Comptes Rendus Mathématique

. In this Note we study a nonlinear system of reaction-di ﬀ usion di ﬀ erential equations consisting of an ordinary di ﬀ erential equation coupled to a fully parabolic chemotaxis system. This system constitutes a mathematicalmodelfortheevolutionofaprey-predatorbiologicalpopulationwithchemotaxisanddormant predators. Under suitable assumptions we prove the global in time existence and boundedness of classical solutions of this system in any space dimension. Résumé. Dans cette Note, nous étudions un système non linéaire d’équations di ﬀ érentielles partielles de type réaction-di ﬀ usion décrivant l’évolution d’un système biologique proie-prédateur avec chimiotaxie et prédateurs dormants. Nous considérons une équation ordinaire couplée à un système parabolique de chimiotaxie.Souscertaineshypothèsesappropriées,nousobtenonsl’existenceglobaleentempsdesolutions classiques du système considéré dans n’importe quelle dimension spatiale.


Introduction
We consider the predator-prey model given by the system with non-negative initial data u( Here Ω is an open bounded domain with smooth boundary ∂Ω of measure |Ω| = 1. Model (1) extends those proposed in [4] and [8]. It is a prey-predator reaction-diffusion system with chemotaxis effect to include predator dormancy. The functions appearing in (1) are smooth and have the following biological meaning.
• u, v and w are the densities of predators, preys and predators with dormant state (resting eggs), respectively. • d ≥ 0 is the diffusion coefficient in the predators term. It is assumed to be small relative to 1, • χ := χ(w) is the function responsible of chemotaxis, • φ(v, x) represents the production of preys, • m u (u, x) is the mortality rate of predators, • f (u, v, x) is related to the predators consumption of preys, • represents the fraction of preys biomass density that can be turned into predator biomass density, • ξ(v) and η(v) are bounded functions related to the distribution of reproduction energy of predators between active and dormant states and verify ξ(v) + η(v) = 1, • a(v) denotes the average dormancy period, • m w (w) represents the mortality rate of dormant predators.
In (1) we assume that f , ξ, m u and m w are increasing functions in (u, v), v, u and w, respectively, while a is non-decreasing in v. Besides, m w is supposed to be small relative to m u .
Based on biological reasons (in accordance with [4], [5], [8] and [9]), we consider the hypotheses: The following theorem states our main result on the global existence and boundedness for solutions of system (1).
The following lemma provides a local in time existence result of a classical solution to (1).
The proof of this fact follows a well-known scheme developed by Amann in [2] and [3] to obtain a maximal weak solution (u, v, w) of (1).
Sketch of the proof. Integrating directly over Ω the second equation (1), using the hypotheses and the Cauchy-Schwarz inequality, we get and hence In order to obtain bounds for u and w in L 1 (Ω) we consider the linear combination u + w + v. Its derivate is Using the assumptions (H 0 )-(H 6 ) we have thus, Sketch of the proof. We have that w t ≥ −m w (w)w. Then using the Maximum Principle we get w(t ) ≥ 0 for all t ∈ (0, T max ). To prove the positivity of the functions u and v we define [7] for more details). After the change of variable u = F (w) u, the first equation in system (1) is reduced to Treating the second equation of the system as a scalar linear equation in v we see, in view of (H 0 ), that v = 0 is a lower solution. Therefore, applying the Maximum Principle for parabolic equations, it holds that v(x, t ) ≥ 0 for all x ∈ Ω and t ∈ [0, T max ]. Further, in view of (H 0 )-(H 4 ), the function G verifies So −G( u, v, w) ≥ 0. Therefore, since u 0 ≥ 0, from the Maximum Principle for parabolic equations, we get that u(x, t ) ≥ 0 and consequently, u(x, t ) ≥ 0 for all x ∈ Ω and t ∈ [0, T max ]. Now note that in account of assumptions (H 1 ) and (H 4 ), the function v Lemma 5. If for each T > 0 there exists a constant M 0 (T ) depending only on T and (u 0 , v 0 , w 0 ) 1,p such that This fact is obtained by applying Theorem 15.5 in [3] to (1).

Proof of Theorem 1
In this section we complete the proof of Theorem 1. First, we prove the L ∞ -boundedness of the functions u and w defined by (1) following a Moser-Alikakos iteration method [1]. Note that for v this property has been already proved in the previous section. Let p > 1 and (u, v, w) be the solution of (1). We assume that (H 0 )-(H 6 ) are satisfied and we proceed en 3 steps: Step 1. For the positive function F (w) given by (12), there exists a constant K > 0, such that Proof. For every p ≥ 1, from the first equation in (1) we have where Since u ≥ 0, F > 0 and p > 1, from (18)  Therefore, we have Now, from (H 0 )-(H 6 ) we can see and Consequently, (20) is reduced to and hence, Step 2. For each u 0 ∈ L ∞ (Ω) the solution u of (1) satisfies for any t < 0, where χ 1 denotes the bounded L 1 [0, ∞) norm.
Proof. Since w verifies the ordinary differential equation in (1), under the hypotheses of the theorem, we have w t ≤ βu − γw ≤ βC ∞ − γw.
Thus, using the Comparison Lemma we get the bound (27).
Proof of Theorem 1. The global existence of (u, v, w) over Ω × (0, ∞) is a direct consequence of the local existence and the uniform boundedness of (u, v, w) in L ∞ established in the previous steps.