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Positive periodic solution for generalized Basener-Ross model
Large time behavior in a predator-prey system with indirect pursuit-evasion interaction
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 |
$ \Omega\subset \mathbb{R}^n $ |
$ \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*} $ |
$ \chi, \xi, \lambda, \mu $ |
$ a $ |
$ b $ |
$ n\le 3 $ |
$ b\lambda<\mu $ |
$ \chi $ |
$ \xi $ |
$ b\lambda>\mu $ |
$ \chi $ |
$ \xi $ |
$ (u, v) $ |
$ u\not\equiv 0 $ |
$ (\lambda, 0) $ |
$ t\to \infty $ |
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. |
[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. |
[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. |
[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. |
[5] |
A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, Inc., New York-Montreal, Que.-London, 1969. |
[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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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 `preytaxis' if individual predators use area-restricted search, The American Naturalist, 130 (1987), 233-270.
doi: 10.1086/284707. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, Inc., New York-Montreal, Que.-London, 1969. |
[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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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 `preytaxis' if individual predators use area-restricted search, The American Naturalist, 130 (1987), 233-270.
doi: 10.1086/284707. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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