On stochastic multi-group Lotka-Volterra ecosystems with regime switching

Focusing on stochastic dynamics involving continuous states as well as discrete events, this paper investigates dynamical behaviors of stochastic multi-group Lotka-Volterra model with regime switching. The contributions of the paper lie on: (a) giving the sufficient conditions of stochastic permanence for generic stochastic multi-group Lotka-Volterra model, which are much weaker than the existing results in the literature; (b) obtaining the stochastic strong permanence and ergodic property for the mutualistic systems; (c) establishing the almost surely asymptotic estimate of solutions. These can specify some realistic recurring phenomena and reveal the fact that regime switching can suppress the impermanence. A couple of examples and numerical simulations are given to illustrate our results.

1. Introduction. The Lotka-Volterra model proposed by Lotka [16] and Volterra [24] describes the interaction of n-species growth bẏ x(t) = diag(x 1 (t), · · · , x n (t)) [b + Ax(t)] , (1) where x(t) = (x 1 (t), · · · , x n (t)) T ∈ R n , each x i (t) represents the concentration size of i-th species, b = (b 1 , · · · , b n ) T ∈ R n is the intrinsic growth rate vector, A = (a ij ) n×n ∈ R n×n is the community matrix in which a ij measures the action of species j on the growth rate of species i. Generally, the intra-specic (if i = j) interaction is competitive as individuals of the same species compete for resources, food, or habitat, thus a ii ≤ 0. If a ij ≤ 0 ( 1 ≤ i, j ≤ n, i = j), (1) is termed competitive system while if a ij ≥ 0 ( 1 ≤ i, j ≤ n, i = j), it is referred to as In fact, ecosystems are always subject to environmental noises. It is necessary to reveal how the noise aects the population ecosystems. Stochastic population ecosystems have been drawn great attention by many researchers depicting more realistic ecosystems. Lotka-Volterra models perturbed by white noise have been investigated in [18,20] while the authors in [5,6,15,25] go further by considering the delay eects on the stochastic population ecosystems. The population habitats are distinguished by circumstance such as nutrition or rainfall [3,22] that at random instants the population growth is subjected to a palpable change. Such random changes can't be described precisely by the white noises induced by Brownian motion. In fact, the model involving both continuous dynamics and discrete events is more pertinent, in which the discrete events are modeled by a continuous-time Markov chain; see [19,23,27]. Our aim is to investigate the stochastic multi-group Lotka-Volterra ecosystem with environmental uctuations described by σ ij (γ(t))dB j (t), (2) or equivalently, in the matrix form, dx(t) = diag(x 1 (t), · · · , x n (t)) [(b(γ(t)) + A(γ(t))x(t)) dt + σ(γ(t))dB(t)] , with initial value x(0) = x 0 ∈ R n + , γ(0) = γ 0 ∈ S := {1, 2, · · · , m}. γ(·) is a continuous-time Markov chain with nite-state space S and the generator Q = (q ij ) m×m ∈ R m×m satisfying q ij ≥ 0 for i = j, j∈S q ij = 0 for each i ∈ S. B(·) is a standard d-dimensional Brownian motion.
The dynamics of ecosystems with regime switching are interesting and amazing. Considering a two-dimensional predator-prey Lotka-Volterra ecosystem switching between two individual systems described by ordinary dierential equations (ODEs), Takeuchi et al. in [23] revealed that both individual systems develop periodically but switching between them makes them become neither permanent nor dissipative. Various large-time behaviors of stochastic Lotka-Volterra ecosystems with regime switching have attracted more and more attention recently. For instance, criteria of extinction and permanence on sample paths to the predatorprey systems are yielded in [1]; The principle of coexistence and exclusion for the two-species competitive system are proved in [21]; More asymptotic properties for competitive systems and generic systems are referred to [3,11,17,28,29]. As far as we know there are few results on mutualism dynamics except for the ergodicity in [14], which is one of our motivations.
Taking into account of the dynamics of ecosystem switching between dierent habitat, we always ask ourselves: Whether the species will be permanent when their individual habitat are impermanent? To illustrate this, consider a stochastic predator-prey system switching randomly between two individual environments described by dx 1 (t) = x 1 (t) (2 − x 2 (t)) dt + x 1 (t)(dB 1 (t) + dB 2 (t)), dx 2 (t) = x 2 (t) 7 4 + x 1 (t) dt + x 2 (t)( √ 2dB 1 (t) + dB 2 (t)), (4) and dx 1 (t) = x 1 (t) 1 − 1 2 x 1 (t) − 2x 2 (t) dt + x 1 (t)( √ 2dB 1 (t) + 2dB 2 (t)), dx 2 (t) = x 2 (t) 2 + x 1 (t) − 1 5 x 2 (t) dt + x 2 (t)(dB 1 (t) + 2dB 2 (t)), modulated by a Markov chain γ(t) with generator Q = −9 9 1 −1 . Note that the sum of community matrix of (4) and its transpose has eigenvalue 0. The existing results in the literatures such as [11,14,20,28] cannot deal with this case. Figure 1 pictures two sample trajectories (resp., colored in black and blue) associated to (4) and (5). We see that the black (resp. blue) trajectory increases to innity (tends to zero). So the species in both environments are impermanent. What will happen to the switched system? This inspires us to formulate this paper. In this paper, we investigate permanence and asymptotic estimate of the sample paths for generic stochastic ecosystems with regime switching, then make further eorts on the mutualism dynamics involving strong permanence and existence of the invariant measure. Our contributions are as follows: • We reveal an interesting feature that the regime switching has average eect on the dynamics and can suppress the impermanence. For instance, the switched ecosystem is stochastically permanent by Theorem 3.8 although both (4) and • Compared with the results on the asymptotic estimates of moments in the literature [11,28] and [19, pp.157-163], we relax the restriction on the coefcients required in [28], and the restriction on the generator of the Markov chain γ(t) which is often required by utilizing the M-Matrix theory in [11] and [19, pp.157-163].
• Combing stochastic Lyapunov analysis and asymptotic analysis, we obtain the criterion on stochastic strong permanence of the mutualism ecosystem; see Theorem 5.4. Moreover, we analyze the almost surely asymptotic properties including the growth rate and sample paths in time average from both the above and below; see Theorems 4.1-4.3.
• Compared with the results on the ergodicity of the mutualism ecosystem in the literature [14,20], we relax the restrictions on the coecients and the generator of the Markov chain γ(t) required in [14] and cover the main result on the ergodicity of the mutualism ecosystem without regime switching in [20].
The above discussions rely on the value of switching rate (generator) which is constant. Hutchinson [7] suggested that suciently frequent switching can reverse Lotka-Volterra system switching between two dierent environment (described by ODEs) that are both favorable to the same species and showed rigorously that the sucient large switching rate depending on time t can lead to extinction of this species or coexistence of the two competing species as t tends to innity. Comparing with the results in [2], we focus on the dynamics of the generic stochastic ecosystem involving several interspecic relationships switched with a constant switching frequency due to the complexity and nonlinearity of stochastic dierential equations (SDEs), then go further to more precise large-time behaviors of the mutualism system. We shed light on the average eect of environmental switching that switching can suppress the impermanence.
The main methods of this paper are stochastic Lyapunov analysis and asymptotic analysis. Yin and Zhu in [27] began to utilize Fredholm alternative result to construct the Lyapunov functions depending on states and yielded stability and instability of linearized switching SDEs [27, pp. 203-207] and linearized switching ODEs [27, pp. 227-230]. Note that (3) is a n-dimensional nonlinear system, especially the interaction among the species brings about more complexity and requires more computational eort. We borrow the Yin and Zhu's idea in [27], construct several Lyapunov functions basing on the special structure, and yield our required dynamics of (3) by combining stochastic Lyapunov analysis with the asymptotic analysis. Moreover, we think this method is an attractive option to investigate more dynamical properties for nonlinear switching systems.
The rest of this paper is organized as follows. We introduce some preliminaries in Section 2. Section 3 is devoted to stochastic permanence of generic Lotka-Volterra ecosystem. Section 4 analyzes the almost surely asymptotic properties of the solutions including the growth rate and the time average. Section 5 begins to investigate mutualistic Lotka-Volterra ecosystem and yields the stochastic strong permanence. Section 6 goes a step further to examine ergodic property and positive recurrence, which accounts for some recurring events of the population ecosystem. Section 7 discusses some examples and carries out numerical simulations to illustrate our main results. Finally, Section 8 is the concluding remarks.
A generator Q or its corresponding Markov chain γ(t) is called irreducible if the following linear equations have a unique solution π = (π 1 , · · · , π m ) ∈ R 1×m (a row vector) satisfying π i > 0 for each i ∈ S. Such a solution is termed a stationary distribution [27]. Throughout this paper, as a standing assumption, we assume γ(t) is irreducible. Note that in the above πQ denotes the usual matrix multiplication. Throughout this paper, we use for example, πU and πu to denote the corresponding matrix multiplications with compatible dimensions. That is, for example, U ∈ R m×m and u ∈ R m := R m×1 , respectively.
Denote the positive cone by R n + = {x ∈ R n : x i > 0 for all 1 ≤ i ≤ n} and the nonnegative positive cone byR n If A is a vector or matrix, denote its transpose by A T and its trace norm by |A| = trace(A T A). If A = (a ij ) n×n is a symmetric n × n matrix, its largest eigenvalue is denoted by λ max (A), and dene λ + max (A) = sup x∈R n + ,|x|=1 x T Ax. Indeed, λ + max (A) ≤ λ max (A). We state a useful lemma which can be found in [19, pp. 58-62].
For any k ∈ S and any V (y, t, k) dened on R n ×R + which is continuously twice dierentiable in y and once in t, dene L y V by L y V (y, t, k) =V t (y, t, k) + V y (y, t, k)f (y, k) + 1 2 trace g T (y, k)V yy (y, t, k)g(y, k) ∂V (y, t, k) ∂y n , and V yy (y, t, k) = ∂ 2 V (y, t, k) ∂y i ∂y j n×n .
3. Stochastic permanence. Stochastic dierential equation (SDE) (3) models the population growth, each component of its solution representing the concentration size of the species. They should not only be nonnegative but also not explode to innity at any nite time. Although the classical theory of SDEs (see e.g. [4,19]) is not applicable directly to SDE (3), the existence of the global positive solution has been obtained by the standard technique in [11]. In this paper we cite this result and its corresponding assumption.
Lemma 3.1. [11] Under Assumption 1, for any initial value x 0 ∈ R n there is a unique solution x(t) to the SDE (3) and the solution will remain in R n + for all t ≥ 0 almost surely.
In this lemma, λ k ≤ 0 for each k ∈ S is necessary for the global existence of solutions because even for n = 1, if some λ k > 0, solutions may explode at nite time [12]. The above result provides us with a great opportunity to analyze the stochastic permanence. For clarity, we cite the denitions of stochastic permanence and its relatives in [11] . Denition 3.2. [11] The solutions of SDE (3) are called stochastically ultimately bounded from the above(resp. the below), if for any ∈ (0, 1), there is a positive constant χ(= χ( )) such that for any initial value x 0 ∈ R n + , γ 0 ∈ S, the solution of SDE (3) has the property that lim sup Denition 3.3. [11] The SDE (3) is called stochastically permanent if its solutions are stochastically ultimately bounded from both the above and below.
We begin with a criterion on asymptotic upper boundedness of the moment, and make use of it to obtain the stochastically ultimate upper boundedness of SDE (3).
Proof. Since the proof is rather technical we divided it into 3 steps.
Step 1. In order for the required assertion (9), we begin to construct an appropriate Lyapunov function depending on the states in S. Under Assumption 1, dene C k := (c 1 (k), · · · , c n (k)) for each k ∈ S. Note that Choose a small constant 0 < p 1 ≤ 1, such that for any 0 < p ≤ p 1 , Then it follows from the inequality a > log a for any a > 0 and the non-positivity of λ k that Step 2. In order to obtain the upper bound of L x V 1 , we compute the limits of V 1 and ϕ as |x| → 0 + and |x| → +∞. For each k ∈ S, and For any l, k ∈ S using Recalling the property of the generator that m l=1 q kl = 0, we have Choose a small constant 0 < p 0 ≤ p 1 , such that for any 0 < p ≤ p 0 , Taking the limit of (14) as |x| → +∞ and using (11), (17), (18), we obtain Step 3. Now, choose a suciently small positive constant ζ = ζ(p) such that Hence it follows from (15), (16), (19) and (20), we have Using the generalized Itô formula, we have where In view of the generalized Itô formula, s. For any t ≥ 0, t∧τ R is a bounded stopping time and t∧τ R → t a.s. The local martingale property implies that E[M V2 (t ∧ τ R )] = 0 and it therefore follows from (21) and (22) that Letting R → ∞, by virtue of the monotonic convergence theorem and the dominated convergence theorem (c.f. [19, pp. 8, 9]) Then we have then the required assertion follows.
Remark 1. Since the proof of Theorem 3.4 is rather technical, now we give an outline to help the reader. In Step 1 basing on the Fredholm alternative result we construct the Lyapunov function V 1 (x, k) depending on the states in S which can reveal the eect of switching randomly between m-environment states more precisely.
We go a step further to compute the limits of L x V 1 (x, k) as |x| → 0 + and |x| → +∞. Finally, we choose an appropriate constant ζ, then estimate L x (e ζt V 1 (x, k)) and analyze the asymptotical properties of E(e ζt V 1 (x(t), γ(t))). Thus the required result follows.
Remark 2. From Theorem 3.4, the sign of πλ is a key for asymptotic upper boundedness of the moment. In fact, πλ can be regarded as weighted arithmetic mean of λ 1 , · · · , λ m with positive weights π 1 , · · · , π m . Therefore some element λ k with a large weight π k contributes more to the weighted mean than how does the element with a low weight. By virtue of the law of large number, roughly, the stationary distribution π k determines the fraction of the time spent by Markov chain γ(t) in state k. The parameters of the k-th subsystem determine λ k representing some character of k-th environment. Therefore this theorem reveals that the asymtotic upper boundedness of moment results from the perform of all environments and uctuation randomly between them. Proof. Fix p > 0 suciently small such that (9) holds. For any ε ∈ (0, 1), let (9). Thus, by virtue of Chebyshev's inequality, The desired assertion follows.
The positivity of x(t) gives hints to the use of reciprocal transformation. Thus we estimate the asymptotic upper bound of the moment of 1/|x(t)| in order to obtain the stochastic ultimate boundedness from the below. For convenience, denȇ Theorem 3.6. Under Assumption 1 and π " β > 0, for θ > 0 suciently small the solution x(t) of (3) with any initial value x 0 ∈ R n + , γ 0 ∈ S has the property that Proof. In order to obtain the upper boundary of the moment of 1/|x(t)| with some order, we dene another Lyapunov function V 3 (x, k) ∈ R n + × S.
Next, similar to the proof of Lemma 3.4, we can obtain the required assertion. To avoid duplication, we omit the remaining part.
Using the method which is similar to that used in Theorem 3.5 we yield the following theorem directly. Theorem 3.7 generalizes the main result in [11]. To complete this section, let us compare our results with that in [11]. We rst cite the aforementioned result in the reference which can be considered as a corollary of ours. Corollary 2. [11] The SDE (3) is stochastically permanent under the following conditions (A.1) The generator Q = (q ij ) m×m satises that for some u ∈ S, q iu > 0 for all i = u.
Remark 5. We highlight our new results in this remark as follows.
1. Note that (A.1) can be regarded as a sucient condition for irreducibility. In [11], (A.1) was used to facilitate the application of M-matrix theory. In this paper, we only assume irreducibility rather than (A.1) and irreducibility. It is readily seen that the condition we used are much weaker than requiring some entire column of the generator to be positive. In fact, from [8], we can see that we do not need each entry of an entire column of the generator but only need each entry of a column after some nite number of iterations to be positive.
2. Note that (A.2) has been improved in the current paper. Assumption 1 implies λ k ≤ 0 for each k ∈ S. Thus πλ < 0 holds even if some λ k < 0.
Therefore the desired assertion follows fromĉ|x| ≤ĉ n i=1 x i ≤ Cx.
Remark 6. By the denitionβ of (26), Theorem 4.1 reveals the fact that the environmental noise can suppress exponential growth of the population and large noise intensity can make all species extinct.
On the other hand we look for the lower bound of the growth rate. For convenience we impose the following assumption.
Assumption 3. There exist positive numbers c 1 , · · · , c n , such that Using a similar argument we get the desired assertion. If there exist positive constants K 1 and K 2 such that
Taking the logarithm of both sides leads to Dividing both sides by t and then taking the supper limit gives lim sup The arbitrariness of ε1 implies lim sup On the other hand, it follows from (29) that log(Cx(t)) |x(s)|ds a.s.
By (35), Similar to the estimate of the superior limit we can obtain lim inf The desired assertion follows from (36) and (38).
Remark 7. Li et al. [11] claimed that the SDE (3)  Assumption 4. For each k ∈ S and i, j = 1, 2, · · · n with i = j, a ii (k) ≤ 0, Although the stochastic permanence implies that |x(t)| can keep away from 0 with large probability in long-time scale, it doesn't guarantee that each component of x(t) keeps away from 0. In this section we concentrate on the survival of each species, which is termed stochastic strong permanence. For clarity we give its denition. Denition 5.1. The SDE (3) is called stochastically strongly permanent if it is stochastically ultimately bounded from the above and below for each component.
Step 1. In order for the required assertion on x i (t) we begin to construct an appropriate Lyapunov function depending on the states in S. It follows from the Fredholm Alternative result (c.f. [27, pp.362-366]) the equation Choose two constants 0 < η Then the direct calculation shows Hence it follows from (2) and the property of the generator that Step 2. In order to obtain the upper bound of LW i , we analyze the limits of W i and ψ i as x i → 0 + and x i → +∞. Obviously, for each k ∈ S Now choose a small constant 0 < η Hence it follows from the denition of β i (k) and (41) that for each k ∈ S Step 3. Choose a positive constant κ i = κ i (η i ) suciently small such that It then follows from (45)-(47) that Thus, for each i, sup Using the generalized Itô formula (c.f. [27, p.29, eq. (2.7)]), we get Recalling the denition of W i we have Corollary 4. Under Assumption 4, if for each k ∈ S, a u (k) ≤ 0, πa u < 0 and each πβ i > 0 for i = 1, · · · , n, where a u (k) and a u are dened in Corollary 1, the SDE (3) is stochastically strongly permanent.
We complete this section by estimating the lower bound of the growth rate on each species.
Theorem 5.5. Under the conditions of Theorem 5.2, for any η i > 0 suciently small, the solution x(t) of (3) satises Proof. Dene U i : Similar to the proof of Theorem 5.2 we can obtain Then by the generalized Itô formula, Inequality (48) in the proof of Theorem 5.2 implies Choose ∆ suciently small such that For any positive integer r, (50) implies that E sup Using the Burkholder-Davis-Gundy inequality (c.f. [18,Theorem 7.3,pp.40 ]), Substituting (53) and (54) into (52) gives For any ε > 0, by the Chebyshev inequality, P ω : sup Thus, by the virtue of the Borel-Cantelli lemma, there exists a set Ω 0 ∈ F, with P(Ω 0 ) = 1 and an integer-valued random variable r 0 (ω) > 1 ∆ + 2, such that for almost all ω ∈ Ω 0 , 1+ε , whenever r > r 0 . (57) The arbitrariness of ε implies lim sup Therefore the desired assertion follows from the denition of U i . (see e.g. [20]) has drawn increasing attention [1,2,12,14,20,21,26]. This section is devoted to obtaining the existence of a unique stationary distribution for the mutualistic Lotka-Volterra ecosystem (3). The positive recurrence of Markovian process (x(t), γ(t)) is often required for its ergodicity in R n + × S, see [9,27]. Now we introduce a useful lemma on the positive recurrence as stated in [27].
(1) For each k ∈ S, with some constant κ 1 ∈ (0, 1] for all y ∈ R n ; (2) There exists a nonempty open set D ⊂ R n with compact closure and a nonnegative function V (·, k) : D c → R for each k ∈ S such that V (·, k) is twice continuously dierentiable and that for some α > 0, Assumption 5. For each k ∈ S, Rank(σ(k)) = n. Theorem 6.2. Under Assumption 5 and the conditions of Theorem 5.4, the process (x(t), γ(t)) given by (3) is positive recurrent in R n + × S . Proof. For each 1 ≤ i ≤ n, dene z i (t) := log x i (t) for t ≥ 0, and z(t) := (z 1 (t), · · · , z n (t)) T . By the generalized Itô formula, for each i, Obviously, the positive recurrence of (x(t), γ(t)) in R n + × S is equivalent to that of (z(t), γ(t)) in R n × S. Thus it suces to prove the process (z(t), γ(t)) given by (58) satises the conditions of Lemma 6.1. Note that σ(k)σ T (k) is positivedenite under Assumption 5, thus condition (1) holds. We now proceed to prove that condition (2) holds. Choose p, η i > 0 satisfying Theorem 3.4 and Theorem 5.2.
Furthermore, using Theorem 4.3 in [27, p.114], we yield the following theorem. Theorem 6.3. Under the conditions of Theorem 6.2, the process (x(t), γ(t)) of (3) has a unique stationary distribution in R n + × S.
By Theorems 4.1, the switching system is extinctive, see Figure 5 . Clearly, the main results in [11,14,20,28] are infeasible for the case that there exists some a ii (k) = 0. However, our results can deal with this case. Therefore, our results have wider range of applications. For mutualistic systems, we give more precise results including the stochastic strong permanence and the existence of the unique stationary distribution. We complete this section by giving another example on a mutualisitc system to illustrate the rest part results.  where each column has a non-diagonal element 0. The main results of [11,14] can't deal with such case. Solving the linear equation (6) we obtain the stationary distribution of γ(t), π = (π 1 , π 2 , π 3 ) = 1 6 , 1 6 , 2 3 .
By Theorems 4.1, the switching system is extinctive, see Figure 12.