LARGE TIME BEHAVIOR IN A PREDATOR-PREY SYSTEM WITH INDIRECT PURSUIT-EVASION INTERACTION

. In a bounded domain Ω ⊂ R n with smooth boundary, this work considers the indirect pursuit-evasion model with positive parameters χ,ξ,λ,µ , a and b . It is ﬁrstly asserted that when n ≤ 3, for any given suitably regular initial data the corresponding homogeneous Neumann initial-boundary problem admits a global and bounded smooth solution. Moreover, it is shown that if bλ < µ and under some explicit smallness conditions on χ and ξ , any nontrival bounded classical solution converges to the spatially homogeneous coexistence state in the large time limit; if bλ > µ , however, then under an explicit smallness assumption on χ but without any restriction on ξ , any bounded classical solution ( u,v ) with u (cid:54)≡ 0 stabilizes to ( λ, 0) as t → ∞ .


Michael Winkler
It is firstly asserted that when n ≤ 3, for any given suitably regular initial data the corresponding homogeneous Neumann initial-boundary problem admits a global and bounded smooth solution. Moreover, it is shown that if bλ < µ and under some explicit smallness conditions on χ and ξ, any nontrival bounded classical solution converges to the spatially homogeneous coexistence state in the large time limit; if bλ > µ, however, then under an explicit smallness assumption on χ but without any restriction on ξ, any bounded classical solution (u, v) with u ≡ 0 stabilizes to (λ, 0) as t → ∞.
To capture even more complex dynamics, and especially the occurrence of wave-like behavior, in yet simple two-component models of predator-prey type, the authors in [23] propose to additionally account for partially directed migration mechanisms reflecting, on the one hand, the ambition of predators to move toward prey-rich regions, and, on the other hand, a certain predisposition of prey individuals to move away from predator-populated areas. In the resulting model, the quantities u = u(x, t) and v = v(x, t) denote the densities of the predator and the prey population, respectively, f and g represent the local kinetics, and the positive parameters χ and ξ measure the strength of attractive and repulsive directed migration, respectively. Indeed, the numerical simulations presented in [23] indicate that even in spatially one-dimensional frameworks and for f and g reflecting functional response of so-called Holling type III, despite their seemingly artless structure such systems can well describe the emergence of soliton-like taxis waves, as observed in experiments involving bacterial populations of E. coli on semi-solid nutrient media ( [24]).
In fact, already the literature on related systems indicates that in fact the introduction of the two cross-diffusion mechanisms in (1.1) may go along with a substantial change of mathematical properties in comparison to those known for the corresponding taxis-free variants in which χ = ξ = 0. For instance, choosing ξ = 0, f ≡ 0 and g(u, v) = u − v shows that the celebrated Keller-Segel chemotaxis system ( [14]), that is, can be viewed as a special case of (1.1), and it is well-known that even finite-time blow-up of solutions to corresponding Neumann problems in bounded n-dimensional domains will occur for suitably large initial data in the cases n = 2 ([9]) and n ≥ 3 ( [27]). The particularly delicate role of the attractive taxis mechanism therein is underlined by the observation that no such drastic aggregation phenomenon occurs when a single cross-diffusion term of the considered form is purely repulsive, such as in the variant of (1.2) given by Indeed, an associated Neumann problem is known to admit global smooth solutions when n = 2, and at least some global weak solutions when n ∈ {3, 4}, and each of these solutions approaches a spatially homogeneous equilibrium in the large time limit ( [3]). Now in contexts of production and degradation processes which are closer to the prototypical choices in Lotka-Volterra systems than those underlying (1.2) and (1.3), the explosion-supporting potential of attractive taxis can partially be compensated by suitably regularizing influences of the respective zero-order expressions involved. Setting ξ = 0 and g(u, v) = −uv + g 0 (v) in (1.1), for instance, reduces (1.1) to the so-called prey-taxis system ( [13]) (1.4) in which analytical findings indicate that at least in two-dimensional settings, the additional dissipation generated by the absorptive term −uv therein may rule out the occurrence of blow-up under various assumptions on χ or on local kinetics functions f and g 0 ([21], [26]; cf. also [28], [12], [8], [16], [21], [29] and [22] for some closely related variants).
In contrast to its subsystems (1.2), (1.3) and (1.4), by simultaneously accounting for two taxis mechanisms the full model (1.1) can no longer be viewed as a triangular cross-diffusion system, which substantially reduces its accessibility to wellestablished analytical techniques, and which is reflected in an apparently complete absence of rigorous results concerned with global solutions to any version of (1.1) involving nontrivial choices of both χ and µ. In order to nevertheless achieve some insight into possible dynamical properties of pursuit-evasion processes, in this work we shall focus on a variant of (1.1) in which the respective tactic movements are oriented along gradients of some indirectly produced stimuli, rather than following individuals directly. In fact, assuming predators and preys to exert speciescharacteristic substances such as pheromones or scent marks, the authors in [25] propose the variant of the pursuit-evasion model (1.1) given by additionally containing the concentrations w = w(x, t) and z = z(x, t) of the respectively emitted chemicals, as well as the positive parameters D w , D z , δ w and δ z . Relying on the circumstance that chemicals diffuse substantially faster than individuals, we shall follow a corresponding and standard quasi-stationary approximation procedure, quite well-established in the context of chemotaxis systems ( [11], [10]), and hence subsequently concentrate on the parabolic-elliptic simplification of (1.5) given by (1.6) Beyond some analytical results addressing questions of global weak solvability and boundedness in two-dimensional boundary value problems ( [7], [1]), numerical evidence indicates that even upon trivial choices of f, g, δ w and δ z , (1.6) may indeed generate various types of patterns ( [6]).
Main results. The purpose of the present work is to investigate possible effects resulting from the interplay of the doubly cross-diffusive and indirectly mediated migration in (1.6) with zero-order kinetics genuinely related to Lotka-Volterra type predator-prey interaction. Accordingly, we shall henceforth consider the indirect pursuit-evasion system in a bounded domain Ω ⊂ R n with smooth boundary, where n ≥ 1, where χ, ξ, λ, µ, a and b are positive parameters, and where u 0 and v 0 are given suitably regular functions.
In this framework, the first of our results asserts global existence of smooth solutions to (1.7) for widely arbitrary initial data in any physically meaningful dimension: let Ω ⊂ R n be a bounded domain with smooth boundary, and suppose that the parameters χ, ξ, λ, µ, a and b are positive. Then for all nonnegative functions which is bounded in the sense that there exists C > 0 satisfying for all t > 0.
Next concerned with the qualitative behavior of these solutions, we recall from well-known facts about the ODE system associated with (1.7) that merely the sign of the number µ−bλ decides about the existence of a spatially homogeneous equilibrium which is positive in both population components and hence reflects coexistence ( [15]). In fact, our second result will reveal that the assumption bλ < µ therefor will retain its sufficiency with regard to asymptotic stability of this steady state, provided that an explicit smallness condition on the taxis coefficients χ and ξ is satisfied. We note that the following statement in this direction actually applies to any global bounded solution to (1.7), regardless of the space dimension n ≥ 1, with unconditional applicability to widely arbitrary solutions when n ≤ 3 due to Proposition 1.1.
Let Ω ⊂ R n be a bounded domain with smooth boundary, let λ > 0, µ > 0, a > 0 and b > 0 satisfy bλ < µ, (1.9) assume that the positive parameter χ is such that and suppose that the positive parameter ξ fulfills Then any nonnegative global bounded classical solution (u, v, w, z) of the boundary value problem in (1.7) with u ≡ 0 and v ≡ 0 satisfies If, conversely, bλ > µ, then under an again explicit restriction on smallness of χ, but now without any constraint on the repulsive taxis coefficient ξ, the predators will asymptotically outcompete the prey population.  Some elementary bounds on the respective total mass functionals will be of substantial importance in the sequel.
Proof. Integrating the first and the second equation of (1.7) with respect to x ∈ Ω, we see that which upon a simple ODE comparison shows that for all t ∈ (0, T max ). Since integrating the third and fourth equations in (1.7) we find that Next making essential use of the assumption n ≤ 3 underlying Proposition 1.1, we shall turn the above L 1 bounds for w and z into L p estimates for u and v. Concerning the latter second solution component v, this can be viewed as a fairly obvious extension of a related result already observed in [20, Theorem 1.1] for a proliferation-free variant of (1.7); the corresponding argument for u, however, will require an additional consideration here, basically reducing to the observation that within the course of a standard L p testing procedure, the nonlinear source term +auv in (1.7) can be compared in strength with the chemotactic term −χ∇ · (u∇w) (see (2.12) below). Lemma 2.3. Let n ≤ 3, χ > 0, ξ > 0, λ > 0, µ > 0, a > 0 and b > 0. Then for any finite p > 1 one can find C(p) > 0 such that the solution of (1.7) satisfies v(·, t) L p (Ω) ≤ C(p) for all t ∈ (0, T max ) (2.5) as well as u(·, t) L p (Ω) ≤ C(p) for all t ∈ (0, T max ). (2.6) Proof. In essence, the proof follows the idea from [20]. So, we only outline the main steps and point out that the two terms of local kinetics will not induce any new technical difficulty in the proof.
Step 2. We multiply the second equation in (1.7) by v p−1 and integrate by parts using the identity ∆z = z − u to obtain that for all t ∈ (0, T max ), thanks to the nonnegativity of u and v. Here since n ≤ 3, we can fix r > 1 such that n 2 < r < n (n − 2) + , (2.9) and thus relying on the Hölder inequality, using (2.7) and invoking the Gagliardo-Nirenberg inequality along with (2.3) we can find c 3 = c 3 (p) > 0 and c 4 = c 4 (p) > 0 such that where r := r r−1 satisfies 2r < 2n (n−2)+ due to the left inequality in (2.9). Inserting this into (2.8) yields for all t ∈ (0, T max ). Now using the Poincaré inequality and noting that v a m for all t ∈ (0, T max ) due to (2.3), we further obtain c 5 = c 5 (p) > 0 such that which combined with (2.10) entails that Upon an ODE comparison, this results in and thereby proves (2.5).
Step 3. By straightforward computation using three integrations by parts, similar to the derivation of (2.8) we have for all t ∈ (0, T max ), where relying on the estimate (2.11) and proceeding as in Step 2 to deal with the first summand on the right hand side of (2.12), we obtain c 7 = c 7 (p) > 0 such that Combining this with (2.12) entails from which (2.6) can be derived, similarly to to reasoning in Step 2, through an application of the Poincaré inequality and (2.2).
We now in a position to complete the proof of global existence and boundedness of solutions to (1.7).
Proof of Proposition 1.1. In view of Lemma 2.3 and standard elliptic regularity theory ( [5]), fixing any p > n we obtain bounds for both w and z in L ∞ ((0, T max ); W 2,p (Ω)), which along with a Sobolev embedding theorem implies ∇w(·, t) L ∞ (Ω) + ∇z(·, t) L ∞ (Ω) ≤ c 1 for all t ∈ (0, T max ) with some c 1 > 0. Using this information together with Lemma 2.3 and performing a Moser-type iteration (cf. [19,Lemma A.1]), we obtain c 2 > 0 such that In conjunction with the extensibility criterion (2.1) in Lemma 2.1, this immediately leads to the statements in Proposition 1.1.

3.
Large time behavior of bounded solutions.

3.1.
A general observation on evolution of functionals involving logarithms. Our qualitative analysis of bounded solutions to (1.7) will rely on the construction of Lyapunov functionals on the basis of the following observations. Lemma 3.1. Any global classical solution of (1.7) with u ≡ 0 and v ≡ 0 satisfies Moreover, and Proof. Since u is positive in Ω × (0, ∞) by the strong maximum principle, we may multiply the first equation in (1.7) by 1 u to obtain using integration by parts that for all t > 0.  7) over Ω. Furthermore, we use (w −v ) as a testing function for the third equation in (1.7) to derive (3.5), whereas (3.6) is a consequence of testing the fourth equation in (1.7) by (z − u ).

3.2.
Coexistence. Proof of Theorem 1.2. Now when the taxis parameters χ and ξ are suitably small, assuming (1.9) enables us to discover a gradient-like structure in (1.7) in the following sense: Lemma 3.2. Let (1.9), (1.10) and (1.11) hold. Then there exist positive numbers α, β, η and γ such that given any global classical solution of (1.7) with u ≡ 0 and v ≡ 0, letting for all t > 0.
Since direct computation shows that for all t > 0 as well as for all t > 0. From this and the identities (3.5) and (3.6) we infer that for all t > 0, (3.11) where Young's inequality implies that for all t > 0 and for all t > 0. This together with (3.11) yields (3.8) with α and β defined as in (3.9) and (3.10), and with η := ε · χ 2 4 u > 0 and γ := ε · aξ 2 4b v > 0. We now assert asymptotic coexistence in the flavor of and under the assumptions of Theorem 1.2.