic cascade, Nat. Commun.7 (2016), 10698.
[52] E.Safitri and D.Aldila, Analyzing effect of harvesting on prey population when prey growth depend on fear‐factor and Allee‐effect, AIP Conf. Proc.2192 (2019), 60017, DOI 10.1063/1.5139163.
[53] S.Halder, J.Bhattacharyya, and S.Pal, Comparative studies on a predator-prey model subjected to fear and Allee effect with type I and type II foraging, J. Appl. Math. Comput.62 (2020), 93-118. · Zbl 1478.92155
[54] J.Kevorkian and J. D.Cole, Perturbation methods in applied mathematics, Springer‐Verlag, New York, 1981. · Zbl 0456.34001
[55] R. E.O’Malley Jr, Singular perturbation methods in ordinary differential equations, Springer‐Verlag, New York, 1991. · Zbl 0743.34059
[56] V. F.Butuzov, N. N.Nefedov, and K. R.Schneider, Singularly perturbed problems in cases of exchange of stabilities, J. Math. Sci.121 (2004), 1973-2079. · Zbl 1081.34051
[57] K.Yadi, Averaging on slow and fast cycles of a three time scale system, J. Math. Anal. Appl.413 (2014), 976-998. · Zbl 1317.34091
[58] M.Aguiar, B.Kooi, A.Pugliese, M.Sensi, and N.Stollenwerk, Time scale separation in the vector borne disease model SIRUV via center manifold analysis. medRxiv.
[59] J.Shen, C. H.Hsu, and T. H.Yang, Fast-slow dynamics for intraguild predation models with evolutionary effects, J. Dyn. Differ. Equ.32 (2020), 895-920, DOI 10.1007/s10884‐019‐09744‐3. · Zbl 1461.34073
[60] L.Owen and J. M.Tuwankotta, On slow-fast dynamics in a classical predator-prey system, Math. Comput. Simul.177 (2020), 306-315. · Zbl 1510.92176
[61] S. M.Salman, Modeling a fast‐slow bitrophic food chain with harvesting, Nonlinear Dynamics Psychol. Life Sci.23 (2019), no. 2, 177-197.
[62] S. M.Salman, A singularly perturbed vector‐bias malaria model incorporating bed net control, Math. Meth. Appl. Sci.46 (2023), no. 5, 5139-5159, DOI 10.1002/mma.8822. · Zbl 1538.34187
[63] B. W.Kooi and J. C. H.Poggiale, Modelling, singular perturbation and bifurcation analyses of bitrophic food chains, Math. Biosci.301 (2018), 93-110. · Zbl 1394.34093
[64] M.Brons and K.Bar‐Eli, Canard explosion and excitation in a model of the Belousov Zhabotinsky reaction, J. Phys. Chem.95 (1991), 8706-8713.
[65] M.Brons and K.Bar‐Eli, Asymptotic analysis of canards in the EOE equations and the role of the inflection line, Proc. R. Soc. London: Math. Phys. Sci.445 (1994), 305322. · Zbl 0807.34067
[66] M.Brons and J.Sturis, Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system, Phys. Rev. E64 (2001), 26209.
[67] J.Moehlis, Canards in a surface oxidation reaction, J. Nonlinear Sci.12 (2002), 319-345. · Zbl 1095.34527
[68] M.Sekikawa, N.Inaba, and T.Tsubouchi, Chaos via duck solution breakdown in a piecewise linear van der Pol oscillator driven by an extremely small periodic perturbation, Phys. D194 (2004), 227-249. · Zbl 1098.34546
[69] E. F.Mishchenko and N. K. h.Rozov, Differential equations with small parameters and relaxation oscillations, Plenum Press, New York, 1980. · Zbl 0482.34004
[70] J. D.Murray, Mathematical biology, Springer‐Verlag, Berlin, 2003. · Zbl 1006.92002
[71] J.Grasman, Asymptotic methods for relaxation oscillations and applications, Springer‐Verlag, New York, 1987. · Zbl 0627.34037
[72] M.Krupa and J. D.Touboul, Canard explosion in delay differential equations, J. Dyn. Differ. Equ.28 (2016), 471-491. · Zbl 1350.34059
[73] H.Jardòn‐Kojakhmetov and C.Kuehn, Controlling canard cycles, J Dyn Control Syst28 (2022), 517-544, DOI 10.1007/s10883‐021‐09553‐2. · Zbl 1506.34074
[74] I.Schurov and N.Solodovnikov, Duck factory on the two‐torus: multiple canard cycles without geometric constraints, J. Dyn. Control Syst.23 (2017), 481-498, DOI 10.1007/s10883‐016‐9335‐6. · Zbl 1382.34063
[75] V. A.Sobolev and E. A.Shchepakina, Duck trajectories in a problem of combustion theory, Differ. Equ.32 (1996), 1177-1186. · Zbl 0893.34054
[76] E.Shchepakina, V.Sobolev, and M. P.Mortell, Singular perturbations, introduction to system order reduction methods with applications, Lecture Notes in Mathematics, Vol. 2114, Springer, Cham-Heidelber-New York-Dordrecht-London, 2014. · Zbl 1397.34003
[77] E.Shchepakina and V.Sobolev, Integral manifolds, canards and black swans, Nonlinear Anal. A44 (2001), 897-908. · Zbl 0985.34039
[78] E.Shchepakina and V.Sobolev, Black swans and canards in laser and combustion models, Singular perturbation and hysteresis, SIAM, Philadelphia, 2005, pp. 207-255. · Zbl 1248.34087
[79] J. S.Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math.20 (1990), 857-872. · Zbl 0725.34049
[80] J.Arino, C. C.McCluskey, and P.vanden Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math.64 (2003), 260-276. · Zbl 1034.92025
[81] C. C.McCluskey and P.vanden Driessche, Global analysis of two tuberculosis models, J. Dyn. Differ. Equ.16 (2004), 139-166. · Zbl 1056.92052
[82] C. K. R. T.Jones, Geometric singular perturbation theory, Dyn. Syst.1609 (1995), 44-118. · Zbl 0840.58040
[83] J.Kevorkian and J.Cole, Multiple scale and singular perturbation methods, Springer‐Verlag, Berlin, Germany, 1995.
[84] E.Benoit, J. L.Callot, F.Diener, and M.Diener, Chasse au canard, Collect. Math.31-32 (1981), 37-119. · Zbl 0529.34046
[85] M.Diener, Nessie Et Les Canards, Publication IRMA, Strasbourg, 1979.
[86] E.Shchepakina, Black swans and canards in two predator-one prey model, Math. Model. Nat Phenom.14 (2019), no. 4, 408. · Zbl 1470.92262
[87] N.Fenichel, Geometric singular perturbation theory, Differ. Equ.31 (1979), 53-98. · Zbl 0476.34034
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