×

Allee-effect-induced instability in a reaction-diffusion predator-prey model. (English) Zbl 1272.37038

Summary: We investigate the spatiotemporal dynamics induced by the Allee effect in a reaction-diffusion predator-prey model. In the case without Allee effect, there is nonexistence of diffusion-driven instability for the model, and in the case with Allee effect, the positive equilibrium may be unstable under certain conditions. This instability is induced both by the Allee effect and diffusion. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee effect is not simple, but rich and complex.

MSC:

37N25 Dynamical systems in biology
92D25 Population dynamics (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences

Software:

PRED_PREY
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Turing, A., The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society B, 237, 37-72 (1952) · Zbl 1403.92034
[2] Levin, S., The problem of pattern and scale in ecology, Ecology, 73, 6, 1943-1967 (1992)
[3] Wang, Z. A.; Hillen, T., Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17, 3 (2007) · Zbl 1163.37383 · doi:10.1063/1.2766864
[4] Britton, N. F., Essential Mathematical Biology (2003), Springer · Zbl 1037.92001 · doi:10.1007/978-1-4471-0049-2
[5] Cross, M. C.; Hohenberg, P. H., Pattern formation outside of equilibrium, Reviews of Modern Physics, 65, 3, 851-1112 (1993) · Zbl 1371.37001
[6] Lee, K. J.; McCormick, W. D.; Ouyang, Q.; Swinney, H. L., Pattern formation by interacting chemical fronts, Science, 261, 5118, 192-194 (1993)
[7] Medvinsky, A. B.; Petrovskii, S. V.; Tikhonova, I. A.; Malchow, H.; Li, B.-L., Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44, 3, 311-370 (2002) · Zbl 1001.92050 · doi:10.1137/S0036144502404442
[8] Ni, W.-M.; Tang, M., Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Transactions of the American Mathematical Society, 357, 10, 3953-3969 (2005) · Zbl 1074.35051 · doi:10.1090/S0002-9947-05-04010-9
[9] Hoyle, R. B., Pattern Formation: An Introduction to Methods (2006), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1087.00001 · doi:10.1017/CBO9780511616051
[10] Ward, M. J.; Wei, J., The existence and stability of asymmetric spike patterns for the Schnakenberg model, Studies in Applied Mathematics, 109, 3, 229-264 (2002) · Zbl 1152.35397 · doi:10.1111/1467-9590.00223
[11] Ward, M. J., Asymptotic methods for reaction-diffusion systems: past and present, Bulletin of Mathematical Biology, 68, 5, 1151-1167 (2006) · Zbl 1334.92004
[12] Wei, J., Pattern formations in two-dimensional Gray-Scott model: existence of single-spot solutions and their stability, Physica D, 148, 1-2, 20-48 (2001) · Zbl 0981.35026 · doi:10.1016/S0167-2789(00)00183-4
[13] Wei, J.; Winter, M., Stationary multiple spots for reaction-diffusion systems, Journal of Mathematical Biology, 57, 1, 53-89 (2008) · Zbl 1141.92007 · doi:10.1007/s00285-007-0146-y
[14] Chen, W.; Ward, M. J., The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model, SIAM Journal on Applied Dynamical Systems, 10, 2, 582-666 (2011) · Zbl 1223.35033 · doi:10.1137/09077357X
[15] McKay, R.; Kolokolnikov, T., Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension, Discrete and Continuous Dynamical Systems B, 17, 1, 191-220 (2012) · Zbl 1257.35037
[16] Segel, L.; Jackson, J., Dissipative structure: an explanation and an ecological example, Journal of Theoretical Biology, 37, 3, 545-559 (1972)
[17] Gierer, A.; Meinhardt, H., A theory of biological pattern formation, Biological Cybernetics, 12, 1, 30-39 (1972) · Zbl 1434.92013
[18] Levin, S.; Segel, L., Hypothesis for origin of planktonic patchiness, Nature, 259, 5545, 659 (1976)
[19] Levin, S. A.; Segel, L. A., Pattern generation in space and aspect, SIAM Review A, 27, 1, 45-67 (1985) · Zbl 0576.92008 · doi:10.1137/1027002
[20] Murray, J. D., Mathematical Biology (2003), Springer · Zbl 1006.92002
[21] Okubo, A.; Levin, S. A., Diffusion and Ecological Problems: Modern Perspectives (2001), New York, NY, USA: Springer, New York, NY, USA · Zbl 1027.92022
[22] Neuhauser, C., Mathematical challenges in spatial ecology, Notices of the American Mathematical Society, 48, 11, 1304-1314 (2001) · Zbl 1128.92328
[23] Cantrell, R. S.; Cosner, C., Spatial Ecology via Reaction-Diffusion Equations (2003), West Sussex, UK: John Wiley & Sons, West Sussex, UK · Zbl 05022052 · doi:10.1002/0470871296
[24] Callahan, T. K.; Knobloch, E., Pattern formation in three-dimensional reaction-diffusion systems, Physica D, 132, 3, 339-362 (1999) · Zbl 0935.35065 · doi:10.1016/S0167-2789(99)00041-X
[25] Bhattacharyay, A., Spirals and targets in reaction-diffusion systems, Physical Review E, 64, 1 (2001)
[26] Leppänen, T.; Karttunen, M.; Kaski, K.; Barrio, R., Dimensionality effects in Turing pattern formation, International Journal of Modern Physics B, 17, 29, 5541-5553 (2003)
[27] He, J., Asymptotic methods for solitary solutions and compactons, Abstract and Applied Analysis, 2012 (2012) · Zbl 1257.35158 · doi:10.1155/2012/916793
[28] Harrison, G. W., Multiple stable equilibria in a predator-prey system, Bulletin of Mathematical Biology, 48, 2, 137-148 (1986) · Zbl 0585.92023 · doi:10.1016/S0092-8240(86)80003-9
[29] Harrison, G. W., Comparing predator-prey models to Luckinbill’s experiment with didinium and paramecium, Ecology, 76, 2, 357-374 (1995)
[30] Alonso, D.; Bartumeus, F.; Catalan, J., Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83, 1, 28-34 (2002)
[31] Baurmann, M.; Gross, T.; Feudel, U., Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, Journal of Theoretical Biology, 245, 2, 220-229 (2007) · Zbl 1451.92248 · doi:10.1016/j.jtbi.2006.09.036
[32] Wang, W.; Liu, Q.-X.; Jin, Z., Spatiotemporal complexity of a ratio-dependent predator-prey system, Physical Review E, 75, 5, 9 (2007) · doi:10.1103/PhysRevE.75.051913
[33] Wang, W.; Zhang, L.; Wang, H.; Li, Z., Pattern formation of a predator-prey system with Ivlev-type functional response, Ecological Modelling, 221, 2, 131-140 (2010)
[34] Upadhyay, R. K.; Wang, W.; Thakur, N. K., Spatiotemporal dynamics in a spatial plankton system, Mathematical Modelling of Natural Phenomena, 5, 5, 102-122 (2010) · Zbl 1197.92049 · doi:10.1051/mmnp/20105507
[35] Wang, J.; Shi, J.; Wei, J., Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, Journal of Differential Equations, 251, 4-5, 1276-1304 (2011) · Zbl 1228.35037 · doi:10.1016/j.jde.2011.03.004
[36] Banerjee, M.; Petrovskii, S., Self-organised spatial patterns and chaos in a ratiodependent predator-prey system, Theoretical Ecology, 4, 1, 37-53 (2011)
[37] Banerjee, M., Spatial pattern formation in ratio-dependent model: higher-order stability analysis, Mathematical Medicine and Biology, 28, 2, 111-128 (2011) · Zbl 1216.92058 · doi:10.1093/imammb/dqq024
[38] Banerjee, M.; Banerjee, S., Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Mathematical Biosciences, 236, 1, 64-76 (2012) · Zbl 1375.92077 · doi:10.1016/j.mbs.2011.12.005
[39] Fasani, S.; Rinaldi, S., Factors promoting or inhibiting Turing instability in spatially extended prey-predator systems, Ecological Modelling, 222, 18, 3449-3452 (2011)
[40] Rodrigues, L. A. D.; Mistro, D. C.; Petrovskii, S., Pattern formation, long-term transients, and the Turing-Hopf bifurcation in a space- and time-discrete predator-prey system, Bulletin of Mathematical Biology, 73, 8, 1812-1840 (2011) · Zbl 1220.92053 · doi:10.1007/s11538-010-9593-5
[41] Allee, W. C., Animal Aggregations: A Study in General Sociology (1978), AMS Press
[42] Courchamp, F.; Berec, J.; Gascoigne, J., Allee Effects in Ecology and Conservation (2008), New York, NY, USA: Oxford University Press, New York, NY, USA
[43] Dennis, B., Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3, 4, 481-538 (1989) · Zbl 0850.92062
[44] McCarthy, M. A., The Allee effect, finding mates and theoretical models, Ecological Modelling, 103, 1, 99-102 (1997)
[45] Wang, G.; Liang, X. G.; Wang, F. Z., The competitive dynamics of populations subject to an Allee effect, Ecological Modelling, 124, 2, 183-192 (1999)
[46] Burgman, M. A.; Ferson, S.; Akçakaya, H. R., Risk Assessment in Conservation Biology (1993), London, UK: Chapman & Hall, London, UK
[47] Lewis, M.; Kareiva, P., Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43, 2, 141-158 (1993) · Zbl 0769.92025
[48] Stephens, P. A.; Sutherland, W. J., Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology and Evolution, 14, 10, 401-405 (1999)
[49] Stephens, P. A.; Sutherland, W. J.; Freckleton, R. P., What is the Allee effect?, Oikos, 87, 1, 185-190 (1999)
[50] Keitt, T. H.; Lewis, M. A.; Holt, R. D., Allee effects, invasion pinning, and Species’ borders, American Naturalist, 157, 2, 203-216 (2002)
[51] Zhou, S. R.; Liu, Y. F.; Wang, G., The stability of predator-prey systems subject to the Allee effects, Theoretical Population Biology, 67, 1, 23-31 (2005) · Zbl 1072.92060
[52] Petrovskii, S.; Morozov, A.; Li, B.-L., Regimes of biological invasion in a predator-prey system with the Allee effect, Bulletin of Mathematical Biology, 67, 3, 637-661 (2005) · Zbl 1334.92363 · doi:10.1016/j.bulm.2004.09.003
[53] Shi, J.; Shivaji, R., Persistence in reaction diffusion models with weak Allee effect, Journal of Mathematical Biology, 52, 6, 807-829 (2006) · Zbl 1110.92055 · doi:10.1007/s00285-006-0373-7
[54] Morozov, A.; Petrovskii, S.; Li, B.-L., Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect, Journal of Theoretical Biology, 238, 1, 18-35 (2006) · Zbl 1445.92248 · doi:10.1016/j.jtbi.2005.05.021
[55] Roques, L.; Roques, A.; Berestycki, H.; Kretzschmar, A., A population facing climate change: joint influences of Allee effects and environmental boundary geometry, Population Ecology, 50, 2, 215-225 (2008)
[56] Çelik, C.; Duman, O., Allee effect in a discrete-time predator-prey system, Chaos, Solitons and Fractals, 40, 4, 1956-1962 (2009) · Zbl 1198.34084 · doi:10.1016/j.chaos.2007.09.077
[57] Zu, J.; Mimura, M., The impact of Allee effect on a predator-prey system with Holling type II functional response, Applied Mathematics and Computation, 217, 7, 3542-3556 (2010) · Zbl 1202.92088 · doi:10.1016/j.amc.2010.09.029
[58] Wang, J.; Shi, J.; Wei, J., Predator-prey system with strong Allee effect in prey, Journal of Mathematical Biology, 62, 3, 291-331 (2011) · Zbl 1232.92076 · doi:10.1007/s00285-010-0332-1
[59] González-Olivares, E.; Meneses-Alcay, H.; González-Yañez, B.; Mena-Lorca, J.; Rojas-Palma, A.; Ramos-Jiliberto, R., Multiple stability and uniqueness of the limit cycle in a Gause-type predator-prey model considering the Allee effect on prey, Nonlinear Analysis: Real World Applicationsl, 12, 6, 2931-2942 (2011) · Zbl 1231.34053 · doi:10.1016/j.nonrwa.2011.04.003
[60] Aguirre, P.; González-Olivares, E.; Sáez, E., Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis. Real World Applications, 10, 3, 1401-1416 (2009) · Zbl 1160.92038 · doi:10.1016/j.nonrwa.2008.01.022
[61] Aguirre, P.; González-Olivares, E.; Sáez, E., Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM Journal on Applied Mathematics, 69, 5, 1244-1262 (2009) · Zbl 1184.92046 · doi:10.1137/070705210
[62] Cai, Y.; Wang, W.; Wang, J., Dynamics of a diffusive predator-prey model with additive Allee effect, International Journal of Biomathematics, 5, 2 (2012) · Zbl 1297.92060 · doi:10.1142/S1793524511001659
[63] Solomon, M. E., The natural control of animal populations, The Journal of Animal Ecology, 18, 1, 1-35 (1949)
[64] Lotka, A. J., Elements of Physical Biology (1925), Williams and Wilkins · JFM 51.0416.06
[65] Jost, C., Comparing Predator-Prey Models Qualitatively and Quantitatively with Ecological Time-Series Data (1998), Institut National Agronomique, Paris-Grignon
[66] Malchow, H.; Petrovskii, S. V.; Venturino, E., Spatiotemporal Patterns in Ecology and Epidemiology (2008), Boca Raton, Fla, USA: Chapman & Hall, Boca Raton, Fla, USA · Zbl 1298.92004
[67] Garvie, M. R., Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB, Bulletin of Mathematical Biology, 69, 3, 931-956 (2007) · Zbl 1298.92081 · doi:10.1007/s11538-006-9062-3
[68] Munteanu, A.; Solé, R., Pattern formation in noisy self-replicating spots, International Journal of Bifurcation and Chaos, 16, 12, 3679-3683 (2006) · Zbl 1113.92007 · doi:10.1142/S0218127406017063
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.