Multiple stationary solutions and global stabilization of reaction-di ﬀ usion Gilpin-Ayala competition model under event-triggered impulsive control

In this paper, the author utilizes Saddle Theorem and variational methods to deduce existence of at least six stationary solutions for reaction-di ﬀ usion Gilpin-Ayala competition model (RDGACM). To obtain the global stabilization of the positive stationary solution of the RDGACM, the author designs a suitable impulsive event triggered mechanism (IETM) to derive the global exponential stability of the the positive stationary solution. It is worth mentioning that the new mechanism can exclude Zeno behavior and e ﬀ ectively reduce the cost of impulse control through event triggering mechanism. Besides, compared with existing literature, the restrictions on the parameters of the RDGACM are relaxed so that the methods used in existing literature can not be applied to the relaxed case of this paper, and so the author makes comprehensive use of Saddle Theorem, orthogonal decomposition of Sobolev space H 10 ( Ω ) and variational methods to overcome the mathematical di ﬃ culty. Numerical examples show the e ﬀ ectiveness of the methods proposed in this paper.

⋆ There is a difference between this paper and [2, Statement 2] with respect to applying the same saddle theorem, for the norm of H 1 0 (Ω) defined in this paper is simpler than that of the proof of [2, Statement 2] so that the methods used in this paper is better than that of [2] .
Assume that u * (x) = (u * 1 (x), u * 2 (x)) T is a positive stationary solution of the system (2.1). Set            (2.2) and the stationary solution (u * 1 (x), u * 2 (x)) T of the system (2.1) corresponds to the zero solution (0, 0) T of the following system: where we denote U = (U 1 , (2.4) The following system is the system (2.3) in form of vector-matrix: Now we consider applying impulse control on the system (2.5): where the sequence {t k , k ∈ Z + } is the set of impulse instants. ∂U(t,x) ∂t denotes the right-hand derivative of U(t, x) on the time variable t, and H k is a constant matrix for k ∈ Z + . The sequence {t k , k ∈ Z + } is the set of impulse instants. The state U(t, x) is assumed to be right continuous and to have left limits on the time variable t. Since the zero solution of the system (2.7) is corresponding to the positive stationary solution u * (x), below we only need to consider the stabilization of the zero solution instead of the stability of positive stationary solution u * (x).
In this paper, the following assumptions are proposed: (A1) For i = 1, 2, set 0 < θ i =ˆθ ǐ θ i < 1 withθ i being an even number, andθ i being an odd number.

Remark 2.
In reality, because the natural resources are limited, it is reasonable to assume a bounded population density u i in the assumption (A2).
Definition 1: Given an impulse instant sequence {t k , k ∈ Z + }, the stationary solution (u * 1 (x), u * 2 (x)) T is globally exponentially stability if the system (2.7) is globally exponentially stability. And the system (2.7) is globally exponentially stability if there exist positive constants λ > 0 and ϱ i < 1 (i = 1, 2) such that for every initial condition (t 0 , η) with where β 0 > 1 is a positive number. (2) J(u) 0 if u ∈ H 2 with ∥u∥ δ; (3) J is bounded below, satisfying inf H J < 0, then J owns at least two non-zero critical points.
Remark 5. Equivalence of norms in finite dimensional spaces yields that there exist two positive constants µ 1 , µ 2 such

Main resuts
Theorem 3.1. Suppose the conditions (A1), (A2) and (2.10) hold. If, in addition, the following two conditions hold: then the system (2.1) possesses at least six stationary solutions, including the positive solution (u * 1 (x), u * 2 (x)) T . To complete the proof of Theorem 3.1, we need to present the following technique lemmas at first. Proof. Let (α(x), 0) T be a stationary solution of the system (2.1), satisfying whose functional is It is obvious that Ψ(0) = 0 and Ψ ∈ C 1 (H 1 0 (Ω), R 1 ), and then a critical point of the functional Ψ is corresponding to the solution of the equation (3.3).
Next, we claim that Ψ satisfies the (PS) condition. and which means that when n is big enough, The condition (A1) yields there is where c 2 is a constant dependent upon c 1 .
It follows by (3.7) that where c 1 is a positive number such that α(x) 2+θ 1 and c 2 is a constant dependent upon c 1 . And (3.9) implies there exists a constant c 3 > 0 such that which verifies the boundedness of {α n } in the Sobolev space H 1 0 (Ω). Further, obviously there exist two positive numbers c 4 , c 5 > 0 big enough such that which means the Caratheodory condition is satisfied. Employing the methods used in the proof of [2, Statement 2] or [3, Theorem 1] results in the existence of a convergent subsequence of the bounded sequence {α n } in the Sobolev space H 1 0 (Ω), and hence the (PS) condition is satisfied. If α ∈ E(λ 1 ) with ∥α∥ δ, then Sobolev embedding theorem or equivalence of norms in finite dimensional spaces yields that there is a constant c 6 > 0 such that which implies that there exists δ > 0 small enough such that Ψ(α) 0.
The proof is similar to that of the boundedness of the (PS) sequence {α n }. For convenience, we still use the above symbols. Let The condition (A1) yields there is c 1 > 0 big enough such that ∫ where c 2 is a constant dependent upon c 1 . 13) where c 1 is a positive number such that α(x) 2+θ 1 Ψ < 0, Lemma 2.1 means two non-zero critical points for Ψ, which completes the proof. Similarly, we can deduce the following Proposition. control technique with fixed impulse instants. Below, we shall design the event-triggered impulsive control, in which impulses only occur when the event which is related to the states of the system is triggered so that the cost of manual control is greatly reduced. This is the innovation of this paper which is different from [5]. But event-triggered impulsive control may brings out Zeno behaviour, and so another difficulty of this paper is to exclude the possibility of Zeno behaviour, for only a well designed impulsive event-triggered mechanism (IETM) can make the possibility eliminated.

It follows by (3.4) that
Below, we design the impulsive event-triggered mechanism (IETM) as follows: For a given diagonal matrix P > 0, and positive constants λ > 0 , we define the impulse instants as follows, where ζ k > 1, satisfying Below, we begin to consider the global stabilization on the ecosystem under event-triggered impulsive control.
then there does not exist Zeno behaviour under IETM(3.14), where x; t 0 , η) be the solution the system (2.7) with the initial value (t 0 , η). Define v(t) = ∫ Ω U T (t, x)PU(t, x)dx. Let t 1 < t 2 < · · · < t n < · · · be the trigger instants, and D be the upper right-hand Dini derivative with respective system (2.7), then Poincare inequality yields It follows by (2.4), differential mean value theorem and the condition (A2) that Combining (3.17),(3.18) and the condition (3.16) results in 19) which together with IETM (3.14) and the condition (3.16) implies that and hence Similarly, we can prove Hence, which completes the proof.
Remark 11. [22,Theorem 3.3] illuminates that the inter-execution time intervals of control task must not be arbitrarily small. In the proof of Proposition 3 of this paper, t n − t n−1 1 c 0 +λ ln(ζ n ), but 1 c 0 +λ ln(ζ n ) may be arbitrarily small, for example, ζ n = e 1 n > 1 and Π ∞ n=1 ζ n = +∞. Thus, Proposition 3 is better than some existing result due to its reducing conservatism, for Proposition 3 can also exclude the Zeno behavior.
Proof. The proof may be completed by considering three cases: Case 1. If the event is never triggered, IETM (3.14) yields Case 2. If the event is triggered in finite times, say N times, we assume t 1 < t 2 < · · · < t N are trigger instants, and and It follows from (3.20) and (3.23) that It follows from (3.27), 3.25) and (3.21) that (3.28)

Combining (3.28) and (3.24) results in
Similarly, we can prove Below, we consider the case of t t N .
Since t N is the last trigger time, IETM (3.14) yields which together with (3.30) means Case 3. If the event is triggered in infinite times, we assume t 1 < t 2 < · · · < t k < · · · are trigger instants.
Proposition 3 exclude the Zeno behaviour. Next, for any given t t 0 , there exists a N ∈ Z + such that t N−1 t < t N .
By way of the similar discussions in Case 2, we can derive (3.34) for all t t 0 , too. That is, (3.34) holds in all cases.
Now it follows from (3.34) that λ min (P)∥U(t)∥ 2 and hence which together with Definition 1 implies that the impulsive system (2.7) is globally exponentially stable under IETM (3.14). Equivalently, the positive stationary solution u * (x) is globally exponentially stable under IETM (3.14).
Remark 12. In Theorem 3.2, the impulse occurs only when the event is triggered. And t n − t n−1 1 c 0 +λ ln(ζ n ) means that if making ζ n become bigger. Thus, the bigger impulse interval can be guaranteed so that the cost of impulse control is smaller than that of existing results involved in the fixed impulsive interval (see [4,5]).  yeilds that the impulsive system (2.7) is globally exponentially stable under IETM (3.14). Equivalently, the positive stationary solution u * (x) is globally exponentially stable under IETM (3.14).

Conclusions and further considerations
In this paper, Saddle Theorem, the orthogonal decomposition of Sobolev space H 1 0 (Ω) and variational methods are applied to deduce the existence of six stationary solutions for RDGACM. Moreover, A new IETM is designed to make the RDGACM stabilized globally. There are many innovations in the ideas and methods (see Remark 6-12 for details). Now, we should consider further work, for example, d 1 λ 2 < b 1 < d 1 λ 3 or b 1 > d 1 λ 3 . Although there are some inspirations from the orthogonal decomposition technique on eigenfunction spaces (see,e.g. [6,8]), it seems to be an interesting problem.