Deterministic and stochastic analysis of a prey–predator model with herd behaviour in both

References

• Ajraldi, V., Pittavino, M., & Venturino, E. (2011). Modeling herd behavior in population systems. Nonlinear Analysis: Real World Applications, 12, 2319–2338. doi: 10.1016/j.nonrwa.2011.02.002
   Web of Science ®Google Scholar
• Bandyopadhyay, M., & Chakrabarti, C. G. (2003). Deterministic and stochastic analysis of a non-linear prey–predator system. Journal of Biological Systems, 11, 161–172. doi: 10.1142/S0218339003000816
   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 Journal of Modelling and Simulation, 11, 3–14.
   Google Scholar
• Bera, S. P., Maiti, A., & Samanta, G. P. (2016). Stochastic analysis of a prey–predator model with herd behaviour of prey. Nonlinear Analysis: Modelling and Control, 21(3), 345–361.
   Web of Science ®Google Scholar
• Braza, P. A. (2012). Predator–prey dynamics with square root functional responses. Nonlinear Analysis: Real World Applications, 13, 1837–1843. doi: 10.1016/j.nonrwa.2011.12.014
   Web of Science ®Google Scholar
• Chattopadhyay, J., Chatterjee, S., & Venturino, E. (2008). Patchy agglomeration as a transition from monospecies to recurrent plankton blooms. Journal of Theoretical Biology, 253, 289–295. doi: 10.1016/j.jtbi.2008.03.008
   PubMed Web of Science ®Google Scholar
• Cosner, C., DeAngelis, D. L., Ault, J. S., & Olson, D. B. (1999). Effects of spatial grouping on the functional response of predators. Theoretical Population Biology, 56, 65–75. doi: 10.1006/tpbi.1999.1414
   PubMed Web of Science ®Google Scholar
• Dimentberg, M. F. (1988). Statistical dynamics of non-linear and time-varying systems. New York, NY: John Wiley & Sons.
   Google Scholar
• Gardiner, C. W. (1983). Handbook of stochastic methods. New York, NY: Springer-Verlag.
   Google Scholar
• Haken, H. (1983). Advanced synergetics. Berlin: Springer-Verlag.
   Google Scholar
• Holling, C. S. (1959a). The components of predation as revealed by a study of small mammal predation of the European pine sawfly. The Canadian Entomologist, 91, 293–320. doi: 10.4039/Ent91293-5
   Google Scholar
• Holling, C. S. (1959b). Some characteristics of simple types of predation and parasitism. The Canadian Entomologist, 91, 385–398. doi: 10.4039/Ent91385-7
   Google Scholar
• Horsthemke, W., & Lefever, R. (1984). Noise induced transitions. Berlin: Springer-Verlag.
   Google Scholar
• Jumarie, G. (1995). A practical variational approach to stochastic optimal control via state moment equations. Journal of the Franklin Institute, 332, 761–772. doi: 10.1016/0016-0032(95)00074-7
   Web of Science ®Google Scholar
• Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130–141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
   Web of Science ®Google Scholar
• Lotka, A. (1925). Elements of physical biology. Baltimore: Williams and Wilkins.
   Google Scholar
• Maiti, A., Jana, M. M., & Samanta, G. P. (2007). Deterministic and stochastic analysis of a ratio-dependent predator–prey system with delay. Nonlinear Analysis: Modelling and Control, 12, 383–398.
   Google Scholar
• Maiti, A., & Samanta, G. P. (2005). Deterministic and stochastic analysis of a prey–dependent predator–prey system. International Journal of Mathematical Education in Science and Technology, 36, 65–83. doi: 10.1080/00207390412331314980
   Google Scholar
• Maiti, A., & Samanta, G. P. (2006). Deterministic and stochastic analysis of a ratio-dependent prey–predator system. International Journal of Systems Science, 37, 817–826. doi: 10.1080/00207720600879252
   Web of Science ®Google Scholar
• Maiti, A., Sen, P., Manna, D., & Samanta, G. P. (2016). A predator–prey system with herd behaviour and strong Allee effect. Nonlinear Dynamics and Systems Theory, 16, 86–101.
   Google Scholar
• Malthus, T. R. (1798). An essay on the principal of population, and a summary view of the principal of population. Harmondsworth: Penguin.
   Google Scholar
• May, R. M. (1974). Stability and complexity in model ecosystems. Princeton: Princeton University Press.
   Google Scholar
• May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467. doi: 10.1038/261459a0
   PubMed Web of Science ®Google Scholar
• Melchionda, D., Pastacaldi, E., Perri, C., & Venturino, E. (2014). Interacting population models with pack behavior. Retrieved from arXiv:1403.4419v1.
   Google Scholar
• Murray, J. D. (1993). Mathematical biology. New York, NY: Springer-Verlag.
   Google Scholar
• Nicolis, G., & Prigogine, I. (1977). Self-organisation in nonequilibrium systems. New York, NY: John Wiley & Sons.
   Google Scholar
• Nisbet, R. M., & Gurney, W. S. C. (1982). Modelling fluctuating populations. New York, NY: John Wiley & Sons.
   Google Scholar
• Pearl, R., & Reed, L. J. (1920). On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proceedings of the National Academy of Sciences, 6, 275–288. doi: 10.1073/pnas.6.6.275
   PubMedGoogle Scholar
• Renshaw, E. (1995). Modelling biological population in space and time. Cambridge: Cambridge University Press.
   Google Scholar
• Valsakumar, M. C., Murthy, K. P. N., & Ananthakrishna, G. (1983). On the linearization of nonlinear Langevin-type stochastic differential equations. Journal of Statistical Physics, 30, 617–631. doi: 10.1007/BF01009680
   Web of Science ®Google Scholar
• Verhulst, P. F. (1838). Notice sur la loi que la population poursuit dans son accroissement. Correspondance mathématique et physique, 10, 113–121.
   Google Scholar
• Volterra, V. (1926). Variazioni e fluttuazioni del numero di individui in specie animali conviventi. Mem. Accd. Linc., 2, 31–113.
   Google Scholar
• Wollkind, D. J., Collings, J. B., & Logan, J. A. (1988). Metastability in a temperature-dependent model system for predator–prey mite outbreak interactions on fruit trees. Bulletin of Mathematical Biology, 50, 379–409. doi: 10.1007/BF02459707
   Web of Science ®Google Scholar
• Xiao, D., & Ruan, S. (2001). Global analysis in a predator–prey system with nonmonotonic functional response. SIAM Journal on Applied Mathematics, 61, 1445–1472. doi: 10.1137/S0036139999361896
   Web of Science ®Google Scholar