Journal of Applied Nonlinear Dynamics
Pattern Formation Scenario through Turing Instability in Interacting ReactionDiffusion Systems with Both Refuge and Nonlinear Harvesting
Journal of Applied Nonlinear Dynamics 9(1) (2020) 121  DOI:10.5890/JAND.2020.03.001
Lakshmi Narayan Guin$^{1}$, Esita Das$^{1}$, Muniyagounder Sambath$^{2}$
$^{1}$ Department of Mathematics, VisvaBharati, Santiniketan731235, West Bengal, India
$^{2}$ Department of Mathematics, Periyar University, Salem636011, Tamil Nadu, India
Download Full Text PDF
Abstract
Of concern in the present theoretical study is to carry out the complex dynamics of a reactiondiffusion predatorprey model incorporating constant proportion of prey refuge and nonlinear prey harvesting with zeroflux boundary conditions. By the method of Lyapunov function, the global stability of the feasible interior equilibrium point for nonspatial model was established. The conditions of diffusiondriven instability were obtained and the Turing space in the parameters space was given as well. Consequently, we present the evolutionary procedure that occupies organism distribution and their interaction of spatially distributed species with diffusion and locate that the model dynamics reveals a diffusioncontrolled formation growth to hole patterns or labyrinthine patterns or holestripe patterns replication over the whole spatial domain. The analytical results are then authenticated with the help of numerical simulations. Our results points out that the diffusion has an immense impact on the prey refuge as well as prey harvesting and extend well the findings of spatiotemporal dynamics in the reactiondiffusion model.
Acknowledgments
The first two authors gratefully acknowledge the financial support in part from Special Assistance Programme (SAPIII) sponsored by the University Grants Commission (UGC), New Delhi, India (Grant No. F.510 / 3 / DRSIII / 2015 (SAPI)). We would like to express thank the anonymous referee and the editor for supportive remarks and ideas.
References

[1]  Murdoch, W.W., Briggs, C.J., and Nisbet, R.M. (2003), Consumerresource dynamics, Princeton University Press, New York. 

[2]  Leslie, P.H. (1948), Some further notes on the use of matrices in population mathematics, Biometrika, 35, 213245. 

[3]  Leslie, P.H. (1958), A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45, 1631. 

[4]  Holling, C.S. (1965), The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 45, 560. 

[5]  Leslie, P.H. and Gower, J.C. (1960), The properties of a stochastic model for the predatorprey type of interaction between two species, Biometrika, 47, 219234. 

[6]  May, R.M. (1974), Stability and complexity in model ecosystems, Princeton University Press, Princeton. 

[7]  AzizAlaoui, M.A. and Okiye, M.D. (2003), Boundedness and global stability for a predatorprey model with modified LeslieGower and Hollingtype II schemes, Applied Mathematics Letters, 16, 10691075. 

[8]  Du, Y., Peng, R., and Wang, M. (2009), Effect of a protection zone in the diffusive Leslie predatorprey model, Journal of Differential Equations, 246, 39323956. 

[9]  Zhu, Y. and Wang, K. (2011), Existence and global attractivity of positive periodic solutions for a predatorprey model with modified LeslieGower Hollingtype II schemes, Journal of Mathematical Analysis and Applications, 384, 400408. 

[10]  Nindjin, A.F., AzizAlaoui, M.A., and Cadivel, M. (2005), Analysis of a predatorprey model with modified LeslieGower and Hollingtype II schemes with time delay, Nonlinear Analysis: Real World Applications, 7, 11041118. 

[11]  Dai, G. and Tang, M. (1998), Coexistence region and global dynamics of a harvested predatorprey system, SIAM Journal on Applied Mathematics, 58, 193210. 

[12]  Kar, T.K. and Ghorai, A. (2005), Dynamic behaviour of a delayed predatorprey model with harvesting, Applied Mathematics and Computation, 217, 90859104. 

[13]  Das, T., Mukherjee, R.N., and Chaudhuri, K.S. (2009), Bioeconomic harvesting of a preypredator fishery, Journal of Biological Dynamics, 3, 447462. 

[14]  Yuan, R., Jiang,W., andWang, Y. (2015), SaddlenodeHopf bifurcation in a modified LeslieGower predatorprey model with timedelay and prey harvesting, Journal of Mathematical Analysis and Applications, 422, 10721090. 

[15]  Liu, W., Li, B., Fu, C., and Chen, B. (2015), Dynamics of a predatorprey ecological system with nonlinear harvesting rate, Wuhan University Journal of Natural Sciences, 20, 2533. 

[16]  Dubey, B., Chandra, P., and Sinha, P. (2002), A resource dependent fishery model with optimal harvesting policy, Journal of Biological Systems, 10, 113. 

[17]  Kar, T.K. and Chaudhuri, K.S. (2004), Harvesting in a twoprey onepredator fishery: a bioeconomic model, Anziam Journal, 45, 443456. 

[18]  Song, X. and Chen, L. (2001), Optimal harvesting and stability for a twospecies competitive system with stage structure, Mathematical Biosciences, 170, 173186. 

[19]  Hu, D. and Cao, H. (2017), Stability and bifurcation analysis in a predatorprey system with MichaelisMenten type predator harvesting, Nonlinear Analysis: Real World Applications, 33, 5882. 

[20]  Gupta, R.P. and Chandra, P. (2013), Bifurcation analysis of modified LeslieGower predatorprey model with MichaelisMenten type prey harvesting, Journal of Mathematical Analysis and Applications, 398, 278295. 

[21]  Hochberg, M.E. and Holt, R.D. (1995), Refuge evolution and the population dynamics of coupled host– arasitoid associations, Evolutionary Ecology, 9, 633661. 

[22]  Huang, Y., Chen, F., and Zhong, L. (2006), Stability analysis of a preypredator model with Holling type III response function incorporating a prey refuge, Applied Mathematics and Computation, 182, 672683. 

[23]  Kar, T.K. (2005), Stability analysis of a prey–predator model incorporating a prey refuge, Communications in Nonlinear Science and Numerical Simulation, 10, 681691. 

[24]  Krivian, V. (1998), Effect of optimal antipredator behaviour of prey on predatorprey dynamics: the role of refuge, Theoretical Population Biology, 53, 131142. 

[25]  Ruxton, G.D. (1995), Short term refuge use and stability of predatorprey models, Theoretical Population Biology, 47, 117. 

[26]  Hassell, M.P. and May, R.M. (1973), Stability in insect hostparasite models, The Journal of Animal Ecology, 42, 693726. 

[27]  Holling, C.S. (1959), The components of predation as revealed by a study of smallmammal predation of the European pine sawfly, The Canadian Entomologist, 91, 293320. 

[28]  Holling, C.S. (1959), Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91, 385398. 

[29]  Hassell, M.P. (1978), The dynamics of arthropod predatorprey systems, Princeton University Press. 

[30]  Gard, T.C. and Hallam, T.G. (1979), Persistence in food webs–I LotkaVolterra food chains, Bulletin of Mathematical Biology, 41, 877891. 

[31]  Hofbauer, J. (1981), A general cooperation theorem for hypercycles, Monatshefte f¨ur Mathematik, 91, 233 240. 

[32]  Hutson, V. and Vickers, G.T. (1983), A criterion for permanent coexistence of species, with an application to a twoprey onepredator system, Mathematical Biosciences, 63, 253269. 

[33]  Cantrell, R.S. and Cosner, C. (2004), Spatial ecology via reactiondiffusion equations, John Wiley & Sons. 

[34]  Turing, A.M. (1952), The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London (Series B), Biological Sciences, 237, 3772. 

[35]  Murray, J.D. (2002), Mathematical Biology, Volumes I and II, SpringerVerlag, Heidelberg. 

[36]  Feng, P. (2014), On a diffusive predatorprey model with nonlinear harvesting, Mathematical Biosciences & Engineering, 11, 807821. 

[37]  Guin, L.N. and Acharya, S. (2017), Dynamic behaviour of a reactiondiffusion predatorprey model with both refuge and harvesting, Nonlinear Dynamics, 88, 15011533. 

[38]  Guin, L.N. and Mandal, P.K. (2014), Spatiotemporal dynamics of reactiondiffusion models of interacting populations, Applied Mathematical Modelling, 38, 44174427. 

[39]  Guin, L.N., Chakravarty, S., and Mandal, P.K. (2015), Existence of spatial patterns in reactiondiffusion systems incorporating a prey refuge, Nonlinear Analysis: Modelling and Control, 20, 509527. 

[40]  Guin, L.N., Mondal, B., and Chakravarty, S. (2016), Existence of spatiotemporal patterns in the reactiondiffusion predatorprey model incorporating prey refuge, International Journal of Biomathematics, 9, 1650085. 

[41]  Strikwerda, J.C. (2004), Finite difference schemes and partial differential equations, SIAM, Philadelphia. 

[42]  Alonso, D., Bartumeus, F., and Catalan, J. (2002), Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83, 2834. 

[43]  Guin, L.N. and Mandal, P.K. (2014), Effect of prey refuge on spatiotemporal dynamics of reactiondiffusion system, Computers and Mathematics with Applications, 68, 13251340. 

[44]  Guin, L.N. (2014), Existence of spatial patterns in a predatorprey model with self and crossdiffusion, Applied Mathematics and Computation, 226, 320335. 