doi: 10.3934/dcdsb.2021237
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Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

1. 

School of Mathematics Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, China

2. 

College of Teacher Education, Harbin Normal University, Harbin, Heilongjiang, 150025, China

* Corresponding author: Jinfeng Wang

Received  April 2021 Revised  August 2021 Early access October 2021

Fund Project: Partially supported by National NSFC grant 11971135, NSFH grant LH2019A017, LH2020A019 and 2018-KYYWF-0999

This paper is concerned with a predator-prey model with stage structure for the predator, with a cross-diffusion term modeling the effect that mature predators move toward the direction of gradient of prey. It is first shown that the corresponding Neumann initial-boundary value problem in an $ n $-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly-in-time bounded for the weak cross-diffusion. It is further shown that, in the presence of cross-diffusion, the model admits threshold-type dynamics in terms of the cross-diffusion coefficient; that is, the homogenous steady state keeps stability for weak attractive prey-taxis, while the stationary patterns will occur for strong attractive prey-taxis. This implies that such cross diffusion does contribute to the rich dynamics of predator-prey model with stage structure for predators.

Citation: Hongfei Xu, Jinfeng Wang, Xuelian Xu. Dynamics and pattern formation in a cross-diffusion model with stage structure for predators. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021237
References:
[1]

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

[2]

H. Amman, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces Differential Operators and Nonlinear Analysis, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathe. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[4]

N. BellomoK. J. PainterY. Tao and M. Winkler, Occurrence vs. absence of taxis-driven instabilities in a May–Nowak model for virus infection, SIAM J. Appl. Math, 79 (2019), 1990-2010.  doi: 10.1137/19M1250261.

[5]

E. X. Dejesus and C. Kaufman, Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equation, Phys. Rev. A, 35 (1987), 5288-5290.  doi: 10.1103/PhysRevA.35.5288.

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A. K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Anal., 13 (1989), 1091-1113.  doi: 10.1016/0362-546X(89)90097-7.

[7]

Y. H. DuP. Y. H. Pang and M. X. Wang, Qualitative analysis of a prey-predator model with stage structure for the predator, SIAM J. Appl. Math., 69 (2008), 596-620.  doi: 10.1137/070684173.

[8]

X. X. Guo and J. F. Wang, Dynamics and pattern formations in diffusive predator-prey models with two prey-taxis, Math. Methods Appl. Sci., 42 (2019), 4197-4212.  doi: 10.1002/mma.5639.

[9]

X. He and S. N. Zheng, Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis, Appl. Math. Lett., 49 (2015), 73-77.  doi: 10.1016/j.aml.2015.04.017.

[10]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[11]

W. J$\ddot{a}$ger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[12]

H. Y. Jin and Z. A. Wang, Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.

[13]

P. Kareiva and G. Odell, Swarms of predators exhibit prey-taxis if individual predators use area-restricted search, American Naturalist, 130 (1987), 233-270. 

[14]

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

[15]

P. LiuJ. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597.

[16]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. 

[17]

G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. 

[18]

Y. S. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.

[19]

Y. S. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.

[20]

Y. S. Tao and M. Winkler, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.  doi: 10.1142/S021820251950043X.

[21]

J. F. WangS. N. Wu and J. P. Shi, Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1273-1289.  doi: 10.3934/dcdsb.2020162.

[22]

J. P. Wang and M. X. Wang, Global bounded solution of the higher-dimensional forager-exploiter model with/without growth sources, Math. Models Methods Appl. Sci., 30 (2020), 1297-1323.  doi: 10.1142/S0218202520500232.

[23]

K. WangQ. Wang and F. Yu, Stationary and time periodic patterns of two-predator and one-prey systems with prey-taxis, Discrete Contin. Dyn. Syst., 37 (2017), 505-543.  doi: 10.3934/dcds.2017021.

[24]

X. L. WangW. D. Wang and G. H. 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.

[25]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1644-1673.  doi: 10.1002/mana.200810838.

[26]

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.

[27]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[28]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

[29]

M. Winkler, Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differential Equations, 263 (2017), 4826-4869.  doi: 10.1016/j.jde.2017.06.002.

[30]

S. N. WuJ. P. Shi and B. Y. 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.

[31]

T. Xiang, Global dynamics for a diffusive predator-prey model with prey-taxis and classical Lotka-Volterra kinetics, Nonlinear Anal. Real World Appl., 39 (2018), 278-299.  doi: 10.1016/j.nonrwa.2017.07.001.

show all references

References:
[1]

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

[2]

H. Amman, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces Differential Operators and Nonlinear Analysis, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathe. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[4]

N. BellomoK. J. PainterY. Tao and M. Winkler, Occurrence vs. absence of taxis-driven instabilities in a May–Nowak model for virus infection, SIAM J. Appl. Math, 79 (2019), 1990-2010.  doi: 10.1137/19M1250261.

[5]

E. X. Dejesus and C. Kaufman, Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equation, Phys. Rev. A, 35 (1987), 5288-5290.  doi: 10.1103/PhysRevA.35.5288.

[6]

A. K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Anal., 13 (1989), 1091-1113.  doi: 10.1016/0362-546X(89)90097-7.

[7]

Y. H. DuP. Y. H. Pang and M. X. Wang, Qualitative analysis of a prey-predator model with stage structure for the predator, SIAM J. Appl. Math., 69 (2008), 596-620.  doi: 10.1137/070684173.

[8]

X. X. Guo and J. F. Wang, Dynamics and pattern formations in diffusive predator-prey models with two prey-taxis, Math. Methods Appl. Sci., 42 (2019), 4197-4212.  doi: 10.1002/mma.5639.

[9]

X. He and S. N. Zheng, Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis, Appl. Math. Lett., 49 (2015), 73-77.  doi: 10.1016/j.aml.2015.04.017.

[10]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[11]

W. J$\ddot{a}$ger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[12]

H. Y. Jin and Z. A. Wang, Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.

[13]

P. Kareiva and G. Odell, Swarms of predators exhibit prey-taxis if individual predators use area-restricted search, American Naturalist, 130 (1987), 233-270. 

[14]

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

[15]

P. LiuJ. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597.

[16]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. 

[17]

G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. 

[18]

Y. S. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.

[19]

Y. S. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.

[20]

Y. S. Tao and M. Winkler, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.  doi: 10.1142/S021820251950043X.

[21]

J. F. WangS. N. Wu and J. P. Shi, Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1273-1289.  doi: 10.3934/dcdsb.2020162.

[22]

J. P. Wang and M. X. Wang, Global bounded solution of the higher-dimensional forager-exploiter model with/without growth sources, Math. Models Methods Appl. Sci., 30 (2020), 1297-1323.  doi: 10.1142/S0218202520500232.

[23]

K. WangQ. Wang and F. Yu, Stationary and time periodic patterns of two-predator and one-prey systems with prey-taxis, Discrete Contin. Dyn. Syst., 37 (2017), 505-543.  doi: 10.3934/dcds.2017021.

[24]

X. L. WangW. D. Wang and G. H. 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.

[25]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1644-1673.  doi: 10.1002/mana.200810838.

[26]

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.

[27]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[28]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

[29]

M. Winkler, Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differential Equations, 263 (2017), 4826-4869.  doi: 10.1016/j.jde.2017.06.002.

[30]

S. N. WuJ. P. Shi and B. Y. 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.

[31]

T. Xiang, Global dynamics for a diffusive predator-prey model with prey-taxis and classical Lotka-Volterra kinetics, Nonlinear Anal. Real World Appl., 39 (2018), 278-299.  doi: 10.1016/j.nonrwa.2017.07.001.

Figure 1.  Change of $ \chi_i $ with non-negative integer $ i $
Figure 2.  $ (\tilde{u}_1,\tilde{u}_2,\tilde{v}) $ is asymptotically stable for $ \chi = 0 $
Figure 3.  $ (\tilde{u}_1,\tilde{u}_2,\tilde{v}) $ remains stable for $ \chi = 200<\hat{\chi} $
Figure 4.  Spatial heterogenous and time-periodic patterns for $ \chi = 308>\hat{\chi} $
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