Global existence and stability in a two-species chemotaxis system

This paper deals with the following two-species chemotaxis system \begin{document}$\left\{ \begin{array}{*{35}{l}} \ \ {{u}_{t}}=\Delta u-{{\chi }_{1}}\nabla \cdot (u\nabla v)+{{\mu }_{1}}u(1-u-{{a}_{1}}w), & x\in \Omega ,t>0, & \\ \ \ {{v}_{t}}=\Delta v-v+h(w), & x\in \Omega ,t>0, & \\ \ \ {{w}_{t}}=\Delta w-{{\chi }_{2}}\nabla \cdot (w\nabla z)+{{\mu }_{2}}w(1-w-{{a}_{2}}u), & x\in \Omega ,t>0, & \\ \ \ {{z}_{t}}=\Delta z-z+h(u),& x\in \Omega ,t>0, & \\\end{array} \right.$ \end{document} under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$Ω\subset\mathbb{R}^{n}$\end{document} with smooth boundary. The parameters in the system are positive and the signal production function h is a prescribed C1-regular function. The main objectives of this paper are two-fold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium \begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document} may be globally attractive in the weak competition case (i.e., \begin{document}$0 ), while the constant stationary solution (0, h(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e., \begin{document}$a_{1}>1>a_{2}>0$\end{document} ). In the fully strong competition case (i.e. \begin{document}$a_{1}, a_{2}>1$\end{document} ), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous \begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document} . The matter which species ultimately wins out depends crucially on the starting advantage each species has.


1.
Introduction. Chemotaxis refers to the movement of cells in response to a chemical signal, a process by which cells change their movement state in the presence of chemical concentration gradient, approaching the chemical favorable environment and avoiding the adverse ones. This phenomenon plays a crucial role in morphogenesis and self-organization of various biological coherent and enthralled structures. Keller and Segel [11] first introduced the following mathematical model to describe the aggregation phase of Dictyostelium discoideum.
Since then, many authors have studied several different models in mathematical biology. And a large body of works have been devoted to determining when blowup occurs or whether globally existing solutions exist. Hillen and Painter [8] studied the formulation from a biological point of view. In contrast with the patterning properties, some key results on the analytical properties are summarised and the solution forms are classified. Horstmann [9] summarized various aspects and results for some general formulations of Keller-Segel models. Naturally, more effects, such as population growth and competition among species, incorporate life forms into complex ecosystems. As such, chemotaxis is not the only effect on the behavior of the cell populations. Often the growth of the population must be taken into account. A typical option to achieve this goal is the addition of logistic growth κu − µu 2 to the first equation (see e.g. [10], [12], [13], [15], [16], [25], [30]). That is, In fact, the presence of such logistic terms, particularly the quadratic term −µu 2 , is sufficient to suppress any blow-up in many relevant situations. In the case where ε = 0, Tello and Winkler [30] proved the existence of global bounded classical solutions under the assumption that either the space dimension does not exceed two, or that the logistic damping effect is strong enough, and also established some multiplicity and bifurcation results for small logistic terms. In presence of certain sub-quadratic degradation terms, even finite-time blow-up is possible, as detected in [34] and [36]. In the fully parabolic case ε > 0, Osaki [24] indicated that in two-dimensional case where x ∈ R 2 , any blow-up phenomenon can be completely suppressed for µ > 0. This result is extended by Winkler [33] to arbitrary space dimensions, who proved the existence and uniqueness of global, smooth, bounded solutions to (2) with large enough µ. Lankeit [13] proved the existence of global weak solutions and showed that in the three-dimensional setting, there exist κ 0 > 0 and T > 0 such that the weak solutions become classical solutions when κ < κ 0 and t > T . Lankeit [13] also proved the attractivity of the trivial steady state when κ ≤ 0 and the existence of an absorbing set when κ > 0 is sufficiently small. However, if the effect of the logistic term is not strong enough, then finite-time blow-up is possible. In the one dimensional case, Winkler [35] proved that if 0 < µ < 1, then there is some criterion on the initial data that ensures the existence of some time up to which any threshold of the population density will be surpassed. Lankeit [14] obtained a criterion guaranteeing some kind of structure formation, which is an extension of the result of Winkler [35] to the higher dimensional case. In [25], a positive Lyapunov exponent together with a rich bifurcation structure was indicated by numerical experiments.
In view of various biological processes, researchers revised the second equation in (2) to εv t = ∆v −v +h(u) and obtained some researches for different representations of h. In the case where h(u) = u(1 + u) β−1 , Nakaguchi and Osaki [21,22] showed the global existence of solutions in L p space under certain relations between the degradation and production orders when 0 < β ≤ 2 and the source term given by µu(1 − u α−1 ) satisfies α > 1. While for a more general function h, Chaplain and Tello [3] studied the asymptotic behavior of solutions under the assumption that 2χ|h (x)| < µ.
However, the situation of single population and single chemoattractant is rare in the real biological systems. So the discussion for a number of populations under the action of variety chemical substances has biological significance. Considering the competition of two species for resources or space (see, e,g. [6,7,37,39]), in this paper we will consider the following boundary value problem with chemotaxis and Lotka-Volterra competition pattern: in a bounded domain Ω ⊂ R n (n ≥ 1) with smooth boundary ∂Ω, where χ 1 , χ 2 , µ 1 , µ 2 , a 1 and a 2 are positive parameters, u = u(x, t) and w = w(x, t) denote the densities of two cell population, whereas v = v(x, t) and z = z(x, t) stand for the concentration of the chemicals produced by w(x, t) and u(x, t), respectively. In model (3), h(s) is a prescribed function on [0, ∞), which represents the production of the chemical substance by the cells.
When h(s) = s, the questions of global existence and large time behavior has been addressed in [2] for weakly competitive species case and for the partially strong competition setting. But there is no relevant work for a general function h yet. Throughout this paper, we always assume that the signal production function h is C 1 -regular and satisfies where L h = sup x≥0 {h (x)} > 0 is a constant. Moreover, we will consider (3) with the initial condition We shall show that the solution of (3) with the initial condition (5) is global and bounded provided that This means that the smallness condition on h suppresses blow-up effects, which is in keeping with the results in one-species Keller-Segel chemotaxis systems; see, for example, [21,22,23]. Furthermore, the long time behavior of the bounded solutions is discussed in three cases. In the weak competition case (that is, a 1 , a 2 < 1), if then any global bounded solution stabilizes to the unique positive spatially homogeneous equilibrium (u * , v * , w * , z * ), where Moreover, there is a positive constant L(µ 1 , µ 2 , χ 1 , χ 2 ) depending on (µ 1 , µ 2 , χ 1 , χ 2 ) such that (u * , v * , w * , z * ) is globally asymptotical stable when In the case where a 2 < 1 < a 1 , if then (0, h(1), 1, 0) is globally asymptotical stable and every global solution converges towards the constant stationary solution (0, h(1), 1, 0) as t → ∞. In the case where a 1 , a 2 > 1, however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous (u * , v * , w * , z * ). The matter which species ultimately wins out depends crucially on the starting advantage each species has. The organization of the remaining part of the paper is as follows. In section 2 we investigate the global existence and boundedness of solutions to (3)-(5). Moreover, we obtain some a priori estimates of u, v, w and z. Section 3 is devoted to the large time behavior in different competition cases. By constructing a Lyapunov functional we obtain some weak convergence information of the solutions, which helps us to prove the convergence assertion. By a routine linearized stability analysis, we investigate the local stability of the two steady states (0, h(1), 1, 0) and (u * , v * , w * , z * ), which, together with global attractivity, can yield the global asymptotical stability.
2. Global existence. We first give the following result on the global existence and boundedness. We first refer to [27] for the following local existence and extensibility result for classical solutions of (3).
Theorem 2.2. Suppose that the parameters µ 1 , µ 2 are positive and χ 1 , χ 2 , a 1 , a 2 are nonnegative. Assume that and h satisfies (4) and (6). Then for any given nonnegative initial values In order to prove this theorem, we shall need the following auxiliary lemma to derive some time-independent estimates for v and z. We refer the details of the proof to [29]. Then for all t ∈ (0, T ).
In order to show the global existence of the solution, it is sufficient to derive the boundedness of u, w, ∇v and ∇z. The following lemma is some basic properties of u and w.  (3): and t+τ t Ω Integrating the first equation of (3) over Ω implies that Since u, w are nonnegative, by the Cauchy-Schwarz inequality we have . By applying the similar steps to the third equation of (3) we obtain the estimates of w and w 2 . (16) and t+τ t Ω where τ = min{1, 1 2 T max } and K 1 , for some η ∈ [0, s]. Then by integrating the second equation of (3) and recalling (12), we have Obviously, we have By multiplying the second equation of (3) with −∆v and integrating by parts we see that 1 2 By using Young's inequality and recalling (18) we have According to the Gagliardo-Nirenberg inequality combined with Theorem 3.4 in [26], there exists K such that for all t ∈ (0, T max ), . From the time integration and the fact that τ ≤ 1, we obtain Similarly, we can obtain the conclusions about z, and so omit the details here. and Proof. Testing the first equation of (3) against u and integrating by parts we have for all t ∈ (0, T max ). Thus we obtain (20) by a direct application of Young's inequality to the formula above. By applying similar steps to the third equation of (3), we obtain (21). Differentiating the second equation of (3) with respect to space, invoking the pointwise identity ∇v · ∇∆v = 1 2 ∆|∇v| 2 − |D 2 v| 2 and integrating by parts we obtain for all t ∈ (0, T max ). By (18) and Young's inequality we have for all t ∈ (0, T max ). Moreover, we have ∂|∇v| 2 ∂ν ≤ 0 for x ∈ ∂Ω and t ∈ (0, T max ) because of the convexity of Ω along with ∂v ∂ν | ∂Ω = 0 ( [17]). Thus we have Similarly we obtain the estimate about z.
Proof. By a straightforward computation and integration by parts we have The last inequality is obtained by Young's inequality. Similarly, we have By a simple linear combination of (20)-(23), (27) and (28), we have This implies that for t ∈ (0, T max ), satisfies y (t) + y(t) By Young's inequality and (6) we have Thus we have In view of (13) and Lemma 2.3 we have y(t) ≤ c 1 , which proves (25) and (26).
Combining with the previous results, now we can prove Theorem 2.2.
3. Global attractivity. As a preparation, we first establish higher regularity of the solution.
The proof of Lemma 3.1 is similar to [1] and hence is omitted. To obtain some weak convergence information, we need to construct a Lyapunov functional and also need to establish the following differential inequality as preparation. and Proof. According to the assumption that u 0 ≡ 0 and the strong maximum principle, we have u > 0 inΩ × (0, ∞). By testing the first equation of (3) against 1 u and intergrating by parts, we obtain for all t > 0. A direct application of Young's inequality to the above formula implies (36). We obtain (37) by applying similar steps to w.

3.1.
Case a 1 < 1 and a 2 < 1. We shall see that (u * , v * , w * , z * ) is attractive to all nontrivial nonnegative solutions of (3)-(5) if the parameter coefficients L h , χ 1 , and χ 2 are large enough and satisfy (7). For this purpose, we start with some preparations. and for all t > 0.
Proof. By testing the second equation of (3) against (v − v * ), we obtain for all t > 0. According to the assumption of h in (4) and Cauchy-Schwarz inequality we have for all t > 0, which yields (38). By applying similar steps to z we obtain (39).
Next we construct a Lyapunov functional to obtain the weak convergence.
Proof. According to (7) and (8), we see that there exist δ 1 , δ 2 > 0 such that δ 1 ∈ I 1 and δ 2 ∈ I 2 , where With these fixed δ 1 and δ 2 , we shall study the property of the function E 1 (t) defined in (41). Set Obviously, H(s) ≥ H(u * ) = 0 for all s > 0. Thus we obtain the nonnegativity of E 1 (t). In view of the differential inequalities established in Lemmas 3.2 and 3.3 we for all t > 0. In view of the decisions of δ 1 , δ 2 and evident choice for ε, we obtain (44). Now, we can state the following result on the attractivity of (u * , v * , w * , z * ).
In what follows, we shall find some sufficient conditions ensuring the global asymptotical stability of the unique spatially homogeneous steady state (u * , v * , w * , z * ). In view of the global attractivity discussed above, we only need to analyze its local stability. By applying some abstract stability results based on analytic semigroup theories (see [5,19]), to get the stability of the spatially homogeneous steady state (u * , v * , w * , z * ), it suffices to prove that the steady state is spectrally stable, i.e., the linearized operator has only eigenvalues with nonnegative real parts (also see [4,31,38]). Linearizing (3) at (u * , v * , w * , z * ) leads to the following system The corresponding characteristic equation is where and 0 = λ 1 < λ 2 < · · · are the eigenvalues of the operator −∆ on Ω under the homogeneous Neumann boundary condition.
Since a 1 a 2 < 1, thus all eigenvalues of A n have negative real parts. If one of h (u * ), h (w * ), χ 1 , and χ 2 is not equal to zero, however, it's really a challenge to solve (49) for ρ and even to analyze the sign of the real parts of ρ. But according to the continuous dependence of eigenvalues with respect to h (u * ) and h (w * ), there exists a positive constant L(µ 1 , µ 2 , χ 1 , χ 2 ) such that all eigenvalues of A n (n ∈ N ∪ {0}) have negative real parts when L h < L(µ 1 , µ 2 , χ 1 , χ 2 ). That is, (u * , v * , w * , z * ) is locally asymptotically stable. Combining with Theorem 3.5, we obtain the following result.

3.2.
Case a 2 < 1 < a 1 . In this case, we shall show that every global solution converges towards the constant stationary solution (0, h(1), 1, 0) when µ 2 is large enough. For this purpose, we introduce the following lemmas. and The proof of Lemma 3.7 is similar to that of Lemma 3.3 and so is omitted here.
Proof. Obviously, the inequity of (9) is a special case of (57) when a 1 = 1 and η = 1+a2 2 . Then we complete the proof by using a similar argument to the proof of Theorem 3.5.

3.3.
Case a 1 , a 2 > 1. According to (50) and (65), we see that the spatially homogeneous equilibrium (u * , v * , w * , z * ) is unstable. The constant stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) are locally asymptotically stable and each of the two steady-states has a domain of attraction. The matter which species ultimately wins out depends crucially on the starting advantage each species has. evident in case (iii). In case (ii), we obtain the local stability of the two semi-trivial steady states (0, h(1), 1, 0) and (1, 0, 0, h(1)), and the instability of (u * , v * , w * , z * ) by a routine linearization analysis. Compared with the results in [20], it's shown that for restrictive chemotaxis coefficients, the stabilization of solutions depends on the dimensionless parameter groupings a 1 and a 2 as well as the initial conditions.