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Large time behavior in a predator-prey system with pursuit-evasion interaction
School of Mathematics Renmin University of China Beijing, 100872, China |
$\begin{equation} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &w_t = \Delta w-w+v,\\ &z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1) \end{equation}$ |
$ \chi $ |
$ \xi $ |
$ \mu $ |
$ \lambda $ |
$ a $ |
$ b $ |
$ \Omega\subset\mathbb{R}^N $ |
$ N $ |
$ a<2 $ |
$ \frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}>\max\{\chi,\xi\} $ |
$ C_{\frac{N}{2}+1} $ |
$ b\mu<\lambda $ |
$ u,v,w,z $ |
$ L^\infty(\Omega) $ |
$ \chi $ |
$ \xi $ |
$ b\mu>\lambda $ |
$ \mu,0,0,\mu $ |
$ L^\infty(\Omega) $ |
$ t\rightarrow \infty $ |
$ \chi $ |
$ \xi $ |
References:
[1] |
P. Amorim and B. Telch, A chemotaxis predator-prey model with indirect pursuit-evasion dynamics and parabolic signal, J. Math. Anal. Appl., 500 (2021), 27pp.
doi: 10.1016/j.jmaa.2021.125128. |
[2] |
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. |
[3] |
X. Cao,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.
doi: 10.1016/j.jmaa.2013.10.061. |
[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] |
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. |
[6] |
M. A. Herrero and J. J. L. Velázques,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.
|
[7] |
M. Hieber and J. Prüss,
Heat kernels and maximal Lp-Lq estimate for parabolic evolution equations, Comm. Par. Differ. Equns., 22 (1997), 1647-1669.
doi: 10.1080/03605309708821314. |
[8] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equns., 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[9] |
Y. Jia and P. Xue,
Effects of the self- and cross-diffusion on positive steady states for a generalized predator-prey system, Nonlinear Anal. Real World Appl., 32 (2016), 229-241.
doi: 10.1016/j.nonrwa.2016.04.012. |
[10] |
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. |
[11] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Bio., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[12] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. "Nauka", Moscow 1967,736 pp. |
[13] |
D. Li,
Global stability in a multi-dimensional predator-prey system with prey-taxis, Discrete Contin. Dyn. Syst., 41 (2021), 1681-1705.
doi: 10.3934/dcds.2020337. |
[14] |
G. Li, Y. Tao and M. Winkler,
Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4383-4396.
doi: 10.3934/dcdsb.2020102. |
[15] |
Y. Lou and W. Ni,
Diffusion, self-diffusion and cross-diffusion, J. Differ. Equns., 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[16] |
Q. Meng and L. Yang,
Steady state in a cross-diffusion predator-prey model with the Beddington-DeAngelis functional response, Nonlinear Anal. Real World Appl., 45 (2019), 401-413.
doi: 10.1016/j.nonrwa.2018.07.012. |
[17] |
Y. Tao and Z. 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 in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equns., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[19] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equns., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[20] |
Y. Tao and M. Winkler,
Global smooth solvability of a parabolic-elliptic nutrient taxis system in domains of arbitrary dimension, J. Differ. Equns., 267 (2019), 388-406.
doi: 10.1016/j.jde.2019.01.014. |
[21] |
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. |
[22] |
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.
|
[23] |
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. |
[24] |
J. Wang, S. Wu and J. 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. |
[25] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Par. Differ. Equns., 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[26] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures. Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[27] |
M. Winkler,
Asymptotic homogenization in a three-dimensional nutrient taxis-system involving food-supported proliferation, J. Differ. Equns., 263 (2017), 4826-4869.
doi: 10.1016/j.jde.2017.06.002. |
[28] |
P. Xue, Y. Jia, C. Ren and X. Li, Non-constant positive solutions of a general Gause-type predator-prey system with self- and cross-diffusions, Math. Model. Nat. Phenom., 16 (2021), 15pp.
doi: 10.1051/mmnp/2021017. |
[29] |
J. Zheng, Y. Li, G. Bao and X. Zou,
A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 462 (2018), 1-25.
doi: 10.1016/j.jmaa.2018.01.064. |
show all references
References:
[1] |
P. Amorim and B. Telch, A chemotaxis predator-prey model with indirect pursuit-evasion dynamics and parabolic signal, J. Math. Anal. Appl., 500 (2021), 27pp.
doi: 10.1016/j.jmaa.2021.125128. |
[2] |
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. |
[3] |
X. Cao,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.
doi: 10.1016/j.jmaa.2013.10.061. |
[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] |
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. |
[6] |
M. A. Herrero and J. J. L. Velázques,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.
|
[7] |
M. Hieber and J. Prüss,
Heat kernels and maximal Lp-Lq estimate for parabolic evolution equations, Comm. Par. Differ. Equns., 22 (1997), 1647-1669.
doi: 10.1080/03605309708821314. |
[8] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equns., 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[9] |
Y. Jia and P. Xue,
Effects of the self- and cross-diffusion on positive steady states for a generalized predator-prey system, Nonlinear Anal. Real World Appl., 32 (2016), 229-241.
doi: 10.1016/j.nonrwa.2016.04.012. |
[10] |
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. |
[11] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Bio., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[12] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. "Nauka", Moscow 1967,736 pp. |
[13] |
D. Li,
Global stability in a multi-dimensional predator-prey system with prey-taxis, Discrete Contin. Dyn. Syst., 41 (2021), 1681-1705.
doi: 10.3934/dcds.2020337. |
[14] |
G. Li, Y. Tao and M. Winkler,
Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4383-4396.
doi: 10.3934/dcdsb.2020102. |
[15] |
Y. Lou and W. Ni,
Diffusion, self-diffusion and cross-diffusion, J. Differ. Equns., 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[16] |
Q. Meng and L. Yang,
Steady state in a cross-diffusion predator-prey model with the Beddington-DeAngelis functional response, Nonlinear Anal. Real World Appl., 45 (2019), 401-413.
doi: 10.1016/j.nonrwa.2018.07.012. |
[17] |
Y. Tao and Z. 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 in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equns., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[19] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equns., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[20] |
Y. Tao and M. Winkler,
Global smooth solvability of a parabolic-elliptic nutrient taxis system in domains of arbitrary dimension, J. Differ. Equns., 267 (2019), 388-406.
doi: 10.1016/j.jde.2019.01.014. |
[21] |
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. |
[22] |
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.
|
[23] |
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. |
[24] |
J. Wang, S. Wu and J. 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. |
[25] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Par. Differ. Equns., 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[26] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures. Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[27] |
M. Winkler,
Asymptotic homogenization in a three-dimensional nutrient taxis-system involving food-supported proliferation, J. Differ. Equns., 263 (2017), 4826-4869.
doi: 10.1016/j.jde.2017.06.002. |
[28] |
P. Xue, Y. Jia, C. Ren and X. Li, Non-constant positive solutions of a general Gause-type predator-prey system with self- and cross-diffusions, Math. Model. Nat. Phenom., 16 (2021), 15pp.
doi: 10.1051/mmnp/2021017. |
[29] |
J. Zheng, Y. Li, G. Bao and X. Zou,
A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 462 (2018), 1-25.
doi: 10.1016/j.jmaa.2018.01.064. |
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