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

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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 and Continuous Dynamical Systems - B, 2020, 25 (9) : 3335-3356. doi: 10.3934/dcdsb.2020064

References:

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[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.
[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.
[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.
[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.
[6]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	V. Hutson, K. Mischaikow and P. Poláčik, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, MD.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[46]	V. Volterra, Variazioni e fluttuazioni del numero d individui in specie animali conviventi, Mem. R. Accad. Naz. Dei Lincei., (1926).
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[6]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	V. Hutson, K. Mischaikow and P. Poláčik, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, MD.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[46]	V. Volterra, Variazioni e fluttuazioni del numero d individui in specie animali conviventi, Mem. R. Accad. Naz. Dei Lincei., (1926).
[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.
[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.
[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.
[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.

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