GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A CHEMOTAXIS SYSTEM WITH CHEMICALS AND PREY-PREDATOR TERMS

. This paper is concerned with a general asymptotic stabilization of arbitrary global positive bounded solutions for the Lotka Volterra reaction diﬀusion systems, with an additional chemotactic inﬂuence and constant coeﬃcients. We consider the dynamics of a mathematical model involving two biological species, both of which move according to random diﬀusion and are attracted/ repulsed by chemical stimulus produced by the other. The bio- logical species present the ability to orientate their movement towards the concentration of the chemical secreted by the other species. The nonlinear system consists of two parabolic equations with Lotka-Volterra-type kinetic terms coupled with chemotactic cross-diﬀusion, along with two elliptic equa- tions describing the behavior of the chemicals. We prove that the solution to the corresponding Neumann initial boundary value problem is global and bounded for regular and positive initial data. Moreover, for diﬀerent ranges of parameters, we show that any positive and bounded solution converges to a spatially constant homogeneous state.


1.
Introduction. Chemotaxis is a biological process through which living organisms orientate their movement along a chemical concentration gradient; such a process is present in different types of biological phenomena as bacteria aggregation, immune system response or angiogenesis in the embryo formation and in tumor development. Many mathematicians study a chemotaxis system which describes a part of the life cycle of cellular slime molds with chemotaxis. For a broad survey on the progress of various chemotaxis models and a rich selection of references, we refer the reader to the survey papers [5], [16], [18], [45], [47]. Multi-species chemotaxis systems have been proposed and investigated after the pioneering work of Keller Segel [24], by e.g., [27], [28], [34], [38], [49], [50] and previous works can be found concerning the two biological species with chemotactic abilities problem.
The predator-prey system of Lotka-Volterra with intra-specific concurrence describes the problem of uncontrolled growth of prey in the absence of predators, so it adds a term that limits this growth and it has been studied in the last hundred years by different authors, from the pioneering works of Lotka 1925 and Volterra in 1926 3336 MIHAELA NEGREANU ([20], [46]) where the evolution of the species is given in terms of a system of two ODEs. The growth of predators is also limited by attaching to the corresponding equation an additional term. The formulation for the two-dimensional case, one of the important models in biology is as follows (see [20], [46]): 1 (u 1 , u 2 ), t > 0, 2 (u 1 , u 2 ), t > 0, with g (p1) 1 (u 1 , u 2 ) = u 1 (a 01 − a 11 u p1 1 − a 12 u 2 ), where the coefficients a ij (for i = 0, 1, 2 and j = 1, 2) are positive given constants, for some positive constants p i , with i = 1, 2.
If diffusion is considered, the problem becomes a PDEs system of two parabolic equations for the species u 1 and u 2 , e.g., the Fisher-KPP equations 1 (u 1 , u 2 ), x ∈ Ω, t > 0, 2 (u 1 , u 2 ), x ∈ Ω, t > 0. ( To describe the spatial effects in the evolution of ecosystems in ecology, this mathematical model has been used. In (3), the functions g (1) i , i = 1, 2, describe intraand inter-specific interactions of two species in an open, bounded and regular domain with smooth boundary Ω ⊂ R n , n ≥ 1. From a mathematical point of view, the system has been already studied for a large range of interactions g (1) 1 , see, for example Pao [35], where the stability of (3) is obtained for a competitive case with g (1) i = u i (a 0i − a ii u i − a ij u j ) for constant coefficients (a ij ) i,j for i, j = 1, 2 and i = j.
It is natural to consider the case where the parameters describing the amount of resources of the environment present some kind of periodicity in its asymptotic behavior. In [9] is studied system (3) with periodic in time coefficients (a 0i ) i=1,2 , satisfying the Gopalsamy condition (see [13]). Ahmad and Lazer in [1] extended these results to the periodic in time dependence of all the coefficients a ij (for i = 0, 1, 2 and j = 1, 2). If the coefficients have a periodic behavior in time and space, the existence of periodic solutions was proved in [11]. A generalization of these results to almost periodic functions for one and also to several species is obtained in [15] and [14]. The competitive case is also analyzed in [19].
The previous non exhaustive review of the two biological species with diffusive movement is only a small part of the large existing literature in the problem which shows the interest for the problem.
In nature, there exist common examples, where the biological species movement is oriented by chemicals gradients, "chemotaxis" where predator moves towards the prey. One finds different situations depending of the ability of the predator and the prey to orient their movement towards these chemical gradients: the predator is able to orient its movement towards the higher concentration of the chemical secreted by the prey; the prey can move away from the higher concentration of the predator (chemorepulsion).
In contrast to previous works studying Lotka-Volterra models featuring diffusion and taxis towards/away from chemical signals secreted by two competing populations [10], the present paper considers a predator-prey type of interactions, still in connection with taxis towards/away from chemical signals produced by the prey and the predator, respectively. Concretely, the following initial-boundary value problem is addressed: where u 1 and u 2 denote population densities of the prey and the predator, respectively, v 1 and v 2 represent concentrations of the (chemical) signals they respectively produce;  (1) and (2), respectively.
Thereby, χ i are the tactic coefficients: χ 1 < 0 and χ 2 > 0 model the situation where the prey is avoiding the predator by moving away from its signal gradient, while the predator is following the prey by following ∇v 1 .
In this work we analyze the more general case where χ i ∈ R (i = 1, 2), although none of the combinations involving χ 1 > 0 or χ 2 < 0 have a reasonable biological interpretation in the predator-prey context, but are interesting from the mathematical viewpoint. Under some specific assumptions on the tactic sensitivity coefficients and those involved in the reaction terms, we prove global existence of adequately defined solutions to (4) and their asymptotic behavior, the latter under the further assumption p i = m i made for i = 1, 2.
We assume that the bounded initial data (u 0 1 , u 0 2 ) satisfy for some γ > 0 and u 0 i > 0 are positive given constant data. We define Ω T := Ω × (0, T ), for any T < ∞ and use this notation through the paper.
In the stated problem it can be seen that the mechanism that limits the growth of prey and predators is given by the terms −a 11 u p1 1 and −a 22 u 2 p2 , respectively. The term −a 11 u p1 1 describes the intra-prey competition for limited external resources while the term −a 22 u 2 p2 reflects competition among predators for the limited number of prey, and they generalize the most frequent case p i = 1. The non-linear nature of the chemotaxis term has been studied in the literature by different authors, see [16] and references therein. The exponents m i indicate nonlinearities with respect to u i in the tactic sensitivity functions; intuitively, there is a reinforcement of movement in direction of ∇v i where the population u i is greater than a normalized value and presents a weaker movement where it is less than it. These terms with p i ≥ 1 induce a negative feedback that slows growth as populations approach their maximum size and a stronger intra-specific concurrence, (via exponents of the involved density).
Our intention is to show that given a ij > 0, i = 0, 1, 2, j = 1, 2, for the real crossdiffusitivies χ 1 and χ 2 , all solutions of (4) will stabilize towards an equilibrium. In Negreanu and Tello [34], the predator prey system with diffusion and taxis is also studied for a general case where the coefficients a ij = a i,j (x, t) (for i = 0, 1, 2 and j = 1, 2) of (1)-(2) are smooth functions in Ω ∞ and present a periodic asymptotic behavior. Under suitable assumptions of converge in a ij to periodic functions, the authors obtained a periodic asymptotic behavior of the solution. In particular if g (p1) 1 = g (p2) 2 = 0, blow up occurs for a range of initial data as it is shown in [39]. In the last decade many types of systems for two biological species with kinetic interaction have been considered. In [42], the authors proved the stability of homogeneous steady states for one chemical (see also [38], [4], [8]). The evolution of the ecosystem for predators following a chemical secreted by the prey which doesn't present chemotactic ability is described by predator-prey models of indirect taxis (see examples in [43], [26], [25], and [40], among others).
In [21], [22] the authors establish sufficient conditions for the existence of solutions and its asymptotic dynamics for competitive systems of two biological species and a chemical with non-constant coefficients and in [23] it is presented the problem for one species with time and space dependence coefficients and growth term.
In [10], systems of two biological species with chemotactic abilities have been studied, i.e., the competitive system is considered for a general case and the global existence and asymptotic behavior are obtained for positive and bounded initial data under the restrictions 2|χ i |β i + a ij < a jj , i, j = 1, 2, i = j. Coexistence and extinction are studied for different parameters and initial data for the fully parabolic problem in [7] for constant coefficients and the global existence of solutions is obtained for µ i > 7 2 χ 2 i . The mathematical model analyzed in this article is related to the systems modeling the competitive interaction between the species with constant and positive coefficients a ij (considered in [10], [49], [48] and [50]), while (4) presents a predatorprey interaction. The system also extends the predator-prey models with indirect taxis (see [43]) to the case where the prey has the ability to orient its movement following a chemical gradient related to the predator. As much as we know, the predator-prey system with two chemicals has not been considered before in the literature from a mathematical point of view.
The basic technical tools in proving the main results here will be the Alikakos method, the fixed point [37] and the Rectangle Method introduced by Pao in [36], which was already exploited in many papers on related models (see [10], [34] for more details). For the case a 11 = a 22 = 0 in (1) and (2), not included in this paper, the system of ordinary differential equations has periodic solutions, not constant and the techniques used here to study the asymptotic behavior of the solutions to the Lotka-Volterra model of a predator-prey interaction with diffusion and chemotactic terms can not possibly be applied directly for technical reasons.
Throughout the present article we work under the following technically motivated assumptions and We find some conditions on the parameters which guarantee the global existence and boundedness of classical solutions with nonnegative initial functions. The main theorem of the paper describing the global existence of solutions of (4) is stated as follows: Theorem 1.1. Assume that Ω ∈ R n , n ≥ 1 is a bounded domain with regular boundary, coefficients a i,j , α j , β j , p j , m j are positive and χ j ∈ R for i = 0, 1, 2 and j = 1, 2. Suppose further that these parameters satisfy (6)- (8). Then for any nonnegative initial data (u 0 1 , u 0 2 ) as in (5), there exists a unique positive solution of (4) globally in time verifying for i = 1, 2 and any T < ∞.
Furthermore, the solutions u i and v i , i = 1, 2 are uniformly bounded, i.e., there exists a positive constant C > 0 such that The second main objective is to prove an asymptotic stabilization property for the solution obtained in Theorem 1.1 for the case p i = m i = 1, with i = 1, 2.
Theorem 1.2. Under the assumptions of Theorem 1.1, the global solution of (4), where u * i are given by 1.
As in the competitive case [10] with , in the first case (10), the species u 1 persists and its density converges to an homogeneous spatial distribution while species u 2 vanishes as t goes to infinity. If (12) holds, both species coexist and the densities stabilize in some constant steady state given by (11).
The results obtained in Theorem 1.2 are valid for particular case χ 1 = χ 2 = 0 where the solutions have the same asymptotic behavior as the ODE system (71). In that case, the results are already known, see for instance [19] and reference therein.
An outline of this paper is as follows: In Section 3 we consider a system of ordinary predator-prey differential equations associated to the nonlinear system of PDE's (4) and deduce qualitative solution properties that will be used in the proofs of our main results. In Section 2, we study the global existence of classical solutions of (4) with given initial data. In Section 4 we study the asymptotic behavior of positive solutions of (4) with p i = m i = 1 using the relation between the solution (u i , u i ) of the ODE system and the solution (u i , v i ) of the PDE system (4). Under some order relation between initial condition, we obtain that such order is preserved, i.e., we bound the solution of (4) between u i (lower bound) and u i (upper bound). The proof follows the rectangle method used in Pao [36] for reaction diffusion systems, (see also Negreanu and Tello [30] and [33] where the method is applied to parabolicelliptic systems with chemotactic terms). We obtain the asymptotic behavior of the solution to (4), the availability of coexistence states, and an extinction phenomenon in the sense that one of the species dies out asymptotically and the other reaches its carrying capacity as time goes to infinity (Theorem 1.2).
Remark 1.1. The predator-prey system with two chemicals has not been considered before in the literature from a mathematical point of view. For the asymptotic behavior, even though the analysis is done in the case p i = m i = 1 therein provided some interest. One issue when attempting to investigate the asymptotic behavior in the general case, is due to the multitude of parameters and the difficulty encountered when applying the comparison method and the known results on the ODE systems. Possibly, for the general case, a Liapunov type energy method could be applied but that is beyond the scope of this article, it is an open problem that we plan to solve.
2. Global existence of solution. We start with the following important result on the local existence of classical solutions of system (4) with nonnegative initial functions. The result is enclosed in the following lemma.
Moreover, for i = 1, 2, Proof. We take T max verifying lim sup The proof follows standard fixed point theory, see for instance Horstmann [17], Biler [6], Horstmann and Winkler [18], or Negreanu and Tello [33], [34] in order to obtain the local existence of the solutions in L 2 (0, T ; H 1 (Ω))∩L ∞ (Ω T ). Uniqueness of solutions is obtained by contradiction, following standard arguments. The nonnegativity of u i is a consequence of the maximum principle for which also implies the non-negativity of v i .
and the following inequalities hold Proof. We prove the lemma for all χ i ∈ R. We consider in detail the first case where both χ i , i = 1, 2 are positive and the second case, χ 1 < 0 (for chemorepellence of prey by its predator) and χ 2 > 0 (for chemoattraction of predator toward its prey).
For the other possible cases, the proof is similar and we emphasize only the terms where the signs of χ i have an effect on the inequalities. By multiplying the first and the second equations of (4) by u γ−1 1 and u γ−1 2 respectively, and integrating by parts over Ω, we get After multiplying by u mi+γ−1 i the third and fourth equations of (4) for i = 1, 2, then by integrating by parts we obtain We have from (18) and (19) that (16) and (17) can be rewritten as follows and In order to study the terms of the above equations we consider two cases (the differences are in the terms in which appear χ i ): • Case χ i > 0.
By removing the non-positive terms on the right side of (20) and (21), we have and similarly, we obtain For m i ≥ 1 and for any γ > 1 with p i ≥ m i , by Young's inequality applied to for arbitrary positive constants ε i , i = 1, 2.
• Case χ 1 < 0 and χ 2 > 0 For this case, we will only write the equations that change with respect to the previous case; instead of (20) we get the expression In order to obtain bounds for Ω u m1+γ−1 1 v 1 , by applying Young's inequality and the regularity results for elliptic equations with Neumann boundary conditions, for the third equation of (4), see [2] and [12], we have the additional inequality
Proof of Theorem 1.1. Solutions u 1 and u 2 verify Since u i are uniformly bounded in L k (Ω) for any k < ∞ and m i , p i , γ verify the hypotheses of Theorem 1.1, we get (u 1 , u 2 ) ∈ L r (Ω T ) for r < 2 and T < T max . The regularity of u i (for i = 1, 2) is a consequence of the parabolic and elliptic regularity of the equations due to the regularity of the coefficients and the boundedness of u i and v i . Then, we have u i ∈ C 2+γ,1+ γ 2 x,t (Ω T ) see Remark 48.3 (ii) in Quittner-Souplet [37]. The well known standard regularity result gives the global existence of solutions.
3. Associated ODE system; super-and sub-solutions; properties. In this section we obtain the asymptotic behavior of the solutions of an auxiliary system of ordinary differential equations related to the original nonlinear system (4) when p i = m i . To do this, we recall some known lemmas on non-autonomous logistic equation and Lotka-Volterra predator prey systems.
Proof. The standard ODE theory gives the local existence of solutions of the system (45)- (46) in (0, T max ) since the right hand side terms of (45) is a second order polynomial. The regularity is a consequence of the nature of coefficients which are constants. We check that u i > 0, for i = 1, 2. Writing the corresponding equations of u i from system (45) as u i = u i f i (u 1 , u 1 , u 2 , u 2 ), f i being smooth functions, one proves easily that u i = 0 is a solution of the previous equation. Taking into account that the initial data is positive u 0 i > 0, by existence and uniqueness of the solution, we claim that u i (t) > 0 for all t > 0. To obtain u i < u i we proceed by contradiction. Suppose (50) is false, i.e., there exists t 0 > 0 such that for all t ∈ (0, t 0 ). Then, one of the following cases occurs The positivity of the solutions gives that in case (51) which contradicts u 1 (t 0 ) = u 1 (t 0 ) and (47). In the same way (52) contradicts u 2 (t 0 ) = u 2 (t 0 ) and (47). To prove case (53), we introduce the functions ϕ 1 and ϕ 2 defined by We have the following differential equations for ϕ i , for i = 1, 2 , which can be rewritten as We turn our attention to the system (54) with initial data ϕ 1 (t 0 ) = 0 ϕ 2 (t 0 ) = 0, where it can be easily checked that (ϕ 1 , ϕ 2 ) = (0, 0) is a solution of the system and by uniqueness of solution together with the initial data, we conclude that ϕ i = 0, for i = 1, 2 in the interval [0, t 0 ), which contradicts the definition of t 0 and ends the proof.
3.1. Case a 02 a 11 − a 01 a 21 > 0. In this subsection we study the case of exclusion of species under assumption (10) for the coefficients (a i,j ) i,j ; we denote by u * 1 and u * 2 the semi-trivial steady state given by (9), i.e., We state the stabilization property result, the extinction phenomena in the sense that two of the species (u 2 , u 2 ) die out asymptotically and the others (u 1 , u 1 ) reach its carrying.
Thus u 2 (t) is a sub-solution of the ordinary differential equation (64) Taking into account that we work under assumption a 01 a 21 − a 11 a 02 < 0 for the coefficients a i,j with i = 0, 1, 2, j = 1, 2, passing to limit for t → ∞ in (64) we obtain lim sup t→∞ u 2 (t) ≤ lim t→∞ y 2 (t) = 0, and the proof ends.
Now we prove that the super-and sub-solutions u 2 and u 2 are comparable in both directions. The result is enclosed in the following lemma Lemma 3.4. There exists a positive constant M > 0 such that, under the assumptions (6)-(7), for each t > 0, we have Proof. The demonstration is similar to the competitive case (see e.g., [10] and [41] for more details), so we present only the sketch of the proof. Equation (57) can be written as with A 1 := 2|χ 2 |β 2 + a 21 − a 11 and A 2 := 2|χ 1 |β 1 + a 12 − a 22 . For each i = 1, 2, both A i are negative due to hypothesis (6)- (7). Taking = min{−A 1 , −A 2 } this implies that d dt ln Thus by integrating over (0, t), by the positivity of ln(u 1 /u 1 ) because u 1 > u 1 , we get with M = e c0 , for some positive constant c 0 > 0.
A direct consequence of Lemmas 3.3 and 3.4 is that u 2 converges to zero as time tends to infinity as follows Lemma 3.5. Let t > 0 be positive. Under the assumptions of Theorem 3.1, the sub-solution u 2 of (45) satisfies (68) Proof. Taking limit when t → ∞ in (64) and by the nonnegativity of u 2 , the proof ends.
Notice that as a consequence of the previous Lemmas, we have that T max = ∞ which implies the global existence of the solutions.
For now, our goal is to obtain that u 1 and u 1 converge to u * 1 = a01 a11 as time tends to infinity. Lemma 3.6. For every t > 0, under assumptions of Theorem 3.1, we have 1.
Proof. The demostration is similar to the competitive case studied in [10] and we suppress the details (see Lemma 2.6 and Lemma 2.10 in [10] for i = 1).
By mean of Lemmas 3.3 and 3.6, we get the following bound and convergence for the sub-and super-solutions u 1 and u 1 , respectively The proof of Theorem 3.1 is done.

3.2.
Case a 01 a 21 − a 02 a 11 > 0. In this section we prove that under assumption a 01 a 21 − a 02 a 11 > 0, there exists a unique globally stable steady with positive coordinates which corresponds to the coexistence of the prey and predator.
Remark that the restrictions (74) are verified in our case and (75) holds due to (6)- (7). Taking into account Lemma 3.1, we can rewrite system (45) as follows and    u 1 ≤ u 1 (a 01 − a 11 u 1 − a 12 u 2 ), with initial data (46) satisfying (47), for t ∈ (0, ∞). As in the previous subsection, we prove that the two pairs of solutions of the ODE's system (45), i.e., (u 1 , u 1 ) and (u 2 , u 2 ) have the same constant limits u * 1 and u * 2 , respectively and, hence, also any function between them.
Lemma 3.7. The pairs of solutions (u 1 , u 2 ) and (u 1 , u 2 ) are super and sub solutions of the prey-predator system (71) if the following relations between the initial data are satisfied 0 < u 0 Thus, we have the ordering Proof. To obtain (80) we take into account the local stability of (45) and we apply a contradiction argument: assuming that ∃ t 0 ∈ (0, ∞) such that , for t < t 0 and i = 1, 2.
Thus, (u 1 , u 2 ) and (u 1 , u 2 ) are super and sub solutions of the prey-predator system (71) for t ∈ (0, ∞). Moreover, the result follows from the fact that (u * 1 , u * 2 ) is an uniformly asymptotically stable solution for the system of ODEs (71), i.e., To demonstrate Theorem 3.2 it is enough to obtain that u i (t) − u i (t) → 0 for i = 1, 2, when t → ∞. (81) Proof. Operating with the equations of (45) as in Lemma 3.4, we obtain for some ε > 0. If we integrate now, an ODE comparison beside (76) and Lemma 3.7 show According to Lemma 3.1, this entails the inequalities (see [41] and [34] for a more detailed demostration):  End of the proof of Theorem 3.2. Theorem 3.2 is a direct consequence of the properties of the solutions obtained in the previous subsection. Note that Lemmas 3.2, 3.8 and relation (76) fulfill under restrictions (6)- (7), independently of (10) and (12). So we conclude that (58) holds.

4.
Comparison principle and asymptotic behavior of solutions. In this section we investigate the asymptotic behavior of the positive solutions of PDE's (4) and we relate its solutions with the solutions of the ODE's system (45). As a preliminary, we state the two-sided pointwise estimates for the solution of (4). The following important theorem provides sufficient conditions for the boundedness of classical solutions of system (4).
We bound the solution of (4) between a lower bound u i and an upper bound u i in order to obtain the same qualitative behavior than u i and u i , for i = 1, 2. The proof is based on the Rectangle Method used in Pao [36], see also Cruz, Negreanu and Tello [10] and Negreanu and Tello [33], [34] where the method is applied to competitive Parabolic-Parabolic-Elliptic-Elliptic, Parabolic-Elliptic systems or Predator Prey Lotka Volterra with periodic coefficients reaction diffusion systems with chemotactic terms.