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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Spatiotemporal patterns of a pursuit-evasion generalist predator-prey model with prey harvesting

Journal of Applied Nonlinear Dynamics 7(2) (2018) 165--177 | DOI:10.5890/JAND.2018.06.005

Lakshmi Narayan Guin; Benukar Mondal; Santabrata Chakravarty

Department of Mathematics, Visva-Bharati, Santiniketan-731235, West Bengal, India

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Abstract

The present investigation deals with a diffusive predator-prey model in order to study the dynamic response of a reaction-diffusion model with linear prey harvesting. The governing equations of the proposed model system subject to the homogeneous Neumann boundary condition provide some qualitative interpretations of solutions to the reaction-diffusion system. The conditions of diffusion-driven instability and the Turing bifurcation region in two parameter space are explored. From the outcome of the present mathematical analysis carried out followed by the numerical simulations based on the model parameters, it reveals that for unequal diffusive coefficients, prey harvesting may induce that diffusion-driven instability resulting in stationary Turing patterns. The choice of parameter values is important to study the effect of prey harvesting and diffusion, while it depends more on the non-linearity of the model system. Moreover, the model dynamics exhibits the influence of both prey harvesting and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, labyrinthine, stripes-spots mixture and spots replication. All these features illustrate that the dynamics of the proposed model with the control of prey harvesting is not straightforward, but rich and complex in nature.

Acknowledgments

The authors are thankful to the learned referees for their valuable comments and suggestions towards an improvement of the present paper. The authors also gratefully acknowledge the financial support in part from Special Assistance Programme (SAP-III) sponsored by the University Grants Commission (UGC), New Delhi, India (Grant No. F.510 / 3 / DRS-III / 2015 (SAP-I)).

References

  1. [1]  Turing, A.M. (1952), The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London (Series B), Biological Sciences, 237, 37-72.
  2. [2]  Meinhardt, H. (1982), Models of biological pattern formation, 6.
  3. [3]  Murray, J.D. (2002), Mathematical Biology II, Springer-Verlag, Heidelberg.
  4. [4]  Okubo, A. (1980), Diffusion and ecological problems: Mathematical models, Springer-Verlag, Berlin (FRG).
  5. [5]  Okubo, A. and Levin, S.A. (2001), Diffusion and ecological problems: modern perspective, Springer-Verlag.
  6. [6]  De Kepper, P., Castets, V., Dulos, E., and Boissonade, J. (1991), Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction, Physica D: Nonlinear Phenomena, 49, 161-169.
  7. [7]  Neuhauser, C. (2001), Mathematical challenges in spatial ecology, Notices of the AMS, 48, 1304-1314.
  8. [8]  Gause, G. F. (2003), The struggle for existence.
  9. [9]  Luckinbill, L.S. (1974), The effects of space and enrichment on a predator-prey system, Ecology, 55, 1142- 1147.
  10. [10]  Luckinbill, L.S. (1973), Coexistence in laboratory populations of Paramecium aurelia and its predator Didinium nasutum, Ecology, 54, 1320-1327.
  11. [11]  Segel, L.A. and Jackson, J.L. (1972), Dissipative structure: an explanation and an ecological example, Journal of Theoretical Biology, 37, 545-559.
  12. [12]  Levin, S.A. and Segel, L.A. (1976), Hypothesis for origin of planktonic patchiness, Nature Publishing Group, 259, 659.
  13. [13]  Guin, L.N. and Mandal, P.K. (2014), Spatiotemporal dynamics of reaction-diffusion models of interacting populations, Applied Mathematical Modelling, 38, 4417-4427.
  14. [14]  Guin, L.N., Chakravarty, S., and Mandal, P.K. (2015), Existence of spatial patterns in reaction-diffusion systems incorporating a prey refuge, Nonlinear Analysis: Modelling and Control, 20, 509-527.
  15. [15]  Guin, L.N., Mondal, B., and Chakravarty, S. (2016), Existence of spatiotemporal patterns in the reactiondiffusion predator-prey model incorporating prey refuge, International Journal of Biomathematics, 9, 1650085.
  16. [16]  Guin, L.N. and Mandal, P.K. (2014), Spatial pattern in a diffusive predator-prey model with sigmoid ratiodependent functional response, International Journal of Biomathematics, 7, 1450047.
  17. [17]  Sun, G.Q., Zhang, G., Jin, Z., and Li, L. (2009), Predator cannibalism can give rise to regular spatial pattern in a predator-prey system, Nonlinear Dynamics, 58, 75-84.
  18. [18]  Sun, G.Q., Sarwardi, S., Pal, P.J., and Rahaman, S. (2010), The spatial patterns through diffusion-driven instability in modified Leslie-Gower and Holling-type II predator-prey model, Journal of Biological Systems, 18, 593-603.
  19. [19]  Xiao, D. and Jennings, L.S. (2005), Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM Journal on Applied Mathematics, 65, 737-753.
  20. [20]  Huang, J., Gong, Y., and Ruan, S. (2013), Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete and Continuous Dynamical Systems-Series B, 18, 2101-2121.
  21. [21]  Dai, G. and Tang, M. (1998), Coexistence region and global dynamics of a harvested predator-prey system, SIAM Journal on Applied Mathematics, 58, 193-210.
  22. [22]  Beddington, J.R. and May, R.M. (1980), Maximum sustainable yields in systems subject to harvesting at more than one trophic level, Mathematical Biosciences, 51, 261-281.
  23. [23]  Beddington, J.R. and Cooke, J.G. (1982), Harvesting from a prey-predator complex, Mathematical Biosciences, 14, 155-177.
  24. [24]  Makinde, O.D. (2007), Solving ratio-dependent predator-prey system with constant effort harvesting using Adomian decomposition method, Applied Mathematics and Computation, 186, 17-22.
  25. [25]  Solow, R.M. and Clark, C.W. (1977), Mathematical Bioeconomics: The Optimal Management of RENEWABLE Resources, JSTOR.
  26. [26]  Hill, S.L., Murphy, E.J., Reid, K., Trathan, P.N., and Constable, A. J. (2006), Modelling Southern Ocean ecosystems: krill, the food-web, and the impacts of harvesting, Biological Reviews, 81, 581-608.
  27. [27]  Christensen, V. (1996), Managing fisheries involving predator and prey species, Reviews in fish Biology and Fisheries, 6, 417-442.
  28. [28]  Chen, L. and Li, Y., and Xiao, D. (1972), Bifurcations in a ratio-dependent predator-prey model with prey harvesting, Canadian Applied Mathematics Quarterly, 19, 293-318.
  29. [29]  Shukla, J.B. and Verma, S. (1981), Effects of convective and dispersive interactions on the stability of two species, Bulletin of Mathematical Biology, 43, 593-610.
  30. [30]  Schreiber, S.J. (1997), Generalist and specialist predators that mediate permanence in ecological communities, Journal of Mathematical Biology, 36, 133-148.
  31. [31]  Lv, Y., Yuan, R., and Pei, Y. (2014), Effect of harvesting, delay and diffusion in a generalist predator-prey model, Applied Mathematics and Computation, 226, 348-366.
  32. [32]  Ermentrout, Bard (1991), Stripes or spots? Non-linear effects in bifurcation of reaction-diffusion equations on the square, Proceedings of Royal Society, London, 434, 413-417.
  33. [33]  Nagorcka, B.N. and Mooney, J.R. (1992), From stripes to spots: prepatterns which can be produced in the skin by a reaction-diffusion system, Mathematical Medicine and Biology, 9, 249-267.
  34. [34]  Shen, J. and Jung, Y. M. (2005), Geometric and stochastic analysis of reaction-diffusion patterns, International Journal of Pure and Applied Mathematics, 19, 195-244.

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