2014, 11(6): 1247-1274. doi: 10.3934/mbe.2014.11.1247

Spatiotemporal complexity in a predator--prey model with weak Allee effects

1. 

Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

2. 

Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India

3. 

Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212

4. 

College of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035, China

Received  March 2014 Revised  July 2014 Published  September 2014

In this article, we study the rich dynamics of a diffusive predator-prey system with Allee effects in the prey growth. Our model assumes a prey-dependent Holling type-II functional response and a density dependent death rate for predator. We investigate the dissipation and persistence property, the stability of nonnegative and positive constant steady state of the model, as well as the existence of Hopf bifurcation at the positive constant solution. In addition, we provide results on the existence and non-existence of positive non-constant solutions of the model. We also demonstrate the Turing instability under some conditions, and find that our model exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. One of the most interesting findings is that Turing instability in the model is induced by the density dependent death rate in predator.
Citation: Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang. Spatiotemporal complexity in a predator--prey model with weak Allee effects. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1247-1274. doi: 10.3934/mbe.2014.11.1247
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show all references

References:
[1]

P. A. Abrams, The fallacies of "ratio-dependent" predation,, Ecology, 75 (1994), 1842. doi: 10.2307/1939644.

[2]

P. A. Abrams and L. R. Ginzburg, The nature of predation: prey dependent, ratio dependent or neither?,, Trends in Ecology & Evolution, 15 (2000), 337. doi: 10.1016/S0169-5347(00)01908-X.

[3]

P. Aguirre, E. González-Olivares and E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect,, Nonlinear Analysis: Real World Applications, 10 (2009), 1401. doi: 10.1016/j.nonrwa.2008.01.022.

[4]

P. Aguirre, E. González-Olivares and E. Sáez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect,, SIAM Journal on Applied Mathematics, 69 (2009), 1244. doi: 10.1137/070705210.

[5]

H. R. Akçakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent predation: An abstraction that works,, Ecology, 76 (1995), 995.

[6]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations,, Journal of Differential Equations, 33 (1979), 201. doi: 10.1016/0022-0396(79)90088-3.

[7]

W. C. Allee, Animal Aggregations, a Study in General Sociology,, University of Chicago Press, (1931).

[8]

D. Alonso, F. Bartumeus and J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns,, Ecology, 83 (2002), 28. doi: 10.2307/2680118.

[9]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence,, Journal of Theoretical Biology, 139 (1989), 311. doi: 10.1016/S0022-5193(89)80211-5.

[10]

A. Ardito and P. Ricciardi, Lyapunov functions for a generalized Gause-type model,, Journal of Mathematical Biology, 33 (1995), 816. doi: 10.1007/BF00187283.

[11]

M. Banerjee and S. Banerjee, Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model,, Mathematical Biosciences, 236 (2012), 64. doi: 10.1016/j.mbs.2011.12.005.

[12]

M. Banerjee and S. Petrovskii, Self-organized spatial patterns and chaos in a ratio-dependent predator-prey system,, Theoretical Ecology, 4 (2011), 37.

[13]

M. Baurmann, T. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations,, Journal of Theoretical Biology, 245 (2007), 220. doi: 10.1016/j.jtbi.2006.09.036.

[14]

D. L. Benson, J. A. Sherratt and P. K. Maini, Diffusion driven instability in an inhomogeneous domain,, Bulletin of Mathematical Biology, 55 (1993), 365. doi: 10.1016/S0092-8240(05)80270-8.

[15]

A. A. Berryman, The orgins and evolution of predator-prey theory,, Ecology, 73 (1992), 1530. doi: 10.2307/1940005.

[16]

C. Bianca, Existence of stationary solutions in kinetic models with Gaussian thermostats,, Mathematical Methods in the Applied Sciences, 36 (2013), 1768. doi: 10.1002/mma.2722.

[17]

D. S Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters,, Journal of Theoretical Biology, 218 (2002), 375. doi: 10.1006/jtbi.2002.3084.

[18]

D. S. Boukal, M. W. Sabelis and L. Berec, How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses,, Theoretical Population Biology, 72 (2007), 136. doi: 10.1016/j.tpb.2006.12.003.

[19]

N. F. Britton, Essential Mathematical Biology,, Springer, (2003). doi: 10.1007/978-1-4471-0049-2.

[20]

K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity,, Journal of Differential Equations, 40 (1981), 232. doi: 10.1016/0022-0396(81)90020-6.

[21]

Y. Cai, W. Wang and J. Wang, Dynamics of a diffusive predator-prey model with additive Allee effect,, International Journal of Biomathematics, 5 (2012). doi: 10.1142/S1793524511001659.

[22]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley, (2003). doi: 10.1002/0470871296.

[23]

E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations,, SIAM Journal of Appllied Mathematics, 35 (1978), 1. doi: 10.1137/0135001.

[24]

L. B. Crowder and W. E. Cooper, Habitat structural complexity and the interaction between bluegills and their prey,, Ecology, 63 (1982), 1802. doi: 10.2307/1940122.

[25]

B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction,, Natural Resource Modeling, 3 (1989), 481.

[26]

X. Ding and J. Jiang, Positive periodic solutions in delayed Gause-type predator-prey systems,, Journal of Mathematical Analysis and Applications, 339 (2008), 1220. doi: 10.1016/j.jmaa.2007.07.079.

[27]

K. Fujii, Complexity-stability relationship of two-prey-one-predator species system model: Local and global stability,, Journal of Theoretical Biology, 69 (1977), 613. doi: 10.1016/0022-5193(77)90370-8.

[28]

M. R. Garvie, Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in Matlab,, Bulletin of Mathematical Biology, 69 (2007), 931. doi: 10.1007/s11538-006-9062-3.

[29]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983). doi: 10.1007/978-3-642-61798-0.

[30]

E. González-Olivares, H. Meneses-Alcay, B. González-Yañez, J. Mena-Lorca, A. Rojas-Palma and R. Ramos-Jiliberto, 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 Applications, 12 (2011), 2931. doi: 10.1016/j.nonrwa.2011.04.003.

[31]

K. Hasík, On a predator-prey system of Gause type,, Journal of Mathematical Biology, 60 (2010), 59. doi: 10.1007/s00285-009-0257-8.

[32]

D. Henry and D. B. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).

[33]

S.-B. Hsu, T.-W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system,, Journal of Mathematical Biology, 42 (2001), 489. doi: 10.1007/s002850100079.

[34]

T.-W. Hwang, Uniqueness of the limit cycle for Gause-type predator-prey systems,, Journal of Mathematical Analysis and Applications, 238 (1999), 179. doi: 10.1006/jmaa.1999.6520.

[35]

Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator,, Journal of Mathematical Biology, 67 (2013), 1227. doi: 10.1007/s00285-012-0584-z.

[36]

W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with constant diffusion rates,, Journal of Mathematical Analysis and Applications, 344 (2008), 217. doi: 10.1016/j.jmaa.2008.03.006.

[37]

Y. Kuang, Global stability of Gause-type predator-prey systems,, Journal of Mathematical Biology, 28 (1990), 463. doi: 10.1007/BF00178329.

[38]

Y. Kuang, Rich dynamics of Gause-type ratio-dependent predator-prey system,, Fields Institute Communication, 21 (1999), 325.

[39]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, Journal of Mathematical Biology, 36 (1998), 389. doi: 10.1007/s002850050105.

[40]

Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems,, Mathematical Biosciences, 88 (1988), 67. doi: 10.1016/0025-5564(88)90049-1.

[41]

S. A. Levin, The problem of pattern and scale in ecology,, Ecology, 73 (1992), 1943. doi: 10.1007/978-1-4615-1769-6_15.

[42]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms,, Theoretical Population Biology, 43 (1993), 141. doi: 10.1006/tpbi.1993.1007.

[43]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, Journal of Differential Equations, 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7.

[44]

Y. Liu, Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems,, Proceedings of the American Mathematical Society, 133 (2005), 3619. doi: 10.1090/S0002-9939-05-08026-3.

[45]

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