Extinction and survival in competitive Lotka-Volterra systems with constant coeﬃcients and inﬁnite delays

. The qualitative properties of a nonautonomous competitive Lotka-Volterra system with inﬁnite delays are studied. By using a result of matrix theory and the ﬂuctuation lemma, we estab-lish a series of easily veriﬁable algebraic conditions on the coeﬃcients and the kernel, which are suﬃcient to ensure the survival and the extinction of a determined number of species. The surviving part is stabilized around a globally stable critical point of a subsystem of the system under study. These conditions also guarantee the asymptotic behavior of the system.

t ≥ 0, i = 1, . . . , r, (3) r being an integer such that 1 < r < n, the initial condition x i (t) = ϕ i (t), t ≤ 0, i = 1, . . . , r, where ϕ i ∈ BC + , Here the initial functions ϕ i (s), i = 1, . . . , n, are the histories of the sizes of the populations in the past. The coefficients b i , a ij , for all 1 ≤ i, j ≤ n, are positive real numbers and the delay kernels K ij : [0, +∞) → [0, +∞), 1 ≤ i, j ≤ n, are continuous functions such that +∞ 0 K ij (t)dt = 1. Integrals represent the hereditary or accumulated inhibitory effect in the past that the population size of one species has had upon the other species.
The following hypotheses are also considered H 1 : For every k = 2, . . . , n, there exists an integer i k < k such that b k a kj < b i k a i k j , j = 1, . . . , k.
H 2 : For every i = 1, . . . r, the inequality b i > r j=1,j =i a ij b j a jj is hold.
H 4 : For every integer k = r + 1, . . . , n, there exists an integer i k < k such that b k a kj < b i k a i k j , j = 1, . . . , k.
Observation 1.1. It is easy to see that there is a natural correspondence between the solutions of the r-dimensional system (3) and those solutions of (1) for which x j (t) = 0 for all j = r + 1, . . . , n. That is, col(x 1 (t), . . . , x r (t)) is a solution of (3) if and only if col(x 1 (t), . . . , x r (t), 0, . . . , 0) is solution of (1).
Observation 1.2. Ahmad and Lazer in [21] proved that if b i > 0, a ij > 0 and if hypothesis H 2 is satisfied; that is, then there exists a unique solution col(u * 1 , . . . , u * r ) of the system of linear equations with u * i > 0, 1 ≤ i ≤ r, and the matrix r × r (a ij ) is invertible or non-singular. Observation 1.3. It can be easily proved that the following points are equilibrium points for the system (1) if and only if The positive critical points are the positive solutions of the linear system of n equations with n unknowns n j=1 It can be easily showed that the following points are equilibrium points for the x * = col(u * 1 , . . . , u * r , 0, . . . , 0) = col(u * , 0) where 0 = col(0, . . . , 0), u * = col(u * 1 , . . . , u * r ), 2 ≤ r < n, and u * is a solution of the linear system (5).
This work is included within the area of mathematical biology, particularly in population dynamics, for we study non-autonomous competitive Lotka-Volterra systems of n-species with infinite delays. These models with delay are also called hereditary models with memory. They are introduced in biology, for example, in biokinetics, which study the theories of biological equilibrium mathematically based on the struggle for existence between species competing on the ground for the same resources. The evolution of these systems is influenced significantly not only by the present but also by its most remote past. There are delayed effects in these systems due to the memory functions, the delay kernel. The use of these models with delay dates back to the time of the Italian mathematician Vito Volterra (1860Volterra ( -1940 and the French mathematician Marcel Brelot (1903Brelot ( -1987, who studied the classical prey-predator model with continuous infinite delay in their publications [25] and [4] respectively. Over the last 50 years and especially the last 20 years, these systems have been and continue to be investigated intensely, as can be seen in the references cited in this work. The system (1) describes the competition between n species, where x i (t) denotes the population density of the i-th species at time t. It is well known, by the fundamental theory of systems of functional differential equations [15], that (1) has only one solution x(t) = col(x 1 (t), . . . , x n (t)) which satisfies the initial conditions (2). It is easy to verify that the solutions of the system (1) corresponding to the initial conditions (2) are defined on [0, +∞) and remain positive for all t ≥ 0.
It should be mentioned that, during the last decade, there were many works on the competitive non-autonomous Lotka-Volterra systems with infinite delay and feedback controls by various researchers (see, e.g., [6], [7], [19], [20] and references cited there).
Montes de Oca and Pérez, by constructing Lyapunov-type functionals, established in [9] and [10] a series of algebraic conditions on the coefficients and kernel which guarantee the survival and extinction of a number of species. In this work the same results are proved without using the construction of Lyapunov functionals. The following results are obtained: (1), with initial condition (2), has the property that Here the point ( b 1 a 11 , 0, . . . , 0) is a global attractor of the system (1).
(2) If H 2 , H 3 and H 4 hold, then any solution (1), with initial condition (2) has the property that x i (t) → 0 exponentially for all i = r + 1, . . . , n as t → ∞ and for all i, is a global attractor of the system (1).

Preliminaries
The following two Lemmas are established in [11].
. . , b r ) and let A = (a ij ) be the matrix with a ij ≥ 0, and a ii , b i > 0, 1 ≤ i, j ≤ r. Suppose that hypothesis H 2 is satisfied. Then the matrix 2D − A, where D = diag (a 11 , . . . , a rr ) is the diagonal matrix of A, has inverse and is given by Proof. See ( [24], Lemma 4.1).
Lemma 2.5. Suppose that hypotheses H 2 and H 3 hold. If is a solution of the system (3) with the initial conditions (4), then for all i, This means that the system (3) is permanently coexistent.
) is a solution of the system (1) with the initial conditions (2), then x i (t) → 0 exponentially for all i, 2 ≤ i ≤ n, as t → +∞. Moreover, there exists a positive number α = α(x) such that x 1 (t) ≥ α for all t ≥ 0.
Proof. The proof of the proposition is analogous to the proof of the proposition 2.1 in [9]. The proof method used in [9] is based on using mathematical induction and proving first that x n → 0 exponentially as t → +∞.
is a solution of the system (1) with the initial conditions (2), then x i (t) → 0 exponentially for all i = r + 1, . . . , n as t → ∞. Moreover, there exists a positive number α = α(x) such that x i (t) > α for all i = 1, . . . , r and t ≥ 0.
Proof. The proof is analogous to the proof of the Proposition 2.3 and 2.4 in [10].

Main theorems
Theorem 3.1. Suppose that hypotheses H 1 and H 3 hold. If x(t) = col (x 1 (t), x 2 (t), . . . , x n (t)) is a solution of the system (1) with the initial conditions (2), then x i (t) → 0 exponentially for all i, 2 ≤ i ≤ n as t → +∞ and is an equilibrium point of the logistics equation ; in other words, the point col b 1 a 11 , 0, . . . , 0 is a global attractor of the system (1).
Proof. By Proposition 2.6, there exists a positive number α = α(x) such that x 1 (t) ≥ α for all t ≥ 0 and x i (t) → 0 exponentially for i, 2 ≤ i ≤ n, as t → +∞. From this fact and by virtue of the lemma 2.2, it follows that On the other hand, by Lemma 2.3 (Fluctuation Lemma), there exists a sequence τ 1 n → +∞ as n → +∞ such that x 1 (τ 1 n ) → 0 and x 1 (τ 1 n ) → x 1 as n → +∞, where x 1 = lim inf t→+∞ x 1 (t). Combining the above with the equation (6) we obtain Replacing t by τ 1 n in the first equation of the system (1), it follows that Letting n → ∞ we get 0 = x 1 [b 1 − a 11 x 1 ]. Since x 1 > 0, we have that On the other hand, From (8) and (9) we obtain Hence lim t→+∞ x 1 (t) = b 1 a 11 . This completes the proof of the theorem.
Proof. By Proposition 2.7, there exists a positive number α = α(x) such that On the other hand, by Lemma 2.5, x i (t) is bounded above for all i, 1 ≤ i ≤ r, and t ≥ 0. Therefore, the components x i (t) for 1 ≤ i ≤ r, are bounded above and below by positive constants. By Lemma 2.3 (Fluctuation Lemma) there are sequences τ i n → ∞ and σ i n → ∞ as n → ∞, such that for all i, 1 ≤ i ≤ r, On the other hand, since t −∞ K ij (t − s)x j (s)ds for i, j, 1 ≤ i, j ≤ r and t ≥ 0 is bounded, without loss of generality we assume that for all i, j, 1 ≤ i, j ≤ r, x j (s)ds both converge. Replacing τ i n and σ i n in the system (3), we obtain for all i, 1 ≤ i ≤ r, As n → +∞ we get By Lemma 2.1 and the definition of upper and lower limit respectively, we have Volumen 54, Número 1, Año 2020 Substituting these inequalities in (10) and (11) we obtain Writing these systems in terms of matrices and vectors as Subtracting the first inequality from the second yields By Lemma 2.4, 2D − A is invertible and its inverse is nonnegative, so (x − x) ≥ 0, this implies that x ≥ x and consequently x = x. From (12) and (13) we have is the unique solution of the system of linear equations (5), we have is a positive equilibrium point that attracts all solutions of the system (3) with initial condition (4). This completes the proof of the theorem. Theorem 3.3. Suppose that hypotheses H 2 , H 3 and H 4 hold. If x(t) = col (x 1 (t), x 2 (t), . . . , x n (t)) is a solution (1), with initial condition of the system (2), then for all i = r + 1, . . . , n, x i (t) → 0 exponentially as t → ∞ and for all is the only equilibrium point of the system which is guaranteed by the hypothesis H 2 , (see observation 1.2). The point col (u * 1 , . . . , u * r , 0, . . . , 0) is a global attractor of the system (1).
Proof. By Proposition 2.7, there exists a positive number α = α(x) such that x i (t) ≥ α > 0 for all i, 1 ≤ i ≤ r, t ≥ 0 and x i (t) → 0 exponentially for i = r + 1, . . . , n, as t → +∞. From this fact and the Lemma 2.2, it follows that On the other hand, by lemma 2.5, x i (t) is bounded above for all i, 1 ≤ i ≤ r, and t ≥ 0. Therefore, the components x i (t) for 1 ≤ i ≤ r, are bounded above and below by positive constants.
By Lemma 2.3 (Fluctuation Lemma) there exist sequences τ i n → ∞ and σ i n → ∞ such that for all i, 1 ≤ i ≤ r, x i (t), as n → ∞, and By (14), and On the other hand, t −∞ K ij (t − s)x j (s)ds for i, j, 1 ≤ i, j ≤ r and t ≥ 0 are bounded, without loss of generality we may assume that for all i, j, 1 ≤ i, j ≤ r, Volumen 54, Número 1, Año 2020 Substituting the sequences τ i n and σ i n into the system (1), we obtain for each i, 1 ≤ i ≤ r, When n → +∞, As in the proof of the theorem 3.2, the desired conclusion is reached. This completes the proof of the theorem.

Examples
In this section we shall give two examples to illustrate the conclusions of Theorems 3.1 and 3.2. In the first one the conclusions of Theorem 3.1 will be illustrated and in the second one, those of Theorem 3.2.
Example 4.1. Consider the system It can be easily verified that the only critical points with non-negative components are those generated by the remark 1.4. By Theorem 3.1 we conclude that for any solution x(t) = col(x 1 (t), x 2 (t), x 3 (t)) of the system (17), with initial conditions (2), it has the property that x 2 (t) and x 3 (t) are extinct and lim t→+∞ x 1 (t) = 9 4 , that is, the point col 9 4 , 0, 0 is the only global attractor of the system (17).
Example 4.2. Consider the system  So the hypothesis H 2 is fulfilled. Clearly µ ij = 1 β ij < ∞ for 1 ≤ i, j ≤ 4 and i = j. This shows that assumption H 3 holds. Therefore all conditions of Theorem 3.3 are satisfied. By observation 1.3, it is known that the equilibrium points for the system (18)  It can be easily verified that the only critical point with non-negative components is col 32 11 , 32 11 , 0, 0 . Therefore, we conclude by Theorem 3.3 that for any solution x(t) = col (x 1 (t), x 2 (t), x 3 (t), x 4 (t)) of the system (18), with initial conditions (2), it has the property that species x 3 (t) and x 4 (t) are extinct and