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November  2020, 25(11): 4383-4396. doi: 10.3934/dcdsb.2020102

Large time behavior in a predator-prey system with indirect pursuit-evasion interaction

Genglin Li 1, , Youshan Tao 2, and  Michael Winkler 3,

1. 

College of Information and Technology, Donghua University, Shanghai 200051, China
2. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China
3. 

Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

Received  September 2019 Revised  November 2019 Published  April 2020

In a bounded domain
$ \Omega\subset \mathbb{R}^n $
 with smooth boundary, this work considers the indirect pursuit-evasion model
$ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u -\chi\nabla \cdot(u \nabla w) +u(\lambda -u+a v), \\ v_t = \Delta v +\xi \nabla \cdot(v\nabla z) +v(\mu-v-b u), \\ 0 = \Delta w -w +v, \\ 0 = \Delta z -z +u, \end{array} \right. \end{eqnarray*} $
with positive parameters
$ \chi, \xi, \lambda, \mu $
,
$ a $
 and
$ b $
.
It is firstly asserted that when
$ n\le 3 $
, for any given suitably regular initial data the corresponding homogeneous Neumann initial-boundary problem admits a global and bounded smooth solution. Moreover, it is shown that if
$ b\lambda<\mu $
 and under some explicit smallness conditions on
$ \chi $
 and
$ \xi $
, any nontrival bounded classical solution converges to the spatially homogeneous coexistence state in the large time limit; if
$ b\lambda>\mu $
, however, then under an explicit smallness assumption on
$ \chi $
 but without any restriction on
$ \xi $
, any bounded classical solution
$ (u, v) $
 with
$ u\not\equiv 0 $
 stabilizes to
$ (\lambda, 0) $
 as
$ t\to \infty $
.
Keywords: chemotaxis, attraction-repulsion, predator-prey, stabilization.
Mathematics Subject Classification: Primary: 5B40, 35B65, 35K57, 35Q92, 92C17.
Citation: Genglin Li, Youshan Tao, Michael Winkler. Large time behavior in a predator-prey system with indirect pursuit-evasion interaction. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4383-4396. doi: 10.3934/dcdsb.2020102
References:
[1]

P. Amorim, B. Telch and L. M. Villada, A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing, Math. Biosci. Eng., 16 (2019), 5114-5145.  doi: 10.3934/mbe.2019257.  Google Scholar

[2]

H. Bréezis and W. A. Strauss, Semilinear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[3]

T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system. Parabolic and Navier-Stokes equations, Part 1, Polish Acad. Sci. Inst. Math., Banach Center Publ., 81 (2008), 105–117. doi: 10.4064/bc81-0-7.  Google Scholar

[4]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[5]

A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, Inc., New York-Montreal, Que.-London, 1969.  Google Scholar

[6]

T. Goudon, B. Nkonga, M. Rascle and M. Ribot, Self-organized populations interacting under pursuit-evasion dynamics, Phys. D, 304/305 (2015), 1-22.  doi: 10.1016/j.physd.2015.03.012.  Google Scholar

[7]

T. Goudon and L. Urrutia, Analysis of kinetic and macroscopic models of pursuit-evasion dynamics, Commun. Math. Sci., 14 (2016), 2253-2286.  doi: 10.4310/CMS.2016.v14.n8.a7.  Google Scholar

[8]

X. He and S. 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.  Google Scholar

[9]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.   Google Scholar

[10]

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

[11]

W. Jäger 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.  Google Scholar

[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.  Google Scholar

[13]

P. Kareiva and G. Odell, Swarms of predators exhibit `preytaxis' if individual predators use area-restricted search, The American Naturalist, 130 (1987), 233-270.  doi: 10.1086/284707.  Google Scholar

[14]

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

[15]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[16]

Y. 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.  Google Scholar

[17]

Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[18]

Y. Tao and M. Winkler, Boundedness and stabilization in a population model with cross-diffusion for one species, Proc. London Math. Soc., 119 (2019), 1598-1632.  doi: 10.1112/plms.12276.  Google Scholar

[19]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[20]

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

[21]

Y. 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.  Google Scholar

[22]

Y. Tao and M. Winkler, Global smooth solvability of a parabolic-elliptic nutrient taxis system in domains of arbitrary dimension, J. Differential Equations, 267 (2019), 388-406.  doi: 10.1016/j.jde.2019.01.014.  Google Scholar

[23]

M. A. Tsyganov, J. Brindley, A. V. Holden and V. N. Biktashev, Quasi-soliton interaction of pursuit-evasion waves in a predator-prey system, Phys. Rev. Lett., 91 (2003), 218102. doi: 10.1103/PhysRevLett.91.218102.  Google Scholar

[24]

M. A. Tsyganov, I. B. Kresteva, A. B. Medvinsky and G. R. Ivanitsky, A novel mode bacterial population wave interaction, Dokl. Akad. Nauk, 333 (1993), 532-536.   Google Scholar

[25]

Y. Tyutyunov, L. Titova and R. Arditi, A minimal model of pursuit-evasion in a predator-prey system, Math. Model. Nat. Phenom., 2 (2007), 122-134.  doi: 10.1051/mmnp:2008028.  Google Scholar

[26]

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.  Google Scholar

[27]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748–767, arXiv: 1112.4156v1. doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[28]

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.  Google Scholar

[29]

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.  Google Scholar

show all references

References:
[1]

P. Amorim, B. Telch and L. M. Villada, A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing, Math. Biosci. Eng., 16 (2019), 5114-5145.  doi: 10.3934/mbe.2019257.  Google Scholar

[2]

H. Bréezis and W. A. Strauss, Semilinear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[3]

T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system. Parabolic and Navier-Stokes equations, Part 1, Polish Acad. Sci. Inst. Math., Banach Center Publ., 81 (2008), 105–117. doi: 10.4064/bc81-0-7.  Google Scholar

[4]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[5]

A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, Inc., New York-Montreal, Que.-London, 1969.  Google Scholar

[6]

T. Goudon, B. Nkonga, M. Rascle and M. Ribot, Self-organized populations interacting under pursuit-evasion dynamics, Phys. D, 304/305 (2015), 1-22.  doi: 10.1016/j.physd.2015.03.012.  Google Scholar

[7]

T. Goudon and L. Urrutia, Analysis of kinetic and macroscopic models of pursuit-evasion dynamics, Commun. Math. Sci., 14 (2016), 2253-2286.  doi: 10.4310/CMS.2016.v14.n8.a7.  Google Scholar

[8]

X. He and S. 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.  Google Scholar

[9]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.   Google Scholar

[10]

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

[11]

W. Jäger 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.  Google Scholar

[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.  Google Scholar

[13]

P. Kareiva and G. Odell, Swarms of predators exhibit `preytaxis' if individual predators use area-restricted search, The American Naturalist, 130 (1987), 233-270.  doi: 10.1086/284707.  Google Scholar

[14]

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

[15]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[16]

Y. 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.  Google Scholar

[17]

Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[18]

Y. Tao and M. Winkler, Boundedness and stabilization in a population model with cross-diffusion for one species, Proc. London Math. Soc., 119 (2019), 1598-1632.  doi: 10.1112/plms.12276.  Google Scholar

[19]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[20]

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

[21]

Y. 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.  Google Scholar

[22]

Y. Tao and M. Winkler, Global smooth solvability of a parabolic-elliptic nutrient taxis system in domains of arbitrary dimension, J. Differential Equations, 267 (2019), 388-406.  doi: 10.1016/j.jde.2019.01.014.  Google Scholar

[23]

M. A. Tsyganov, J. Brindley, A. V. Holden and V. N. Biktashev, Quasi-soliton interaction of pursuit-evasion waves in a predator-prey system, Phys. Rev. Lett., 91 (2003), 218102. doi: 10.1103/PhysRevLett.91.218102.  Google Scholar

[24]

M. A. Tsyganov, I. B. Kresteva, A. B. Medvinsky and G. R. Ivanitsky, A novel mode bacterial population wave interaction, Dokl. Akad. Nauk, 333 (1993), 532-536.   Google Scholar

[25]

Y. Tyutyunov, L. Titova and R. Arditi, A minimal model of pursuit-evasion in a predator-prey system, Math. Model. Nat. Phenom., 2 (2007), 122-134.  doi: 10.1051/mmnp:2008028.  Google Scholar

[26]

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.  Google Scholar

[27]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748–767, arXiv: 1112.4156v1. doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[28]

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.  Google Scholar

[29]

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.  Google Scholar

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