On the practical stability with respect to h -manifolds of hybrid Kolmogorov systems with variable impulsive perturbations

Nonlinear Analysis xxx (xxxx) xxx

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On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations Ivanka M. Stamova ∗, Gani Tr. Stamov The University of Texas at San Antonio, Department of Mathematics, San Antonio, TX, 78249, USA

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Article history: Received 5 November 2019 Accepted 20 January 2020 Communicated by Enrico Valdinoci MSC: 34D35 34A37 92D25

abstract The present paper is devoted to the problems of practical stability with respect to h-manifolds for impulsive Kolmogorov systems. Variable impulsive perturbations are considered. We will investigate these problems in the light of the vector Lyapunov’s method of piecewise continuous functions. As an example, the case of Lotka–Volterra systems is elaborated. The effect of uncertain parameters on the stability behavior of the model is also examined. The obtained results extend and complement the existing stability results for Kolmogorov and related systems. © 2020 Elsevier Ltd. All rights reserved.

Keywords: Practical stability h-manifolds Impulsive Kolmogorov systems

1. Introduction The proposed research focuses on a main direction of qualitative analysis of Kolmogorov-type models, using methods of stability and control theories. The classical Kolmogorov systems u˙ i (t) = ui (t)fi (t, u(t)), ui ≥ 0, i = 1, 2, . . . , N ,

(1)

have been widely used to model community of n interacting species, where ui is the density and fi is the per capita growth rate of species i [37–39,48,51]. As researches emphasized (see, for example, [38]) fi not only depends on the densities of the interacting populations, but also fluctuates with time or is subject to certain seasonal variations. Note that Kolmogorov-type systems (1) include as particular cases Lotka–Volterra and related systems in population dynamics, as well as, some models in mathematical finance that have been a subject of extended research activity [1,2,5,6,8,10,13]. The study of the qualitative properties of Kolmogorovtype systems is still a very active area of research [3,18,23,24,46]. It is also worth to emphasize that some important results on this topic of interest have been proposed by S. Ahmad and his co-authors [1,2,5,6,8,10]. ∗ Corresponding author. E-mail addresses: [email protected] (I.M. Stamova), [email protected] (G.T. Stamov).

https://doi.org/10.1016/j.na.2020.111775 0362-546X/© 2020 Elsevier Ltd. All rights reserved.

Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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On the other side, the fact that a real population system is often affected by some short-term perturbations that can be described as impulsive effects inspired many researchers to create qualitative methods for dynamic analysis of Kolmogorov and related systems under impulsive perturbations. See, for example, [4,7,9,11,20,30,45,49] and the bibliography therein. Indeed, the impulsive mathematical models are intensively applied as a natural description of the observed evolution phenomenon in population dynamics models. In addition, considering impulsive perturbations (shocks) will give opportunities to control the dynamic processes of the model. Note that, in general, a dynamical system under impulsive perturbations is represented by a two-component system consisting of a continuous and discrete component, and is essentially, a hybrid system under specific assumptions on the dynamical properties of both components (see [14,17,22,29,47] and the references therein). However, all cited above existing results on impulsive Kolmogorov models considered impulsive perturbations at fixed moments of time. Since in the case when the impulses take place at variable impulsive moments a number of phenomena may appear, such as “beating” of the solutions, bifurcation, merging of the solutions, loss of the property of autonomy, etc. [15,19,21,25,44], the results on systems with variable impulsive perturbations are very rare. However, considering variable impulsive perturbations in competitive systems is more general and reflects our understanding that in reality, the communication between subsystems can be state-dependent. That is why in the present research we will investigate the effect of variable impulsive perturbations on some stability properties of Kolmogorov-type models. To the best of our knowledge, such systems have not been considered in the existing literature. The stability and permanence of solutions are the most investigated issues for Kolmogorov and related systems [1,6,7,9–11,20,30,45,46,48]. Both issues are important in order to understand and/or predict the behavior in time. For instance, asymptotic stability results are used to quantify the rate of convergence of a given system to its equilibrium, and they can also be used to understand the extent to which the system changes under the influence of external factors. But, in many cases, though a system is stable or asymptotically stable in the Lyapunov sense, it is actually useless in practice because of undesirable transient characteristics (e.g., the stability domain or the attraction domain is not large enough to allow the desired deviation to cancel out). The concept of practical stability is much more efficient in such situations. It generalizes the permanence concept by investigating the behavior of a system contained within specified bounds during a fixed time interval when a state of the system may be even unstable in the classical Lyapunov sense, but its performance may be acceptable. Many problems fall into this category including the travel of a space vehicle between two points and the problem, in a chemical process, of keeping the temperature within certain bounds [12,26,33,50]. Despite the high importance of the practical stability notion for theory and applications it is not developed for Kolmogorov-type impulsive models and this is the basic aim of our research. Moreover, in the present paper we consider a generalization of the practical stability notion, namely, practical stability of h-manifolds. This notion contains as a special case practical stability of certain solutions (zero solutions, equilibrium states, periodic solutions, etc.) and deals with a specific set of solutions defined by a function h [16,31,35,40]. Therefore, with this research we generalize and complement some qualitative results for Kolmogorov-type models governed by impulsive differential equations and unify various stability concepts for such models. The main results in this paper are proved by the use of vector Lyapunov piecewise continuous functions. The method of vector functions in the stability theory of dynamic equations is described in the book [28]. The role and advantages of such Lyapunov-type functions in the stability theory of differential equations are well known [25,27,28,36,43]. The obtained practical stability results are applied to a specific Kolmogorov system of Lotka–Volterra type. In addition, the uncertain case is also investigated. Indeed, a real system always involves uncertainties due to inaccuracy in model parameter measurements, data input, and any kind of unpredictability [32]. Such Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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uncertainties may give rise to instability of the system. That is why the analysis of the effect of uncertain values of the parameters in some of the components is of a significant interest for applications. See, for example, [32,34,41,42] and the references therein. The paper is organized according to the following plan. In Section 2 we introduce the hybrid Kolmogorovtype system with variable impulsive perturbations. The practical stability notion with respect to h-manifolds and the class of vector Lyapunov functions are defined. Some preliminary results are also given. Section 3 establishes the main practical stability conditions with respect to h-manifolds. The conditions are based on the comparison result given in Section 2. In Section 4, the main practical stability results are applied to a Kolmogorov system of Lotka–Volterra type. The uncertain case is considered in Section 5. Section 6 is devoted to some comments on the obtained results and future research in this direction. 2. Preliminaries Let RN be the N -dimensional Euclidean space with norm ∥ . ∥, R+ = [0, ∞), t0 ∈ R+ and let Ω be a positive cone Ω = {u ∈ RN , ui ≥ 0} containing the origin. Consider the following Kolmogorov system of impulsive differential equations with variable impulsive perturbations { u˙ i (t) = ui (t)fi (t, u(t)), t ̸= τk (u(t)), (2) ∆ui (t) = Iik (ui (t)), t = τk (u(t)), k = 1, 2, . . . , where fi : [t0 , ∞) × Ω → R, Iik : R+ → R, i = 1, 2, . . . , N , τk : Ω → R, k = 1, 2, . . . . Let f = col(f1 , f2 , . . . , fN ), Ik = col(I1k , I2k , . . . , IN k ) and let u0 ∈ Ω . Denote by u(t) = u(t; t0 , u0 ) the solution of (2) that satisfies the initial condition u(t+ 0 ; t0 , u0 ) = u0 .

(3)

The solutions u(t) of (2) are piecewise continuous functions with points of discontinuity of the first kind at which they are left continuous [14,15,47], i.e., at the moments tlk when the integral curve of the solution u(t) meets the hypersurfaces { } σk = (t, u) ∈ [t0 , ∞) × Ω : t = τk (u) the following relations are satisfied: + u(t− lk ) = u(tlk ), u(tlk ) = u(tlk ) + Ilk (u(tlk )).

The points tl1 , tl2 , . . . , (t0 < tl1 < tl2 ) are the impulsive moments. Let us note that, in general, k ̸= lk . In other words, it is possible that the integral curve of the problem under consideration does not meet the hypersurface σk at the moment tk . Let τ0 (u) ≡ t0 for u ∈ Ω . We assume that the functions τk (u) are continuous and the following relations hold: t0 < τ1 (u) < τ2 (u) < . . . , τk (u) → ∞ as k → ∞ uniformly on u ∈ Ω . We also suppose that the functions f , Ik and τk are smooth enough on [t0 , ∞) × Ω and Ω , respectively, to guarantee existence, uniqueness and continuability of the solution u(t) = u(t; t0 , u0 ) of the initial value problem (IVP) (2), (3) on the interval [t0 , ∞) for each u0 ∈ Ω , and t0 ∈ R+ and absence of the phenomenon “beating”. For more results about such systems, we refer the reader to [14,15,47]. We also assume that solutions of (2) with initial conditions of the type (3) are nonnegative, and if ui0 > 0 for some i, then ui (t) > 0 for all t ≥ t0 . If, moreover, t = τk (ui ) and (t, ui ) ∈ (t0 , ∞) × (0, ∞), then ui (t) + Iik (ui (t)) > 0 for all i = 1, 2, . . . , N and k = 1, 2, . . . . Note that these assumptions are natural from an applied point of view. Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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Let h = h(t, u), h : [t0 , ∞) × Ω → R, be a function. The next manifolds we will call h-manif olds defined by the function h: Mt = {u ∈ Ω : h(t, u) = 0, t ∈ [t0 , ∞)}, Mt (λ) = {u ∈ Ω : ∥h(t, u)∥ < λ, t ∈ [t0 , ∞)}, λ > 0. We will consider the above manifolds under the following assumptions: H2.1. The function h is continuous on [t0 , ∞)×Ω and the sets Mt , Mt (λ) are (N −1)-dimensional manifolds in RN . H2.2. Each solution u(t; t0 , u0 ) of the IVP (2), (3) satisfying ∥h(t, u(t; t0 , u0 ))∥ ≤ L < ∞ is defined on the interval [t0 , ∞). Next, motivated by [37], for an h-manifold related to the Kolmogorov system (2) we will define the following generalizations of the practical stability notions. Definition 2.1. The system (2) is said to be: (a) practically stable with respect to the function h, if for given (λ, A) with 0 < λ < A, we have u0 ∈ Mt+ (λ) implies u(t; t0 , u0 ) ∈ Mt (A), t ≥ t0 for some t0 ∈ R+ ; 0 (b) uniformly practically stable with respect to the function h if (a) holds for every t0 ∈ R+ ; (c) practically globally exponentially stable with respect to the function h, if for given (λ, A) with 0 < λ < A and u0 ∈ Mt+ (λ) there exist constants γ, µ > 0 0 ( ) u(t; t0 , u0 ) ∈ Mt A + γ∥h(t0 , u0 )∥e−µ(t−t0 ) for some t0 ∈ R+ . Remark 2.2. Practical stability of the system (2) with respect to the function h guarantees that the set {(t, u) : t ∈ [t0 , ∞), u ∈ Mt } is a positively invariant set of (2). Let t1 , t2 , . . . (t0 < t1 < t2 < · · · ) be the impulsive control instants at which the integral curve (t, u(t; t0 , u0 )) of the IVP (2), (3) meets the hypersurfaces σk , k = 1, 2, . . ., i.e. each of the points tk is a solution of some of the equations t = τk (u(t)), k = 1, 2, . . .. In the proofs of our main results we will use the impulsive vector comparison principle [7,9,14,47] and for this reason together with (2), we consider the following comparison system { v(t) ˙ = F (t, v(t)), t ̸= tk , (4) ∆v(tk ) = v(t+ k ) − v(tk ) = Jk (v(tk )), tk > t0 , m m m where F : [t0 , ∞) × Rm + → R , Jk : R+ → R , k = 1, 2, . . . and F (t, 0) = 0, t ∈ [t0 , ∞), Jk (0) = 0, k = 1, 2, . . . . + Let v0 ∈ Rm + . Denote by v = v(t; t0 , v0 ) the solution of (4) satisfying the initial condition v(t0 ) = v0 , and by J + (t0 , v0 ) the maximal interval of type [t0 , ω) in which the solution v(t; t0 , v0 ) is defined. In Rm we consider the natural partial ordering defined as: for two vectors v, v¯ ∈ Rm we will say that v ≥ v¯ if vj ≥ v¯j for each j = 1, 2, . . . , m, and v > v¯ if vj > v¯j for each j = 1, 2, . . . , m. m Definition 2.3. A function F : [t0 , ∞) × Rm + → R , F = col(F1 , . . . , Fm ), is said to be quasi-monotone m increasing in [t0 , ∞)×R+ if for each pair of points (t, v) and (t, v¯) from [t0 , ∞)×Rm + and for i ∈ {1, 2, . . . , m}, the inequality Fi (t, v) ≥ Fi (t, v¯) holds whenever vi = v¯i and vj ≥ v¯j for j = 1, 2, . . . , m, i ̸= j, i.e. for any fixed t ∈ [t0 , ∞) and any i ∈ {1, 2, . . . , m}, the function Fi (t, v) is non-decreasing with respect to (v1 , v2 , . . . , vi−1 , vi+1 , . . . , vm ). Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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m Note that [28,43,47], in the case when the function F : [t0 , ∞) × Rm is continuous and quasi+ → R monotone increasing all solutions of problem (4) starting from the point (t0 , v0 ) ∈ (t0 , ∞) × Rm + lie between two singular solutions — the maximal and the minimal ones. + + Definition 2.4. The solution v + : J + (t0 , v0 ) → Rm + of the system (4) for which v (t0 ; t0 , v0 ) = v0 is said + m to be a maximal solution if any other solution v : [t0 , ω ˜ ) → R+ , for which v(t0 ) = v0 satisfies the inequality v + (t) ≥ v(t) for t ∈ J + (t0 , v0 ) ∩ [t0 , ω ˜ ).

Analogously, the minimal solution of system (4) is defined. In what follows we will consider only such solutions v(t) of (4) for which v(t) ∈ Rm +. Depending on the measure that we will use for the solutions, we can modify the definitions of practical stability for the comparison system (4). In this paper we will use the following set and practical stability definitions for (4). Definition 2.5. The system (4) is said to be: (a) practically stable, if for given (λ, A) with 0 < λ < A and v0 = (v10 , v20 , . . . , vm0 )T ∈ Rm + , we have max1≤j≤m vj0 < λ implies max1≤j≤m vj+ (t; t0 , v0 ) < A, t ≥ t0 for some t0 ∈ R+ ; (b) uniformly practically stable, if the point (a) holds for every t0 ∈ R+ ; (c) practically globally exponentially stable, if for given (λ, A) with 0 < λ < A and max1≤j≤m vj0 < λ there exist constants γ, µ > 0 ( ) max vj+ (t; t0 , v0 ) < A + γ max vj0 e−µ(t−t0 ) 1≤j≤m

1≤j≤m

for some t0 ∈ R+ . ⋃∞ Let Gk = {(t, u) : τk−1 (u) < t < τk (u), u ∈ Ω }, k = 1, 2, . . . , and G = k=1 Gk . In the further considerations, we use vector piecewise continuous auxiliary functions (see, for example, [25,27,28,36,43,47] and the references therein). Definition 2.6. A function V : [t0 , ∞) × Ω → Rm + , belongs to the class V0 if: 1. V (t, u) is continuous in G, locally Lipschitz continuous with respect to its second argument on each of the sets Gk , and V (t, 0) = 0 for t ≥ t0 ; 2. For each k = 1, 2, . . . and (t∗0 , u∗0 ) ∈ σk there exist the finite limits V (t∗0 − 0, u∗0 ) =

lim

V (t, u), V (t∗0 + 0, u∗0 ) =

(t,u)→(t∗ ,u∗ ) 0 0 (t,u)∈Gk

lim

V (t, u)

(t,u)→(t∗ ,u∗ ) 0 0 (t,u)∈Gk+1

and the equality V (t∗0 − 0, u∗0 ) = V (t∗0 , u∗0 ) holds. For t > t0 , t = ̸ τk (u(t)), k = 1, 2, . . . , and u ∈ Ω , we define the upper right-hand Dini derivative of a function V ∈ V0 with respect to system (2) by 1 D+ V (t, u) = lim sup [V (t + h, u(t + h)) − V (t, u(t))]. + h h→0 The next vector comparison lemma will be very useful in the proof of our main results. Similar vector and scalar comparison results are given in [4,7,9,25–28,33–36,43,45]. Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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Lemma 2.7 ([47]). Assume that: 1. The function F is quasi-monotone increasing, continuous in the sets (tk , tk+1 ] × Rm + , k ∈ N ∪ {0}, and for k = 1, 2, . . . and ξ ∈ Rm there exists the finite limit + lim

F (t, v).

(t,v)→(t,ξ) t>tk

2. The maximal solution v + (t; t0 , v0 ) ∈ Rm + of the system (4) is defined for t ≥ t0 . m 3. The functions φk : Rm → R , φ (u) = u + Jk (u), k = 1, 2, . . ., are non-decreasing in Rm k + + +. m 5. The function V : [t0 , ∞) × Ω → R+ , V = col(V1 , V2 , . . . , Vm ), V ∈ V0 , is such that V (t+ , u + Ik (u)) ≤ φk (V (t, u)), u ∈ Ω , t = tk , k = 1, 2, . . . , D+ V (t, u) ≤ F (t, V (t, u)), (t, u) ∈ G. Then V (t+ 0 , u0 ) ≤ v0 implies V (t, u(t; t0 , u0 )) ≤ v + (t; t0 , v0 ),

t ∈ [t0 , ∞).

3. Main results In our main theorems we will use the Hahn class K = {w ∈ C[R+ , R+ ] : w is strictly increasing and w(0) = 0} of functions w, which are called wedges. Theorem 3.1. Assume that 0 < λ < A are given and: 1. Conditions H2.1–H2.2 hold; 2. Assumptions of Lemma 2.7 hold; 3. The functions Ik are such that Ik (0) = 0, k = 1, 2, . . . ; 4. The function V ∈ V0 is such that the inequalities w1 (∥h(t, u)∥) ≤ max Vj (t, u) ≤ χ(t)w2 (∥h(t, u)∥), 1≤j≤m

(5)

hold, where (t, u) ∈ [t0 , ∞) × Ω , w1 , w2 ∈ K and the function χ(t) ≥ 1 is defined and continuous for t ∈ [t0 , ∞); 5. The inequality χ(t+ 0 )w2 (λ) < w1 (A) holds. Then: (a) If the system (4) is practically stable, then the system (2) is practically stable with respect to the function h; (b) If the system (4) is uniformly practically stable, then the system (2) is uniformly practically stable with respect to the function h. Proof . (a) From the practical stability of the system (4) and condition 5 of Theorem 3.1, for the maximal ˆ implies max1≤j≤m v + (t; t0 , v0 ) < A, ˆ t ≥ t0 , where solution v + (t; t0 , v0 ) we have max1≤j≤m vj0 < λ j + ˆ ˆ ˆ ˆ λ = χ(t0 )w2 (λ), A = w1 (A), λ < A for some t0 ∈ R+ . On the other hand, for t0 ∈ R+ and u0 ∈ Mt+ (λ) it follows 0

+ ˆ χ(t+ 0 )w2 (∥h(t0 , u0 )∥) < λ. Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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Then, from (5) we get + + ˆ max Vj (t+ 0 , u0 ) ≤ χ(t0 )w2 (∥h(t0 , u0 )∥) < λ.

1≤j≤m

Hence, ˆ max vj+ (t; t0 , V (t+ 0 , u0 )) < w1 (A),

t ≥ t0 .

1≤j≤m

Let u = u(t; t0 , u0 ) be the solution of the IVP (2), (3). Then from Lemma 2.7 for t ≥ t0 , we have ( ) max Vj (t, u(t; t0 , u0 )) ≤ max vj+ t; t0 , V (t+ 0 , u0 ) . 1≤j≤m

1≤j≤m

(6)

(7)

From (5)–(7), we get the inequalities w1 (∥h(t, u(t; t0 , u0 ))∥) ≤ max Vj (t, u(t; t0 , u0 )) 1≤j≤m

( +

) ≤ max vj t; t0 , V (t+ 0 , u0 ) < w1 (A), 1≤j≤m

t ∈ [t0 , ∞).

Therefore, ∥h(t, u(t; t0 , u0 ))∥ < A for t ≥ t0 , i.e., the system (2) is practically stable with respect to the function h. (b) From the uniform practical stability of the system (4) and condition 5 of Theorem 3.1, for the maximal solution v + (t; t0 , v0 ) we have ˆ implies max v + (t; t0 , v0 ) < A, ˆ t ≥ t0 , max vj0 < λ j

1≤j≤m

1≤j≤m

(8)

ˆ = χ(t+ )w2 (λ), Aˆ = w1 (A), λ ˆ < Aˆ for every t0 ∈ R+ . where λ 0 We will prove that u0 ∈ Mt+ (λ) implies u(t; t0 , u0 ) ∈ Mt (A), t ≥ t0 for every t0 ∈ R+ . Suppose that this 0 is not true. Then, there exists a t∗ > t0 such that ∥h(t∗ , u(t∗ ))∥ ≥ A and ∥h(t, u(t; t0 , u0 ))∥ < A, t ∈ [t0 , tk ), where tk < t∗ ≤ tk+1 for some k. Then, from condition 5 of Lemma 2.7 and the assumptions about the nonnegativeness of the solutions of (2) after an impulsive perturbation, it follows that there exists t0 such that tk < t0 < t∗ , and ∥h(t0 , u(t0 ))∥ ≥ A and u(t0 ; t0 , u0 ) ∈ Ω .

(9)

Hence, setting v0 = V (t+ 0 , u0 ), since all the conditions of Lemma 2.7 are satisfied, we get V (t, u(t; t0 , u0 )) ≤ v + (t; t0 , V (t+ 0 , u0 )),

t ∈ [t0 , t0 ].

(10)

From (9), condition 4 of Theorem 3.1, (8) and (10), it follows that w1 (A) ≤ w1 (∥h(t0 , u(t0 ))∥) ≤ max Vj (t0 , u(t0 ; t0 , u0 )) 1≤j≤m

( +

≤ max vj 1≤j≤m

) t0 ; t0 , V (t+ 0 , u0 ) < w1 (A),

t ∈ [t0 , ∞).

The contradiction obtained proves that (2) is uniformly practically stable with respect to the function h. The proof of Theorem 3.1 is complete. Theorem 3.2. If in Theorem 3.1, the inequality (5) is replaced by the condition ∥h(t, u)∥ ≤ max Vj (t, u) ≤ H(L)∥h(t, u)∥, 1≤j≤m

(11)

where H(L) ≥ 1 exists for any 0 < L ≤ ∞, then, the practical global exponential stability of system (4) implies practical global exponential stability of the model (2) with respect to the function h. Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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Proof . Let t0 ∈ R+ and max1≤j≤m vj0 < λ. The practical global exponential stability of system (4) implies the existence of constants γ, µ > 0 such that ( ) max vj+ (t; t0 , v0 ) < A + γ max vj0 e−µ(t−t0 ) t ≥ t0 . 1≤j≤m

1≤j≤m

Let u0 ∈ Mt (λ) and u = u(t; t0 , u0 ) be the solution of the IVP (2), (3). For v0 = V (t+ 0 , u0 ), from Lemma 2.7, we get (7). Then, from (7) and (11), we have ∥h(t, u(t; t0 , u0 ))∥ ≤ max Vj (t, u(t; t0 , u0 )) ≤ max vj+ (t; t0 , V (t+ 0 , u0 )) 1≤j≤m


1≤j≤m

(

)

max vj0 e−µ(t−t0 )

1≤j≤m

< A + γH(L)∥h(t0 , u0 )∥e−µ(t−t0 ) ,

t ∈ [t0 , ∞).

Hence, ( ) u(t; t0 , u0 ) ∈ Mt A + γH(L)∥h(t0 , u0 )∥e−µ(t−t0 ) for t ≥ t0 , i.e., the system (2) is globally practically exponentially stable with respect to the function h and the proof is complete. Remark 3.3. A series of practical stability criteria can be obtained similarly to Theorems 3.1 and 3.2, if instead of max1≤j≤m Vj (t, u) other specific measures are used. Some appropriate measures are the following: m ∑

Vj (t, u);

j=1 m ∑

αj Vj (t, u), αj = const > 0;

j=1

L(V (t, u)), where L : Rm + → R+ is a nondecreasing function. In any of the above cases corresponding modifications of Definition 2.5 and Lemma 2.7 are required. 4. The Lotka–Volterra case The goal of this section is to apply our main results and to investigate the practical stability with respect to a function h for a specific choice of the function f in (2) of a particular importance. To this end, we set in (2) N ∑ fi = ri − aij uj (t), j=1

where ri , aij ∈ R+ and we consider the following system of Lotka–Volterra impulsive differential equations with variable impulsive perturbations. [ ] ⎧ N ∑ ⎪ ⎪ ⎪ aij uj (t) , t ̸= τk (u(t)), ⎪ ⎨ u˙ i (t) = ui (t) ri − j=1 (12) ⎪ ⎪ ui (t+ ) = ui (t) + Iik (ui (t)), t = τk (u(t)), ⎪ ⎪ ⎩ ui0 = ui (t+ 0 ), for t0 ∈ R+ , i = 1, 2, . . . , N and k = 1, 2, . . . . Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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The motivation to consider the model (12) came from the fact that Lotka–Volterra and related systems are essential models of multi-species population dynamics that form a well-known class of Kolmogorov-type systems. The stability properties of such systems have been extensively studied in the existing literature. For important stability results on (12) when Iik = 0 (continuous case), see [1,10] and the references therein; and for stability results on (12) in the framework of fixed moments of impulsive perturbations, see [7,9,10,45,47]. The case of variable impulsive perturbations for type (2) models is still not reported in the literature. In addition, results on practical stability of such systems do not exist so far. The next theorem, will offer such results as a corollary of Theorem 3.2. We again consider only nonnegative solutions of (12) that remain nonnegative after an impulsive perturbation. It follows then that for any closed interval contained in (τk−1 (u), τk (u)], k = 1, 2, . . . , u ∈ Ω , there exist positive numbers r and R such that r ≤ ui (t) ≤ R for i = 1, 2, . . . , N [4,7,9–11,47,49]. Theorem 4.1. Assume that 0 < λ < A are given and: 1. Conditions H2.1–H2.2 hold; 2. τ0 (u) ≡ t0 for u ∈ Ω , the functions τk are continuous, and t0 < τ1 (u) < τ2 (u) < · · · < τk (u) → ∞ as k → ∞ unif ormly on Ω ; 3. lk < lk+1 < · · · < lk+p < . . . , where lk is the number of hypersurfaces met by the integral curve (t, u(t)) of (12) at the moment tk , where k, lk , p = 1, 2, . . . ; 4. The integral curves of (12) meet each hypersurface σ1 , σ2 , . . . at most once; 5. The system parameters satisfy (∫ ) t

1≤i≤N

¯

eA(t−s) r+ ds

max

< A,

t0 ¯

+ R , eAt is the fundamental matrix of the system where r+ = col(r1+ , r2+ , . . . , rN ) = col(r1 , r2 , . . . , rN ) 1+r ¯ v(t) ˙ = −A∗ v(t), A∗ = r(1+r) 1+R A, ⎛ ⎞ a11 a12 a13 . . . a1N ⎜ a21 a22 a23 . . . a2N ⎟ ⎟ A¯ = (aij )N ×N = ⎜ ⎝. . . . . . . . . . . . . . . . . . . . . . . . . . . .⎠ ; aN 1 aN 2 aN 3 . . . aN N

6. The functions Ik = col(I1k , I2k , . . . , IN k ) are such that Iik (ui (tk )) = −γik ui (tk ),

0 < γik < 1, i = 1, 2, . . . , N , k = 1, 2, . . . .

Then (12) is practically globally exponentially stable with respect to the function h = max1≤i≤N ui . Proof . Let we suppose, without loss of generality that λ > 1. Let t0 ∈ R+ , u0 ∈ Mt+ (λ) and let u(t) = 0

col(u1 (t, u2 (t), . . . , uN (t))) be the solution of (12) that satisfies u(t+ 0 ) = u0 , u0 = col(u10 , u20 , . . . , uN 0 ). We define a Lyapunov function ( ) V (t, u(t)) = col ln(1 + u1 ), ln(1 + u2 ), . . . , ln(1 + uN ) . Then, for t ≥ t0 and t = τk (u), k = 1, 2, . . . , u ∈ Ω , we have that u(t) ∈ Ω implies u(t+ ) = u(t)+Ik (u(t)) ∈ Ω . Also, from condition 6 of Theorem 4.1, we obtain ( ) V (t+ , u(t+ )) = col ln(1 + u1 (t+ )), ln(1 + u2 (t+ )), . . . , ln(1 + uN (t+ )) Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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( ( ) ( ) ( )) = col ln 1 + (1 − γ1k )u1 (t) , ln 1 + (1 − γ2k )u2 (t) , . . . , ln 1 + (1 − γN k )uN (t) ≤ V (t, u(t)). On the other hand for t ̸= τk (u), k = 1, 2, . . . , u ∈ Ω along (12) we get [ ] N ∑ 1 1 + D Vi (t, u(t)) = u˙ i (t) = ri − aij uj (t) 1 + ui (t) 1 + ui (t) j=1 n r(1 + r) ∑ R ri − aij ln(1 + uj (t)). ≤ 1+r 1 + R j=1 Consider the comparison problem ⎧ ˙ = −A∗ v + r+ , t ̸= tk , ⎪ ⎨ v(t) + v(tk ) = v(tk ), tk > t0 , ⎪ ⎩ v(t0 ) = v0 ∈ RN +,

(13)

where tk , (t0 < t1 < t2 < . . . ) are the points at which the integral curve (t, u(t; t0 , u0 )) of the IVP (12) meets + the hypersurfaces σk , k = 1, 2, . . . , r+ = col(r1+ , r2+ , . . . , rN ). From conditions 1–5 of Theorem 4.1, we have that for any initial v0 ∈ RN + with max1≤j≤N vj0 < λ there exists a positive constant µ such that ( ) max vj+ (t; t0 , v0 ) < A + max vj0 e−µ(t−t0 ) , t ≥ t0 , 1≤j≤N

1≤j≤N

i.e. the comparison system (13) is practically globally exponentially stable. Hence, according to Theorem 3.2, the system (12) is practically globally exponentially stable with respect to the function h = max1≤i≤N ui . Remark 4.2. Theorem 4.1 offers sufficient conditions that guarantee the practical global exponential stability with respect to a function of the impulsive Lotka–Volterra model (12). Note that, for sufficiently large A the conditions of Theorem 4.1 may not guarantee the Lyapunov-type global exponential stability of (12). See, for example, [4,7,9–11,47,49] and the references therein. Therefore, the concept of practical stability is independent of that of Lyapunov stability. Moreover, since the practical stability can be achieved in a setting time (finite or infinite), it is very efficient in real applications. But, the problem for practical stability of Kolmogorov and related models has not been studied previously. 5. The uncertain case Now we will consider the uncertain case with the corresponding to (12) system given by ⎧ [ ] N ⎪ ∑ ( ) ⎪ ⎨ u˙ (t) = u (t) r + r˜ − aij + a ˜ij uj (t) , t ̸= τk (u(t)), i i i i j=1 ⎪ ⎪ ⎩ ui (t+ ) = ui (t) − γik ui (t) − γ˜ik ui (t), t = τk (u(t)),

(14)

where the constants r˜i , a ˜ij , γ˜ik , i, j = 1, . . . , N , k = 1, 2, . . . , represent the uncertainty of the system [32]. In the case, when all of these constants are zeros, we will receive the “nominal system” (12). Definition 5.1. The system (12) is said to be practically globally robustly exponentially stable with respect to the function h if for t0 ∈ R+ , u0 ∈ Mt0 (λ) and for any r˜i , a ˜ji , γ˜ik , i, j = 1, . . . , N the system (14) is practically globally exponentially stable with respect to the function h. Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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The proof of the next theorem follows directly from Theorem 4.1. Theorem 5.2. Assume that: 1. The conditions of Theorem 4.1 are met. 2. The parameters r˜i and a ˜ij are bounded and 0 < max

1≤i≤N

R (r+ + r˜i ) < A. 1+r i

3. The uncertain constants γ˜ik are such that 0 < γ˜ik < 1 − γik i = 1, 2, . . . , N , k = 1, 2, . . . . Then the system (12) is practically globally robustly exponentially stable with respect to the function h = max1≤i≤N ui . Remark 5.3. Different from the existing results on uncertain systems [32,34,41,42], this paper is concerned with the effect of uncertain terms on the practical stability behavior of Lotka–Volterra models. Theorem 5.2 shows that an added advantage of the practical stability control is in the simplification and universality of the conditions on the uncertain parameters of the system. 6. Concluding remarks The paper introduces a hybrid Kolmogorov-type system with variable impulsive perturbations. Sufficient conditions for practical stability of the solutions with respect to h-manifolds are established. The obtained results are applied to an impulsive Lotka–Volterra system. In addition, the effect of uncertain values of the system’s parameters is evaluated. This research extends and complements the existing stability results on Kolmogorov and related systems. Moreover, the advantages of practical stability control can greatly improve its effectiveness in real applications. Some extensions of the proposed results to different classes of Kolmogorov-type models, including stochastic models, models on time-scales, and some others, are important topics for future research. References [1] S. Ahmad, A.C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka–Volterra system, Nonlinear Anal. 40 (2000) 37–49. [2] S. Ahmad, A.C. Lazer, Average growth and extinction in a competitive Lotka–Volterra system, Nonlinear Anal. 62 (3) (2005) 545–557. [3] S. Ahmad, A.C. Lazer, On a property of a generalized Kolmogorov population model, Discrete Contin. Dyn. Syst. 33 (2013) 1–6. [4] S. Ahmad, G.Tr. Stamov, Almost periodic solutions of N -dimensional impulsive competitive systems, Nonlinear Anal. RWA 10 (2009) 1846–1853. [5] S. Ahmad, I.M. Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal. RWA 5 (2004) 219–229. [6] S. Ahmad, I.M. Stamova, Partial persistence and extinction in N-dimensional competitive systems, Nonlinear Anal. 60 (2005) 821–836. [7] S. Ahmad, I.M. Stamova, Asymptotic stability of an N -dimensional impulsive competitive system, Nonlinear Anal. RWA 8 (2007) 654–663. [8] S. Ahmad, I.M. Stamova, Survival and extinction in competitive systems, Nonlinear Anal. RWA 9 (2008) 708–717. [9] S. Ahmad, I. Stamova, Stability criteria for impulsive Kolmogorov-type systems of nonautonomous differential equations, Rend. Istit. Mat. Univ. Trieste 44 (2012) 19–32. [10] S. Ahmad, I.M. Stamova (Eds.), Lotka–Volterra and Related Systems: Recent Developments in Population Dynamics, Walter de Gruyter, Berlin, 2013. [11] G. Ballinger, X. Liu, Permanence of population growth models with impulsive effects, Math. Comput. Modelling 26 (1997) 59–72. [12] G. Ballinger, X. Liu, Practical stability of impulsive delay differential equations and applications to control problems, in: X. Yang, K.L. Teo, L. Caccetta (Eds.), Optimization Methods and Applications, in: Applied Optimization, Kluwer, Dordrecht, 2001, pp. 3–21. Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.

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Please cite this article as: I.M. Stamova and G.T. Stamov, On the practical stability with respect to h-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Analysis (2020) 111775, https://doi.org/10.1016/j.na.2020.111775.