August  2020, 25(8): 2895-2921. doi: 10.3934/dcdsb.2020045

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

Received  October 2018 Revised  September 2019 Published  August 2020 Early access  February 2020

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.

Citation: Evan C. Haskell, Jonathan Bell. Pattern formation in a predator-mediated coexistence model with prey-taxis. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2895-2921. doi: 10.3934/dcdsb.2020045
References:
[1]

B. E. AinsebaM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ: Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. ChertockA. KurganovX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

C. GaiQ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Hsu, Predator-mediated coexistence and extinction, Math. Biosci., 54 (1981), 231-248.  doi: 10.1016/0025-5564(81)90088-2.  Google Scholar

[20]

G. E. Hutchinson, The paradox of the plankton, Amer. Naturalist, 95 (1961), 137-145.  doi: 10.1086/282171.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. LeeT. Hillen and M. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.  doi: 10.1080/17513750802716112.  Google Scholar

[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.  Google Scholar

[28]

Y. LiK. Lin and C. Mu, Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system, Electronic J. Differential Equations, 2015 (2015), 1-13.   Google Scholar

[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.  Google Scholar

[30]

R. M. May, Stability in multispecies community models, Math. Biosci., 12 (1971), 59-79.  doi: 10.1016/0025-5564(71)90074-5.  Google Scholar

[31]

J. Murray, Mathematical Biology. I: An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[32]

R. T. Paine, Food web complexity and species diversity, Amer. Naturalist, 100 (1966), 65-75.  doi: 10.1086/282400.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

X. WangW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

S. WuJ. 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.  Google Scholar

show all references

References:
[1]

B. E. AinsebaM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ: Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. ChertockA. KurganovX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

C. GaiQ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Hsu, Predator-mediated coexistence and extinction, Math. Biosci., 54 (1981), 231-248.  doi: 10.1016/0025-5564(81)90088-2.  Google Scholar

[20]

G. E. Hutchinson, The paradox of the plankton, Amer. Naturalist, 95 (1961), 137-145.  doi: 10.1086/282171.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. LeeT. Hillen and M. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.  doi: 10.1080/17513750802716112.  Google Scholar

[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.  Google Scholar

[28]

Y. LiK. Lin and C. Mu, Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system, Electronic J. Differential Equations, 2015 (2015), 1-13.   Google Scholar

[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.  Google Scholar

[30]

R. M. May, Stability in multispecies community models, Math. Biosci., 12 (1971), 59-79.  doi: 10.1016/0025-5564(71)90074-5.  Google Scholar

[31]

J. Murray, Mathematical Biology. I: An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[32]

R. T. Paine, Food web complexity and species diversity, Amer. Naturalist, 100 (1966), 65-75.  doi: 10.1086/282400.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

X. WangW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

S. WuJ. 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.  Google Scholar

Figure 1.  A plot of $ \chi $ versus $ k^{2} $. The red line is $ \chi_{c} $ as a function of $ k^{2} $. The blue dashed line shows $ a(k^{2})b(k^{2}, \chi)-c(k^{2}, \chi) = 0 $. Values of ($ k^{2}, \chi $) above (below) the blue dashed line correspond to $ a(k^{2})b(k^{2}, \chi)-c(k^{2}, \chi) $ positive (negative). This is an example where we use the parameter set: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, d) = (0.70, 0.70, 0.10, 0.30, 1.39, 0.30, 0.70, 0.25) $ Hence, $ a_{1}b_{2}-rb_{1} = 0.07 $, so $ k_{min}^{2} = 0.26 $
Figure 2.  A plot of $ \chi_{c_{1}} $ and $ \chi_{c_{2}} $ versus $ k^{2} $. $ \chi_{c_{1}} $ (red line) and $ \chi_{c_{2}} $ (blue dashed line) as functions of $ k^{2} $ are plotted for the parameter set: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu) = (0.10, 0.20, 0.10, 1.50, 3.0, 0.6, 0.1) $
$ u $ "wins". Note that while parameters are such that in the absence of a repulsive prey defense mechanism by $ u $, a predator mediated coexistence is expected; yet, once this state is disrupted $ v $ is driven to extinction in the presence of this taxis mechanism. Parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (0.25, 0.75, 0.75, 0.25, 0.6875, 0.5, 0.25, 15, 0.5) $">Figure 3.  Case 1: $ u $ "wins": In this example the parameters are chosen so that in the pure competition system $ u $ "wins". Note that while parameters are such that in the absence of a repulsive prey defense mechanism by $ u $, a predator mediated coexistence is expected; yet, once this state is disrupted $ v $ is driven to extinction in the presence of this taxis mechanism. Parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (0.25, 0.75, 0.75, 0.25, 0.6875, 0.5, 0.25, 15, 0.5) $
$ v $ "wins". Note that while parameters are such that in the absence of a repulsive prey defense mechanism by $ u $, a predator mediated coexistence is expected; yet, once this state is disrupted, the prey coexist with peak densities out of phase with the predator in the presence of this taxis mechanism. Parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.1, 0.1, 1.4, 0.3, 0.4, 0.4, 0.2, 50, 0.25) $">Figure 4.  Case 2: $ v $ "wins": In this example the parameters are chosen so that in the pure competition system $ v $ "wins". Note that while parameters are such that in the absence of a repulsive prey defense mechanism by $ u $, a predator mediated coexistence is expected; yet, once this state is disrupted, the prey coexist with peak densities out of phase with the predator in the presence of this taxis mechanism. Parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.1, 0.1, 1.4, 0.3, 0.4, 0.4, 0.2, 50, 0.25) $
$ u(x, t) $, $ v(x, t) $, and $ w(x, t) $ are given by the blue, green, and red lines respectively. The parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.25, 1.0, 0.25, 0.5, 1.25, 0.5, 0.25, 25, 0.1) $">Figure 5.  Spatio-temporal patterns: $ v $ "wins" case. The first column shows the initial dynamics from the perturbation of the homogeneous steady state and the second column shows the spatio-temporal pattern that is achieved. In the last row $ u(x, t) $, $ v(x, t) $, and $ w(x, t) $ are given by the blue, green, and red lines respectively. The parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.25, 1.0, 0.25, 0.5, 1.25, 0.5, 0.25, 25, 0.1) $
$ u(x, t) $, $ v(x, t) $, and $ w(x, t) $ are given by the blue, green, and red lines respectively. The parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (0.9, 1.1, 0.2, 0.2, 1.09, 0.5, 1.0, 16, 0.25) $">Figure 6.  Spatio-temporal patterns: $ u $ "wins" case. The first column shows the initial dynamics from the perturbation of the homogeneous steady state and the second column shows the spatio-temporal pattern that is achieved. In the last row $ u(x, t) $, $ v(x, t) $, and $ w(x, t) $ are given by the blue, green, and red lines respectively. The parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (0.9, 1.1, 0.2, 0.2, 1.09, 0.5, 1.0, 16, 0.25) $
$ u $, a predator mediated coexistence is expected; yet, once this state is disrupted, the prey coexist with peak densities out of phase with the predator in the presence of this taxis mechanism. Parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.25, 1.00, 0.25, 0.50, 1.25, 0.5, 0.25, -0.1, 10) $">Figure 7.  Attracting ($ \chi < 0 $) case: In this example the parameters are chosen so that in the pure competition system $ v $ "wins". Note that while parameters are such that in the absence of a attracing prey-taxis mechanism by $ u $, a predator mediated coexistence is expected; yet, once this state is disrupted, the prey coexist with peak densities out of phase with the predator in the presence of this taxis mechanism. Parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.25, 1.00, 0.25, 0.50, 1.25, 0.5, 0.25, -0.1, 10) $
[1]

Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162

[2]

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