This work explores the predator-prey chemotaxis system with two chemicals
\begin{eqnarray*}
\left\{
\begin{array}{llll}
u_t=\Delta u+\chi\nabla\cdot(u\nabla v)+\mu_1u(1-u-a_1w),\quad &x\in \Omega, t>0,\\
v_t=\Delta v-\alpha_1 v+\beta_1w,\quad &x\in \Omega, t>0,\\
w_t=\Delta w-\xi\nabla \cdot(w\nabla z)+\mu_2 w(1+a_2u-w),\quad &x\in\Omega, t>0,\\
z_t=\Delta z-\alpha_2 z+\beta_2u,\quad &x\in \Omega, t>0,\\
\end{array}
\right.
\end{eqnarray*}
in an arbitrary smooth bounded domain $\Omega\subset \mathbb{R}^n$
under homogeneous Neumann boundary conditions. The parameters in the
system are positive.
We first prove that if $n\leq3$, the corresponding initial-boundary
value problem admits a unique global bounded classical solution,
under the assumption that $\chi, \xi$, $\mu_i, a_i, \alpha_i$ and
$\beta_i(i=1,2)$ satisfy some suitable conditions. Subsequently, we
also analyse the asymptotic behavior of solutions to the above
system and show that
$\bullet$ when $a_1<1$ and both $\frac{\mu_1}{\chi^2}$ and
$\frac{\mu_2}{\xi^2}$ are sufficiently large, the
global solution $(u, v, w, z)$ of this system exponentially converges to $(\frac{1-a_1}{1+a_1a_2}, \frac{\beta_1(1+a_2)}{\alpha_1(1+a_1a_2)}, \frac{1+a_2}{1+a_1a_2}, \frac{\beta_2(1-a_1)}{\alpha_2(1+a_1a_2)})$ as $t\rightarrow \infty$;
$\bullet$ when $a_1>1$ and $\frac{\mu_2}{\xi^2}$ is sufficiently
large, the global bounded classical solution $(u, v, w, z)$ of this
system exponentially converges to $(0, \frac{\alpha_1}{\beta_1}, 1,
0)$ as $t\rightarrow \infty$;
$\bullet$ when $a_1=1$ and $\frac{\mu_2}{\xi^2}$ is sufficiently
large, the global bounded classical solution $(u, v, w, z)$ of this
system polynomially converges to $(0, \frac{\alpha_1}{\beta_1}, 1,
0)$ as $t\rightarrow \infty$.
[1] Amann H. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems//Haroske D, Runst T, Schmeisser H -J. Function Spaces. Differential Operators and Nonlinear Analysis, 1993:9-126
[2] Amorim P, Telch B, Villada L M. A reaction-diffusion predator-prey model with pursuit, evasion and nonlocal sensing. Mathematical Biosciences and Engineering, 2019, 16(5):5114-5145
[3] Bai X, Winkler M. Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ Math J, 2016, 65(2):553-583
[4] Barbălat I. Systèmes d'equations diff'erentielles d'oscillations non linéaires. Rev Roumaine Math Pures Appl, 1959, 4:267-270
[5] Black T. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete Contin Dyn Syst Ser B, 2017, 22(4):1253-1272
[6] Black T, Lankeit J, Mizukami M. On the weakly competitive case in a two-species chemotaxis model. IMA J Appl Math, 2016, 81(5):860-876
[7] Cieślak T, Laurencot P, Morales-Rodrigo C. Global existence and convergence to steady states in a chemorepulsion system//Parabolic and Navier-Stokes Equations. Banach Center Publ Polish Acad Sci Inst Math, 2008, 81:105-117
[8] Miao L, Yang H, Fu S. Global boundedness in a two-species predator-prey chemotaxis model. Appl Math Lett, 2021, 111:106639
[9] Hillen T, Painter K J. A user's guide to PDE models for chemotaxis. J Math Biol, 2009, 58(1/2):183-217
[10] Horstmann D, Winkler M. Boundedness vs. blow-up in a chemotaxis system. J Differential Equations, 2005, 215(1):52-107
[11] Fu S, Miao L. Global existence and asymptotic stability in a predator-prey chemotaxis model. Nonlinear Anal RWA, 2020, 54:103079
[12] Goudon T, Urrutia L. Analysis of kinetic and macroscopic models of pursuit-evasion dynamics. Comm Math Sci, 2016, 14(8):2253-2286
[13] Keller E F, Segel L A. Initiation of slime mold aggregation viewed as an instability. J Theor Biol, 1970, 26(3):399-415
[14] Pan X, Wang L, Zhang J, Wang J. Boundedness in a three-dimensional two-species chemotaxis system with two chemicals. Z Angew Math Phys, 2020, 71:26
[15] Lankeit J. Chemotaxis can prevent thresholds on population density. Discrete Contin Dyn Syst Ser B, 2015, 20(5):1499-1527
[16] Lin K, Mu C. Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete Contin Dyn Syst Ser B, 2017, 22(6):2233-2260
[17] Lin K, Mu C, Wang L. Boundedness in a two-species chemotaxis system. Math Methods Appl Sci, 2015, 38(18):5085-5096
[18] Li D, Mu C, Lin K, Wang L. Convergence rate estimates of a two-species chemotaxis system with two indirect signal production and logistic source in three dimensions. Z Angew Math Phys, 2017, 68(3)
[19] Mizukami M. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signaldependent sensitivity. Discrete Contin Dyn Syst Ser B, 2017, 22:2301-2319
[20] Mizukami M. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete Contin Dyn Syst Ser S, 2020, 13(2):269-278
[21] Mizukami M, Yokota T. Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion. J Differ Equ, 2016, 261(5):2650-2669
[22] Mizukami M. Boundedness and stabilization in a two-species chemotaxis-competiton system of parabolic-parabolicelliptic type. Math Methods Appl Sci, 2018, 41:234-249
[23] Negreanu M, Tello J I. Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant. J Differ Equ, 2015, 258(5):1592-1617
[24] Negreanu M, Tello J I. Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals. J Math Anal Appl, 2019, 474(2):1116-1131
[25] Stinner C, Tello J I, Winkler M. Competitive exclusion in a two-species chemotaxis model. J Math Biol, 2014, 68(7):1607-1626
[26] Tao Y, Wang M. Global solution for a chemotactic-haptotactic model of cancer invasion. Nonlinearity, 2008, 21(10):2221-2238
[27] Tao Y, Wang Z A. Competing effects of attraction vs. repulsion in chemotaxis. Math Models Methods Appl Sci, 2013, 23(1):1-36
[28] Tao Y, Winkler M. A chemotactic-haptotactic model:the role of porous medium diffusion and logistic source. SIAM J Math Anal, 2011, 43:685-704
[29] Tao Y. Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete Contin Dyn Syst Ser B, 2013, 18(10):2705-2722
[30] Tao Y, Winkler M. Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals. Discrete Contin Dyn Syst Ser B, 2015, 20(9):3165-3183
[31] Tello J I, Winkler M. A chemotaxis system with logistic source. Commun Partial Diff Eqns, 2007, 32:849-877
[32] Tello J I, Winkler M. Stabilization in a two-species chemotaxis system with a logistic source. Nonlinearity, 2012, 25(5):1413-1425
[33] Tu X, Mu C, Zheng P, Lin K. Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete Contin Dyn Syst, 2018, 38(7):3617-3636
[34] Wang L, Zhang J, Mu C, Hu X. Boundedness and atabilization in a two-species chemotaxis system with two chemicals. Discrete Contin Dyn Syst Ser B, 2020, 25(1):191-221
[35] Winkler M. Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system. J Math Pures Appl, 2013, 100(5):748-767
[36] Winkler M. Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Commun Partial Diff Eqns, 2010, 35(8):1516-1537
[37] Winkler M. How far can chemotactic cross-diffusion enforce exceeding carrying capacities?. J Nonlinear Sci, 2014, 24(5):809-855
[38] Winkler M. Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening. J Differ Equ, 2014, 257(4):1056-1077
[39] Winkler M. Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems. Discrete Contin Dyn Syst Ser B, 2017, 22(7):2777-2793
[40] Wu S, Shi J, Wu B. Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis. J Differ Equ, 2016, 260(7):5847-5874
[41] Xie L, Wang Y. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete Contin Dyn Syst Ser B, 2017, 22(7):2717-2729
[42] Xie L, Wang Y. On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics. J Math Anal Appl, 2019, 471(1/2):584-598
[43] Yu H, Wang W, Zheng S. Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals. Nonlinearity, 2018, 31(2):502-514
[44] Zhang Q, Li Y. Global boundedness of solutions to a two-species chemotaxis system. Z Angew Math Phys, 2015, 66(1):83-93
[45] Zhang Q. Boundedness in chemotaxis systems with rotational flux terms. Math Nachr, 2016, 289(17/18):2323-2334
[46] Zhang Q, Liu X, Yang X. Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals. J Math Phys, 2017, 58(11):111504
[47] Zheng P, Mu C. Global boundedness in a two-competing-species chemotaxis system with two chemicals. Acta Appl Math, 2017, 148(1):157-177
[48] Zheng P, Mu C, Mi Y. Global stability in a two-competing-species chemotaxis system with two chemicals. Differ Integral Equ, 2018, 31(7/8):547-558
