Rich dynamics in a spatial predator-prey model with delay. (English) Zbl 1338.92096
Summary: We study the spatiotemporal dynamics of a diffusive Holling-Tanner predator-prey model with discrete time delay. Via analytically and numerically analysis, we unveil six types of patterns with and without time delay. Among them, of particular novel is the observation of linear pattern (consisting of a series of parallel lines), whose formation is closely related with the temporal Hopf bifurcation threshold. Moreover, we also find that larger time delay or diffusion of predator may induce the extinction of both prey and predator. Theoretical analysis and numerical simulations validate the well-known conclusion: diffusion is usually beneficial for stabilizing pattern formation, yet discrete time delay plays a destabilizing role in the generation of pattern.
MSC:
| 92D25 | Population dynamics (general) |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 37N25 | Dynamical systems in biology |
Keywords:
pattern formation; reaction-diffusion equation; predator-prey system; spatiotemporal model; discrete time delayReferences:
| [1] | Turing, A. M., The chemical basis of morphogenesis, Philos. Trans. R. Soc. London B, 237, 37-72 (1952) · Zbl 1403.92034 |
| [2] | Biktashev, V. N.; Tsyganov, M. A., Envelope quasi-solitons in dissipative systems with cross-diffusion, Phys. Rev. Lett., 107, 134101 (2011) |
| [3] | Dufiet, V.; Boissonade, J., Dynamics of Turing pattern monolayers close to onset, Phys. Rev. E, 53, 4883-4892 (1996) |
| [4] | Gurney, W. S.C.; Veitch, A. R.; Cruickshank, I.; McGeachin, G., Circles and spirals: population persistence in a spatially explicit predator-prey model, Ecology, 79, 2516-2530 (1998) |
| [5] | Hassell, M. P.; Comins, H. N.; May, R. M., Spatial structure and chaos in insect population dynamics, Nature, 353, 255-258 (1991), (London) |
| [6] | Kondo, S.; Asai, R., A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376, 765-768 (1995), (London) |
| [7] | Murray, J. D., Mathematical Biology (1993), Springer-Verlag: Springer-Verlag Berlin, London · Zbl 0779.92001 |
| [8] | Ouyang, Q.; Swinney, H. L., Transition from a uniform state to hexagonal and striped Turing patterns, Nature, 352, 610-612 (1991), (London) |
| [9] | Saunoriene, L.; Ragulskis, M., Secure steganographic communication algorithm based on self-organizing patterns, Phys. Rev. E, 84, 056213 (2011) |
| [10] | Varea, V.; Aragon, J. L.; Barrio, R. A., Confined Turing patterns in growing systems, Phys. Rev. E, 56, 1250-1253 (1997) |
| [11] | Woolley, T. E.; Baker, R. E.; Gaffney, E. A.; Maini, P. K., Stochastic reaction and diffusion on growing domains: understanding the breakdown of robust pattern formation, Phys. Rev. E, 84, 046216 (2011) |
| [12] | Segel, L.; Jackson, J., Dissipative structure: an explanation and an ecological example, J. Theor. Biol., 37, 545-559 (1972) |
| [13] | Gierer, A.; Meinhardt, H., A thoery of Biological pattern formation, Kybernetik, 12, 30-39 (1972) · Zbl 1434.92013 |
| [14] | Levin, S.; Segel, L., Hypothesis for origin of planktonic patchiness, Nature, 259, 659 (1976), (London) |
| [15] | Sun, G. Q.; Jin, Z.; Liu, Q. X.; Li, L., Dynamical complexity of a spatial predator-prey model with migration, Ecol. Model., 219, 248-255 (2008) |
| [16] | Sengupta, A.; Kruppa, T.; Lowen, H., Chemotactic predator-prey dynamics, Phys. Rev. E, 83, 031914 (2011) |
| [17] | Lai, Y. M.; Newby, J.; Bressloff, P. C., Effects of demographic noise on the synchronization of a metapopulation in a fluctuating environment, Phys. Rev. Lett., 107, 118102 (2011) |
| [18] | May, R. M., Stability and complexity in model ecosystems (1978), Princeton University Press: Princeton University Press Princeton, NJ |
| [19] | Tanner, J. T., The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56, 855-867 (1975) |
| [20] | Wollkind, D. J.; Collings, J. B.; Logan, J. A., Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees, Bull. Math. Biol., 50, 379-409 (1988) · Zbl 0652.92019 |
| [21] | May, R. M., Limit cycles in predator-prey communities, Science, 177, 900-902 (1972) |
| [22] | Hsu, S. B.; Hwang, T. W., Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55, 763-783 (1995) · Zbl 0832.34035 |
| [23] | Sáez, E.; Gonzá-Olivares, E., Dynamics of a predator-prey model, SIAM J. Appl. Math., 59, 1867-1878 (1999) · Zbl 0934.92027 |
| [24] | Wang, Y. X.; Li, W. T., Spatial patterns of the Holling-Tanner predator-prey model with nonlinear diffusion effects, Appl. Anal., 92, 2168-2181 (2013) · Zbl 1274.35392 |
| [25] | Choudhury, B. S.; Nasipuri, B., Spatio-temporal chaos in a Holling-Tanner predator-prey model with Holling type-IV functional response, Int. J. Ecol. Econ. Stat., 31, 19-40 (2013) |
| [26] | Choudhury, B. S.; Nasipuri, B., Self-organized spatial patterns due to diffusion in a Holling-Tanner predator-prey model, Comput. Appl. Math. (2014) · Zbl 1337.92172 |
| [27] | Wang, M. X.; Pang, P. Y.H.; Chen, W. Y., Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA J. Appl. Math., 73, 815-835 (2008) · Zbl 1180.35290 |
| [28] | Faria, T., Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254, 433-463 (2001) · Zbl 0973.35034 |
| [29] | Hadeler, K. P.; Ruan, S., Interaction of diffusion and delay, Discrete Contin. Dyn. Syst. B, 8, 95-105 (2007) · Zbl 1128.92001 |
| [30] | Yang, X. P., Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects, Appl. Math. Comput., 192, 552-566 (2007) · Zbl 1193.35098 |
| [31] | Yan, S.; Lian, X. Z.; Wang, W. M., Spatiotemporal dynamics in a delayed diffusive predator model, Appl. Math. Comput., 224, 524-534 (2013) · Zbl 1334.92382 |
| [32] | Sen, S.; Ghosh, P.; Riaz, S. S.; Ray, D. S., Time-delay induced instabilities in reaction-diffusion systems, Phys. Rev. E, 80, 046212 (2009) |
| [33] | Ghosh, P., Control of the Hopf-Turing transition by time-delayed global feedback in a reaction-diffusion system, Phys. Rev. E, 84, 016222 (2011) |
| [34] | Piotrowska, M. J., Activator-inhibitor system with delay and pattern formation, Math. Comp. Model., 42, 123-131 (2005) · Zbl 1080.92013 |
| [35] | Banerjee, M.; Zhang, L., Influence of discrete delay on pattern formation in a ratio-dependent prey-predator model, Chaos, Solitons Fractals, 67, 73-81 (2014) · Zbl 1349.92115 |
| [36] | Kunag, Y., Delay Differential Equations with Application in Population Dynamics (1993), Academic Press: Academic Press Boston, MA · Zbl 0777.34002 |
| [37] | Foryś, U., Biological delay systems and Mikhailov criterion of stability, J. Biol. Syst., 12, 1-16 (2004) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
