# American Institute of Mathematical Sciences

August  2022, 27(8): 4531-4549. doi: 10.3934/dcdsb.2021240

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

 School of Mathematics Renmin University of China Beijing, 100872, China

* Corresponding author: Yuanyuan Ke

Received  June 2021 Revised  August 2021 Published  August 2022 Early access  October 2021

This work considers a pursuit-evasion model
 $$$\left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &w_t = \Delta w-w+v,\\ &z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1)$$$
with positive parameters
 $\chi$
,
 $\xi$
,
 $\mu$
,
 $\lambda$
,
 $a$
and
 $b$
in a bounded domain
 $\Omega\subset\mathbb{R}^N$
(
 $N$
is the dimension of the space) with smooth boundary. We prove that if
 $a<2$
and
 $\frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}>\max\{\chi,\xi\}$
, (1) possesses a global bounded classical solution with a positive constant
 $C_{\frac{N}{2}+1}$
corresponding to the maximal Sobolev regularity. Moreover, it is shown that if
 $b\mu<\lambda$
, the solution (
 $u,v,w,z$
) converges to a spatially homogeneous coexistence state with respect to the norm in
 $L^\infty(\Omega)$
in the large time limit under some exact smallness conditions on
 $\chi$
and
 $\xi$
. If
 $b\mu>\lambda$
, the solution converges to (
 $\mu,0,0,\mu$
) with respect to the norm in
 $L^\infty(\Omega)$
as
 $t\rightarrow \infty$
under some smallness assumption on
 $\chi$
with arbitrary
 $\xi$
.
Citation: Dayong Qi, Yuanyuan Ke. Large time behavior in a predator-prey system with pursuit-evasion interaction. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4531-4549. doi: 10.3934/dcdsb.2021240
##### 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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##### 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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