Classical and generalized solutions of an alarm-taxis model

In bounded, spatially two-dimensional domains, the system \begin{equation*} \left\lbrace\begin{alignedat}{3} u_t&= d_1 \Delta u&&&&+ u(\lambda_1 - \mu_1 u - a_1 v - a_2 w), \\ v_t&= d_2 \Delta v&&- \xi \nabla \cdot (v \nabla u)&&+ v(\lambda_2 - \mu_2 v + b_1 u - a_3 w),\\ w_t&= d_3 \Delta w&&- \chi \nabla \cdot (w \nabla (uv))&&+ w(\lambda_3 - \mu_3 w + b_2 u + b_3 v), \end{alignedat}\right. \end{equation*} complemented with initial and homogeneous Neumann boundary conditions, models the interaction between prey (with density $u$), predator (with density $v$) and superpredator (with density $w$), which preys on both other populations. Apart from random motion and prey-tactical behavior of the primary predator, the key aspect of this system is that the secondary predator reacts to alarm calls of the prey, issued by the latter whenever attacked by the primary predator. We first show in the pure alarm-taxis model, i.e. if $\xi = 0$, that global classical solutions exist. For the full model (with $\xi>0$), the taxis terms and the presence of the term $-a_2 uw$ in the first equation apparently hinder certain bootstrap procedures, meaning that the available regularity information is rather limited. Nonetheless, we are able to obtain global generalized solutions. An important technical challenge is to guarantee strong convergence of (weighted) gradients of the first two solution components in order to conclude that approximate solutions converge to a generalized solution of the limit problem.


Introduction
An essential step in understanding ecosystems is shedding light on the interactions of the participating species with one another or with the environment.The first among such interactions studied in mathematical models are predator-prey, competitive or symbiotic relations between two species, as modeled by Lotka-Volterra type systems [34,41,18], these three types differing from each other by the signs of their 'reaction terms'.Incorporating spatial dependence admits the description of motion, usually in form of random motion [3,23] which can already alter the relative success of competing species [5].But also directed advances towards, e.g.stationary food sources (e.g.[4]) are treated (and shown to be advantageous).A particular of such sources may be given by individuals of prey, resulting in prey-taxis, as studied by Kareiva and Odell [24] for ladybug beetles and goldenrod aphids and mathematically investigated (with regards to solvability in classical or weak settings) in, e.g.[47,20,49,17,35,45].On the opposite side of this phenomenon, the prey may react to the presence of predators by attempting to evade, leading to predator-taxis models (for a mathematical treatment, see [48]).
The combination of both types of taxis (predator-prey taxis, pursuit-evasion dynamics; [39,14,40]) leads to mathematically more challenging systems, due to their highly cross-diffusive structure.Nevertheless, some results on weak solvability [37,38,10] or stability of stationary states [9] are available.
The step to more than two species can make a significant difference: Even in ODE models, suddenly chaotic behaviour can be observed [16].Regarding mechanisms of interaction, it opens up further possibilities (besides additional sets of predator/prey relationships).For example, it becomes possible for prey to engage in anti-predator behaviour by way of attracting a secondary predator that attacks the primary predator, thus aiding the prey (even though, at the same time, it may still prey on the original prey).An alarm call model capturing this behaviour has been suggested by [15].We consider the following system (with u denoting the density of prey, v that of the predator und w that of the superpredator): in Ω × (0, ∞), in Ω × (0, ∞), where d i , χ, λ i , µ i , a i , b i > 0 for i ∈ {1, 2, 3} and ξ ≥ 0. (1.5) The most notable feature of (1.1) is the alarm-taxis term, −χ∇ • (w∇(uv)).Compared to other taxis models, the 'signal' therein consists in the nonlinear expression uv arising from the clash of prey and primary predator.Another key challenge for the mathematical analysis stems from the fact that the directed motion of the secondary predator is (at least partially) influenced by another component which itself undergoes some taxis, a feature (1.1) shares with certain food-chain models (e.g.[21,19,31]) and forager-exploiter systems (cf.[46,36,2,42,33,30], for instance).
For one-dimensional domains, global existence of classical solutions to (1.1) was established in [15], where also pattern formation was discussed.
In two-dimensional settings, replacing the summand w(1 − w) in h by w(1 − w σ ) for any σ > 1 ensures global boundedness of solutions [32], but for σ = 1, a crucial gradient estimate no longer works (cf.[32,Remark 1.2]).If, instead, the term uw in f and h is replaced by a functional response of the form uw u+w , [22] proves global boundedness of classical solutions.In both of these settings, smallness of the taxis coefficients ξ and χ leads to exponential convergence of solutions to a constant steady state.
Higher-dimensional domains have been treated in [51] (it seems worth noting that also three-dimensional domains are biologically relevant; for an overview of alarm signals in aquatic ecosystems, see e.g.[7]) and it was shown there that the system admits globally bounded classical solutions as long as the coefficients of the logistic terms are sufficiently large.
At first glance, the main difference between (1.1) and the three-species food-chain model in [21] seems to be that between alarm-taxis and prey-taxis in the third equation.Actually, however, at least for questions of global solvability, this inclusion of another function in the taxis term does not constitute an overwhelmingly large difference -after all, this function is rather soon seen to be bounded.A larger change is the interaction between first and third trophic level, i.e. the insertion of the −uw term in the first equation.Its absence enables the authors of [21] to apply suitable testing procedures eventually obtain uniform-in-time L 4 (Ω) bounds for ∇u, which, as 4 is larger than the spatial dimension of Ω, allows for semigroup arguments yielding boundedness of v, even without g containing logistic terms.Biologically, this term models a direct predator-prey relationship between secondary predator and prey (that is, its presence turns "the prey calls for help" into "the prey calls for help, despite detrimental effects").
The full coupling term uw is contained in the alarm-taxis model in [32] and the food-chain model in [50], but both results require stronger than quadratic absorption terms in the equation for w.
A weaker form of this effect is considered for instance in the above-mentioned [22], where an alarm-taxis system with 'ratio-dependent' functional response ( uw u+w instead of uw in (1.1)) is studied.However, although such a term can be biologically motivated, mathematically the change from uw is rather drastic.After all, it enables the estimate | uw u+w | ≤ u and due to a priori estimates for u and v (see Lemma 4.4 and Lemma 4.5), the function f (u, v, w) is readily seen to be uniformly in time bounded in L 2 (Ω), which in turn implies an L ∞ -L 4 bound for ∇u and then, as above, boundedness of v (cf.[22, Lemma 3.3 and Lemma 3.5]).In contrast, for (1.1), we are only able to bound ∇u in L ∞ -L 2 , which appears to be (barely) insufficient to start any bootstrap procedures for v.
The results of the present article are twofold.First, we consider a pure alarm-taxis system without preytactic effects on the intermediate trophic level.Here it turns out that classical solutions are global.
For the full model, including prey-taxis of the primary and alarm-taxis of the secondary predator, we obtain global generalized solutions under mere positivity assumptions (and without largeness conditions) on the parameters.
The solution concept will be made precise in Definition 4.1.It is a relative of renormalized solutions, as established in the context of the Boltzmann equation in [6] and used in context of chemotaxis models (see, e.g. the survey [28] or [11, Section 1.2] for an overview).
Plan of the paper.We start by stating a local existence result and collecting first basic estimates in Section 2. As observed at the beginning of Section 3, for ξ = 0 the system (1.1) resembles a chemotaxis-consumption system with a logistic source term, for which global existence of classical solutions in two-dimensional settings is to be expected.That is, by choosing a proof which does not rely on intricate energy-type arguments based on cancellations of worrisome terms, we rather rapidly obtain Theorem 1.1.
The proof of Theorem 1.2 is considerably more delicate.Mainly due to the fact that u and uv are contained in the cross-diffusive terms in the second and third equation in (1.1), respectively, the available a priori estimates for u, v and w get steadily worse.While the quadratic dampening term in the (second and) third equation implies a uniform-in-time bound for Ω |∇v| 2 , the exponent 2 does not exceed the spatial dimension and hence well-established semigroup arguments yielding boundedness of v seem to be unavailable.Nonetheless, in Subsection 4.2 we are able to collect sufficient a priori estimates to conclude that a pointwise limit of solutions to approximate problems is sufficiently regular for all integral terms in Definition 4.1 of generalized solutions to make sense.
However, as some of the integrands there are quadratic in (∇u, ∇v, ∇w), the weak convergence obtained by boundedness in certain reflexive spaces turns out to be insufficient.To overcome this issue in Subsection 4.3, we crucially make use of the main results in [11] which inter alia assert strong convergence of weighted gradients as long as the right-hand side of the considered parabolic equation converges weakly in space-time L 1 .This allows us prove in Subsection 4.4 that the limit (u, v, w) is indeed a global generalized solution of (1.1), i.e. to prove Theorem 1.2.

Preliminaries
As first step, we state a local existence result, including an extensibility criterion.The function σ is included so that we can use the same lemma for both the classical solutions of Theorem 1.1 and for solutions to the approximate system used during the proof of Theorem 1.2 in Section 4.
Then there exists T max = T max (u 0 , v 0 , w 0 ) ∈ (0, ∞] and a nonnegative, unique maximal classical solution (2.2) Proof.This follows by a straightforward adaptation of well-established fixed point arguments as employed for instance in [1,Lemma 3.1].
Essentially because f , g and h model the interaction of a food chain, there is a positive linear combination of these functions which grows at most linearly.Thus, as first yet very basic a priori estimates for solutions of (2.2) we obtain uniform-in-time L 1 bounds and space-time 2) given by Lemma 2.1 fulfils where T := min{T 0 , T max (u 0 , v 0 , w 0 )}.
Then two applications of Jensen's inequality imply in [0, T ).Thus, by an ODE comparison argument, .
Next, we state two consequences of the Gagliardo-Nirenberg inequality.
As the second and third subproblem in (2.2) share some structural properties, it appears sensible to study the quite general convection-diffusion equation (2.9) Provided that d > 0 and that the given functions σ, ψ 1 , ψ 2 , z 0 are bounded in appropriate spaces, we obtain a priori estimates for solutions of (2.9) by employing a testing procedure and Ladyzhenskaya's trick (cf.[25]).In less general settings, corresponding estimates have for instance been obtained in [22,Lemma 3.2] by a similar method.
Proof.Testing (2.9) with z and applying Young's inequality yields Here, Lemma 2.3 provides c 1 > 0 such that where so that an ODE comparison argument and the variation-of-constants formula assert for all t ∈ (0, T ).An application of the monotone convergence theorem then yields (2.11) for C := max{ 2 d , 1} max{M, 2}e c2(M+T ) > 0.
3 Global classical solutions for the system without prey-taxis Throughout this section, we fix a smooth, bounded domain Ω ⊂ R 2 , functions, parameters and initial data as in (1.2)-(1.6).Importantly, we also set ξ = 0, that is, we consider the setting without prey-taxis.Then Lemma 2.1 asserts that there is a nonnegative, unique maximal classical solution (u, v, w) of regularity (2.1) of (1.1).We fix this solution as well as its maximal existence time T max = T max (u 0 , v 0 , w 0 ).
The goal of this section is to prove Theorem 1.1, i.e. that this solution is global in time.To that end, we now collect several a priori estimates which will eventually show that (2.3) does not hold, which according to Lemma 2.1 can only happen if T max = ∞.
The first such bound going beyond Lemma 2.2 makes use of the structure of f and g, the comparison principle, and the assumption ξ = 0.
} is a supersolution of the second equation in (1.1) so that also v is bounded from above.Finally, nonnegativity of u and v has already been asserted in Lemma 2.1.
Since the shape of f and g rather directly yield boundedness of u and v (as evidenced by Lemma 3.1), the situation of ξ = 0 presents itself as similar to chemotaxis-consumption systems with one equation resembling for some bounded function z and, in this case, essentially logistic source terms h.For this setting, global existence in two-dimensional domains is not surprising (cf.[29], [27]), although some arguments relying on delicate energy-type arguments may not be transferable.
As ξ = 0, the L 2 space-time estimates provided by Lemma 2.2 rapidly imply a priori estimates for certain spatial derivatives of u and v.
Lemma 3.2.Let T ∈ (0, ∞) ∩ (0, T max ].Then there is C > 0 such that Proof.Testing the first equation with −∆u gives so that due to Lemma 2.2 and Lemma 3.1 the statement for the first solutions component follows upon an integration in time.The estimate for the second one follows analogously.
The previous two lemmata make Lemma 2.4 applicable and thus yield a uniform-in-time L 2 bound for w.
Proof.We set ψ 1 = χuv and ψ 2 = h, then Lemma 3.1, Lemma 3.2 and continuity of w 0 imply (2.10) for some M > 0, so that Lemma 2.4 asserts The right-hand side herein is bounded by (2.5) and Lemma 3.1.
With these estimates at hand, showing T max = ∞ already comes down to employing a rather standard bootstrap procedure.
Fixing q ∈ (2, θ), we make again use of the variations-of-constants formula to obtain ds for all t ∈ (0, T max ).Here we make use of Hölder's inequality to obtain with r := qθ θ−q > 0 that for all s ∈ (0, T max ).Therefore, there is c 7 > 0 such that and hence M 1 max{4,r} (t) ≤ 2c 7 for all t ∈ (0, T max ).
In particular, there is c 8 > 0 such that In combination, (3.1), (3.2), (3.3) and the extensibility criterion in Lemma 2.1 show that our assumption T max < ∞ must be false.
Proof of Theorem 1.1.All claims have been established in Lemma 3.4.
4 Global generalized solutions for the system with prey-taxis

Solution concept
In this section, we will construct global generalized solution of (1.1) with ξ > 0. We begin by introducing our solution concept.
Definition 4.1.Let Ω ⊂ R 2 be a smooth, bounded domain, assume (1.2)-(1.4)and let u 0 , v 0 , w 0 ∈ L 1 (Ω) be nonnegative.A triple (u, v, w) ∈ L 2 loc (Ω × [0, ∞)) of nonnegative functions with ∇u, ∇v, 1 {w≤k} ∇w ∈ L 2 loc (Ω × [0, ∞)) for all k ∈ N (where we denote the characteristic function of a set A by 1 A ) is called a global generalized solution of (1.1) if • u and v are weak solutions of the respective subproblems in (1.1), that is, and ) is a weak φ-supersolution of the corresponding subproblem in (1.1) in the sense that This concept is consistent with the notion of classical solutions.
Remark 4.3.For the integral terms in Definition 4.1 to be well-defined, slightly less regularity would suffice; for example, one could require ∇v ∈ L 1 and 1 {v≤k} ∇v ∈ L 2 instead of ∇v ∈ L 2 .While we rely on the strong L 2 convergence of 1 {vε≤k} ∇v ε (cf.Lemma 4.11), it is even more easily obtained that ∇v (without the additional cutoff) belongs to L 2 loc (Ω × [0, ∞)) (see (4.17)).On the other hand, posing an integrability condition on 1 {w≤k} ∇w and not on ∇w is crucial, as this allows to conclude the needed boundedness from an estimate of ∇ ln(w + 1) instead of ∇w itself.(See Lemma 4.8 and (4.20).)
Again for each ε ∈ (0, 1), Lemma 2.1 then asserts that there exist T max,ε ∈ (0, ∞] and a nonnegative maximal classical solution in Ω. (4.8) As in Section 3, the lack of a cross-diffusive term together with the structure of f implies several a priori estimates for the first solution component.
Proof.This can be shown as in Lemma 3.1 and Lemma 3.2.
When ξ is positive, however, bounds for the second solution component are not as easily obtained as in Section 3; both the comparison principle used in Lemma 3.1 and the testing procedure employed in Lemma 3.2 are no longer applicable.Fortunately, we can at least make use of Lemma 2.4 to obtain the following Lemma 4.5.For all finite T ∈ (0, T max,ε ], there exists C > 0 such that for all ε ∈ (0, 1).
Proof.As Lemma 4.4 and (4.5) provide ε-independent bounds for the quantities in (2.10) (with ψ 1 = u ε and z 0 = v 0ε ), we can apply Lemma 2.4 to obtain c 1 > 0 such that for all ε ∈ (0, 1).As the right-hand side is bounded by Lemma 2.2 and Lemma 4.4, this yields the desired estimates.
Before collecting further ε-independent a priori estimates, we briefly state that the approximate solutions exist globally.To that end, we already make use of Lemma 4.4.
Proof.Suppose there is ε ∈ (0, 1) such that T max,ε is finite.According to Lemma 4.4, there is } is a supersolution to the second equation in (4.8).Similarly, } is a supersolution to the third equation in (4.8), so that all solution components and thus also the zeroth order terms in (4.8) are bounded in L ∞ (Ω × (0, T max,ε )).
As a consequence of Lemma 4.4 and Lemma 4.5, we see that the second equation in (4.8) can be written as a heat equation with a force term uniformly bounded in Lemma 4.7.For all T > 0, there exists C > 0 such that Proof.By Hölder's and Young's inequalities as well as (4.7), we have for all ε ∈ (0, 1) and some c 1 , c 2 > 0, where Q T := Ω × (0, T ).As moreover for all ε ∈ (0, 1) and some c 3 , c 4 > 0 by Young's inequality and (2.7), we conclude (4.9) upon applying Lemma 4.4, Lemma 4.5 and Lemma 2.2.
Although the second and third equation in (4.8) are structurally similar in some sense, we cannot repeat the reasoning in Lemma 4.5 (nor the one in Lemma 3.3) -our known ε-independent bounds for uv are much worse than for u.Instead, we shall employ a testing procedure in order to obtain a priori estimates for the gradient of w ε which, while weaker than those for the other solution components, will still turn out to be sufficient for our purposes.Lemma 4.8.For all T ∈ (0, ∞), there exists C > 0 such that for all ε ∈ (0, 1).
In order to prepare applications of the Aubin-Lions lemma, we next collect estimates for the time derivatives which rapidly follow from the bounds obtained above.Lemma 4.9.For all T ∈ (0, ∞), there is C > 0 such that and for all ε ∈ (0, 1).

The limit process ε ց 0: Obtaining solution candidates
With the a priori estimates obtained in the previous subsection at hand, we are now able to take the limit of (u ε , v ε , w ε ) (along some null sequence) in appropriate spaces.
In order to show that the triple (u, v, w) obtained in Lemma 4.10 is actually a generalized solution of (1.1) in the sense of Definition 4.1, we in particular need to show that φ(u, w) is a weak φ-supersolution for certain choices of φ; that is, that (4.3) holds.However, the latter contains quadratic expressions of (weighted) gradients of all solution components, meaning that the weak convergence of gradient terms asserted by Lemma 4.10 is yet insufficient.Thus, we are interested in strong convergence of (weighted) gradients of u ε and v ε .(The term in (4.3) which is quadratic in ∇w has a favorable sign and can be treated by making use of the weak lower semicontinuity of the norm.)According to [11], this follows if the respective equations can be written as a heat equation with a force term converging in sufficiently strong topologies.Fortunately, the latter is contained in the above analysis and thus we obtain Lemma 4.11.Let (u, v, w) and (ε j ) j∈N ⊂ (0, 1) be as given by Lemma 4.10.Then there exists a subsequence of (ε j ) j∈N , which we do not relabel, such that ) as ε = ε j ց 0.
The convergence of ∇ ln(w ε + 1) in (4.20) seems much too weak for a treatment of ∇w ε .If, however, the latter term is combined with some cutoff, these two a priori different modes of convergence become indistinguishable.We make this observation precise in the following form allowing for rather direct application to the equations at hand.Lemma 4.12.Let (u, v, w) and (ε j ) j∈N be as given by Lemma 4.10 and Lemma 4.11, respectively.Moreover, let as ε = ε j ց 0.

Lemma 4 . 13 .
The triple (u, v, w) constructed in Lemma 4.10 is a global generalized solution of (1.1) in the sense of Definition 4.1.