Pattern formation of a predator-prey model with the cost of anti-predator behaviors

Xiaoying Wang; Xingfu Zou; Xiaoying Wang; Xingfu Zou

doi:10.3934/mbe.2018035

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 3: 775-805. doi: 10.3934/mbe.2018035

Previous Article Next Article

Pattern formation of a predator-prey model with the cost of anti-predator behaviors

Xiaoying Wang ,
Xingfu Zou ^,

Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7

Received: 21 September 2016 Revised: 07 June 2017 Published: 01 June 2018
MSC : 92D25, 34C23, 92D40

We propose and analyse a reaction-diffusion-advection predator-prey model in which we assume that predators move randomly but prey avoid predation by perceiving a repulsion along predator density gradient. Based on recent experimental evidence that anti-predator behaviors alone lead to a 40% reduction on prey reproduction rate, we also incorporate the cost of anti-predator responses into the local reaction terms in the model. Sufficient and necessary conditions of spatial pattern formation are obtained for various functional responses between prey and predators. By mathematical and numerical analyses, we find that small prey sensitivity to predation risk may lead to pattern formation if the Holling type Ⅱ functional response or the Beddington-DeAngelis functional response is adopted while large cost of anti-predator behaviors homogenises the system by excluding pattern formation. However, the ratio-dependent functional response gives an opposite result where large predator-taxis may lead to pattern formation but small cost of anti-predator behaviors inhibits the emergence of spatial heterogeneous solutions.
- Pattern formation,
- stability,
- predator-prey model,
- anti-predator behaviors,
- bifurcation,
- global stability
Citation: Xiaoying Wang, Xingfu Zou. Pattern formation of a predator-prey model with the cost of anti-predator behaviors[J]. Mathematical Biosciences and Engineering, 2018, 15(3): 775-805. doi: 10.3934/mbe.2018035

Related Papers:

Abstract

We propose and analyse a reaction-diffusion-advection predator-prey model in which we assume that predators move randomly but prey avoid predation by perceiving a repulsion along predator density gradient. Based on recent experimental evidence that anti-predator behaviors alone lead to a 40% reduction on prey reproduction rate, we also incorporate the cost of anti-predator responses into the local reaction terms in the model. Sufficient and necessary conditions of spatial pattern formation are obtained for various functional responses between prey and predators. By mathematical and numerical analyses, we find that small prey sensitivity to predation risk may lead to pattern formation if the Holling type Ⅱ functional response or the Beddington-DeAngelis functional response is adopted while large cost of anti-predator behaviors homogenises the system by excluding pattern formation. However, the ratio-dependent functional response gives an opposite result where large predator-taxis may lead to pattern formation but small cost of anti-predator behaviors inhibits the emergence of spatial heterogeneous solutions.

References

[1]	[ B. E. Ainseba,M. Bendahmane,A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Analysis: Real World Applications, 9 (2008): 2086-2105.
[2]	[ N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, Journal of Differential Equations, 33 (1979): 201-225.
[3]	[ D. Alonso,F. Bartumeus,J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83 (2002): 28-34.
[4]	[ H. Amann, Dynamic theory of quasilinear parabolic equations Ⅱ: Reaction-diffusion systems, Differential Integral Equations, 3 (1990): 13-75.
[5]	[ H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ: Global existence, Mathematische Zeitschrift, 202 (1989): 219-250.
[6]	[ H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, 133 (1993): 9-126.
[7]	[ J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, Journal of Animal Ecology, 44 (1975): 331-340.
[8]	[ V. N. Biktashev,J. Brindley,A. V. Holden,M. A. Tsyganov, Pursuit-evasion predator-prey waves in two spatial dimensions, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14 (2004): 988-994.
[9]	[ X. Cao, Y. Song and T. Zhang, Hopf bifurcation and delay-induced Turing instability in a diffusive lac operon model, International Journal of Bifurcation and Chaos, 26 (2016), 1650167, 22pp.
[10]	[ S. Creel,D. Christianson, Relationships between direct predation and risk effects, Trends in Ecology & Evolution, 23 (2008): 194-201.
[11]	[ W. Cresswell, Predation in bird populations, Journal of Ornithology, 152 (2011): 251-263.
[12]	[ D. L. DeAngelis,R. A. Goldstein,R. V. O'neill, A model for tropic interaction, Ecology, 56 (1975): 881-892.
[13]	[ T. Hillen,K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009): 183-217.
[14]	[ T. Hillen,K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Advances in Applied Mathematics, 26 (2001): 280-301.
[15]	[ C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, The Canadian Entomologist, 91 (1959): 293-320.
[16]	[ C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91 (1959): 385-398.
[17]	[ H. Jin,Z. Wang, Global stability of prey-taxis systems, Journal of Differential Equations, 262 (2017): 1257-1290.
[18]	[ J. M. Lee,T. Hillen,M. A. Lewis, Pattern formation in prey-taxis systems, Journal of Biological Dynamics, 3 (2009): 551-573.
[19]	[ S. A. Levin, The problem of pattern and scale in ecology, Ecology, 73 (1992): 1943-1967.
[20]	[ E. A. McGehee,E. Peacock-López, Turing patterns in a modified Lotka-Volterra model, Physics Letters A, 342 (2005): 90-98.
[21]	[ A. B. Medvinsky,S. V. Petrovskii,I. A. Tikhonova,H. Malchow,B. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44 (2002): 311-370.
[22]	[ J. D. Meiss, Differential Dynamical Systems, SIAM, 2007.
[23]	[ M. Mimura,J. D. Murray, On a diffusive prey-predator model which exhibits patchiness, Journal of Theoretical Biology, 75 (1978): 249-262.
[24]	[ M. Mimura, Asymptotic behavior of a parabolic system related to a planktonic prey and predator system, SIAM Journal on Applied Mathematics, 37 (1979): 499-512.
[25]	[ A. Morozov,S. Petrovskii,B. Li, Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect, Journal of Theoretical Biology, 238 (2006): 18-35.
[26]	[ J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.
[27]	[ W. Ni,M. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, Journal of Differential Equations, 261 (2016): 4244-4274.
[28]	[ K. J. Painter,T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart, 10 (2002): 501-543.
[29]	[ S. V. Petrovskii,A. Y. Morozov,E. Venturino, Allee effect makes possible patchy invasion in a predator-prey system, Ecology Letters, 5 (2002): 345-352.
[30]	[ D. Ryan,R. Cantrell, Avoidance behavior in intraguild predation communities: A cross-diffusion model, Discrete and Continuous Dynamical Systems, 35 (2015): 1641-1663.
[31]	[ H. Shi,S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA Journal of Applied Mathematics, 80 (2015): 1534-1568.
[32]	[ H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.
[33]	[ Y. Song,X. Zou, Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a Hopf-Turing bifurcation point, Computers and Mathematics with Applications, 37 (2014): 1978-1997.
[34]	[ Y. Song,T. Zhang,Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Communications in Nonlinear Science and Numerical Simulation, 33 (2016): 229-258.
[35]	[ Y. Song and X. Tang, Stability, steady-state bifurcations, and Turing patterns in a predator-prey model with herd behavior and prey-taxis, Studies in Applied Mathematics,(2017).
[36]	[ J. H. Steele, Spatial Pattern in Plankton Communities (Vol. 3), Springer Science & Business Media, 1978.
[37]	[ Y. Tao, Global existence of classical solutions to a predator-prey model with nolinear prey-taxis, Nonlinear Analysis: Real World Applications, 11 (2010): 2056-2064.
[38]	[ J. Wang,J. Shi,J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, Journal of Differential Equations, 251 (2011): 1276-1304.
[39]	[ J. Wang,J. Wei,J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, Journal of Differential Equations, 260 (2016): 3495-3523.
[40]	[ X. Wang,L. Y. Zanette,X. Zou, Modelling the fear effect in predator-prey interactions, Journal of Mathematical Biology, 73 (2016): 1179-1204.
[41]	[ X. Wang,W. Wang,G. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Mathematical Methods in the Applied Sciences, 38 (2015): 431-443.
[42]	[ Z. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 037108, 13 pp.
[43]	[ S. Wu,J. Shi,B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, Journal of Differential Equations, 260 (2016): 5847-5874.
[44]	[ J. Xu,G. Yang,H. Xi,J. Su, Pattern dynamics of a predator-prey reaction-diffusion model with spatiotemporal delay, Nonlinear Dynamics, 81 (2015): 2155-2163.
[45]	[ F. Yi,J. Wei,J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009): 1944-1977.
[46]	[ L. Y. Zanette,A. F. White,M. C. Allen,M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011): 1398-1401.
[47]	[ T. Zhang and H. Zang, Delay-induced Turing instability in reaction-diffusion equations, Physical Review E, 90 (2014), 052908.

Reader Comments

Your name:*

Email:*
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)