A Beddington-DeAngelis type one-predator two-prey competitive system with help. (English) Zbl 1412.92266
Summary: In this paper, we present and study a two competing prey and one predator system where during predation both the teams of prey help each other and the rate of predation on both the teams are different. The prey consumption rate per individual of predator is considered as Beddington-DeAngelis type. Beddington-DeAngelis type functional response describes predator feeding rate depending on both the prey and predator density. Ecological significance of considering this type of functional response depends on how frequently a predator encounters another predator while feeding on the same type of prey, resulting in strong intraspecific competition and thereby reducing the predator feeding rate. Firstly we establish persistence and permanence of the system and examine the role of cooperation coefficient. Different conditions for the coexistence of equilibrium solutions are discussed. Then the local asymptotic stability of various equilibrium points and the existence of Hopf bifurcation at the positive equilibrium are carried out to understand the dynamics of the model system. The Kolmogorov analysis of subsystems suggests for the existence of transcritical bifurcation. Sufficient conditions for the nonexistence of periodic orbits of the considered model system have been presented. The analytical results found in this paper are illustrated with the help of numerical examples.
MSC:
| 92D25 | Population dynamics (general) |
| 37N25 | Dynamical systems in biology |
| 34C23 | Bifurcation theory for ordinary differential equations |
Keywords:
transcritical bifurcation; periodic solution; competition; two-parameter bifurcation; Beddington-DeAngelis functional responseReferences:
| [1] | Armstrong, R.A., McGehee, R.: Competitive exclusion. Am. Nat. 115(2), 151-170 (1980) · doi:10.1086/283553 |
| [2] | Jana, D., Banerjee, A., Samanta, G.P.: Degree of prey refuges: control the competition among prey and foraging ability of predator. Chaos Solitons Fractals 104, 350-362 (2017) · Zbl 1380.92054 · doi:10.1016/j.chaos.2017.08.031 |
| [3] | Lou, Y., Wu, C.-H.: Global dynamics of a tritrophic model for two patches with cost of dispersal. SIAM J. Appl. Math. 71(5), 1801-1820 (2011) · Zbl 1254.34068 · doi:10.1137/100817954 |
| [4] | Upadhyay, R.K., Rai, V.: Crisis-limited chaotic dynamics in ecological systems. Chaos Solitons Fractals 12, 205-218 (2001) · Zbl 0977.92033 · doi:10.1016/S0960-0779(00)00141-7 |
| [5] | Kesh, D., Sarkar, A.K., Roy, A.B.: Persistence of two prey-one predator system with ratio-dependent predator influence. Math. Method Appl. Sci. 23(4), 347-356 (2000) · Zbl 0951.92029 · doi:10.1002/(SICI)1099-1476(20000310)23:4<347::AID-MMA117>3.0.CO;2-F |
| [6] | Kar, T.K., Chaudhuri, K.S.: Harvesting in a two-prey one-predator fishery: a bioeconomic model. ANZIAM J. 45, 443-456 (2004) · Zbl 1052.92052 · doi:10.1017/S144618110001347X |
| [7] | Tripathi, J.P., Abbas, S., Thakur, M.: Local and global stability analysis of a two prey one predator model with help. Commun. Nonlinear Sci. Numer. Simul. 19, 3284-3297 (2014) · Zbl 1510.92187 · doi:10.1016/j.cnsns.2014.02.003 |
| [8] | Gakkhar, S., Singh, B.: The dynamics of a food web consisting of two preys and a harvesting predator. Chaos Solitons Fractals 34, 1346-1356 (2007) · Zbl 1142.34335 · doi:10.1016/j.chaos.2006.04.067 |
| [9] | Fujii, K.: Complexity-stability relationship of two-prey-one-predator species system model: local and global stability. J. Theor. Biol. 69(4), 613-623 (1977) · doi:10.1016/0022-5193(77)90370-8 |
| [10] | Gakkhar, S., Singh, B.: Complex dynamic behavior in a food web consisting of two preys and a predator. Chaos Solitons Fractals 24(3), 789-801 (2005) · Zbl 1081.37060 · doi:10.1016/j.chaos.2004.09.095 |
| [11] | Lv, Y., Cao, J., Song, J., Yuan, R., Pei, Y.: Global stability and Hopf-bifurcation in a zooplankton-phytoplankton model. Nonlinear Dyn. 76, 345-366 (2014) · Zbl 1319.92045 · doi:10.1007/s11071-013-1130-2 |
| [12] | Hsu, S.-B., Ruan, S., Yang, T.-H.: On the dynamics of two-consumers one-resource competing systems with Beddington DeAngelis functional response. Discrete Contin. Dyn. Syst. 19, 3284-3297 (2014) |
| [13] | Rogers, D.J., Hassell, M.P.: General models for insect parasite and predator searching behavior: interference. J. Anim. Ecol. 43, 239-253 (1974) · doi:10.2307/3170 |
| [14] | Freedman, H.I., Rao, S.H.R.: The trade-off between mutual interference and time lags in predator-prey systems. Bull. Math. Biol. 45(6), 991-1004 (1983) · Zbl 0535.92024 · doi:10.1007/BF02458826 |
| [15] | Erbe, L.H., Freedman, H.I.: Three-species food-chain models with mutual interference and time delays. Math. Biosci. 80, 57-80 (1986) · Zbl 0592.92024 · doi:10.1016/0025-5564(86)90067-2 |
| [16] | Jana, D., Agrawal, R., Upadhyay, R.K.: Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain. Chaos Solitons Fractals 69, 50-63 (2014) · Zbl 1352.92126 · doi:10.1016/j.chaos.2014.09.001 |
| [17] | Jana, D., Tripathi, J.P.: Impact of generalist type sexually reproductive top predator interference on the dynamics of a food chain model. J. Dyn. Control, Int (2016). https://doi.org/10.1007/s40435-016-0255-9 · doi:10.1007/s40435-016-0255-9 |
| [18] | Andersson, M., Erlinge, S.: Influence of Predation on Rodent Populations, pp. 591-597. Wiley, Oikos (1977) |
| [19] | Mukherjee, D., Roy, A.B.: Uniform persistence and global stability of two prey-predator pairs linked by competition. Math. Biosci. 99(1), 31-45 (1990) · Zbl 0709.92021 · doi:10.1016/0025-5564(90)90137-N |
| [20] | Parrish, J.D., Saila, S.B.: Interspecific competition, predation and species diversity. J. Theor. Biol. 27, 207-220 (1970) · doi:10.1016/0022-5193(70)90138-4 |
| [21] | Cramer, N.F., May, R.M.: Interspecific competition, predation and species diversity: a comment. J. Theor. Biol. 34, 289-293 (1972) · doi:10.1016/0022-5193(72)90162-2 |
| [22] | Kang, Y., Wedekin, L.: Dynamics of a intraguild predation model with generalist or specialist predator. J. Math. Biol. 67(5), 1227-1259 (2013) · Zbl 1286.34071 · doi:10.1007/s00285-012-0584-z |
| [23] | Sih, A.: Prey refuges and predator-prey stability. Theor. Popul. Biol. 31, 1-12 (1987) · doi:10.1016/0040-5809(87)90019-0 |
| [24] | Sih, A., Crowley, P., McPeek, M., Petranka, J., Strohmeier, K.: Predation, competition, and prey communities: a review of field experiments. Ann. Rev. Ecol. Syst. 16, 269-311 (1985) · doi:10.1146/annurev.es.16.110185.001413 |
| [25] | Sih, A., Kats, L.B., Moore, R.D.: Effect of predatory sunfish on the density, drift and refuge use of stream salamander larvae. Ecology 73(4), 1418-1430 (1992) · doi:10.2307/1940687 |
| [26] | Parshad, R.D., Basheer, A., Jana, D., Tripathi, J.P.: Do prey handling predators really matter: Subtle effects of a Crowley-Martin functional response. Chaos Solitons Fractals 103, 410-421 (2017) · Zbl 1375.35039 · doi:10.1016/j.chaos.2017.06.027 |
| [27] | Beddington, J.R.: Mutual interference between parasites or predators and it’s effect on searching efficiency. J. Anim. Ecol. 44(1), 331-340 (1975) · doi:10.2307/3866 |
| [28] | DeAngelis, D.L., Goldstein, R.A., O’Neill, R.V.: A model for tropic interaction. Ecology 56(4), 881-892 (1975) · doi:10.2307/1936298 |
| [29] | Huisman, G., De Boer, R.J.: A formal derivation of the Beddington functional response. J. Theor. Biol. 185, 389-400 (1997) · doi:10.1006/jtbi.1996.0318 |
| [30] | Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385-398 (1959) · doi:10.4039/Ent91385-7 |
| [31] | Lahrouz, A., Settati, A., Mandal, P.S.: Dynamics of a switching diffusion modified Leslie-Gower predator-prey system with Beddington-DeAngelis functional response. Nonlinear Dyn. 85(2), 853-870 (2016) · Zbl 1355.37101 · doi:10.1007/s11071-016-2728-y |
| [32] | Tripathi, J.P., Tyagi, S., Abbas, S.: Global analysis of a delayed density dependent predator-prey model with Crowley-Martin functional response. Commun. Nonlinear Sci. Numer. Simul. 30, 45-69 (2016) · Zbl 1489.92125 · doi:10.1016/j.cnsns.2015.06.008 |
| [33] | Tripathi, J.P., Abbas, S., Thakur, M.: Dynamical analysis of a prey-predator model with Beddington-DeAngelis type function response incorporating a prey refuge. Nonlinear Dyn. 80, 177-196 (2015) · Zbl 1345.92125 · doi:10.1007/s11071-014-1859-2 |
| [34] | Upadhyay, R.K., Agrawal, R.: Dynamical analysis of a prey-predator model with Beddington-DeAngelis type function response incorporating a prey refuge. Nonlinear Dyn. 83(1-2), 821-837 (2016) · Zbl 1349.37092 · doi:10.1007/s11071-015-2370-0 |
| [35] | Abbas, S., Bahuguna, D., Banerjee, M.: Effect of stochastic perturbation on a two species competitive model. Nonlinear Anal. Hybrid Syst. 3(3), 195-206 (2009) · Zbl 1192.34097 · doi:10.1016/j.nahs.2009.01.001 |
| [36] | Dugatkin, L.A.: Co-operation Among Animals: A Evolutionary Prospective. Oxford University Press, New York (1997) |
| [37] | Martin, N.M., Mitani, J.C.: Conflict and Co-operation in Wild Life Chimpanzees. Advances in the Study of Behaviour, vol. 35. Academic Press, Cambridge (2005) |
| [38] | Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge (1998) · Zbl 0914.90287 · doi:10.1017/CBO9781139173179 |
| [39] | Tripathi, J.P., Abbas, S., Thakur, M.: Stability analysis of two prey one predator model. AIP Conf. Proc. 1479, 905-909 (2012) · doi:10.1063/1.4756288 |
| [40] | Elettreby, M.F.: Two prey one-predator model. Chaos Solitons Fractals 39, 2018-2027 (2009) · Zbl 1197.34095 · doi:10.1016/j.chaos.2007.06.058 |
| [41] | Vance, R.R.: Predation and resource partitioning in one predator-two prey model communities. Am. Nat. 112, 797-813 (1978) · doi:10.1086/283324 |
| [42] | Jana, D., Agrawal, R., Upadhyay, R.K.: Dynamics of generalist predator in a stochastic environment: effect of delayed growth and prey refuge. Appl. Math. Comput. 268, 1072-1094 (2015) · Zbl 1410.34136 |
| [43] | Sun, G.Q., Wu, Z.Y., Wang, Z., Jin, Z.: Influence of isolation degree of spatial patterns on persistence of populations. Nonlinear Dyn. 83(1-2), 811-819 (2016) · doi:10.1007/s11071-015-2369-6 |
| [44] | Brikhoff, G., Rota, G.C.: Ordinary Differential Equations. Ginn, Boston (1982) |
| [45] | Ahmad, S., Rao, M.R.M.: The Theory of Ordinary Differential Equations with Applications in Biology and Engineering. Affliated East- West Press Private Limited, New Delhi (1999) |
| [46] | Tripathi, J.P., Abbas, S., Thakur, M.: A density dependent delayed predator-prey model with Beddington-DeAngelis type function response incorporating a prey refuge. Commun. Nonlinar Sci. Numer. Simul. 22, 427-450 (2015) · Zbl 1329.92112 · doi:10.1016/j.cnsns.2014.08.018 |
| [47] | Liu, B., Teng, Z., Chen, L.: Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy. J. Comput. Appl. Math. 193, 347-362 (2006) · Zbl 1089.92060 · doi:10.1016/j.cam.2005.06.023 |
| [48] | Gakkhar, S., Naji, R.K.: Existence of chaos in two-prey, one-predator system. Chaos Solitons Fractals 17, 639-649 (2003) · Zbl 1034.92033 · doi:10.1016/S0960-0779(02)00473-3 |
| [49] | Arditi, R., Michalski, J.: Nonlinear Food Web Models and Their Response to Increased Basal Productivity, Food Webs, pp. 122-133. Springer, New York (1996) |
| [50] | May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1973) |
| [51] | Gupta, R.P., Chandra, P.: Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting. J. Math. Anal. Appl. 398, 278-295 (2013) · Zbl 1259.34035 · doi:10.1016/j.jmaa.2012.08.057 |
| [52] | Jana, D., Bairagi, N.: Habitat complexity, dispersal and metapopulations: macroscopic study of a predator-prey system. Ecol. Complex. 17, 131-139 (2014) · doi:10.1016/j.ecocom.2013.11.006 |
| [53] | Jana, D., Ray, S.: Impact of physical and behavioral prey refuge on the stability and bifurcation of Gause type Filippov prey-predator system. Model. Earth Syst. Environ. 2, 24 (2016). https://doi.org/10.1007/s40808-016-0077-y · doi:10.1007/s40808-016-0077-y |
| [54] | Busenberg, S., Vandendriessche, P.: A method for proving the non-existence of limit cycles. J. Math. Anal. Appl. 172, 463-479 (1993) · Zbl 0779.34026 · doi:10.1006/jmaa.1993.1037 |
| [55] | Chattopadhyay, J., Pal, S.: Viral infection on phytoplankton-zooplankton system—a mathematical model. Ecol. Model. 151, 15-28 (2002) · doi:10.1016/S0304-3800(01)00415-X |
| [56] | Hutson, V., Vickers, G.T.: A criterion for permanent coexistence of species, with an application to a two-prey one-predator system. Math. Biosci. 63(2), 253-269 (1983) · Zbl 0524.92023 · doi:10.1016/0025-5564(82)90042-6 |
| [57] | Butler, G., Freedman, H.I., Waltman, P.: Uniformly persistent systems. Proc. Am. Math. Soc. 96(3), 425-430 (1986) · Zbl 0603.34043 · doi:10.1090/S0002-9939-1986-0822433-4 |
| [58] | Freedman, H.I., Ruan, S.: Uniform persistence in functional differential equations. J. Differ. Equ. 115, 173-192 (1995) · Zbl 0814.34064 · doi:10.1006/jdeq.1995.1011 |
| [59] | Freedman, H.I., So, J.H.: Global stability and persistence of simple food chains. Math. Biosci. 76, 69-86 (1985) · Zbl 0572.92025 · doi:10.1016/0025-5564(85)90047-1 |
| [60] | Gard, T.C., Hallam, T.G.: Persistence in food webs-I Lotka-Volterra food chains. Bull. Math. Biol. 41(6), 877-891 (1979) · Zbl 0422.92017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
