Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

We present a biological fractional n-species delayed cooperation model of Lotka-Volterra type. The considered fractional derivatives are in the Caputo sense. Impulsive control strategies are applied for several stability properties of the states, namely Mittag-Leffler stability, practical stability and stability with respect to sets. The proposed results extend the existing stability results for integer-order n−species delayed Lotka-Volterra cooperation models to the fractional-order case under impulsive control.


Introduction
The dynamics of Lotka-Volterra and related systems are traditionally applied to problems in population dynamics [1][2][3][4]. For example, the work in Reference [5] studied a delayed cooperation ecosystem composed of a number (n) of species using the following Lotka-Volterra model: where i, j = 1, ..., n; t ≥ 0, x i (t) represents the density of the species i at time t, r i (t) are the intrinsic growth rates at time t, a i (t), b i (t) and c i (t) are the cooperative coefficients, τ ii denote the time delay due to gestation period of species i, τ ij , j = i denote time delays related to the maturation of the species i and j, respectively, 0 ≤ τ ij ≤ τ, τ = const, (i = 1, 2, ..., n). It is assumed that only adult individuals of species i can benefit the species j and vice versa. The model (1) has been originally introduced in Reference [6] for n = 2 and τ ij = 0, i, j = 1, 2, to describe the cooperative interaction between two species. Since then, it has been developed by several authors. See, for example, References [7][8][9].
An important objective in understanding biological systems is to determine their approach to equilibrium. Relative entropy, part of an information-theoretic approach, can be used to determine approach to equilibrium in biological systems, including the Lotka-Volterra type [10]. In addition, entropy methods are increasingly applied to biological models as robustness markers [11], as well as to determine information content in artificial neural networks [12].
Research on impulsive Lotka-Volterra and related systems has also increased because of their application to the study of population densities that are affected by some impulsive factors that

Fractional Lotka-Volterra-Type Model and Preliminary Notes
We will use the following notations: R + = [0, ∞), R n is the n-dimensional Euclidean space and ||x|| = |x 1 | + ... + |x n | denote the norm of x ∈ R n .
Let u ∈ C 1 [[0, b], R], b > 0. For any 0 < q < 1, following Reference [36], we consider the Caputo fractional derivative of order q with the lower limit 0 for a function u, defined as where Γ is the Gamma function and u is the first-order ordinary derivative of the function u.
We will consider the following initial condition associated with the model (2), where the initial function ϕ 0 : [−τ, 0] → R n , ϕ 0 = (ϕ 10 , ϕ 20 , ..., ϕ n0 ) T is continuous, positive and bounded on [−τ, 0]. For a continuous function g(t) defined on [0, ∞), we denote The goal of our research is to investigate the effects of impulsive external perturbations on the stability behavior of the model (2) at some fixed points t 1 , t 2 , ... such that 0 < t 1 < t 2 < ... < t k < ... and lim k→∞ t k = ∞. To this end we will add impulsive controllers that can be used to synchronize the densities of all species onto that of system (2), and introduce the following impulsive control system where are the control inputs, δ(t) is the Dirac impulsive function, y i : R + → R + , I ik : R + → R, i = 1, ..., n, k = 1, 2, . . . . Due to the incorporation of the Delta impulsive function the controller η(t) has an effect on sudden changes in the states of the model (4) at the time instants t k , that is, η(t) is an impulsive control of (4). For t > 0 and i = 1, 2, ..., n, we denote From (4), we have that [35,36,39,61], η i (y i (t)) = 0 for t = t k , k = 1, 2, ...., i = 1, 2, ..., n, and where h > 0 is sufficiently small. As h → 0 + , we obtain Therefore, the correspondent n-dimensional fractional-order driven system is given by where the quantities y i (t k ) and y i (t + k ) = lim h→0 + y i (t k + h) represent, respectively, the population densities of species i before and after an impulsive perturbation at the moment t k and the continuous in R + functions I ik characterize the impact of the impulse effect on the species i at the moments t k . For more results on fractional-order models under impulsive control where the Dirac delta function is incorporated in a similar sense, we refer the reader to References [39,61] and the references therein.
For some results on positive solutions of fractional Lotka-Volterra-type models we refer to References [32,33].
In addition, we assume that, if the functions I ik are such that −y i ≤ I ik (y i ) ≤ 0 for y i ∈ R + , i = 1, 2, ..., n, k = 1, 2, ..., then there exist positive constants m and M < ∞ such that For integer-order Lotka-Volterra models with and without impulses, similar results can be found in References [1,5,[7][8][9]21,23]. Since one of the main advantages of Caputo's fractional derivatives is that the initial conditions for fractional differential equations with such derivatives have a similar form as for integer-order differential equations, the validity of (7) can be verified using similar steps as in the integer-order models.
Define the norm ||φ|| τ = sup It is well known [1,2,4,[7][8][9]13,21,23] that Lyapunov functions can be a successful instrument in the investigation of the stability properties of a Lotka-Volterra type system. To apply the Lyapunov function method, we introduce the following notations: For V ∈ V 0 and t ∈ [t k−1 , t k ), we will use the following fractional derivative of V of order q, 0 < q < 1 with respect to system (1) defined [39] by We also recall the following basic comparison result of fractional calculus from Reference [39].

Lemma 1.
Assume that the function V ∈ V 0 is such that for t ∈ R + , φ ∈ P C, and for a constant µ > 0 the inequality For more comparison results and properties on Mittag-Leffler functions we refer to References [35][36][37]39].

Mittag-Leffler Stability Results
.., y * n (t; 0, φ * 0 )) T be the solution of the impulsive control system (5) with initial function φ * 0 . In the next, we shall suppose that We will use the following definition for Mittag-Leffler stability of the state y * (t) of the model (5) [45][46][47][48][49][50][51][52][53][54]. Definition 1. The state y * (t) of system (5) is said to be globally Mittag-Leffler stable, if where E q is the corresponding Mittag-Leffler function, q ∈ (0, 1), In some of the literature [50], the constant µ is also called the degree of Mittag-Leffler state estimator, which can be considered as an equivalence of the convergence rate as state estimator error tends to zero when time t goes to infinity.
Our main result in this section is the next theorem.
2. The impulsive functions I ik are such that Then, the state y * (t) of system (5) is globally Mittag-Leffler stable.
Proof. The proof is based on the Lyapunov-Razumikhin technique [39]. We consider the following Lyapunov function candidate For k = 1, ..., and φ ∈ PCB, according to (8), we have Also, for t ≥ 0, t = t k , k = 1, 2, ..., and φ ∈ PCB, we estimate the derivative C D q + V(t, φ(0)) along the system (5), and get that is valid whenever In view of (10) and condition 1 of Theorem 1, there exists a positive µ such that From (9) and (11), by Lemma 1, it follows that and this completes the proof of the theorem.

Remark 1.
The assumptions of Theorem 1 guarantee the Mittag-Leffler stability of the states of the impulsive control system (5). The impulsive control strategy is designed by (8). Therefore, the proposed structure of the control law can be applied to design impulsive based observers to estimate the states of (5) and globally Mittag-Leffler synchronize the states of the system (5) onto that of system (2). Note that, the synchronization scheme is independent on the lengths of the impulsive intervals.
Since I ik (y * i ) = 0, i = 1, 2, ..., n, k = 1, 2, ..., the state y * is also a steady state for the controlled model (2). Hence, in the case when y * (t) = y * = const, the providing results in Theorem 1 can be used so by means of convenient impulsive control strategy to keep the Mittag-Leffler stability properties of the equilibrium states of system (2). Note that the Mittag-Leffler stability for fractional-order models corresponds to the exponential stability for integer-order systems, which guarantees the fast convergence rate.

Practical Stability Results
In this section, following Reference [56] we will provide practical stability results for the impulsive control model (5).
We introduce the following definition.

Definition 2.
The state y * (t) of the fractional impulsive control system (5) is said to be: (a) practically uniformly globally stable with respect to (λ, A), if given (λ, (b) practically uniformly globally Mittag-Leffler stable with respect to (λ, A), if given (λ, A) with Condition 1 of Theorem 1 guarantees the existence of a positive constant µ such that (11) holds. The application of (11) together with the appropriate choice of the Lyapunov-candidate function V imply the global Mittag-Leffler stability of the state y * (t) of the control system (5) by means of the designed impulsive control law (8). This section will firstly address the case, when µ = 0 and the state y * (t) is not globally Mittag-Leffler stable. We will prove that in such case the impulsive controller (8) may assure its practical uniform global stability.
We will use again the Lyapunov function candidate and functions of the following class: K = {a ∈ C[R + , R + ] : a(u) is strictly increasing in u and a(0) = 0 }.

b(λ) < A holds.
Then, the state y * (t) of system (5) is practically uniformly globally stable with respect to (λ, A).

Remark 3.
In Theorem 2 we proved that even in some concrete cases, when the system (5) is not globally Mittag-Leffler stable, by the impulsive controller (8) we can made the vicinity of a state y * arbitrarily small so that all other states to remain arbitrarily close to it provided that their initial functions were sufficiently near the initial data of the particular state y * . The property of practical stability is very important in many applications when the Mittag-Leffler stability is conservative [58,60,61]. For type (2) models from the population biology this can compensate a not stable initial population by keeping the population densities between particular bounds for t ≥ 0. Theorem 3. Assume that: 1. There exists a function V ∈ V 0 for which conditions of Lemma 1 are met.

The function V(t, y) is such that
where the continuous function a(u) ≥ 1 exists for any u > 0.
From Lemma 1, we have that where µ > 0. From (14) and condition 2 of Theorem 3, we have Therefore, for any ν ≥ a(u), u > 0, we have which shows that system (5) is practically uniformly globally Mittag-Leffler stable with respect to (λ, A).

Remark 4.
For A = 0 Theorem 3 implies global Mittag-Leffler stability. However, Theorem 3 can be applied when this stability notion is pointless from the applied point of view. In such cases the designed impulsive controller allows the practical uniform global Mittag-Leffler stability.

Remark 5.
More detailed explanations about the importance of the practical stability notion for applied problems, as well as, for relations between stability and practical stability concepts can be found in References [56,61,64].

Stability of Sets
In this Section, we extend the Lyapunov function method to stability of sets criteria for the impulsive control system (5). In fact, considering closed sets of states as attractors instead of equilibrium states is common in population dynamics [1].
Let B ⊂ R n be an arbitrary set (compact or not compact). Define the norms: ||y|| B = inf{||y − z|| : z ∈ B} is the distance from y ∈ R n to B; ||φ|| B τ = sup

Remark 6.
The notion of stability of sets with respect to system (5) generalizes the stability of single states concepts of system (5). For example, when B = {y ∈ R n : y = y * (t), t ≥ 0}, Definition 3 reduces to Definition 1 for the global Mittag-Leffler stability of the state y * (t), t ≥ 0.

Remark 7.
The requirement for uniform boundedness of all solutions of (5) with respect to B is required by the fact that for noncompact sets B there is an opportunity that some solutions may tend to infinity in finite time within the set. If the set B is compact (for example, the equilibrium state), then the boundedness condition is surplus.
Theorem 4. Assume that: 1. There exists a function V ∈ V 0 such that V(t, y) = 0 for y ∈ B and V(t, y) > 0 for y ∈ R n + \ B, t ≥ 0. 2. There exists a function b ∈ K such that 3. For the function V conditions of Lemma 1 are met. Then, the set B is globally Mittag-Leffler stable with respect to system (5).
Proof. We will first prove the uniform boundedness of the solutions of (5) with respect to the set B.
Let η > 0 be given and let Let y(t; 0, φ 0 ) be the solution of the system (5) corresponding to the initial function φ 0 .
Since the conditions of Lemma 1 are met, then for µ = 0, we have Now, condition 2 of Theorem 4 and (15) imply the inequalities Since the solution y(t; 0, φ 0 ) is arbitrary the above proves the uniform boundedness of the solutions of (5) with respect to the set B.
In a similar way, as above, using Lemma 1 and condition 2 of Theorem 4, we obtain , t ≥ 0, which according to Definition 4 means that the set B is globally Mittag-Leffler stable set with respect to (5).
Remark 8. Theorem 4 shows that the impulsive control law (8) is designed so that gets global Mittag-Leffler stability of a set B with respect to the fractional model (5). For the integer-order systems this is equivalent to the fact that all trajectories arrive at a given target set B. Hence, the proposed result is of a great importance in the cases when the information on the system's parameters is incomplete and ones has to conduct a robust stability analysis.

Remark 9.
It is well know that if B is globally asymptotically stable with respect to (5), then the trivial solution of (5) is practically stable [65].

Remark 10.
We can also define the hybrid concept of practical stability of an arbitrary set B with respect to the impulsive control system (5). In fact, this concept is more natural than that of a neighbourhood of a state. In this case, one important question will be how well the stability behavior can be impulsively controlled, which is a subject for our future research.

Examples and Simulations
Example 1. Consider the following two-species fractional cooperation Lotka-Volterra system We can easily check that the point ( 2 7 , 1 5 ) is an equilibrium for the model (16). The corresponding fractional control Lotka-Volterra system is given by under impulsive control law where 0 < t 1 < t 2 < ... and lim k→∞ t k = ∞.
If λ and A are such that k < λ < A, then for an initial function φ 0 = (φ 01 , φ 02 ) T such that the state ( 2 7 , 1 5 ) will not be practically uniformly globally Mittag-Leffler stable. Note that in the particular example, condition 2 of Theorem 2 is not satisfied.
The example again shows that Mittag-Leffler stability is not enough for practical Mittag-Leffler stability.

Example 3.
Consider the two-species fractional cooperation Lotka-Volterra model We can easily check that the point ( 2 7 , 3 5 ) is an equilibrium for the model (19). However, since we cannot make any conclusion about its global Mittag-Leffler stability.
Following the steps in the proof of Theorem 1 using the Lyapunov function V(t, y) = ||y|| B it is trivial to show that the set B is globally Mittag-Leffler stable with respect to (17), (18).

Conclusions
In this paper, we introduce a fractional-order Lotka-Volterra type cooperation model. Based on Lyapunov function theory we present an impulsive control strategy that can be applied as a Mittag-Leffler stability and synchronization mechanism. In addition, more general practical stability results and stability of sets results with respect to the impulsive control system are established. Examples are presented to demonstrate the obtained criteria and the relations between the investigated stability behaviors. It is shown that the proposed impulsive control technique provides a useful tool for the design of appropriate population models. It can be also beneficial to models with reaction-diffusion terms which are the subject of our future research. Funding: F.S. and R.T. partially funded by National Institutes of Health BRAIN Theories NIMH-NIBIB R01EB026939.

Conflicts of Interest:
The authors declare no conflict of interest.