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Spatiotemporal Model of a Predator–Prey System with Herd Behavior and Quadratic Mortality

Teekam Singh

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247 667, Uttarakhand, India

E-mail Address: tsingh@ma.iitr.ac.in

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and

Sandip Banerjee

http://orcid.org/0000-0003-2817-5729

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247 667, Uttarakhand, India

E-mail Address: sandofma@iitr.ac.in

Corresponding author.

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https://doi.org/10.1142/S0218127419500494Cited by:18 (Source: Crossref)

Abstract

In this paper, we have investigated a diffusive predator–prey model with herd behavior. Also, we considered that the mortality of predators is linear as well as quadratic. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of control parameters. Using extensive numerical simulations, we obtain non-Turing spatiotemporal patterns in the model with linear mortality of predators, and Turing pattern formation, namely, spotted pattern and mixed pattern (spots-stripes) in model with quadratic mortality of predators. The results focus on the effect of the changing mortality rates of predator in pattern dynamics of a diffusive predator–prey model and help us in the better understanding of the dynamics of the predator–prey interaction in real environment.

Keywords:

References

Ajraldi, V., Pittavino, M. & Venturino, E. [2011] “ Modeling herd behavior in population systems,” Nonlin. Anal.: Real World Appl. 12, 2319–2338. Crossref, Web of Science, Google Scholar
Baurmann, M., Gross, T. & Feudel, U. [2007] “ Instabilities in spatially extended predator–prey systems: Spatiotemporal patterns in the neighborhood of Turing–Hopf bifurcations,” J. Theor. Biol. 245, 220–229. Crossref, Web of Science, Google Scholar
Beddington, J. R. [1975] “ Mutual interference between parasites or predators and its effect on searching efficiency,” J. Animal Ecol. 44, 331–340. Crossref, Web of Science, Google Scholar
Belvisi, S. & Venturino, E. [2013] “ An ecoepidemic model with diseased predators and prey group defense,” Simul. Model. Pract. Th. 34, 144–155. Crossref, Web of Science, Google Scholar
Bera, S. P., Maiti, A. & Samanta, G. P. [2015] “ Modelling herd behavior of prey: Analysis of a prey–predator model,” World J. Model. Simul. 11, 3–14. Google Scholar
Braza, P. A. [2012] “ Predator–prey dynamics with square root functional responses,” Nonlin. Anal.: Real World Appl. 13, 1837–1843. Crossref, Web of Science, Google Scholar
Brentnall, S. J., Richards, K. J., Brindley, J. & Murphy, E. [2003] “ Plankton patchiness and its effect on larger-scale productivity,” J. Plankton Res. 25, 121–140. Crossref, Web of Science, Google Scholar
Crowley, P. H. & Martin, E. K. [1989] “ Functional responses and interference within and between year classes of a dragonfly population,” J. North Am. Benthol. Soc. 8, 211–221. Crossref, Web of Science, Google Scholar
DeAngelis, D. L., Goldstein, R. A. & O’Neill, R. V. [1975] “ A model for tropic interaction,” Ecology 56, 881–892. Crossref, Web of Science, Google Scholar
Descombes, S. & Massot, M. [2004] “ Operator splitting for nonlinear reaction–diffusion systems with an entropic structure: Singular perturbation and order reduction,” Numer. Math. 97, 667–698. Crossref, Web of Science, Google Scholar
Fulton, E. A., Smith, A. D. M. & Johnson, C. R. [2003] “ Mortality and predation in ecosystem models: Is it important how these are expressed?” Ecol. Model. 169, 157–178. Crossref, Web of Science, Google Scholar
Gambino, G., Lombardo, M. C. & Sammartino, M. [2018] “ Cross-diffusion-induced subharmonic spatial resonances in a predator–prey system,” Phys. Rev. E 97, 012220–012230. Crossref, Web of Science, Google Scholar
Gimmelli, G., Kooi, B. W. & Venturino, E. [2015] “ Stability, ecoepidemic models with prey group defense and feeding saturation,” Ecol. Complex. 22, 50–58. Crossref, Web of Science, Google Scholar
Holling, C. S. [1959a] “ The components of predation as revealed by a study of small-mammal predation of the European pine sawfly,” Can. Entomol. 91, 293–320. Crossref, Google Scholar
Holling, C. S. [1959b] “ Some characteristics of simple types of predation and parasitism,” Can. Entomol. 91, 385–398. Crossref, Google Scholar
Ji, C. Y., Jiang, D. Q. & Li, X. Y. [2011] “ Qualitative analysis of a stochastic ratio-dependent predator–prey system,” J. Comput. Appl. Math. 235, 1326–1341. Crossref, Web of Science, Google Scholar
Kuang, Y. & Beretta, E. [1998] “ Global qualitative analysis of a ratio-dependent predator–prey system,” J. Math. Biol. 36, 389–406. Crossref, Web of Science, Google Scholar
LeVeque, R. J. [1982] “Time-split methods for partial differential equations,” PhD thesis, Stanford University, CA, Department of Computer Science, pp. 1–107. Google Scholar
Ma, X., Shao, Y., Wang, Z., Luo, M., Fang, X. & Ju, Z. [2016] “ An impulsive two-stage predator–prey model with stage-structure and square root functional responses,” Math. Comput. Simul. 119, 91–107. Crossref, Web of Science, Google Scholar
Matia, S. N. & Alam, S. [2013] “ Prey–predator dynamics under herd behavior of prey,” Univ. J. Appl. Math. 1, 251–257. Crossref, Google Scholar
McGehee, E. A., Schutt, N., Vasquez, D. A. & Peacock, E. L. [2008] “ Bifurcations, and temporal and spatial patterns of a modified Lotka–Volterra model,” Int. J. Bifurcation and Chaos 18, 2223–2248. Link, Web of Science, Google Scholar
Meinhardt, H. [2008] “ Models of biological pattern formation: From elementary steps to the organization of embryonic axes,” Curr. Top. Dev. Biol. 81, 1–63. Crossref, Web of Science, Google Scholar
Owolabi, K. M. & Atangana, A. [2018] “ Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios,” Eur. Phys. J. Plus 133, 43–55. Crossref, Web of Science, Google Scholar
Segel, L. A. & Jackson, J. L. [1972] “ Dissipative structure: An explanation and an ecological example,” J. Theor. Biol. 37, 545–559. Crossref, Web of Science, Google Scholar
Tang, X. & Song, Y. [2015] “ Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator–prey model with herd behavior,” Appl. Math. Comput. 254, 375–391. Crossref, Web of Science, Google Scholar
Tanner, J. T. [1975] “ The stability and the intrinsic growth rates of prey and predator populations,” Ecology 56, 855–867. Crossref, Web of Science, Google Scholar
Turing, A. M. [1952] “ The chemical basis of morphogenesis,” Philos. Trans. Roy. Soc. Lond. B Biol. Sci. 237, 37–72. Crossref, Web of Science, Google Scholar
Xu, C., Yuan, S. & Zhang, T. [2016] “ Global dynamics of a predator–prey model with defense mechanism for prey,” Appl. Math. Lett. 62, 42–48. Crossref, Web of Science, Google Scholar
Yuan, S., Xu, C. & Zhang, T. [2013] “ Spatial dynamics in a predator–prey model with herd behavior,” Chaos 23, 033102-1–10. Crossref, Web of Science, Google Scholar
Zhang, X. A., Chen, L. & Neumann, A. U. [2000] “ The stage-structured predator–prey model and optimal harvesting policy,” Math. Biosci. 168, 201–210. Crossref, Web of Science, Google Scholar