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Pattern formation in a predator-mediated coexistence model with prey-taxis
1. | Department of Mathematics, Nova Southeastern University, Fort Lauderdale, FL, USA |
2. | Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD, USA |
Can foraging by predators or a repulsive prey defense mechanism upset predator-mediated coexistence? This paper investigates one scenario involving a prey-taxis by a prey species. We study a system of three populations involving two competing species with a common predator. All three populations are mobile via random dispersal within a bounded spatial domain $ \Omega $, but the predator's movement is influenced by one prey's gradient representing a repulsive effect on the predator. We prove existence of positive solutions, and investigate pattern formation through bifurcation analysis and numerical simulation.
References:
[1] |
B. E. Ainseba, M. Bendahmane and A. Noussair,
A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.
doi: 10.1016/j.nonrwa.2007.06.017. |
[2] |
N. D. Alikakos,
$L^{p}$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[3] |
M. S. Alnæs, J. Blechta, J. Hake, A. Johansson and B. Kehlet, et al., The FEniCS project, version 1.5, Archive Numerical Software, 3.
doi: 10.11588/ans.2015.100.20553. |
[4] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis, Teubner-Texte Math., 133, Teubner, Stuttgart, 1993, 9–126.
doi: 10.1007/978-3-663-11336-2_1. |
[5] |
H. Amann,
Dynamic theory of quasilinear parabolic equations. Ⅱ: Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.
|
[6] |
B. Ayuso and L. D. Marini,
Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 47 (2009), 1391-1420.
doi: 10.1137/080719583. |
[7] |
M. Bendahmane,
Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis, Netw. Heterog. Media, 3 (2008), 863-879.
doi: 10.3934/nhm.2008.3.863. |
[8] |
M. Burger and J.-F. Pietschmann,
Flow characteristics in a crowded transport model, Nonlinearity, 29 (2016), 3528-3550.
doi: 10.1088/0951-7715/29/11/3528. |
[9] |
G. Caristi, K. P. Rybakowski and T. Wessolek, Persistence and spatial patterns in a onepredator-two-prey Lotka-Volterra model with diffusion, Ann. Mat. Pura Appl. (4), 161
(1992), 345–377.
doi: 10.1007/BF01759645. |
[10] |
J. M. Chase, P. A. Abrams, J. P. Grover, S. Diehl and P. Chesson, et al., The interaction
between predation and competition: A review and synthesis, Ecology Lett., 5 (2002), 302–315.
doi: 10.1046/j.1461-0248.2002.00315.x. |
[11] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[12] |
C. Cosner,
Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
[13] |
N. Cramer and R. May,
Interspecific competition, predation and species diversity: A comment, J. Theoretical Biology, 34 (1972), 289-293.
doi: 10.1016/0022-5193(72)90162-2. |
[14] |
D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Mathematics & Applications, 69, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-22980-0. |
[15] |
W. Feng,
Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179 (1993), 592-609.
doi: 10.1006/jmaa.1993.1371. |
[16] |
C. Gai, Q. Wang and J. Yan,
Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.
doi: 10.3934/dcds.2015.35.1239. |
[17] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[18] |
D. Horstmann et al., From 1970 until present: The Keller-Segel model in chemotaxis and its
consequences, I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103–165. |
[19] |
S. Hsu,
Predator-mediated coexistence and extinction, Math. Biosci., 54 (1981), 231-248.
doi: 10.1016/0025-5564(81)90088-2. |
[20] |
G. E. Hutchinson,
The paradox of the plankton, Amer. Naturalist, 95 (1961), 137-145.
doi: 10.1086/282171. |
[21] |
H.-Y. Jin and Z.-A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[22] |
P. Kareiva and G. Odell,
Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Naturalist, 130 (1987), 233-270.
doi: 10.1086/284707. |
[23] |
P. Korman and A. W. Leung,
A general monotone scheme for elliptic systems with applications to ecological models, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 315-325.
doi: 10.1017/S0308210500026391. |
[24] |
A. Kurganov and M. Lukáčová-Medvid'ová,
Numerical study of two-species chemotaxis models, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 131-152.
doi: 10.3934/dcdsb.2014.19.131. |
[25] |
N. Lakos,
Existence of steady-state solutions for a one-predator-two-prey system, SIAM J. Math. Anal., 21 (1990), 647-659.
doi: 10.1137/0521034. |
[26] |
J. Lee, T. Hillen and M. Lewis,
Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.
doi: 10.1080/17513750802716112. |
[27] |
C. Li and Z. Hong, Global existence of classical solutions to a three-species predator-prey model with two prey-taxes, J. Appl. Math., 2012, 12pp.
doi: 10.1155/2012/702603. |
[28] |
Y. Li, K. Lin and C. Mu,
Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system, Electronic J. Differential Equations, 2015 (2015), 1-13.
|
[29] |
A. Logg, K.-A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method. The FEniCS Book, Lecture Notes in Computational Science and Engineering, 84, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-23099-8. |
[30] |
R. M. May,
Stability in multispecies community models, Math. Biosci., 12 (1971), 59-79.
doi: 10.1016/0025-5564(71)90074-5. |
[31] |
J. Murray, Mathematical Biology. I: An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.
doi: 10.1007/b98868. |
[32] |
R. T. Paine,
Food web complexity and species diversity, Amer. Naturalist, 100 (1966), 65-75.
doi: 10.1086/282400. |
[33] |
J. Parrish and S. Saila,
Interspecific competition, predation and species diversity, J. Theoretical Biology, 27 (1970), 207-220.
doi: 10.1016/0022-5193(70)90138-4. |
[34] |
J. I. Tello and D. Wrzosek,
Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.
doi: 10.1142/S0218202516400108. |
[35] |
J. I. Tello and D. Wrzosek,
Inter-species competition and chemorepulsion, J. Math. Anal. Appl., 459 (2018), 1233-1250.
doi: 10.1016/j.jmaa.2017.11.021. |
[36] |
S. Vȧge, G. Bratbak, J. Egge, M. Heldal and A. Larsen, et al., Simple models combining
competition, defence and resource availability have broad implications in pelagic microbial
food webs, Ecology Lett., 21 (2018), 1440–1452.
doi: 10.1111/ele.13122. |
[37] |
X. Wang, W. Wang and G. Zhang,
Global bifurcation of solutions for a predator–prey model with prey-taxis, Math. Methods Appl. Sci., 38 (2015), 431-443.
doi: 10.1002/mma.3079. |
[38] |
X. Wang and X. Zou,
Pattern formation of a predator-prey model with the cost of anti-predator behaviors, Math. Biosci. Eng., 15 (2018), 775-805.
doi: 10.3934/mbe.2018035. |
[39] |
Z. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 13pp.
doi: 10.1063/1.2766864. |
[40] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[41] |
D. Wrzosek,
Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal., 59 (2004), 1293-1310.
doi: 10.1016/j.na.2004.08.015. |
[42] |
S. Wu, J. Shi and B. Wu,
Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations, 260 (2016), 5847-5874.
doi: 10.1016/j.jde.2015.12.024. |
show all references
References:
[1] |
B. E. Ainseba, M. Bendahmane and A. Noussair,
A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.
doi: 10.1016/j.nonrwa.2007.06.017. |
[2] |
N. D. Alikakos,
$L^{p}$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[3] |
M. S. Alnæs, J. Blechta, J. Hake, A. Johansson and B. Kehlet, et al., The FEniCS project, version 1.5, Archive Numerical Software, 3.
doi: 10.11588/ans.2015.100.20553. |
[4] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis, Teubner-Texte Math., 133, Teubner, Stuttgart, 1993, 9–126.
doi: 10.1007/978-3-663-11336-2_1. |
[5] |
H. Amann,
Dynamic theory of quasilinear parabolic equations. Ⅱ: Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.
|
[6] |
B. Ayuso and L. D. Marini,
Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 47 (2009), 1391-1420.
doi: 10.1137/080719583. |
[7] |
M. Bendahmane,
Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis, Netw. Heterog. Media, 3 (2008), 863-879.
doi: 10.3934/nhm.2008.3.863. |
[8] |
M. Burger and J.-F. Pietschmann,
Flow characteristics in a crowded transport model, Nonlinearity, 29 (2016), 3528-3550.
doi: 10.1088/0951-7715/29/11/3528. |
[9] |
G. Caristi, K. P. Rybakowski and T. Wessolek, Persistence and spatial patterns in a onepredator-two-prey Lotka-Volterra model with diffusion, Ann. Mat. Pura Appl. (4), 161
(1992), 345–377.
doi: 10.1007/BF01759645. |
[10] |
J. M. Chase, P. A. Abrams, J. P. Grover, S. Diehl and P. Chesson, et al., The interaction
between predation and competition: A review and synthesis, Ecology Lett., 5 (2002), 302–315.
doi: 10.1046/j.1461-0248.2002.00315.x. |
[11] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[12] |
C. Cosner,
Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
[13] |
N. Cramer and R. May,
Interspecific competition, predation and species diversity: A comment, J. Theoretical Biology, 34 (1972), 289-293.
doi: 10.1016/0022-5193(72)90162-2. |
[14] |
D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Mathematics & Applications, 69, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-22980-0. |
[15] |
W. Feng,
Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179 (1993), 592-609.
doi: 10.1006/jmaa.1993.1371. |
[16] |
C. Gai, Q. Wang and J. Yan,
Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.
doi: 10.3934/dcds.2015.35.1239. |
[17] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[18] |
D. Horstmann et al., From 1970 until present: The Keller-Segel model in chemotaxis and its
consequences, I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103–165. |
[19] |
S. Hsu,
Predator-mediated coexistence and extinction, Math. Biosci., 54 (1981), 231-248.
doi: 10.1016/0025-5564(81)90088-2. |
[20] |
G. E. Hutchinson,
The paradox of the plankton, Amer. Naturalist, 95 (1961), 137-145.
doi: 10.1086/282171. |
[21] |
H.-Y. Jin and Z.-A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[22] |
P. Kareiva and G. Odell,
Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Naturalist, 130 (1987), 233-270.
doi: 10.1086/284707. |
[23] |
P. Korman and A. W. Leung,
A general monotone scheme for elliptic systems with applications to ecological models, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 315-325.
doi: 10.1017/S0308210500026391. |
[24] |
A. Kurganov and M. Lukáčová-Medvid'ová,
Numerical study of two-species chemotaxis models, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 131-152.
doi: 10.3934/dcdsb.2014.19.131. |
[25] |
N. Lakos,
Existence of steady-state solutions for a one-predator-two-prey system, SIAM J. Math. Anal., 21 (1990), 647-659.
doi: 10.1137/0521034. |
[26] |
J. Lee, T. Hillen and M. Lewis,
Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.
doi: 10.1080/17513750802716112. |
[27] |
C. Li and Z. Hong, Global existence of classical solutions to a three-species predator-prey model with two prey-taxes, J. Appl. Math., 2012, 12pp.
doi: 10.1155/2012/702603. |
[28] |
Y. Li, K. Lin and C. Mu,
Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system, Electronic J. Differential Equations, 2015 (2015), 1-13.
|
[29] |
A. Logg, K.-A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method. The FEniCS Book, Lecture Notes in Computational Science and Engineering, 84, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-23099-8. |
[30] |
R. M. May,
Stability in multispecies community models, Math. Biosci., 12 (1971), 59-79.
doi: 10.1016/0025-5564(71)90074-5. |
[31] |
J. Murray, Mathematical Biology. I: An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.
doi: 10.1007/b98868. |
[32] |
R. T. Paine,
Food web complexity and species diversity, Amer. Naturalist, 100 (1966), 65-75.
doi: 10.1086/282400. |
[33] |
J. Parrish and S. Saila,
Interspecific competition, predation and species diversity, J. Theoretical Biology, 27 (1970), 207-220.
doi: 10.1016/0022-5193(70)90138-4. |
[34] |
J. I. Tello and D. Wrzosek,
Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.
doi: 10.1142/S0218202516400108. |
[35] |
J. I. Tello and D. Wrzosek,
Inter-species competition and chemorepulsion, J. Math. Anal. Appl., 459 (2018), 1233-1250.
doi: 10.1016/j.jmaa.2017.11.021. |
[36] |
S. Vȧge, G. Bratbak, J. Egge, M. Heldal and A. Larsen, et al., Simple models combining
competition, defence and resource availability have broad implications in pelagic microbial
food webs, Ecology Lett., 21 (2018), 1440–1452.
doi: 10.1111/ele.13122. |
[37] |
X. Wang, W. Wang and G. Zhang,
Global bifurcation of solutions for a predator–prey model with prey-taxis, Math. Methods Appl. Sci., 38 (2015), 431-443.
doi: 10.1002/mma.3079. |
[38] |
X. Wang and X. Zou,
Pattern formation of a predator-prey model with the cost of anti-predator behaviors, Math. Biosci. Eng., 15 (2018), 775-805.
doi: 10.3934/mbe.2018035. |
[39] |
Z. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 13pp.
doi: 10.1063/1.2766864. |
[40] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[41] |
D. Wrzosek,
Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal., 59 (2004), 1293-1310.
doi: 10.1016/j.na.2004.08.015. |
[42] |
S. Wu, J. Shi and B. Wu,
Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations, 260 (2016), 5847-5874.
doi: 10.1016/j.jde.2015.12.024. |







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