Stability analysis of a single-species Markovian jumping ecosystem

: In this paper, impulsive control on a single-species Markovian jumping ecosystem leads to a stability criterion, and the newly-obtained theorems improve the related existing results. Numerical examples illuminate the effectiveness.


Introduction
As pointed out in [1], the following Logistic system has been widely concerned and studied due to its importance in the development of ecology: where Z(t) represents the density or quantity of population Z at time t, R > 0 and K represent the intrinsic growth rate of population and environmental capacity, respectively. Because all solutions of nonlinear ecosystem are difficult to be given accurately, people pay more attention to the long-term dynamic trend of ecosystem, i.e., the long-term trend of population density (see, e.g. [1][2][3][4][5]). People especially want to know whether the population will tend to a positive constant after a long time, which is related to the final long-term existence of the population. For example, the authors of [2] investigated the long time behavior of the following stochastic ecosystem for a single-species: Animal populations will inevitably spread because of climate, foraging and random walking. And hence the reaction-diffusion ecological models well simulate the real ecosystem, and ( [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). Particularly, reaction-diffusion ecosystem were studied in [11][12][13][14][15][16][17][18][19][20][21]. For example, in [12], a single-species Markovian jumping ecosystem with diffusion and delayed feedback under Dirichlet boundary value was investigated: u(t, x) =0, x ∈ ∂Ω, t 0. (3) Markov systems often occurred in various engineering technologies (see, e.g. [24][25][26]). Particularly, Markovian jumping delayed feedback model reflects well the influence of stochastic factors on time delays of the changes of populations, such as weather, temperature, humidity, ventilation status, and so on. But the case of Neumann boundary value on a single-species ecosystem is seldom researched. In fact, Neumann zero boundary value model well simulates the biosphere boundary without population migration. For example, freshwater fish do not enter the sea through rivers. Inspired by some ideas or methods of the related literature [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]30], the author is to investigate the stability of a single-species Markovian jumping ecosystem with diffusion and delayed feedback under Neumann zero boundary value.

System descriptions
Denote by (Υ, F , P) the complete probability space with a natural filtration {F t } t≥0 . Let S = {1, 2, · · · , n 0 } and the random form process {r(t) : [0, +∞) → S} be a homogeneous, finite-state Markovian process with right continuous trajectories with generator Π = (γ ij ) n 0 ×n 0 and transition probability from mode i at time t to mode j at time t + δ, i, j ∈ S, where γ ij 0 is transition probability rate from i to j(j = i) and γ ii = − ∑ n 0 j=1,j =i γ ij , δ > 0 and lim Consider the following ecosystem with diffusion and delayed feedback where a > 0 and b > 0 describe the growth rate and the intra-specific competition. Besides, the initial value function Γ(s, x) is bounded on [−τ, 0] × Ω. For convenience, c(r(t)) is denoted simply by c r for r ∈ S. In addition, due to the limited resources of nature, the population density should have an upper limit. At the same time, the population density should also have a lower limit. For example, if the population density of whales is less than a certain degree, the population will become extinct, because male whales and female whales cannot meet in the sea. So the following assumption is suitable: (H1) There exist two positive constants N 1 and N 2 with N 1 a b N 2 such that Definition 1. Set u * (t, x) ≡ u * (x), ∀ (t, x) ∈ [−τ, +∞) × Ω, then u * (x) is said to be a stationary solution of the ecosystem system (5) if u * (x) satisfies the boundedness assumption (H1), and Now, people can easily conclude from Definition 1 that u * ≡ a b is a stationary solution of the ecosystem system (5). Moreover, setting U(t, x) = u(t, x) − u * , then the system (5) is translated into where the zero solution of the system (8) is corresponding to the positive stationary solution u * ≡ a b of the ecosystem (5). And hence, the stability of the above-mentioned zero solution will be investigated below. Furthermore, applying an impulse control on the natural ecosystem (5) whose zero solution is corresponding to the equilibrium point u * ≡ a b of the following system:

Definition 2. For an arbitrarily given
, and the following integral equations hold for any t ∈ [0, T] and x ∈ Ω, and Remark 1. Definition 2 is well defined in view of [22] and [23]. In this paper, the following condition is also required: e t∆ w (see [22]).
Lemma 1.(see, e.g. [30]). Let Ω be a bounded domain of R m with a smooth boundary ∂Ω of class C 2 by Ω. v(x) is a real-valued function belonging to H 1 0 (Ω) and ∂v(x) ∂ν | ∂Ω = 0. Then which λ 1 is the smallest positive eigenvalue of the Neumann boundary problem where ν represents the external normal direction of ∂Ω.
Lemma 2 (Banach contraction mapping principle [29]) Let Θ be a contraction operator on a complete metric space , then there exists a unique point u ∈ for which Θ(u) = u.

Main result
Firstly, the unique existence of the stationary solution of the ecosystem (5) should be considered. Moreover, the unique stationary solution of the ecosystem should be positive. Based on the two point, the author presents the following unique existence theorem: Theorem 1. Suppose (H1) holds. For all r ∈ S, the system (5) possesses a positive stationary solution If, in addition, the following condition is satisfied: then the positive solution u * is the unique stationary solution of the system (5).
Thus, Definition 1 yields that u * > 0 defined in Theorem 1 is the unique stationary solution of the system (5). Below, the author claims that u * is the unique stationary solution of the ecosystem (5). Indeed, if u * and v * (x) are two different stationary solutions of the system (5), then Poincare inequality and the boundary condition yield The condition (12), Definition 2 and the continuity of u * and v * lead to which contradicts the inequality (13). This completes the proof.

Remark 2.
As far as I am concerned, Theorem 1 is the first theorem to give the unique existence of the equilibrium point of a single-species ecosystem under Neumann boundary value. Next, the global stability of the stationary solution u * ≡ a b should be investigated.
then zero solution of the system (9) is globally exponential stability in the pth moment, equivalently, u * ≡ a b of the system (10) is globally exponential stability in the pth moment. where Proof. Banach contraction mapping principle will play a important role in this proof, so the author formulates a contraction mapping on a suitable complete metric space firstly.
Let be the normed space consisting of functions g(t, x) : , where g satisfies: (A1) g is pth moment continuous at t 0 with t = t k (k ∈ N) ; (A2) for any given x ∈ Ω, lim t→t − k g(t, x) and lim t→t + k g(t, x) exist, and lim It is not difficult to verify that the normed space is a complete metric space if it is equipped with the following metric: Construct an operator Θ such that for any given U ∈ , Below, it is necessary to show that Θ : → , which may require four steps to achieve the goal.
Step 1. The author claim that for U ∈ , Θ(U) must be pth moment continuous at t 0 with ] means the boundedness of U, and let δ be small enough scalar, a routine proof yields that if δ → 0, for t ∈ [0, +∞) \ {t k } ∞ k=1 ,  which proves the claim. And then (A1) is verified.
Step 4. Verifying (A4), i.e., for U ∈ , verifying Indeed, Moreover, and Holder inequality yield On the other hard, e αt U i (t) p → 0 means that for any ε > 0, there exists t * > 0 such that all e αt U i (t) p < ε. And so, which together with the arbitrariness of ε implies that Combining (20)-(28) yields (19). It follows from the above four steps that Finally, the author claims that Θ is a contractive mapping on . Indeed, for any U, V ∈ , Holder inequality and (H2) yield Similarly, Suppose t j−1 < t t j , then the definition of Riemann integral b a e s ds yields It follows from (30)-(33) that where r satisfies 0 < r < 1. This shows that Θ : → is a contraction mapping such that there exists the fixed point U of Θ in , which implies that U is the solution of the system (9), satisfying e αt U p → 0, t → +∞ so that e αt u − u * p → 0, t → +∞ . Therefore, zero solution of the system (9) is globally exponential stability in the pth moment, equivalently, u * ≡ a b of the system (10) is globally exponential stability in the pth moment.

Remark 3.
As far as I am concerned, it is the first paper to apply the Laplacian semigroup theory to deal with the stability of a single-species ecosystem with Markovian jumping and delayed feedback.
Example 2.Suppose all the data of Example are applied to this example. Assume, in addition, p = 1.5, M k ≡ 1.02, µ = 5, Obviously the condition (H2) holds in Example 1, and direct calculations yield: And hence the condition (14) is satisfied. Thereby, Theorem 2 yields that zero solution of the system (9) is globally exponential stability in the pth moment, equivalently, u * ≡ 2.5 of the system (10) is globally exponential stability in the pth moment.

Conclusions and further considerations
In this paper, there are some improvements on mathematical methods, for it is the first paper to employ fixed point theory, Laplacian semigroup theory and variational methods to solve the unique existence of the globally stable positive equilibrium point of a single-species ecosystem with Markovian jumping and delayed feedback. Numerical examples are given to show the feasibility of artificial management of nature by way of impulse control.
As pointed out in [28, Definition 1], the author originally proposed a class of global asymptotical stability in the meaning of switching in [27,Definition 3]. Particularly, in the case of one subsystem, the switched system becomes non-switched common system, the global asymptotical stability in the meaning of switching becomes that in the classical meaning. The author actually gave both of the mentioned two classes of global stability for the unique (positive) stationary solution in [27]. Now the author wants to know whether the two classes of global stability can be applied to ecosystem, and what meanings about the stability in the meaning of switching for an ecosystem ? This is an interesting problem. Besides, in [27,Statement 2], the author originally design an example to show that under the influence of diffusion, the unique equilibrium point of ordinary differential system with Lipschitz continuous activation function becomes at least three equilibrium points of its corresponding partial differential system. Now the author wants to know whether the unique equilibrium point is globally stable. Furthermore, the author in [27,Section 5] originally proposed four problems, particularly [27, Problem 1 ] and [27,Problem 4 ] can also be suitable for the case of Neumann boundary value in this paper. Such problems are also interesting. Moreover, how to consider the case of infinite delays in [10, Problem 6] to ecosystem ? It is also an interesting problem. To sum up, the following problems are more interesting: Problem 1. Is the zero solution of the ordinary differential equation in [21,Theorem 3] or [27,Statement 2] global stable? Problem 2. How to design another example somewhat similar to [21,Theorem 3] or [27,Statement 2] with Lipschitz continuous activation function, where the ordinary differential equation possesses a globally stable equilibrium point, but its corresponding partial differential equation owns at least two stationary solutions.