Andrey Polyakov

Denis Efimov

Bernard Brogliato

Markus Reichhartinger

Consistent Discretization of Locally Homogeneous Finite-time Stable Control Systems

An algorithm of consistent implicit discretization for locally homogeneous finite-time stable system having discontinuity only at the origin is developed. It preserves finite-time stability property in the discrete-time models. The homogeneous domination approach is utilized for analysis of the discretized model. The scheme is demonstrated for a simplified model of a quadrotor control system.

I. INTRODUCTION

Discretization issues are important for a digital implementation of estimation and control algorithms as well as for a computer simulation of control processes. Construction of a so-called consistent stable discretization is a non-trivial problem for essentially non-linear ordinary differential equations (ODEs), which do not satisfy some classical regularity assumptions. In our case consistent discretization means that the resulting discrete-time model approximates solutions and preserves stability/convergence properties of the original continuous-time system. An inconsistent discretization of non-Lipschitz feedback algorithms results in degradation of control precision [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], chattering effect [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], or even instability [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF]. For example, the sliding mode algorithms are known to be difficult in practical realization [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], [START_REF] Acary | Chattering-free digital slidingmode control with state observer and disturbance rejection[END_REF], [START_REF] Kikuuwe | Proxybased sliding mode control: A safer extension of pid position control[END_REF] due to their discontinuous (set-valued) nature, which may involve chattering caused by the discretization (the so-called numerical chattering). The mentioned papers have discovered that the implicit discretization technique is useful for practical implementation of non-smooth and discontinuous control and estimation algorithms. In particular, numerical chattering suppression in both input and output, as well as a good closed-loop performance have been confirmed experimentally in [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], [START_REF] Huber | Recent Trends in Sliding Mode Control[END_REF].

A symmetry of functions or vector fields with respect to a certain group of transformations (dilations) is known as homogeneity [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Rosier | Etude de quelques problèmes de stabilization[END_REF], [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF]. Nonlinear homogeneous differential equations/inclusions form an important class of control systems [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF]. They can be utilized as local approximations [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] or set-valued extensions [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] of nonlinear control systems. For example, some models of process control [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF], nonholonomic mechanical systems [START_REF] Pomet | Time-varying exponential stabilization of nonholonomic systems in power form[END_REF] and systems with frictions [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] are homogeneous or at least locally homogeneous. It is well known that any 1 Andrey Polyakov and Denis Efimov are with Inria Lille, Univ. Lille, CNRS, UMR 9189 -CRIStAL, (F-59000 Lille, France) and with ITMO University, 49 av. Kronverskiy, Saint-Petersburg, 197101, Russia 2 Bernard Brogliato is with Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, 38000, France bernard.brogliato@inria.fr 3 Markus Reichhartinger is with Graz University of Technology, Austria

ẋ = f (x) y=Φ(x) ⇔ ẏ = f (y) ← a consistent discrete-time approximation ← → a consistent discrete-time approximation → x i+1 = Φ -1 (Φ(x i )+h f (Φ(x i+1 ))) x=Φ -1 (y) ⇔ y i+1 = y i +h f (y i+1 )
Fig. 1. The scheme of the consistent implicit discretization of ẋ = f (x) asymptotically stable homogeneous system with a negative degree is finite-time stable

The type of homogeneity is identified by the dilation group, which can be linear [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] or nonlinear [START_REF] Rosier | Etude de quelques problèmes de stabilization[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]. In this paper, we deal with the linear geometric homogeneity studied originally in [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF] for infinite dimensional models. In the finite dimensional case the corresponding dilation group (see [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] for more details) is given by the matrix exponential function e sGd , where s ∈ R is the group parameter and G d ∈ R n×n is an anti-Hurwitz matrix known also as a generator of the dilation.

Recently [START_REF] Polyakov | Consistent discretization of finite-time and fixed-time stable systems[END_REF], a problem of consistent discretization of homogeneous finite-dimensional systems with negative degree has been studied. In particular, a discretization algorithm preserving the finite-time stability in the obtained discretetime model has been developed. It is essentially based on topological equivalence of any stable homogeneous system to a quadratically stable one (see Fig. 1 and [START_REF] Polyakov | Consistent discretization of finite-time and fixed-time stable systems[END_REF], [START_REF] Polyakov | Quadratic-like stabilizability of homogeneous systems[END_REF]).

It is shown [START_REF] Polyakov | Consistent discretization of finite-time stable homogeneous systems[END_REF] that an implementation of finite-time controllers using the developed scheme rejects numerical chattering and essentially improves the control precision. This paper continues this research direction and proposes a consistent discretization scheme for non-homogeneous finitetime stable systems, which admit homogeneous approximation at the origin. Notation A = sup x =1 Ax and A = inf x =1 Ax for A ∈ R n×n ; I n ∈ R n×n is the identity matrix; 0 denotes zero element; diag{λ 1 ,.., λ n } -diagonal matrix with elements λ i ; P 0 means positive definiteness of P = P ∈ R n×n ; λ max (P ) and λ min (P ) denote maximal and minimal eigenvalues of P = P ∈ R n×n ; (λ) denotes the real part of a complex number λ; the notation P 1 2 means that P 1 2 = M is such that P = M 2 ; the class K consists of continuous strictly increasing functions σ : [0, +∞) → [0, +∞), σ(0) = 0; B(r) = {x ∈ R n : x < r} is an open ball of the radius r > 0; the closed ball is denoted by B(r); a mapping R n ⇒ R m is set-valued if it associates a set of R m with each x ∈ R n .

II. PROBLEM STATEMENT

Let us consider the non-linear system

ẋ = f (x), t > 0, x(0) = x 0 , (1)
where x(t) ∈ R n is the system state and the nonlinear function f : R n → R n is continuous on R n \{0}, i.e., the only possible discontinuity point of f is the origin. System (1) is assumed to be forward complete for solutions understood in the sense of Filippov [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF]: An absolutely continuous function φ(•, x 0 ) : [0, +∞) → R n is a solution to (1) if φ(0, x 0 ) = x 0 and for almost all t > 0 it satisfies the differential inclusion

ẋ ∈ F (x) = ε>0 µ(N )=0 cof (x + B(ε)\{N }), (2)
where B(ε) denotes the ball in R n of the radius ε, co denotes convex closure and µ(N ) means that the Lebesgue measure of the set N ⊂ R n is zero. In our case,

F (x) = {f (x)} is a singleton for x ∈ R n \{0}, but F (0) = ε>0 cof (B(ε)\{0})
is a set, each time f is discontinuous at 0. Recall [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF] that the origin of system ( 1) is said to be locally uniformly finite-time stable, if it is Lyapunov stable, and there exist a neighborhood U of the origin and a locally bounded function T : U → [0, +∞) such that any solution φ(•, x 0 ) to (1) satisfies φ(t, x 0 ) = 0 for t ≥ T (x 0 ), x 0 ∈ U .

We understand the consistency of a discretization scheme for the finite-time stable system (1) in the sense of the following definition.

Definition 1: [24] A (possibly) set-valued mapping

Q : R + × R n × R n ⇒ R n
is said to be a consistent discrete-time approximation of the locally uniformly finite-time stable system (1) in a neighborhood U of the origin if ∃h 0 > 0 :

• Existence property: for any x ∈ U and any h ∈ (0, h 0 ), there exists xh ∈ U :

0 ∈ Q(h, x, xh ), (3)
and xh = 0 is the unique solution to 0 ∈ Q(h, 0, xh ). • Finite-time convergence property: for any h ∈ (0, h 0 ) each sequence {x i } +∞ i=0 generated by the inclusion 0 ∈ Q(h, x i , x i+1 ), i = 0, 1, 2, .... (4)
converges to zero in a finite number of steps, i.e., ∀x 0 ∈ U \{0} i * > 0 such that x i = 0 for i ≥ i * . • Approximation property: for any ε > 0 and any R > ε, there exists a function ω ∈ K such that any sequence

{x i } +∞ i=0 generated by (4) satisfies φ(h, x i ) -x i+1 ≤ h ω(h), (5)
provided that x i+1 , x i ∈ U and x i+1 , x i ∈ [ε, R],
where φ(•, x i ) is a solution to (1) with the initial condition x(0) = x i .

Notice that the last property in this definition requires the existence of the conventional estimate (5) for the discretization error on any compact set from U \{0} (since ε > 0 and R > ε can be selected arbitrary small and arbitrary large, respectively). The origin is excluded because of a possible singularity of the vector field f at zero (in particular, it can be discontinuous at the origin). The inequality (5) describes a local (one-step) approximation error. An approximation error on the time interval [0, T (x 0 )] is O(ω(h)) provided that h = T (x 0 )/N with N ∈ N. This error tends to zero as h → 0, i.e., as N → +∞.

The aim of the paper is to design a consistent (in the sense of Definition 1) discretization scheme for locally finite-time stable system (1), under the assumption that the vector field f admits a local homogeneous approximation at the origin (see [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] or the next section for the details).

III. PRELIMINARIES

A. Generalized Homogeneity

The generalized homogeneity [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF], [START_REF] Polyakov | Quadratic-like stabilizability of homogeneous systems[END_REF], [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF], [START_REF] Polyakov | On homogeneous finitetime control for linear evolution equation in hilbert space[END_REF] deals with the group of linear transformations (linear dilations).

Definition 2: A map

d : R → R n×n is called dilation in R n if it satisfies • Group property: d(0) = I n and d(t + s) = d(t)d(s) = d(s)d(t) for all t, s ∈ R; • Continuity property: d is a continuous map, i.e., ∀t ∈ R, ∀ε > 0, ∃δ > 0 : |s-t| < δ ⇒ d(s)-d(t) ≤ ε; • Limit property: lim
= G d d(s), d(s) = e Gds = +∞ i=0 s i G i d i! , s ∈ R.
The most popular dilations in R n are [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] the uniform (or standard) dilation (L. Euler 17th century) : d 1 (s) = e s I n , s ∈ R; and the weighted dilation (Zubov 1958, [7]):

d 2 (s) = diag{e ris }, s ∈ R, r i > 0, i = 1, ..., n. They satisfy Definition 2 with G d1 = I n and G d2 = diag{r i }, respectively. In fact, any anti-Hurwitz 1 matrix G d ∈ R n×n defines a dilation d(s) = e Gds in R n . Definition 3: [21]
The dilation d is strictly monotone if there exists β > 0 such that d(s) ≤ e βs as s < 0. Obviously, the monotonicity of a dilation may depend on the norm • in R n .

Theorem 1: [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] Let d be a dilation in R n , then 1) all eigenvalues λ i of the matrix G d are placed in the right complex half-plane, i.e., (λ i ) > 0, i = 1, 2, ..., n; 2) there exists a matrix P ∈ R n×n such that

P G d + G d P 0, P = P 0; (6)
3) the dilation d is strictly monotone with respect to the weighted Euclidean norm • = •, • induced by the inner product x, z = x P z with P satisfying (6). Moreover,

e αs ≤ d(s) ≤ d(s) ≤ e βs if s ≤ 0, e βs ≤ d(s) ≤ d(s) ≤ e αs if s ≥ 0, (7)
where α = 1 2 λ max P

1 2 G d P -1 2 + P -1 2 G d P 1 2 and β = 1 2 λ min P 1 2 G d P -1 2 + P -1 2 G d P 1 2
. The latter theorem proves that any dilation d is strictly monotone if R n is equipped with the norm x =

√

x P x, provided that the matrix P 0 satisfies [START_REF] Huber | Recent Trends in Sliding Mode Control[END_REF].

Definition 4: [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] A continuous function p : R n → R + is said to be a d-homogeneous norm if p(x) → 0 as x → 0 and p(d(s)x) = e s p(x) > 0 for x ∈ R n \{0} and s ∈ R.

Obviously, the d-homogeneous norm is neither a norm nor semi-norm in the general case, since the triangle inequality may not hold. However, many authors (see e.g. [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Polyakov | On homogeneous finitetime control for linear evolution equation in hilbert space[END_REF] and references therein) call a function satisfying the latter definition a homogeneous norm. We follow this tradition.

The canonical homogeneous norm • d : R n → R + is defined as

x d = e sx where s x ∈ R : d(-s x )x = 1. (8)
The map • d : R n → [0, +∞) is well defined and single-valued for monotone dilations [START_REF] Polyakov | On homogeneous finitetime control for linear evolution equation in hilbert space[END_REF]. In [START_REF] Polyakov | On finite-time stabilization of evolution equations: A homogeneous approach[END_REF] such a homogeneous norm was called canonical because it is induced by a (canonical) norm in R n . Notice that

d(ln x d ) ≤ x ≤ d(ln x d ) for x ∈ R n ,
and, due to (7), • d is continuous at zero. Proposition 1: [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] If d is a strictly monotone dilation then

• the canonical homogeneous norm • d is Lipschitz continuous on R n \{0}; • if the norm • is smooth outside the origin then the homogeneous norm • d is also smooth outside the origin, d d(-s)x ds < 0 if s ∈ R, x ∈ R n \{0} and ∂ x d ∂x = x d ∂ z ∂z | z=d(-s)x ∂ z ∂z | z=d(-s)x G d d(-s)x s=ln x d (9)
Below we use the notation • d only for the canonical homogeneous norm induced by the weighted Euclidean norm x =

√

x P x with a matrix P 0 satisfying (6). The unit sphere S is defined using the same norm.

Vector fields, which are homogeneous with respect to dilation d, have many properties useful for control design and state estimation of linear and nonlinear plants as well as for analysis of convergence rates [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF].

Definition 5: [21] A vector field f : R n → R n (resp. a function h : R n → R) is said to be d-homogeneous if there exists ν ∈ R f (d(s)x) = e νs d(s)f (x), ∀x ∈ R n \{0}, ∀s ∈ R. (10) (resp. h(d(s)x) = e νs h(x), ∀x ∈ R n \{0}, ∀s ∈ R.) (11)
The number ν ∈ R is called the homogeneity degree of f (resp. h).

Let F d (R n ) (resp. H d (R n )) be the set of vector fields R n → R n (resp. functions R n → R) satisfying the identity (10) (resp. (11)), which are continuous on R n \{0}. Let deg F d (f ) (resp. deg H d (h)) denote the homogeneity degree of f ∈ F d (R n ) (resp. h ∈ H d (R n )).
The homogeneity allows local properties (e.g., smoothness) of vector fields (functions) to be extended globally [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF]. For instance [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF], the vector field f ∈ F d (R n ) is Lipschitz continuous on R n \{0} if and only if it satisfies Lipschitz condition on the unit sphere S. Similarly, since the map s → d(s)x is locally uniformly continuous, then uniform continuity of f ∈ F d (R n ) on the unit sphere implies its local uniform continuity on R n \{0} (see [START_REF] Polyakov | Consistent discretization of finite-time stable homogeneous systems[END_REF]).

B. Quadratic Stability of Nonlinear Homogeneous Systems

Homogeneity may simplify the analysis of differential equations. The most important property of d-homogeneous systems is the symmetry of solutions [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Grune | Homogeneous state feedback stabilization of homogeneous systems[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]. Namely, if

ϕ ξ0 : [0, T ) → R n is a solution to ξ = f (ξ), f ∈ F d (R n ) ( 12
)
with the initial condition ξ(0

) = ξ 0 ∈ R n , then ϕ d(s)ξ0 : [0, e -νs T ) → R n defined as ϕ d(s)ξ0 (t) = d(s)ϕ ξ0 (te νs ), s ∈ R is a solution to (12) with the initial condition ξ(0) = d(s)ξ 0 , where ν = deg F d (f ).
The latter property implies many corollaries. In this paper we use the next one.

Theorem 2: [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] The following five claims are equivalent: 1) The origin of the system (12) is asymptotically stable.

2) There exists a Lyapunov function V for the system [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] such that

V ∈ H d (R n ) ∩ C ∞ (R n ).
3) The origin of the system

ż = z 1+deg F d (f ) (In-G d )zz P z P G d z + I n f z z (13)
is asymptotically stable, where z = √ z P z, and the positive definite matrix P ∈ R n×n satisfies (6). 4) For any positive definite matrix P ∈ R n×n satisfying (6) there exists

Ψ ∈ F d (R n ) ∩ C ∞ (R n \{0}), deg F d (Ψ) = 0, such that Ψ is a diffeomorphism on R n \{0}, a homeo- morphism on R n , Ψ(0) = 0 and ∂(Ψ (ξ)P Ψ(ξ)) ∂ξ f (ξ) < 0 if Ψ (ξ)P Ψ(ξ) = 1. ( 14
)
Moreover, Ψ d ∈ H d (R n ) ∩ C ∞ (R n \{0}
) is a Lyapunov function for the system (12). 5) For any matrix P ∈ R n×n satisfying (6), there exists a smooth mapping

Ξ ∈ C ∞ (R n \{0}, R n×n ), det(Ξ(z)) = 0, ∂Ξ(z) ∂zi z = 0, Ξ(e s z) = Ξ(z) for z = (z 1 , ..., z n ) ∈ R n \{0}, s ∈ R, i = 1, ..., n z Ξ (z)P Ξ(z) (In-G d )zz P z P G d z +I n f z z < 0.
(15) This theorem proves two important facts:

• Any generalized homogeneous system ( 12) is homeomorphic on R n and diffeomorphic on R n \{0} to a standard homogeneous one [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF]. The corresponding change of coordinates is given by

z = Φ(ξ) := ξ d d(-ln ξ d )ξ (16)
and ξ = Φ -1 (z) := d(ln z ) z z . • Any asymptotically stable generalized homogeneous system is homeomorphic on R n and diffeomorphic on R n \{0} to a quadratically stable one. Indeed, making the change of variables z = Ψ(ξ) we derive

ż = f (z) = ∂Ψ(ξ) ∂ξ f (ξ) ξ=Ψ -1 (z)
, but the criterion ( 14) implies that z P ż < 0 if z P z = 1, so the homogeneous norm • d is the Lyapunov function to the latter system. Finally, the change of variable x = z d d(ln z d )z gives z d = x , so the transformed system ẋ = f (x) is quadratically stable with the Lyapunov function V defined as V (x) = x 2 = x P x, where

f (x) = x 1+deg F d (f ) (In-Gd)xx P x P Gdx +I n f x x .
Below we show that the transformations Φ and Ψ can be utilized for the design of a consistent discretization scheme for locally homogeneous systems.

Recall [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF] that if the homogeneous system ( 12) is asymptotically stable and deg Fd (f ) < 0, then it is globally uniformly finite-time stable.

Remark 1: If d is a dilation with the generator G d , then for any fixed α > 0, the group d α defined as d α (s) := d(αs), s ∈ R n , is the dilation with the generator

G d α = αG d . If f ∈ F d (R n ), then f ∈ F d α (R n ) and deg F d α (f ) = α deg F d (f ). In other words, if deg F d (f ) < 0, then a new dilation d α can be selected such that deg F d α (f ) = -1.

C. Local Homogeneity

Local homogeneous approximations has been studied in [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF] in the case of for weighted dilations. In the generalized case local homogeneity is defined analogously. Definition 6: A vector field

f 0 ∈ F d (R n ) is said to be a d- homogeneous approximation of a vector field f : R n → R n at 0 if lim s→-∞ e -deg F d (f0)s d(-s)f (d(s)x) -f 0 (x) = 0 uniformly on x ∈ S.
Local homogeneity at ∞ can be introduced similarly considering the limit s → +∞.

The next simple result follows the well-known ideas of homogeneous domination approach [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF].

Lemma 1: Let f 0 ∈ F d (R n ) be a local d-homogeneous approximation of a vector field f : R n → R n at 0 and deg Fd (f 0 ) < 0. If the system ẋ = f 0 (x) (17)
is asymptotically stable then the system (1) is locally finitetime stable.

A consistent discretization scheme for finite-time stable homogeneous systems is developed [START_REF] Polyakov | Consistent discretization of finite-time stable homogeneous systems[END_REF] based on Theorem 2. Below we use similar ideas in order to design a consistent discretization scheme locally (close to 0), where the dhomogeneous approximation f 0 is a dominating nonlinearity. The next trivial corollary plays a key role in the later constructions.

Corollary 1: If all conditions of Lemma 1 hold, then for any matrix P ∈ R n×n satisfying (6), there exist r > 0 and a smooth mapping

Ξ ∈ C ∞ (R n \{0}, R n×n ), det(Ξ(z)) = 0, ∂Ξ(z) ∂zi z = 0, Ξ(e s z) = Ξ(z), for z = (z 1 , ..., z n ) ∈ R n \{0}, s ∈ R, i = 1, ..., n, such that the inequality z Ξ (z)P Ξ(z) (In-G d )zz P z P G d z +I n d(-ln z )f d(ln z )z z < 0
holds for any z ∈ B(r)\{0}.

IV. FINITE-TIME STABLE IMPLICIT DISCRETIZATION

The main idea of the design of a finite-time stable discretization for homogeneous systems, is to use the coordinate transformation [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF].

Theorem 3: Let a vector field f : R n → R n be locally uniformly continuous on any ring Then there exists r > 0 such that the map The latter theorem is based on the fact that the system ẏ = f0 (y) admits a quadratic Lyapunov function (the condition [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF] with Ξ =const). However, as it was shown in Theorem 2, any stable homogeneous system is equivalent to a quadratically stable one. If f in Theorem 3 is replaced with the equivalent one:

K(r 1 , r 2 ) = {x ∈ R n : r 1 ≤ x ≤ r 2 }, 0 < r 1 < r 2 < r max < +∞ and admit a d-homogeneous approximation f 0 ∈ F d (R n ) at 0 such that deg F d (f 0 ) = -
Q : R + × R n × R n ⇒ R n defined as Q(h, x i , x i+1 ) = Q(h, Φ(x i ), Φ(x i+1 )), (18)
Φ(x) = x d d(-ln x d )x, Q(h, y i , y i+1 ) = y i+1 -y i -h F (y i+1 ), F (y) = ε>0 co f (y + B(ε)\{0}), (19) f (y
f new (x) = ∂Ψ(ξ) ∂ξ f (ξ) ξ=Ψ -1 (x) , x ∈ R n , ξ ∈ R n ,
where

Ψ ∈ F d (R n ) is a diffeomorphism on R n \{0}
V ∈ H d (R n ) ∩ C ∞ (R n ) with degree deg H (V ) = µ > 0
can always be found for any asymptotically stable system with a homogeneous vector field f ∈ F d (R n ). In this case, the required transformation Ψ can be defined as follows

Ψ(ξ) = d ln V 1/µ (ξ) ξ d ξ.
(see [START_REF] Polyakov | Quadratic-like stabilizability of homogeneous systems[END_REF] for more details).

In other words, if we know a homogeneous Lyapunov function for a finite-time stable homogeneous system then, we can easily design a consistent (in the sense of Definition 1) implicit discretization scheme.

Therefore, any finite-time stable system allowing a finitetime stable homogeneous approximation at 0 admits a consistent implicit approximation, at least locally.

V. EXAMPLE: DISCRETE-TIME APPROXIMATION OF A QUADROTOR CONTROL SYSTEM

Let us consider the system

ẋ = A(x)x + Bu, A(x) = 0 I 0 0 0 0 E(x) 0 0 0 0 I 0 0 0 0 , B = 0 0 0 I , where x ∈ R 8 , u ∈ R 2 , E(x) = sin(x 5 ) x 5 0 0 sin(x 6 ) cos(x 5 ) x 6
, I = ( 1 0 0 1 ). The system of this form appears as a simplified model for the control of horizontal position of a quadrotor, when the vertical position and a yaw angle are already stabilized in some set-points.

Obviously, A(x)x admits d-homogeneous approximation A 0 x at 0 with the homogeneity degree -1, d(s) = e sGd and G d = 4I 0 0 0 0 3I 0 0 0 0 2I 0 0 0 0 I , A 0 = 0 I 0 0 0 0 I 0 0 0 0 I 0 0 0 0 . Let the control law is defined as

u(x) = Kd(-ln x d )x (20)
where K ∈ R 2×8 such that

(A 0 + BK) P + P (A 0 + BK) + (G d P + P G d ) = 0, G d P + P G d 0, P 0.
Such a selection of K is always possible (see [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF]). Moreover, it can be shown that the control u has globally bounded and discontinuity only at the origin. The closed-loop system is locally finite-time stable, V = x d is a Lyapunov function and f 0 (x) = A 0 x + Bu(x) is a d-homogeneous approximation of the closed-loop system at zero, deg .

Obviously, F (y) = f (y) if y = 0 and

F (0) = (A 0 + BK + G d -I)B(1) ⊂ R n .
Hence, if y i ∈ h F (0) then the inclusion y i+1 ∈ y i + h F (y i+1 ) has a solution y i+1 = 0. For the case y i = h F (y i+1 ) some properly adapted numerical procedures (like Newton method) is required for solving of the obtained discrete-time inclusion y i ∈ y i+1 -h F (y i+1 ). Notice, that f is smooth outside the origin, so the Newton method can be applied while, for example, y

[k] i+1 / ∈ h F (0),
where k denotes the number of Newton's iteration. The latter inclusion can be utilized as one of conditions for termination of the Newton iterations, since next the state of the system converges to zero in one step.

Let us denote q i+1 = y i+1 , z i+1 = yi+1 yi+1 and à = A 0 + BK + G d . Similarly to [START_REF] Polyakov | Consistent discretization of finite-time stable homogeneous systems[END_REF] the discretized control can be defined as u i = u(x i+1 ) = u Φ -1 (y) = Kz i+1 . If y i (I n -Ã) -P (I n -Ã) -1 y i ≤ h 2 then y i ∈ h F (0), q i+1 = 0 and z i+1 = h -1 (I n -Ã) -1 y i ;

Otherwise, according the scheme proposed above the value of y i+1 (as well as q i+1 = 0 and z i+1 ) are assumed to be derived by means of a Newton method. Notice that the matrix I -Ã is invertible [START_REF] Polyakov | Consistent discretization of finite-time stable homogeneous systems[END_REF], so the control u i is well defined. The original state is given by x i+1 = d(ln y i+1 )y i+1 / y i+1 . The simulation results for the consistent discretization scheme with h = 0.04 are given in Fig. 2 and the results for the explicit Euler discretization with the same sampling period are depicted in Fig. 3. They confirm finite-time convergence of the consistent discretization scheme and the numerical chattering of the explicit one.

VI. DISCUSSIONS AND CONCLUSIONS

In the paper it is shown that any homogeneous finitetime stable system allowing finite-time stable homogeneous approximation at 0 always admits an implicit discretization scheme preserving finite-time convergence. For practical implementation, implicit discretization scheme needs to be supported with an appropriated algorithm of solving the corresponding implicit equation/inclusion [START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF]. Development of such algorithms is planned as future research.

  s→-∞ d(s)x = 0 and lim s→+∞ d(s)x = +∞ uniformly on the unit sphere S := {x : x = 1}. The dilation d is a continuous group of invertible linear maps d(s) ∈ R n×n , d(-s) = [d(s)] -1 . The matrix G d ∈ R n×n G d = lim s→0 d(s)-I s is known [28, Chapter 1] as the generator of the group d. It satisfies the following properties d d(s) ds

1

 1 and the condition (15) with Ξ = I n holds for f 0 .

  ) := y (I-G d )yy P y G d P y +I n d(ln y )f d(ln y ) y y , is a consistent discrete-time approximation in U = {x ∈ R n : x d ≤ r/2} of the locally uniformly finite-time stable system (1) with the maximum step size h 0 = r 2 sup y∈R n : y ≤r f (y) .

  given in Theorem 2, then the condition Ξ = I n required for Theorem 3 is fulfilled.According to Theorem 2, a homogeneous Lyapunov function

Fd (f 0 ) = - 1 .

 1 All conditions of Theorem 3 are fulfilled. Applying the coordinate transformation y = Φ(x) = x d d(ln x d )x we derive f (y) = A 0 +BK + 1-y P ∆A(y)y y GdP y (G d -I)+∆A(y) y y , y y 6 ) cos( y y 5 ) y y 6

Fig. 2 .

 2 Fig. 2. Simulation results for the consistently discretized model.

Fig. 3 .

 3 Fig. 3. Simulation results for the explicit Euler method.

The matrix G d ∈ R n is anti-Hurwitz if -G d is Hurwitz.

VII. ACKNOWLEDGMENTS

The authors thank the support of the French National Research Agency (ANR), Project ANR-18-CE40-0008 "DIG-ITSLID". The first and the second authors also acknowledge the support of the Government of Russian Federation (Grant 08-08) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).


Olivier Pays
email: olivier.pays@univ-angers.fr

Alice Ekori

‡ Herv E Fritz

On the Advantages of Mixed-Species Groups: Impalas Adjust Their Vigilance When Associated With Larger Prey Herbivores

Keywords: antipredator behaviour, group living, group size, herbivores, mixed-species group, predator detection, vigilance

Prey can obtain valuable benefits from associating with other species if heterospecifics help to detect predators or locate good food patches. In mixed-species groups, how species respond to the presence of other species remains a poorly explored question although it might give crucial insights into mechanisms underlying the interspecific coexistence. We studied temporary mixed-species groups of large herbivores in Hwange National Park (Zimbabwe) between the common impala (Aepyceros melampus), the focal species here, and bigger species including the plains zebra (Equus quagga), the greater kudu (Tragelaphus strepsiceros) or the blue wildebeest (Connochaetes taurinus). In the Hwange savanna, the focal and smaller species are exposed to a larger range of predators than the associated species. In this context, we investigated how impalas adjusted their vigilance with group size comparing impala-only and mixed-species groups and whether the identity of heterospecifics affected vigilance of impalas. Our study showed that the time impalas spent in vigilance significantly decreased with group size when they formed impala-only groups, whereas it did not significantly vary with group size in mixed-species groups. Moreover, in mixed-species groups, impalas did not adjust their time spent in vigilance with the proportion of conspecifics and the identity of the associated species. Thus, the mechanism underlying the difference of impalas' behavioural adjustment of vigilance with group size between single-and mixed-species groups seemed to be related to the presence but not to the number and the identity of heteropecifics. Finally, we discuss the concept that larger and dominant heterospecifics were likely to increase competition for food access, thereby forcing higher vigilance of impalas, outweighing any reduction from collective vigilance.

Introduction

While animals can acquire information about the environment from their own perception, they can also acquire information from other conspecifics [START_REF] Dall | Information and its use by animals in evolutionary ecology[END_REF]; see [START_REF] Goodale | Interspecific information transfer influences animal community structure[END_REF] for a review). In this context, animals can also obtain valuable benefits by being associated with other species (forming temporary or stable mixed-species groups) if the presence and behaviours of heterospecifics provide information facilitating the detection of predators [START_REF] Rainey | Hornbills can distinguish between primate alarm calls[END_REF][START_REF] Templeton | Nuthatches eavesdrop on variations in heterospecific chickadee mobbing alarm calls[END_REF][START_REF] Aplin | Social networks predict patch discovery in a wild population of songbirds[END_REF] or the location of potentially good foraging patches [START_REF] Richter | Resource choice of social wasps: influence of presence, size and species of resident wasps[END_REF][START_REF] Stout | The use of conspecific and interspecific scent marks by foraging bumblebees and honeybees[END_REF]; but see the concept of public information in [START_REF] Danchin | Public information: from nosy neighbors to cultural evolution[END_REF][START_REF] Valone | From eavesdropping on performance to copying the behavior of others: a review of public information use[END_REF]. Indeed, the focal and associated species may be sensitive to different cues, and the sum of information processed in a mixed-species group could make individuals more aware of their environment than they would be in single-species group [START_REF] Gautier-Hion | Monospecific vs polyspecific life: a comparative study of foraging and antipredatory tactics in a community of Cercopithecus monkeys[END_REF][START_REF] Goodale | Response to conspecific and heterospecific alarm calls in mixed-species bird flocks of a Sri Lankan rainforest[END_REF]. Moreover, it can be profitable if the focal and associated species are of the same trophic level, as the two associated species need to avoid the same predators and find similar resources. Such mixedspecies groups might be even more profitable when the associated species shares with the focal species several common predators but forages upon different resources. In this context, individuals of the focal species can benefit in predator detection while suffering little competition for food access [START_REF] Morse | Ecological aspects of some mixed-species foraging flocks of birds[END_REF].

Stable mixed-species groups that persist regardless of the heterogeneity of the distribution of threats and resources are formed by many taxa including birds [START_REF] Jullien | Multi-species territoriality and dynamics in neotropical forest understorey bird flocks[END_REF][START_REF] Sridhar | Why do birds participate in mixed-species foraging flocks? A large-scale synthesis[END_REF], fish [START_REF] Ward | Mixed-species shoaling in fish: the sensory mechanisms and costs of shoal choice[END_REF], insects [START_REF] Menzel | Selective interspecific tolerance in tropical Crematogaster-Camponotus associations[END_REF], mammals [START_REF] Stensland | Mixed species groups in mammals[END_REF] or even involving several taxa [START_REF] Rasa | Dwarf mongoose and hornbill mutualism in the Taru Desert, Kenya[END_REF]. Here, in contrast to stable groups, this article examines mixed-species groups involving two species temporarily associated while foraging. While several studies have concentrated on the antipredator advantages of mixed-species grouping in birds, ungulate associations have received relatively little attention although ungulates are often found in mixed-species groups [START_REF] Washburn | The social life of baboons[END_REF][START_REF] Altmann | Baboon Ecology[END_REF][START_REF] Elder | Social groupings and primate associations of the bushbuck (Tragelaphus scriptus)[END_REF][START_REF] Leuthold | African Ungulates: a Comparative Review of Their Ethology and Behavioral Ecology[END_REF][START_REF] Gosling | Defense guilds of savannah ungulates as a context for scent communication[END_REF][START_REF] Sinclair | Does interspecific competition or predation shape the African ungulate community?[END_REF][START_REF] Fitzgibbon | Mixed-species grouping in Thomson's and Grant's gazelles: the antipredator benefits[END_REF]. Indeed, in large herbivores, how the focal species responds to the presence of the associated species remains a poorly explored question although it might give crucial information on the existence of such mixed-species groups and even more on mechanisms underlying the interspecific coexistence.

Several reasons have been proposed in the literature to explain mixed-species groups [START_REF] Sridhar | Why do birds participate in mixed-species foraging flocks? A large-scale synthesis[END_REF][START_REF] Harrison | Mixed-species flocks: an example of niche construction?[END_REF][START_REF] Farine | Social network analysis of mixed-species flocks: exploring the structure and evolution of interspecific social behaviour[END_REF]. Studies have reported that many mammals and birds have learned to associate heterospecific alarm calls with the presence of a predator [START_REF] Hauser | How infant vervet monkeys learn to recognize starling alarm callsthe role of experience[END_REF][START_REF] Shriner | Yellow-bellied marmot and goldenmantled ground squirrel responses to heterospecific alarm calls[END_REF][START_REF] Fichtel | Reciprocal recognition of sifaka (Propithecus verreauxi verreauxi) and red-fronted lemur (Eulemur fulvus rufus) alarm calls[END_REF][START_REF] Rainey | Hornbills can distinguish between primate alarm calls[END_REF][START_REF] Magrath | A mutual understanding? Interspecific responses by birds to each other's aerial alarm calls[END_REF]. Moreover, species that normally form small groups can increase group size by heterospecific association and hence enhance the effectiveness of antipredator behaviour (as the dilution effect or collective detection) [START_REF] Goodale | Interspecific information transfer influences animal community structure[END_REF]). An additional benefit of mixed-species grouping may arise if predators show a preference for a particular prey species. [START_REF] Sinclair | Does interspecific competition or predation shape the African ungulate community?[END_REF], for example, suggested that plains zebra, Equus quagga, in the Serengeti can reduce their risk of predation by staying close to Serengeti white-bearded wildebeest, Connochaetes mearnsi, the preferred prey of large carnivores in the area. Fitz-Gibbon (1990) reported that Grant's gazelles Nanger granti benefited from the association with Thomson's gazelles Eudorcas thomsonii because the cheetahs Acinonyx jubatus preference for the smaller Thomson's gazelles reduced the rate at which the Grant's gazelles were attacked when they formed mixed-species groups. However, while antipredator benefits might bring crucial support to our understanding of the existence of such mixed-species groups, several disadvantages can also arise. Indeed, if the associated species are socially or morphologically dominant and can increase their foraging success by kleptoparasitism of focal species or by local enhancement, heterospecific association may lead to a higher resource competition, both in terms of resource exploitation and interference competition. Accordingly, [START_REF] Krause | Living in Groups[END_REF] suggested that there were two main reasons why focal species would tolerate associated species:

(1) all species would compensate by being able to reduce antipredator vigilance when associated with others; or (2) the cost of avoiding these species might be too great compared with the loss of foraging opportunities.

We studied temporary mixed-species groups of two large herbivores in Hwange National Park, Zimbabwe. In our study area, the impala (Aepyceros melampus), the focal species in this study, is temporarily associated while foraging with different associated species including the plains zebra, the greater kudu (Tragelaphus strepsiceros) or the blue wildebeest (Connochaetes taurinus). Temporary mixed-species groups are particularly interested to study from the impalas' perspective (i.e. that of the focal species) as the focal species were associated with larger body-size herbivores. Adult (male/female) impala, zebra, greater kudu and wildebeest weigh on average 60/45, 250/220, 257/ 170 and 200/163 kg, respectively [START_REF] Estes | The Behavior Guide to African Mammals[END_REF]). In the Hwange context, all of these medium-size herbivores are preyed upon by a large range of predators [START_REF] Fritz | A food web perspective on large herbivore community limitation[END_REF]) but impalas, the focal and smallest species, were expected to suffer from a larger range of predators than the associated species [START_REF] Sinclair | Patterns of predation in a diverse predator-prey system[END_REF]. Here, the aim of this study was to investigate whether impalas adjusted their vigilance when they formed mixed-species groups (i.e. when they were associated with one of the other species cited above). We thus compared the relationship between vigilance and group size in impalas when they formed impala-only and mixed-species groups. We also tested whether impalas adjusted their vigilance with the identity of associated species and the number of heterospecifics they were associated with.

Our predictions were the following. We hypothesized that impalas in impala-only groups should decrease their vigilance with group size if the main function of vigilance in this prey species was ascribed to predator detection [START_REF] Pays | Detecting predators and locating competitors while foraging: an experimental study of a medium-sized herbivore in an African savanna[END_REF]. As the risk of predation to individuals may be reduced in large groups as a result of dilution [START_REF] Bednekoff | Randomness, chaos and confusion in the study of antipredator vigilance[END_REF], an individual could benefit from an increase in group size by reducing its own rate of antipredator vigilance, thus increasing its time available for feeding, without reducing the group's probability of detecting potential danger [START_REF] Ale | The contingencies of group size and vigilance[END_REF]Pays et al. 2007b). If heterospecifics were perceived as 'supplementary impalas', the mixed-species group would lead to an increase of the perceived group size (i.e. the overall group size). Thus, we would expect a negative trend between group size and vigilance, indicating a dilution-effect process [START_REF] Hamilton | Geometry for the selfish herd[END_REF]. If heterospecifics relaxed impalas in their level of perception of predation risk, we would expect under this assumption that individuals in mixed-species groups were on average less vigilant than in impala-only groups. The third assumption would consider that, while heterospecifics would relax impalas allowing them to reduce their antipredator vigilance, the associated species, morphologically dominant, would affect impalas in their food access. Indeed, the arrival of larger grazers might force impalas to leave from their current patch, if these patches are situated on the larger grazers' path. Thus, to avoid this passive interference (i.e. without aggression), impalas would increase their social monitoring in mixed-species groups. Thus, under this third assumption, we would observe two compensating mechanisms. The proportion of time impalas spent monitoring other conspecifics (to maintain social cohesion) and heterospecifics (to limit interference) would increase, whereas the proportion of time they spent scanning the environment (i.e. antipredator vigilance) would decrease with group size; as a result, overall vigilance level of impalas might not change with group size in mixed-species groups [START_REF] Favreau | Interactions among social monitoring, anti-predator vigilance and group size in eastern grey kangaroos[END_REF]. Finally, a similar pattern would be observed in mixed-species groups if impalas increased their vigilance in large groups of heterospecifics as a consequence of larger heterospecifics in mixed-species groups increasing predation risk, which may occur as lions prefer to catch larger-bodied animals (Loveridge et al. 2007a). Thus, impalas would experience an increase in the risk of being opportunistically preyed upon in the confusion of the chase if lions attack larger mixed-species groups including zebras, wildebeests or kudus.

Methods

Study Area and Animals

The fieldwork was conducted in the Main Camp area of Hwange National Park (HNP) in South-Western Zimbabwe (19°00 0 S, 26°30 0 E) from mid-Feb. to mid-Apr. 2010. The study site is an open grassland area of ca. 64 ha surrounded by Acacia/Combretum bush, one of the plant communities characteristic of the mixedbushed grassland of the eastern Kalahari sands region [START_REF] Rogers | A Woody Vegetation Survey of Hwange National Park[END_REF]). Long-term mean annual rainfall is around 606 mm, which mainly occurs during a rainy season from the end of Oct. to the end of Apr. [START_REF] Chamaill E-Jammes | Detecting climate changes of concerns in highly variable environment: quantiles regressions reveal that droughts worsen in Hwange National Park, Zimbabwe[END_REF].

Observations were performed near Main Camp. In this area, the main predators of impalas comprised spotted hyenas, Crocuta crocuta, wild dogs, Lycaon pictus, leopards, Panthera pardus, cheetahs and occasionally lions, Panthera leo [START_REF] Bourgarel | Approche de la dynamique des populations de grands herbivores dans une aire prot eg ee: L'exemple de l'impala (Aepyceros melampus) au Zimbabwe[END_REF]). During the wet seasons, densities of impalas, zebras, kudus and wildebeests were estimated to be 0.98 AE 0.36, 1.08 AE 0.53, 0.60 AE 0.22 and 0.23 AE 0.11 ind/km², respectively (Chamaill e-Jammes et al. 2009).

We observed 50-100 female impalas daily at the study site, 3-20 zebras, 2-12 kudus and 6-17 wildebeests mainly foraging (mostly grazing, licking saltrich soil or browsing for kudus) in the open. All female impalas in the study area formed a single clan that was divided into a variable number of groups with marked fusion-fission dynamics; this was known because about 30 adult females were ear-tagged. On the grassland, impalas occasionally form mixedspecies groups when foraging, being temporarily associated with zebras or kudus or wildebeests.

Ethics Statement

Our study was conducted under permits from the Director General of the Zimbabwe Parks and approved by the Wildlife Management Authority (Ref: D/M/ Gen/(T), Permit: 23(l) (c) (ii) 01/2010), and it complied with the current laws of Zimbabwe. The study was based only on focal sampling, and our disturbance to wildlife was consequently very limited. We did not capture and ear-tag impalas for the purpose of this study.

Behavioural Data

Behavioural data were collected by focal sampling from vehicles, respecting a minimal distance of 100 m between the focal group and the observer to minimize disturbance. Animals were chosen at random for a 5-min focal sample, during the daytime. We recorded only groups whose predominant activity was foraging and which did not move far during the recording, ensuring that group size and composition did not change during the 5-min focal sample. Spatial groups were identified on the basis of a maximal separation between adjacent members of 50 m, and on the maintenance of social and spatial cohesion of the group members during all focal sampling. There were no ambiguities in defining a group using these criteria as interindividual distances were very small. Group size was therefore taken as being equal to the size of the spatial group. We distinguished impala-only from mixed-species groups in which impalas were associated with at least one heterospecific (zebras, kudus or wildebeests). For each observation, we determined group size, the numbers of impalas and heterospecifics and the species impalas were associated with during the focal sample. The mixed-species groups that we sampled included impalas and only one of the species cited above (i.e. we sampled no mixed-species groups with three or more species).

Observations were performed in two periods only, the 3 h after sunrise and the 3 h before sundown, for two main reasons: (1) these are the daylight periods within which the main predators are active [START_REF] Hunter | Vigilance behavior in african ungulates: the role of predation pressure[END_REF], and we therefore expected that impalas and other studied prey would then be likely to exhibit antipredator strategies; and (2) impalas are mainly engaged in foraging activity in these periods [START_REF] Jarman | Daily activity of impala[END_REF]). Although we tended to limit the number of observations during any single morning or afternoon session, several individuals were sampled during some sessions. In such cases, the observer took care to avoid studying the same individual twice per session by preferentially observing tagged animals. However, although re-sampling was possible on untagged animals, we strongly believe that it represented a negligible part of our data set.

Distance to cover (D ≤ 50, 50 < D ≤ 100, D > 100 m) was estimated while sampling individuals. It is commonly used as a proxy for predation risk [START_REF] Lima | Protective cover and the use of space: different strategies in finches[END_REF][START_REF] Burger | Factors affecting vigilance in springbok: importance of vegetative cover, location in herd, and herd size[END_REF][START_REF] Blumstein | Group size but not distance to cover influences agile wallaby (Macropus agilis) time allocation[END_REF]), but its role (i.e. obstructive or protective) is ambiguous for prey, and depends on many factors concerning both the predator and the prey. Expectations for the effect of distance to cover on individual vigilance are therefore not clear. However, impalas and other larger studied prey in the area including zebras and wildebeests graze in open areas, and as their main predators spend the daytime in cover and hunt from cover (e.g. [START_REF] Loarie | Lion hunting behaviour and vegetation structure in an African savanna[END_REF] for lions), we can assume that they should perceive closeness to cover as risky.

Visibility around foraging individuals provides direct information on their visual obstruction [START_REF] Whittingham | The effect of limited visibility on vigilance behaviour and speed of predator detection: implications for the conservation of granivorous passerines[END_REF]). [START_REF] Pays | Detecting predators and locating competitors while foraging: an experimental study of a medium-sized herbivore in an African savanna[END_REF] showed that impalas adjusted their vigilance in relation to grass height as visibility around them varied. Therefore, we estimated the grass height of the patch in which the focal individual was foraging. We considered grass to be short when it was below the hooves of the focal impala, medium, below the upper part of the metacarpals, and tall when grass height reached the tibia, which is above their eyes when feeding. In tall grass, visibility is strongly reduced when feeding in the herb layer. Observations were not conducted when wind speeds were high.

We studied adult females of impalas to avoid any sex or age effect [START_REF] Pays | Does sex affect individual and collective vigilance in the eastern grey kangaroo?[END_REF]. From each observation, we determined the time during which an impala was vigilant. We defined vigilance in an animal as being when it raised its head above the horizontal with marked orientation of ears, scanning its surroundings, without moving its feet. No ambiguities were encountered in distinguishing a vigilant from a non-vigilant animal. The position of the individuals within a group (i.e. peripheral or central; e.g. [START_REF] Blanchard | Within-group spatial position and vigilance: a role also for competition? The case of impalas with a controlled food supply[END_REF]) was impossible to determine during the sampling because the animals moved frequently when foraging. For all focal individuals, we determined the distance between the focal individual and the nearest impala (DNearImpala) (D ≤ 5, D > 5 m) and, in mixed-species groups, the distance to the nearest heterospecific (D ≤ 10, D > 10 m). Such interindividual classes allowed us to define unambiguous situations when animals were foraging during 5-min focal samples.

We sampled the behaviours of 56 impalas, 25 in impala-only groups and 31 in mixed-species groups comprising 11, 7 and 13 groups with zebras, wildebeests and kudus, respectively. The ranges of group sizes of impala-only and mixed-species groups were similar (i.e. from 10 to 76 and 10 to 73 individuals, respectively). In mixed-species groups, the proportion of impalas ranged from 0.16 to 0.93.

Data Analyses

As the major aim of this study was to investigate whether impalas adjusted their vigilance with group size differently when associated with other larger prey, we used a linear fixed-effects model approach and conducted a model selection starting with a model including group size, group type (impala-only vs. mixed-species groups) and the interaction between these two variables (Table 1). To achieve normality and homoscedasticity and to reach linearity between our variables in all statistical procedures described above, we arcsine square root-transformed the proportion of time spent in vigilance and logtransformed the group size. To test for significance of effects, we used a backward selection procedure, beginning with the full model including the two-way interaction Log(group size) 9 group type (model 1, Table 1) and comparing it to the model without the interaction (model 2, Table 1) using a F test. The same procedure was applied to test for the effect of group size (model 2 vs. model 4, Table 1) and group type (model 2 vs. model 3, Table 1). We also verified that the selected model had the lowest Akaike's corrected criterion (AICc) and a DAICc > 2 compared with the closest competing ones, including the null model in the comparison.

Using the same procedure (i.e. model comparison using F-value and AICc difference), we tested whether distance to the nearest impala (D ≤ 5 vs. D > 5 m), distance to cover (D ≤ 50, 50 < D ≤ 100 vs. D > 100 m) and grass height (short, medium vs. tall) affected the proportion of time an impala spent in vigilance. Here, model 1 including Log(group size), group type and the two-way interaction (Table 1) was compared with the same model including one of each of the factors cited above (model 5, 6, 7, Table 1).

Finally, we investigated whether impalas in mixedspecies groups adjusted their vigilance to the proportion of impalas, the species they were associated with, and distance to the nearest heterospecific. Using the same procedure, we compared the null model (without including the Log(group size) as it did not significantly affect vigilance in mixed-species groups; see Table 3) and a model including one of the tested variable cited above (Table 3).

All analyses were performed in R 2.15.1 (R Development Core Team 2012).

Results

Model comparison indicated that group size and group type significantly affected the time an impala spent in vigilance (model 2, Table 1). According to the coefficients derived for each factor and the interaction group size 9 group type (Table 2), the time impalas spent in vigilance significantly decreased with group size when they formed impala-only groups but did not significantly vary with group size in mixedspecies groups, with a lower level of vigilance in mixed-groups (Fig. 1). However, Table 1 showed that neither distance to the nearest impala nor grass height significantly affected the time an impala spent in vigilance. Moreover, considering the two outcomes for model comparison between models 1 and 6 (p = 0.077 and DAICc = 1.73, Table 1), we did not consider that distance to cover had a significant effect on the time an impala spent in vigilance.

Finally, in mixed-species groups, impalas did not change their time spent in vigilance with the proportion of conspecifics, the species they were associated with and the distance to the nearest heterospecific (Table 3).

Discussion

Our main result indicated that the time impalas spent in vigilance significantly decreased with group size when impalas formed single-species groups, whereas it did not significantly vary with group size in mixedspecies groups. This negative trend in single-species groups has been reported in many species including birds and mammals (Pays et al. 2007a;[START_REF] Beauchamp | What is the magnitude of the groupsize effect on vigilance?[END_REF]. As the risk of predation to individuals was reduced in large groups as a result of dilution [START_REF] Bednekoff | Randomness, chaos and confusion in the study of antipredator vigilance[END_REF], impalas tended to reduce their own rate of antipredator vigilance when group size increased, thus increasing their time available for feeding, without reducing the group's probability of detecting potential danger [START_REF] Ale | The contingencies of group size and vigilance[END_REF]Pays et al. 2007b). Although vigilance might also serve other functions, including detection of potential competitors [START_REF] Slotow | Influence of social status, distance from cover, and group size on feeding and vigilance in White-crowned Sparrows[END_REF][START_REF] Cresswell | Interference competition at low competitor densities in blackbirds Turdus merula[END_REF], assessment of patch quality [START_REF] Templeton | Vicarious sampling: the use of personal and public information by starlings foraging in a simple patchy environment[END_REF][START_REF] Smith | The use and misuse of public information by foraging red crossbills[END_REF], searching for mates [START_REF] Burger | Effects of group size and sex on vigilance in ostriches (Struthio camelus): antipredator strategy or mate competition?[END_REF], and location of food patches discovered by companions [START_REF] Coolen | Head position as an indicator of producer and scrounger tactics in a ground feeding bird[END_REF], this negative trend of vigilance with group size indicated that vigilance in prey species of such body size was mainly used for antipredator detection [START_REF] Pulliam | On the advantages of flocking[END_REF]. This result was expected as (1) populations of African ungulates like impala, in communities rich in predators, are usually top-down regulated, rather than bottom-up [START_REF] Sinclair | Patterns of predation in a diverse predator-prey system[END_REF], ( 2) predator off-take on mesoherbivores in the Hwange system were abundant [START_REF] Fritz | A food web perspective on large herbivore community limitation[END_REF], and (3) impalas were preyed upon by a large range of predators including spotted hyaenas, lions, leopards, cheetahs, African wild dogs as well as black-backed jackals, Canis mesomelas, for young impalas [START_REF] Bourgarel | Approche de la dynamique des populations de grands herbivores dans une aire prot eg ee: L'exemple de l'impala (Aepyceros melampus) au Zimbabwe[END_REF][START_REF] Pays | Detecting predators and locating competitors while foraging: an experimental study of a medium-sized herbivore in an African savanna[END_REF].

The time impalas spent in vigilance did not significantly vary with group size in mixed-species groups. This result does not the support the assumption that heterospecifics were simply perceived as 'supplementary impalas'. As we observed that impalas spent more time in vigilance in small-to medium-impala-only groups than in small-to medium-mixed-species groups (Fig. 1), our results support the assumption that heterospecifics relaxed impalas in their level of perception of predation risk. However, the result of the interaction between group size and group type suggests that this pattern is valid only up to 21 individuals (i.e. when Log(group size) = 1.32 in Fig. 1) and apparently reversed beyond this threshold group Group size of mixed-species groups (GS) was log-transformed. PropImpala was the proportion of impalas in mixed-species groups. Mixed-species groups (Species) (Impalas associated with zebras, wildebeests or kudus) and distance to the nearest heterospecific (DNearHeteroSp) (D ≤ 10, D > 10 m) were considered as categorical. Levels used as the references in the models are italicized in the legends.

size. While other species would relax impalas in their investment in vigilance in small-to medium-size groups, they would tend to make impalas maintain a higher vigilance level in large mixed-species groups. This result supported previous theoretical models [START_REF] Beauchamp | Should vigilance always decrease with group size?[END_REF]) and empirical observations in single-species groups of mammals [START_REF] Treves | Within-group vigilance in red colobus and redtail monkeys[END_REF]. For instance, it has been reported that although eastern grey kangaroos, Macropus giganteus, spent on average most of their vigilance time in antipredator vigilance (about 75% of the overall vigilance), their proportion of time spent looking at other group members increased with group size, while the proportion of time they spent scanning the environment decreased with group size; as a result, overall vigilance levels did not change with group size [START_REF] Favreau | Interactions among social monitoring, anti-predator vigilance and group size in eastern grey kangaroos[END_REF]. Our result here showed that such a mechanism would apply for the focal species in mixed-species groups. In future studies, while impalas are less vigilant in large groups of conspecifics, it would be interesting to sample more small mixed-species groups (even rare in nature) to confirm that impalas would be less vigilant in this context. Moreover, it would be interesting to investigate the target of each vigilance act exhibited by individuals [following Favreau et al.'s method (2010)] to support our suggestion that the lack of a group-size effect on overall vigilance in multispecies groups is caused by an increase of time allocation to social vigilance with group size.

In mixed-species groups, impalas did not adjust their time spent in vigilance with the proportion of conspecifics. Thus, the mechanism underlying the difference of behavioural adjustment of vigilance with group size between single-and mixed-species groups in impalas seemed to be related to the presence but not to the number of heteropecifics. Forming mixedspecies groups for impalas might be profitable as it would allow impalas to reduce their level of risk endured as (1) the presence and behaviour of heterospecifics might improve predator detection (Templeton & Greene 2007) particularly if the impala population is regulated by predation [START_REF] Sinclair | Patterns of predation in a diverse predator-prey system[END_REF], (2) collective detection with heterospecifics was expected to be higher than the one observed in an impala-only group (of a similar group size) particularly if the associated species is sensitive to different cues [START_REF] Goodale | Response to conspecific and heterospecific alarm calls in mixed-species bird flocks of a Sri Lankan rainforest[END_REF], and (3) previous studies on the Hwange system have reported that zebras, wildebeests and kudus were the preferred prey of lions (Loveridge et al. 2007a,b). [START_REF] Sinclair | Does interspecific competition or predation shape the African ungulate community?[END_REF] suggested that zebra in the Serengeti tended to stay close to wildebeest to reduce their own risk of predation as wildebeest were the preferred prey of all the large car-nivore species [START_REF] Kruuk | The Spotted Hyena[END_REF][START_REF] Schaller | The Serengeti Lion[END_REF], and he concluded that predation pressure should cause the mixed-species herding. However, associating with a preferred prey species of lion, impalas may be exposed to more lion attacks and therefore increased the risk of being opportunistically preyed upon in the confusion of the chase. [START_REF] Fitzgibbon | Mixed-species grouping in Thomson's and Grant's gazelles: the antipredator benefits[END_REF] reported that Thomson's gazelles in mixed-species groups shared vigilance to some extent with Grant's gazelles, leaving more time available for feeding. However, compared with joining conspecifics, FitzGibbon (1990) suggested that joining Grant's gazelles to form mixedspecies groups had few additional advantages and such associations were likely to be beneficial only when the number of conspecifics in the resulting mixed-species groups was relatively high. Thus, when they have the opportunity (i.e. when other species are in the surroundings), impalas might form multispecies groups to limit the effect of predation; more work is however needed to understand the proximate mechanisms promoting mixed-species grouping.

The lack of group-size effect on vigilance in multispecies groups, possibly caused by an increase of social monitoring with group size in multispecies groups, might explain why not all impalas form mixed-species groups with larger body-size herbivores when foraging despite an increased advantage in detecting predators. Although we did not detect any effect of species and distance to the nearest heterospecific in mixedspecies groups, impalas may suffer from being close to a heterospecific because of disturbance and competition while foraging. Impalas, because of their lower daily intake, can tolerate shorter swards and more dispersed bites than zebras, wildebeest and kudus because of body-mass allometry [START_REF] Wilmshurst | What constrains daily intake in Thomson's gazelles?[END_REF][START_REF] Owen-Smith | Adaptive Herbivore Ecology. From Resources to Populations in Variable Environments[END_REF]. Nevertheless, both zebras and wildebeests can graze swards as short as or even shorter than impalas can [START_REF] Arsenault | Resource partitioning by grass height among grazing ungulates does not follow body size relation[END_REF]) and hence would be strong competitors for small patches of short green flush. However, we might not expect impalas to suffer from severe competition for forage from these much larger heterospecifics if (1) impalas with their much smaller body size have much lower absolute food intake demands thereby being able to meet maintenance requirements more easily than larger-bodied herbivores on short grass [START_REF] Illius | The allometry of food intake in grazing ruminants[END_REF], (2) impala have more selective mouths than the other species thereby enabling them to more efficiently access quality forage that the larger species cannot, and (3) the larger species may facilitate grazing for impala by removing stems and litter thereby increasing access to basal leaves. Thus more investigations are needed here to examine the likelihood of the hypothesis of food competition as in our context, kudus and impalas should be dietarily more separate (than impalas and zebras or wildebeests), so the risk of foraging interference would have been less in impala-kudu groups. Further, if the mechanism underlying mixed-species grouping were not shaped by predation pressure but by resource distribution with species searching for the same food in a heterogeneously food-distributed landscape, impalas might pay a foraging cost due to interference in the temporary association with morphologically dominant associated species. It might be particularly true when food is rare in the environment or rapidly depleting within feeding patches. Consequently, even if increasing social monitoring with group size might be profitable for resource detection, impalas may be more constrained in their resource access in mixedspecies groups. However, impalas might tolerate the presence of other species as they would compensate by being able to reduce antipredator vigilance and/or the cost of avoiding associates might be too great compared with the cost of lost foraging opportunities in the patch or of movement to another patch [START_REF] Krause | Living in Groups[END_REF]. In conclusion, more studies are needed to investigate (1) whether temporary mixed-species grouping allows impalas to maximize their food intake and ultimately their survival, and (2) both interference costs and predation risks advantages from mixed-species groups.

  The presence of ear-tagged impalas on the area is explained by a long-term individual-based study on population dynamics in HNP started in 2007 supervised by Dr. H. Fritz under permits from the same authorities cited above (Ref: D/M/Gen/(T): Permit: 23(l) (c) (ii) 12/2007; 23(l) (c) (ii) 31/2007; 23(l) (c) (ii) 04/2009; 23(l) (c) (ii) 01/2010).

Fig. 1 :

 1 Fig. 1: Relationship between the arcsine square root-transformed proportion of time an impala spent in vigilance and the log-transformed group size in impala groups (black dots, full line) and mixed-species groups (open symbols, dashed line). Fitted curves are derived from the model 1 in theTable 1 and coefficients in Table 2.

Table 1 :

 1 Model comparison allowing to test for the effect of group size, group type, distance to the nearest impala, distance to cover and grass height on the arcsine square root-transformed proportion of time an impala spent in vigilance

	ID Model	Model structure	LogLike	AICc	df	Model comparison	F	df	p
	0	h 0	37.79	À71.58	2
	1	h 0 + h 1 ÁLogGS + h 2 ÁGT + h 3 ÁGS9GT	44.86	À79.72	5
	2	h 0 + h 1 ÁLogGS + h 2 ÁGT	42.02	À76.03	4	1-2	5.556	1	0.022
	3	h 0 + h 1 ÁLogGS	41.07	À76.15	3	2-3	1.811	1	0.184
	4	h 0 + h 2 ÁLogGT	38.24	À70.47	3	2-4	7.656	1	0.008
	5	h 0 + h 1 ÁLogGS + h 2 ÁGT + h 3 ÁGS9GT + h 4 ÁDNearImpala	44.90	À77.81	6	1-5	0.085	1	0.772
	6	h 0 + h 1 ÁLogGS + h 2 ÁGT + h 3 ÁGS9GT + h 5 ÁDCover	47.73	À81.45	7	1-6	2.693	2	0.077
	7	h 45.12	À76.23	7	1-7	0.232	2	0.794

0 + h 1 ÁLogGS + h 2 ÁGT + h 3 ÁGS9GT + h 6 ÁDGrassHeight

Group size (GS) was log-transformed. Group type (GT) (impala group, mixed-species group), Distance to the nearest impala (DNearImpala) (D ≤ 5, D > 5 m), distance to cover (DCover) (D ≤ 50, 50 < D ≤ 100, D > 100 m) and grass height (GrassHeight) (short, medium and tall) were considered as categorical. Levels used as the references in the models are italicized in the legends.

Table 2 :

 2 Coefficients (AESE) of the factors influencing the arcsine square root-transformed proportion of time an impala spent in vigilance Group size (GS) was log-transformed. Group type (GT) (impala-only group, mixed-species group) was considered as categorical. Level used as the reference in the models is italicized in the legend. *,** and ns indicate 0.05 ≤ p < 0.01, 0.01 ≤ p < 0.001 and non-significant, respectively (see Table1for statistical details).

		ID coefficient
		of model 1
	Variables	in Table 1	Coefficient SE
	Intercept h 0	0.731	0.133
	LogGS	h 1	À0.339*	0.091
	GT	h 2	À0.405 ns	0.192 When impala-only
				group is the reference
	LogGS	h 3	0.307** 0.130 When impala-only
	9 GT			group is the reference

  Table 1 and coefficients in Table 2.

Table 3 :

 3 Model comparison allowing to test for the effect of group size, proportion of impalas, the species with which impalas were associated and distance to the nearest heterospecific on the arcsine square root-transformed proportion of time an impala spent in vigilance in mixed-species groups

	ID Model	Model structure	LogLik	AICc	df	Model comparison	F	df	p
	8	h 0	23.34	À42.68	2
	9	h 0 + h 1 ÁLogGS	23.40	À40.79	3	8-9	0.114	1	0.738
	10	h 0 + h 2 ÁPropImpala	23.44	À38.88	4	8-10	0.071	1	0.793
	11	h 0 + h 3 ÁSpecies	25.92	À39.84	5	8-11	1.532	2	0.230
	12	h 0 + h 2 ÁDNearHeteroSp	23.51	À39.01	4	8-12	0.195	1	0.664

Acknowledgements

The Director General of the Zimbabwe Parks and Wildlife Management Authority is acknowledged for providing the opportunity to carry out this research. We are particularly grateful to Dr. H. Madzikanda for his long-term support and friendship, Mr. A. Musakwa, the Area Manager, for his support and understanding and to the Parks Senior ecologist G. Mtare. This study was carried out in the framework of the HERD program (Hwange Environmental Research Development), funded by the CNRS INEE ('Institut Ecologie et Environnement'), 'Zones Ateliers' programme, the Agence National de la Recherche FEAR project (ANR-08-BLAN-0022), the CIRAD and the French 'Minist ere des Affaires Etrang eres', and the 'Ambassade de France au Zimbabwe'. We would also like to thank the whole HERD team including Arnold Winston Tshipa for his crucial help in collecting data, Peter Jarman and Damien Farine for helpful comments on an earlier version of the manuscript and John Fryxell and one anonymous referee for their helpful comments.