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September  2020, 25(9): 3335-3356. doi: 10.3934/dcdsb.2020064

Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms

Mihaela Negreanu

Depto. de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

Received  February 2019 Revised  October 2019 Published  September 2020 Early access  April 2020

Fund Project: The first author is supported by by project MTM2017- 83391-P MICINN (Spain)

This paper is concerned with a general asymptotic stabilization of arbitrary global positive bounded solutions for the Lotka Volterra reaction diffusion systems, with an additional chemotactic influence and constant coefficients. We consider the dynamics of a mathematical model involving two biological species, both of which move according to random diffusion and are attracted/ repulsed by chemical stimulus produced by the other. The biological species present the ability to orientate their movement towards the concentration of the chemical secreted by the other species. The nonlinear system consists of two parabolic equations with Lotka-Volterra-type kinetic terms coupled with chemotactic cross-diffusion, along with two elliptic equations describing the behavior of the chemicals. We prove that the solution to the corresponding Neumann initial boundary value problem is global and bounded for regular and positive initial data. Moreover, for different ranges of parameters, we show that any positive and bounded solution converges to a spatially constant homogeneous state.

Keywords: Chemotaxis, maximum principle, global existence, steady state solutions, stability of solutions, coexistence states, extinction, asymptotic behavior, comparition principle, Lotka Volterra prey predator system.
Mathematics Subject Classification: Primary: 92C17, 35K55, 35B35; Secondary: 35B51.
Citation: Mihaela Negreanu. Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3335-3356. doi: 10.3934/dcdsb.2020064
References:
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S. Ahmad and A. Lazer, Asymptotic behavior of solutions of periodic competition diffusion systems, Nonlinear Anal., 13 (1989), 263-284.  doi: 10.1016/0362-546X(89)90054-0.  Google Scholar

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S. Agmon, A. Douglas and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.  Google Scholar

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Z. Amine and R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl., 185 (1994), 477-489.  doi: 10.1006/jmaa.1994.1262.  Google Scholar

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X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

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N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

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P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.   Google Scholar

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T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.  Google Scholar

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T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.  doi: 10.1093/imamat/hxw036.  Google Scholar

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C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44 (1984), 1112-1132.  doi: 10.1137/0144080.  Google Scholar

[10]

E. Cruz, M. Negreanu and J. I. Tello, Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 69 (2018), 20pp. doi: 10.1007/s00033-018-1002-1.  Google Scholar

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S. M. Fu and M. Ruyun, Existence of a global coexistence state for periodic competition model systems, Nonlinear Anal., 28 (1997), 1265-1271.  doi: 10.1016/S0362-546X(97)82873-8.  Google Scholar

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K. Gopalsamy, Exchange of equilibria in two species Lotka-Volterra competition models, J. Austral. Math. Soc. Ser. B, 24 (1982), 160-170.  doi: 10.1017/S0334270000003659.  Google Scholar

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G. Hetzer and W. Shen, Convergence in almost periodic competition diffusion systems, J. Math. Anal. Appl., 262 (2001), 307-338.  doi: 10.1006/jmaa.2001.7582.  Google Scholar

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G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34 (2002), 204-227.  doi: 10.1137/S0036141001390695.  Google Scholar

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D. Horstmann, Generalizing the Keller–Segel Model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

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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.  Google Scholar

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V. Hutson, K. Mischaikow and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.  doi: 10.1007/s002850100106.  Google Scholar

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V. Hutson, K. Mischaikow and P. Poláčik, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, MD. Google Scholar

[21]

T. B. Issa and W. Shen, Uniqueness and stability of coexistence states in two species models with/without chemotaxis on bounded heterogeneous environments, J. Dynam. Differential Equations, 31 (2019), 2305–-2338. doi: 10.1007/s10884-018-9706-7.  Google Scholar

[22]

T. B. Issa and W. Shen, Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments, J. Dynam. Differential Equations, 31 (2019), 1839–-1871. doi: 10.1007/s10884-018-9686-7.  Google Scholar

[23]

T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973.  doi: 10.1137/16M1092428.  Google Scholar

[24]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[25]

F. Kentarou and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.  doi: 10.1016/j.jde.2017.02.031.  Google Scholar

[26]

H. Li and Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.  doi: 10.1016/j.aml.2017.10.006.  Google Scholar

[27]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar

[28]

M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.  doi: 10.1016/j.jde.2016.05.008.  Google Scholar

[29]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.  Google Scholar

[30]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.  Google Scholar

[31]

M. Negreanu and J. I. Tello, On a parabolic-elliptic chemotactic system with non-constant chemotactic sensivity, Nonlinear Anal., 80 (2013), 1-13.  doi: 10.1016/j.na.2012.12.004.  Google Scholar

[32]

M. Negreanu and J. I. Tello, On a competitive system under chemotactic effects with non-local terms, Nonlinearity, 26 (2013), 1083-1103.  doi: 10.1088/0951-7715/26/4/1083.  Google Scholar

[33]

M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2669-2688.  doi: 10.3934/dcdsb.2013.18.2669.  Google Scholar

[34]

M. Negreanu and J. I. Tello, Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.  doi: 10.1016/j.jmaa.2019.02.007.  Google Scholar

[35]

C. V. Pao, Coexistence and statbility of a competition-diffusion system in population dynamics, J. Math. Anal. Appl., 83 (1981), 54-76.  doi: 10.1016/0022-247X(81)90246-8.  Google Scholar

[36]

C. V. Pao, Comparison methods and stability analysis of reaction-diffusion systems, in Comparison Methods and Stability Theory, Lecture Notes in Pure and Appl. Math., 162, Dekker, New York, 1994, 277–292.  Google Scholar

[37]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhauser Advanced Texts, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-8442-5.  Google Scholar

[38]

C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar

[39]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar

[40]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc. (JEMS), 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.  Google Scholar

[41]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[42]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar

[43]

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

[44]

A. Tineo, On the asymptotic behavior of some population models, J. Math. Anal. Appl., 167 (1992), 516-529.  doi: 10.1016/0022-247X(92)90222-Y.  Google Scholar

[45]

A. M. Turing, The chemical basis of morphogenesis,, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[46]

V. Volterra, Variazioni e fluttuazioni del numero d individui in specie animali conviventi, Mem. R. Accad. Naz. Dei Lincei., (1926). Google Scholar

[47]

M. Winkler, Finite time blow-up in th higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[48]

Q. Zhang, Competitive exclusion for a two-species chemotaxis system with two chemicals., Appl. Math. Lett., 83 (2018), 27-32.  doi: 10.1016/j.aml.2018.03.012.  Google Scholar

[49]

Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys, 58 (2017), 9pp. doi: 10.1063/1.5011725.  Google Scholar

[50]

P. Zheng, C. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals, J. Math. Phys, 58 (2017), 17pp. doi: 10.1063/1.5010681.  Google Scholar

show all references

References:
[1]

S. Ahmad and A. Lazer, Asymptotic behavior of solutions of periodic competition diffusion systems, Nonlinear Anal., 13 (1989), 263-284.  doi: 10.1016/0362-546X(89)90054-0.  Google Scholar

[2]

S. Agmon, A. Douglas and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.  Google Scholar

[3]

Z. Amine and R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl., 185 (1994), 477-489.  doi: 10.1006/jmaa.1994.1262.  Google Scholar

[4]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

[5]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[6]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.   Google Scholar

[7]

T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.  Google Scholar

[8]

T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.  doi: 10.1093/imamat/hxw036.  Google Scholar

[9]

C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44 (1984), 1112-1132.  doi: 10.1137/0144080.  Google Scholar

[10]

E. Cruz, M. Negreanu and J. I. Tello, Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 69 (2018), 20pp. doi: 10.1007/s00033-018-1002-1.  Google Scholar

[11]

S. M. Fu and M. Ruyun, Existence of a global coexistence state for periodic competition model systems, Nonlinear Anal., 28 (1997), 1265-1271.  doi: 10.1016/S0362-546X(97)82873-8.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[13]

K. Gopalsamy, Exchange of equilibria in two species Lotka-Volterra competition models, J. Austral. Math. Soc. Ser. B, 24 (1982), 160-170.  doi: 10.1017/S0334270000003659.  Google Scholar

[14]

G. Hetzer and W. Shen, Convergence in almost periodic competition diffusion systems, J. Math. Anal. Appl., 262 (2001), 307-338.  doi: 10.1006/jmaa.2001.7582.  Google Scholar

[15]

G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34 (2002), 204-227.  doi: 10.1137/S0036141001390695.  Google Scholar

[16]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[17]

D. Horstmann, Generalizing the Keller–Segel Model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[18]

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.  Google Scholar

[19]

V. Hutson, K. Mischaikow and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.  doi: 10.1007/s002850100106.  Google Scholar

[20]

V. Hutson, K. Mischaikow and P. Poláčik, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, MD. Google Scholar

[21]

T. B. Issa and W. Shen, Uniqueness and stability of coexistence states in two species models with/without chemotaxis on bounded heterogeneous environments, J. Dynam. Differential Equations, 31 (2019), 2305–-2338. doi: 10.1007/s10884-018-9706-7.  Google Scholar

[22]

T. B. Issa and W. Shen, Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments, J. Dynam. Differential Equations, 31 (2019), 1839–-1871. doi: 10.1007/s10884-018-9686-7.  Google Scholar

[23]

T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973.  doi: 10.1137/16M1092428.  Google Scholar

[24]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[25]

F. Kentarou and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.  doi: 10.1016/j.jde.2017.02.031.  Google Scholar

[26]

H. Li and Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.  doi: 10.1016/j.aml.2017.10.006.  Google Scholar

[27]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar

[28]

M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.  doi: 10.1016/j.jde.2016.05.008.  Google Scholar

[29]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.  Google Scholar

[30]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.  Google Scholar

[31]

M. Negreanu and J. I. Tello, On a parabolic-elliptic chemotactic system with non-constant chemotactic sensivity, Nonlinear Anal., 80 (2013), 1-13.  doi: 10.1016/j.na.2012.12.004.  Google Scholar

[32]

M. Negreanu and J. I. Tello, On a competitive system under chemotactic effects with non-local terms, Nonlinearity, 26 (2013), 1083-1103.  doi: 10.1088/0951-7715/26/4/1083.  Google Scholar

[33]

M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2669-2688.  doi: 10.3934/dcdsb.2013.18.2669.  Google Scholar

[34]

M. Negreanu and J. I. Tello, Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.  doi: 10.1016/j.jmaa.2019.02.007.  Google Scholar

[35]

C. V. Pao, Coexistence and statbility of a competition-diffusion system in population dynamics, J. Math. Anal. Appl., 83 (1981), 54-76.  doi: 10.1016/0022-247X(81)90246-8.  Google Scholar

[36]

C. V. Pao, Comparison methods and stability analysis of reaction-diffusion systems, in Comparison Methods and Stability Theory, Lecture Notes in Pure and Appl. Math., 162, Dekker, New York, 1994, 277–292.  Google Scholar

[37]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhauser Advanced Texts, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-8442-5.  Google Scholar

[38]

C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar

[39]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar

[40]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc. (JEMS), 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.  Google Scholar

[41]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[42]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar

[43]

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

[44]

A. Tineo, On the asymptotic behavior of some population models, J. Math. Anal. Appl., 167 (1992), 516-529.  doi: 10.1016/0022-247X(92)90222-Y.  Google Scholar

[45]

A. M. Turing, The chemical basis of morphogenesis,, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[46]

V. Volterra, Variazioni e fluttuazioni del numero d individui in specie animali conviventi, Mem. R. Accad. Naz. Dei Lincei., (1926). Google Scholar

[47]

M. Winkler, Finite time blow-up in th higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[48]

Q. Zhang, Competitive exclusion for a two-species chemotaxis system with two chemicals., Appl. Math. Lett., 83 (2018), 27-32.  doi: 10.1016/j.aml.2018.03.012.  Google Scholar

[49]

Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys, 58 (2017), 9pp. doi: 10.1063/1.5011725.  Google Scholar

[50]

P. Zheng, C. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals, J. Math. Phys, 58 (2017), 17pp. doi: 10.1063/1.5010681.  Google Scholar

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