Discontinuity, Nonlinearity, and Complexity
Incorporating Prey Refuge in a PreyPredator Model with BeddingtonDeAngelis Type Functional Response: A Comparative Study on IntraSpecific Competition
Discontinuity, Nonlinearity, and Complexity 9(3) (2020) 395419  DOI:10.5890/DNC.2020.09.005
Hafizul Molla$^{1}$, Md. Sabiar Rahman$^{2}$, Sahabuddin Sarwardi$^{3}$
$^{1}$ Department of Mathematics, Manbhum Mahavidyalaya, Purulia  723 131, West Bengal, India
$^{2}$ Department of Mathematics, Gobordanga Hindu College, North 24 Parganas 743 273, West Bengal, India
$^{3}$ Department of Mathematics & Statistics, Aliah University, IIA/27, New Town, Kolkata  700 160, India
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Abstract
The present study deals with a preypredator system with prey refuge depending on both species with the BeddingtonDeAngelis response function. We propose a mathematical model for predatorprey interactions, allowing prey refuge in the absence of intraspecific competition and in the presence of intraspecific competition among the predators. We have analyzed the models in terms of boundedness, persistence, existence of equilibria and their stability and Hopf bifurcation. Existence of paradox of enrichments
are examined well in both the cases. The analytical findings of this study are substantially validated by sufficient numerical simulations. The ecological implications of the obtained results are discussed as well.
Acknowledgments
The corresponding author Dr. Sarwardi is thankful to the Department of Mathematics & Statistics, Aliah University, for providing opportunities to perform the present work.
References

[1]  Rashevsky, N. (1951),Mathematical biology of social behavior, Univ. Chicago Press. 

[2]  May, R.M. (1973), Stability and Complexity in Model Ecosystems, Prin. Univ. Press, Princeton. 

[3]  Rubinow, S.I. (1975), Introduction to mathematical biology, Wiley. 

[4]  May, R.M. and McLean, A.R. (1976), Theoretical ecology: principles and applications, blackwell Scientific Publications, Oxf. Univ. Press. 

[5]  Freedman, H.I. (1980), Deterministic mathematical models in population ecology, Marcel Debber, New York. 

[6]  Epstein, J.M. (1997), Nonlinear dynamics, mathematical biology and social science, Westview Press. 

[7]  Kot, M. (2001), Elements of Mathematical Ecology, Camb. Univ. Press, New York. 

[8]  Britton, N. (2012), Essential mathematical biology, Sprin. Sc. & busi. Media. 

[9]  Lotka, A.J. (1962), Elements of Mathematical Biology, Dover, New York. 

[10]  Volterra, V. (1957), Opere matematiche: memorie e note pubblicate a cura dell’Accademia nazionale dei Lincei col concorso del Consiglio nazionale delle ricerche, 3. 

[11]  Holling, C.S. (1965), The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Canada, 45, 1–60. 

[12]  Cantrell, R.S. and Cosner, C. (2001), On the dynamics of predatorprey models with the BeddingtonDeAngelis functional response, J. Math. Anal. Appl., 257, 206–222. 

[13]  Cui, J. and Takeuchi, Y. (2006), Permanence, extinction and periodic solution of predatorprey systemwith Beddington DeAngelis functional response. J. Math. Anal. Appl., 317, 464–474. 

[14]  Huo, H.F., Li, W.T. and Nieto, J.J. (2007), Periodic solutions of delayed predatorprey model with the Beddington DeAngelis functional response. Chaos Sol. Frac., 33, 505–512. 

[15]  Hwang, T.W. (2003), Global analysis of the predatorprey system with BeddingtonDeAngelis functional response. J. Math. Anal. Appl., 281, 395–401. 

[16]  Hainzl, J. (1988), Stability and Hopf bifurcation a predatorprey system with several parameters. SIAM J Appl. Math., 48, 170–180. 

[17]  He, X. (1996), Stability and delays in a predatorprey system. J. Math. Anal. Appl., 198, 355–370. 

[18]  Murray, J.D. (1993),Mathematical Biology, SpringerVerlag, Berlin, Germany. 

[19]  Curds, C.R. and Cockburn, A. (1968), Studies on the growth and feeding of Tetrahymena pyriformis in axenic and monoxenic culture. J. Gen. Microbiol., 54, 343–358. 

[20]  Hassell,M.P. and Varley, G.C. (1969), New inductive populationmodel for insect parasites and its bearing on biological control. Nature, 223, 1133–1137. 

[21]  Salt, G.W. (1974), Predator and prey densities as controls of the rate of capture by the predator Didinium nasutum, Ecol., 55, 434–439. 

[22]  Gutierrez, A.P. (1992), The physiological basis of ratiodependent predatorprey theory: a metabolic pool model of Nicholson blowflies as an example, Ecol., 73, 1552–1563. 

[23]  Abrams, A. and Ginzburg, L.R. (2000), The nature of predation: prey dependent, ratio dependent or neither?, Trends Ecol. Evol., 15, 337–341. 

[24]  Arditi, R., Ginzburg, L.R. and Akcakaya, H.R. (1991), Variation in plankton densities among lakes: a case for ratio dependent models, Amer. Naturalist, 138, 1287–1296. 

[25]  Arditi, R., Perrin, N. and Saiah, H. (1991), Functional response and heterogeneities: an experimental test with cladocerans, OIKOS, 60, 69–75. 

[26]  Arditi, R. and Saiah, H. (1992), Empirical evidence of the role of heterogeneity in ratiodependent consumption, Ecol., 73 , 1544–1551. 

[27]  Cosner, C., DeAngelis, D.L., Ault, J.S. and Olson, D.B. (1999), Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56, 65–75. 

[28]  Arditi, R. and Ginzburg, L.R. (1989), Coupling in predatorprey dynamics: ratiodependence, J. Theor. Biol., 139, 311–326. 

[29]  Freedman, H.I. and Mathsen, R.M. (1993), Persistence in predatorprey systems with ratiodependent predator influence, Bull. Math. Biol., 55, 817–827. 

[30]  Hsu, S.B., Hwang, T.W. and Kuang, Y. (2003), Global analysis of MichaelisMenten type ratiodependent predatorprey system, J. Math. Biol., 42, 489–506. 

[31]  Jost, C., Arino, O. and Arditi, R. (1999), About deterministic extinction in ratiodependent predatorprey models, Bull. Math. Biol., 61, 19–32. 

[32]  Fan, M., Wang, Q. and Zou, X.F. (2003), Dynamics of a nonautonomous ratiodependent predatorprey system, Proc. Roy. Soc. Edin. Sect., 133, 97–118. 

[33]  Wang, Q., Fan, M. and Wang, K. (2003), Dynamics of a class of nonautonomous semi ratiodependent predatorprey systems with functional responses, J. Math. Anal. Appl., 278, 443–471. 

[34]  Fan, M. and Wang, K. (2002), Periodic solutions of a discrete time nonautonomous ratiodependent predatorprey system, Math. Comput. Mod., 35, 951–961. 

[35]  Kuang, Y. (1999), Rich dynamics of Gause type ratiodependent predatorprey system, Fields Inst. Commun., 21, 325–337. 

[36]  Haque,M. (2009), Ratiodependent predatorpreymodels of interacting populations, Bull. Math. Biol., 71(2), 430–452 . 

[37]  Beddington, J.R. (1975), Mutual interference between parasites or predators and its effect on searching efficiency, J.Ani. Ecol., 44, 331–340. 

[38]  DeAngelis, D.L., Goldstein, R.A. and O’Neill, R.V. (1975), A model for trophic interaction, Ecol., 56, 881–892. 

[39]  Ruxton, G., Gurney, W.S.C. and DeRoos, A. (1992), Interference and generation cycles, Theor. Popul. Biol., 42, 235– 253. 

[40]  Thieme, H. and Yang, J.L. (2000), On the complex formation approach in modeling predatorprey relations, mating and sexual disease transmission, Electron. J. Diff. Equ. Conf., 5, 255–283. 

[41]  Liao, X., Zhou, S. and Chen, Y. (2007), On permanence and global stability in a general GilpinAyala competition predatorprey discrete system, Appl. Math. Comput., 190, 500–509. 

[42]  Naji, R. and Balasim, A. (2007), Dynamical behavior of a three species food chain model with BeddingtonDeAngelis functional response, Chaos Soli. Frac., 32, 1853–1866. 

[43]  Redheffer, R. (1996), Nonautonomous LotkaVolterra systems, I. J. Diff. Equ., 127, 519–541. 

[44]  Zhang, S. Tan, D. and Chen, L. (2006), Dynamic complexities of a food chain model with impulsive perturbations and BeddingtonDeAngelis functional response, Chaos Soli. Frac, 27, 768–777. 

[45]  Mukherjee, D. (2016), The effect of refuge and immigration in a predatorprey system in the presence of a competitor for the prey, Nonl. Anal.: Real World Appl., 31, 277–287. 

[46]  Tripathi, J.P., Abbas, S. and Thakur, M. (2015), Dynamical analysis of a preypredator model with Beddington DeAngelis type function response incorporating a prey refuge, Nonl. Dyn., 80, 177–196. 

[47]  Sarwardi, S., Mandal, P.K. and Ray, S. (2012), Analysis of a competitive preypredator system with a prey refuge, Bios., 110(3), 133–148. 

[48]  Kar, T.K. (2005), Stability analysis of a preypredator model incorporating a prey refuge, Commu. Nonl. Sc. Num. Simu., 10, 681–691 

[49]  Murdoch,W.W. and Oaten, A. (1975), Predation and population stability, Adv. Ecol. Res., 9, 1–131. 

[50]  Stein, R.A. (1977), Selective predation, optimal foraging and the predatorprey interaction between fish and crayfish, Ecol., 58(6), 1237–1253. 

[51]  Haque, M. (2011), A detailed study of the BeddingtonDeAngelis predatorprey model, Math. Bios., 234, 1–16. 

[52]  Haque, M., Rahman, Md. S., Venturino, E. and Li, BL. (2014), Effect of a functional responsedependent prey refuge in a predatorprey model, Ecol. Comp., 20, 248–256. 

[53]  GonzálezOlivares, E., GonzalezYanez, B., BecerraKlix, R. and RamosJiliberto, R. (2017),Multiple stable states in a model based on predatorinduced defenses, Ecol. Complexity, 32, 111–120. 

[54]  Haque, Md. M. and Sarwardi, S. (2018), Dynamics of a Harvested PreyPredator Model with Prey Refuge Dependent on Both Species, I. J. B. C., 28(12), 1830–40. 

[55]  Molla, H., Rahman, Md. S. and Sarwardi, S. (2019), Dynamics of a PredatorPrey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species, IJNSNS, De Gruyter, 20(1), 1–16. 

[56]  Gard, T.C. and Hallam, T.G. (1979), Persistence in Food web1, LotkaVolterra food chains, Bull. Math. Biol., 41, 877891. 

[57]  Freedman, H. and Waltman, P. (1984), Persistence in models of three interacting predatorprey populations, Math. Bios., 68(2), 213–231. 

[58]  Birkhoff, G. and Rota, G.C. (1978), Ordinary Differential Equations, New York, Wiley. 

[59]  Collings, J.B. (1995), Bifurcation and stability analysis of a temperaturedependent mite predatorprey interaction model incorporating a prey refuge, Bull. Math. Biol., 57(1), 63–76. 

[60]  Alaoui, A.M. and Okiye, D.M. (2003), Boundedness and global stability for a predatorprey model with modified LeslieGower and Hollingtype II schemes, Appl. Math. Letters, 16(7), 1069–1075. 

[61]  Hale, J. (1989), Ordinary differential equation, Klieger Publishing Company, Malabar. 

[62]  Sharma, S. and Samanta, G. (2015), A LeslieGower predatorprey model with disease in prey incorporating a prey refuge, Chaos, Soli. & Fract., 70, 69–84. 

[63]  Wei,. F. and Fu, Q. (2016), Globally asymptotic stability of a predatorprey model with stage structure incorporating prey refuge, Int. J. Biomath., 9(04). 

[64]  Wei, F. and Fu, Q. (2016), Hopf bifurcation and stability for predatorprey systems with BeddingtonDeAngelis type functional response and stage structure for prey incorporating refuge, Appl. Math. Model., 40(1), 126–134. 

[65]  Li, H.L., Zhang, L., Hu, C., Jiang Y.L. and Teng, Z. (2017), Dynamical analysis of a fractionalorder predatorprey model incorporating a prey refuge, J. Appl. Math. Comput., 54(12), 435–449. 

[66]  Guin, L.N., Mondal, B., and Chakravarty, S. (2017), Stationary patterns induced by selfand crossdiffusion in a BeddingtonDeAngelis predatorprey model, Int. J. Dyn. Contrl., 5(4), 1051–1062. 

[67]  Wang, H., Thanarajah, S. and Gaudreau, P. (2018), Refugemediated predatorprey dynamics and biomass pyramids, Math. Biosc., 298, 29–45. 

[68]  Ray, S. and Stra ˘ skraba, M. (2001), The impact of detritivorous fishes on a mangrove estuarine system, Ecol. Model., 140 (3), 207–218. 