JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.1 (1-24) Available online at www.sciencedirect.com ScienceDirect J Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise N.H Dang a,1 , N.H Du b,2 , G Yin a,∗,1 a Department of Mathematics, Wayne State University, Detroit, MI 48202, USA b Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam Received 24 July 2013; revised 27 March 2014 Abstract This work focuses on population dynamics of two species described by Kolmogorov systems of competitive type under telegraph noise that is formulated as a continuous-time Markov chain with two states Our main effort is on establishing the existence of an invariant (or a stationary) probability measure In addition, the convergence in total variation of the instantaneous measure to the stationary measure is demonstrated under suitable conditions Moreover, the Ω-limit set of a model in which each species is dominant in a state of the telegraph noise is examined in detail © 2014 Elsevier Inc All rights reserved MSC: 34C12; 60H10; 92D25 Keywords: Kolmogorov systems of competitive type; Telegraph noise; Stationary distribution; Ω-limit set; Attractor; Markovian switching * Corresponding author E-mail addresses: dangnh.maths@gmail.com (N.H Dang), dunh@vnu.edu.vn (N.H Du), gyin@math.wayne.edu (G Yin) This research was supported in part by the National Science Foundation DMS-1207667 This research was supported in part by NAFOSTED No 101.03.2014.58 http://dx.doi.org/10.1016/j.jde.2014.05.029 0022-0396/© 2014 Elsevier Inc All rights reserved JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.2 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• Introduction Kolmogorov systems of differential equations have been used to model the evolution of many biological and ecological systems An example is the well-known competitive Lotka–Volterra model, which represents the dynamics of the population sizes of different species in an ecosystem [2,5–7,10,18] It has been well recognized that the traditional models are often not adequate to describe the reality due to random environment and other random factors Recently, resurgent attention has been drawn to treat systems that involve both continuous dynamics and discrete events For example, using the common terminology of ecology, we considered a Lotka–Volterra model in which the growth rates and the carrying capacities are subject to environmental noise; see related models and formulations as well as various definitions of terms in [18] (see also [17], also related works [13] and [14]) It was noted that the qualitative changes of the growth rates and the carrying capacities form an essential aspect of the dynamics of the ecosystem These changes usually cannot be described by the traditional deterministic population models For instance, the growth rates of some species in the rainy season will be much different from those in the dry season Note that the carrying capacities often vary according to the changes in nutrition and/or food resources Likewise, the interspecific or intraspecific interactions differ in different environments The environment changes cannot be modeled as solutions of differential equations in the traditional setup They are random discrete events that happen at random epochs A convenient formulation is to use a continuous-time Markov chain taking values in a finite set The result dynamic systems become nowadays popular so-called regime-switching differential equations In this work, we consider a two-dimensional system that is modulated by a Markov chain taking values in M = {1, 2} Individual equations corresponding to the states and are different Thus in lieu of one system of Kolmogorov equations, one needs to deal with systems of equations correspond to each state in M Our focus in this work is devoted to analyzing ergodic behavior of such systems In the recent work [3], assuming that the random environment is represented by a continuous-time, two-state Markov chain, we obtained certain limit results and depicted the Ω-limit set for systems that have a globally stable positive equilibrium In this paper, we concentrate on the problem: What are sufficient conditions that ensure the ergodicity of the Kolmogorov systems? That is, what are conditions to ensure the existence of stationary distributions for such systems It is well known that the coupling owing to the Markov chain makes the underlying systems more difficult to analyze For example, in the study of stability, it has been known that a system resulted from two individual stable differential equations coupled by a Markov chain may be unstable So our intuition may not always give the correct conclusion By carefully analyzing such systems, this paper provides sufficient conditions for existence of stationary distributions of competitive type Kolmogorov systems The rest of the paper is arranged as follows The formulation of the problem is given in Section Then Section takes up the issue of the existence of an invariant probability measure Section continues the investigation by focusing on Ω-limit sets and properties of the invariant measure Section deals with dynamics of the system when each species dominates in one state Finally, Section concludes the paper with further remarks Problem formulation Let (Ω, F, P) be a complete probability space and {ξ(t) : t ≥ 0} be a continuous-time Markov chain defined on (Ω, F, P), whose state space is a two-element set M = {1, 2} and whose genq q12 −α α = β −β with α > and β > Note that we use Ω in lieu erator is given by Q = q11 21 q22 JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.3 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• of the usual Ω to denote the sample space, denote an element of Ω by ω, and reserve Ω for the notion of omega-limit set to avoid notional conflict It follows that, = (p1 , p2 ), the stationary distribution of {ξ(t) : t ≥ 0} satisfying the system of equations Q=0 p1 + p2 = 1, is given by β , t→∞ α+β α p2 = lim P ξ(t) = = t→∞ α+β p1 = lim P ξ(t) = = (2.1) Such a two-state Markov chain is commonly referred to as telegraph noise because of the graph of its sample paths Let Ft be the σ -algebra generated by ξ(·) up to time t and P-null sets The filtration {Ft }t≥0 satisfies the usual condition That is, it is increasing and right continuous while F0 contains all P-null sets Then (Ω, F, {Ft }, P) is a complete filtered probability space In this paper, we focus on a Kolmogorov system under telegraph noise given by x(t) ˙ = x(t)a ξ(t), x(t), y(t) y(t) ˙ = y(t)b ξ(t), x(t), y(t) , (2.2) where (x, y) and bi (x, y) are real-valued functions defined for i ∈ M and (x, y) ∈ R2+ , and are continuously differentiable in (x, y) ∈ R2+ = {(x, y) : x ≥ 0, y ≥ 0} Note that in the above and henceforth, we write (x, y) instead of a(i, x, y) to distinguish the discrete state i with the continuous state (x, y) Because of the telegraph noise ξ(t), the system switches randomly between two deterministic Kolmogorov systems x(t) ˙ = x(t)a1 x(t), y(t) y(t) ˙ = y(t)b1 x(t), y(t) , x(t) ˙ = x(t)a2 x(t), y(t) y(t) ˙ = y(t)b2 x(t), y(t) (2.3) (2.4) Stemming from models in classical competitive ecosystems, throughout this paper, we impose the following two assumptions Assumption 2.1 For each i ∈ M, (x, y) and bi (x, y) are continuously differentiable in (x, y) ∈ R2+ Moreover, (x,0) ∂ai∂x < ∀x > and i ∈ M (0, 0) > 0, lim supx→∞ (x, 0) < (0,y) ∂bi∂y < ∀y > and i ∈ M bi (0, 0) > 0, lim supy→∞ bi (0, y) < JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.4 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• Assumption 2.2 For any (x0 , y0 ) ∈ R2+ , there is a compact set D = D(x0 , y0 ) ⊂ R2+ containing (x0 , y0 ) such that D is an invariant set under both systems (2.3) and (2.4) These conditions are satisfied for all well-known competitive models in ecology Under these assumptions, we can derive the existence and uniqueness of a global positive solution given 2,◦ 2,◦ the initial value (x(0), y(0)) = (x0 , y0 ) ∈ R+ , where R+ = {(x, y) : x > 0, y > 0} denotes the interior of the set R+ Moreover, it is noted in Assumption 2.1 that we only impose conditions on the boundary so that this assumption can be satisfied by an even wider range of models That is, not only are the conditions satisfied for competitive models, but also for some cooperative ones Consider two equations on the boundary u(t) ˙ = u(t)a ξ(t), u(t), , u(0) ∈ (0, ∞) (2.5) v(t) ˙ = v(t)b ξ(t), 0, v(t) , v(0) ∈ (0, ∞) (2.6) For (2.5), it is easily seen that the pair of processes (ξ(t), u(t)) is Markovian and the associated operator of the Markov process is given by Lgi (u) = qij gj (u) + ua1 (u, 0) j =1 d gi (u), du for i ∈ M, and for each gi (u) defined on (0, ∞) and continuously differentiable in u By Assumption 2.1, there is a unique pair (u1 , u2 ) satisfying a1 (u1 , 0) = and a2 (u2 , 0) = In case u1 = u2 , without loss of generality, assume u1 < u2 Under Assumption 2.1, the process (ξ(t), u(t)) has a unique invariant probability measure concentrated on M × [u1 , u2 ] The stationary density (μ1 , μ2 ) of (ξ(t), u(t)) can be obtained from the solution of the system of Fokker–Planck equations L∗ μi (u) = for i ∈ M (L∗ denotes the adjoint of L) ⎧ d ⎪ ⎪ ua1 (u, 0)μ1 (u) = 0, ⎨ −αμ1 (u) + βμ2 (u) − du ⎪ d ⎪ ⎩ αμ1 (u) − βμ2 (u) − ua2 (u, 0)μ2 (u) = du (2.7) Solving the system of equations above, we obtain μ1 (u) = θ F (u) , u|a1 (u, 0)| μ2 (u) = θ F (u) , u|a2 (u, 0)| where u β α + dτ , τ a1 (τ, 0) τ a2 (τ, 0) F (u) = exp − u ∈ [u1 , u2 ], u = u u2 θ= p1 u1 F (u) F (u) + p2 du u|a1 (u, 0)| u|a2 (u, 0)| −1 u1 + u2 , JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.5 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• Likewise, under Assumption 2.1, for (2.6), there exist v1 , v2 such that for v1 < v2 , the stationary density of (ξ(t), v(t)), say (ν1 , ν2 ), is given by ν1 (v) = ζ G(v) , v|b1 (0, v)| ν2 (u) = ζ G(v) , v|b2 (0, v)| where v G(v) = exp − v v2 ζ= v1 β α + dτ , τ b1 (0, τ ) τ b2 (0, τ ) G(v) G(v) + p2 dv p1 v|b1 (0, v)| v|b2 (0, v)| v ∈ [v1 , v2 ], v = v + v2 , −1 In case u1 = u2 (resp v1 = v2 ), (μ1 (u), μ2 (u)) = (δu−u1 , δu−u1 ) (resp (ν1 (u), ν2 (u) = (δv−v1 , δv−v1 ))) is a generalized density given by δu−u1 (resp δv−v1 ), where δ· is the Dirac function Remark 2.1 Analyzing the dynamics on the boundary provides us with important properties of positive solutions To gain insight, let us first look at the deterministic system (2.3) On the boundary, there are three equilibria (0, 0), (u1 , 0), and (0, v1 ) Under Assumption 2.1, the origin is a source and other solutions cannot approach it On the other hand, we note that the eigenvalues of the Jacobian matrix ∂(xa1 (x,y)) ∂(xa1 (x,y)) ∂x ∂y ∂(yb1 (x,y)) ∂(yb1 (x,y)) ∂x ∂y are v1 ∂b ∂x (0, v1 ) and a1 (0, v1 ) In view of Assumption 2.1, ∂b ∂x (0, v1 ) < Therefore, a sufficient condition for (0, v1 ) to repel positive solutions is that it is a saddle point or equivalently a1 (0, v1 ) > In the same manner, we need b1 (u1 , 0) > to guarantee that (u1 , 0) does not attract positive solution Using basic results from the dynamical systems theory, it is not hard to show that under theses conditions, system (2.3) is permanent, that is any positive solution will never approach the boundary The idea used above can be generalized to treat our random system (2.2) in which invariant measures take the role of equilibria Moreover, the values a1 (0, v1 ), b1 (u1 , 0) now need to be replaced with the expected values of a(ξ(t), 0, v(t)) and b(ξ, u(t), 0) with respect to their corresponding invariant measures, respectively For this reason, we introduce λ1 and λ2 , which play a crucial role to determine the dynamical behaviors of (2.2) λ1 = p1 a1 (v, 0)ν1 (v) + p2 a2 (v, 0)ν2 (v) dv, [v1 ,v2 ] λ2 = p1 b2 (u, 0)μ1 (u) + q2 b2 (u, 0)μ2 (u) du (2.8) [u1 ,u2 ] In our recent paper [3], imposing the condition λ1 , λ2 > 0, we have given the Ω-limit set of positive solutions to (2.2) in some cases However, the questions whether the positivity of λ1 and λ2 implies the existence of a finite invariant measure for the process (ξ(t), x(t), y(t)) and what is the behavior of the omega-limit set if neither (2.3) nor (2.4) has a positive equilibrium are still JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.6 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• open The purpose of this paper is to address these questions In Section 3, we prove the existence of an invariant probability measure, provided that λ1 > and λ2 > 0, which is assumed throughout this paper Section is an improvement of the results in [3] In particular, we described the Ω-limit set of system (2.2) requiring only that either system (2.3) or system (2.4) has a globally stable positive equilibrium Stability in total variation of solution to (2.2) is obtained Furthermore, in Section 5, we consider (2.2) in the case where each species is dominant in one state The existence of an invariant probability measure The trajectory of ξ(t) is piecewise-constant and cadlag (right continuous having left limits) functions Let = τ < τ1 < τ2 < · · · < τn < · · · be its jump times Put σ = τ − τ0 , σ = τ − τ1 , ., σn = τn − τn−1 , σ1 = τ1 is the first jump time from the initial state; σ2 is the time that process ξ(t) sojourns in the state ξ(τ1 ) It is known that {σk }∞ k=1 are conditionally independent given the sequence Note that if ξ(0) is given, then ξ(τ {ξτk }∞ n ) is known because the process ξ(t) takes only two k=1 is a sequence of independent random variables taking values in [0, ∞) values Hence, {σk }∞ n=1 Moreover, if ξ(0) = 1, then σ2n+1 has the exponential density α1[0,∞) (t) exp(−αt) and σ2n has the density β1[0,∞) (t) exp(−βt) Conversely, if ξ(0) = 2, then σ2n has the exponential density α1[0,∞) (t) exp(−αt) and σ2n+1 has the density β1[0,∞) (t) exp(−βt) (see [4, vol 2, p 217]) Here 1[0,∞) = for t ≥ (= for t < 0) For a positive initial value (x0 , y0 ), we denote by (x(t, ω, x0 , y0 ), y(t, ω, x0 , y0 )) the solution to Eq (2.2) at time t , starting in (x0 , y0 ) (or (x(t, x0 , y0 ), y(t, x0 , y0 )), (x(t), y(t)) whenever there is no ambiguity) Remark 3.1 We note that, as seen in [3, p 394], under Assumptions 2.1 and 2.2, there is a < δ < M such that we can suppose without loss of generality that (x(t, x0 , y0 ), y(t, x0 , y0 )) ∈ [0, M]2 \ [0, δ]2 ∀t ≥ Note that δ, M are chosen such that (δ, 0), bi (0, δ) > and (M, 0), bi (0, M) < for all i ∈ M We now state the main theorem of this section Theorem 3.1 If λ1 and λ2 are positive, the Markov process (ξ(t), x(t), y(t)) has an invariant 2,◦ probability measure π ∗ on the state space M × R+ To prove this theorem, we need to estimate the average time that (x(t), y(t)) spends on some 2,◦ compact subset of M × R+ In the proof of [3, Theorem 2.1], we showed that (x(t), y(t)) cannot stay near the boundary for a long time However, the method in that proof failed to uniformly estimate the sojourn time In this paper, making use of a suitable stationary process, we can estimate the sojourn time uniformly with a large probability Hence, the existence of an invariant probability measure will be shown To proceed, we need some auxillary results We JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.7 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• begin with the initial data P{ξ(0) = 1} = p1 , P{ξ(0) = 2} = p2 It follows that for all subsequent time t ≥ 0, P{ξ(t) = 1} = p1 , P{ξ(t) = 2} = p2 , which implies that ξ(t) is a stationary process Therefore, there exists a semigroup of P-measure preserving transformations θ t satisfying ξ(t + s, ω) = ξ(t, θ s ω) ∀t, s > Let u(ω, t, u0 ) and v(ω, t, v0 ) be solutions of (2.5) and (2.6) with initial values u0 , v0 respectively Fix u∗ , v∗ ∈ [δ, M], denote u∗ (ω, t) = u(ω, t, u∗ ) and v ∗ (ω, t) = v(ω, t, v∗ ) The following lemma holds Lemma 3.1 For any ε > 0, there exists a non-random positive number T1 = T1 (ε) such that with probability 1, |u(t, ω, u0 ) − u∗ (t, ω)| < ε and |v(t, ω, u0 ) − v ∗ (t, ω)| < ε for all t > T1 , provided u0 , v0 ∈ [δ, M] Proof For simplicity, in this proof, we denote u(t) = u(ω, t, u0 ) and drop ω from u∗ (ω, t) Without loss of generality, suppose that u0 < u∗ Owing to the uniqueness of the solutions, we have δ ≤ u(t) < u∗ (t) ≤ M ∀t ≥ (0,y) (x,0) Let m be a positive number such that ∂ai∂x ≤ −m and ∂ai∂y ≤ −m ∀δ ≤ x, y ≤ M It is clear that d ln u∗ (t) − ln u(t) = a ξ(t), u∗ (t), − a ξ(t), u(t), dt ≤ −m u∗ (t) − u(t) (3.1) The mean value theorem yields ∗ ∗ u (t) − u(t) ≥ ln u∗ (t) − ln u(t) ≥ u (t) − u(t) δ M Hence, d ln u∗ (t) − ln u(t) ≤ −m u∗ (t) − u(t) ≤ −mδ ln u∗ (t) − ln u(t) dt In view of the comparison theorem, we obtain ln u∗ (t) − ln u(t) ≤ (ln u∗ − ln u0 )e−mδt , which implies that u∗ (t) − u(t) ≤ M(ln u∗ − ln u0 )e−mδt ≤ Mδ(ln M − ln δ)e−mδt Similarly, v ∗ (t) − v(t) ≤ Mδ(ln M − ln δ)e−mδt Letting t → ∞ obtains the desired result The proof is complete ✷ Lemma 3.2 For any ε > 0, there exists a T2 = T2 (ε) > and a subset A ∈ F∞ with P(A) > −ε such that ∀t > T2 and ω ∈ A, JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.8 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• t t b ξ(s), u∗ (s), ds − λ1 < ε, t t a ξ(s), 0, v ∗ (s) ds − λ2 < ε (3.2) Proof Since (ξ(t), v(t)) has a unique invariant distribution (whose density is (ν1 (v), ν2 (v))), t it follows from the Birkhoff Ergodic theorem that limt→∞ 1t a(ξ(s), 0, v(s))ds = λ1 almost surely given that v(0) admits (ν1 (v), ν2 (v)) as its density function Moreover, it is not diffi˜ v) cult to show that for any given initial value v(0) > 0, and any (i, ˜ ∈ M × (v1 , v2 ), we have ˜ v) P{(ξ(t), v(t)) = (i, ˜ for some t ≥ 0} = As a result, we obtain the strong law of large numt bers, that is, for any initial value v(0) > 0, P{limt→∞ 1t a(ξ(s), 0, v(s))ds = λ1 } = (see [1, t p 169] or [11] for more details) In particular, P{limt→∞ 1t a(ξ(s), 0, v ∗ (s))ds = λ1 } = 1 t ∗ Similarly, P{limt→∞ t b(ξ(s), u (s), 0)ds = λ1 } = The rest of this proof is now straightforward ✷ As a result of Lemmas 3.1 and 3.2, we obtain the following proposition Proposition 3.1 We can find a T3 = T3 (ε) ≥ T2 (ε) such that ∀t > T3 and δ ≤ u0 , v0 ≤ M, t t b ξ(s, ω), u(s, ω, u0 ), ds − λ1 < ε, t t a ξ(s, ω), 0, v(s, ω, v0 ) ds − λ2 < ε, (3.3) for almost all ω ∈ A, where A is the set mentioned in Lemma 3.2 (0,y) (x,0) Let L = max{| ∂ai∂x |; | ∂bi∂y | : ≤ x, y ≤ M} > 0, we have the following lemma Lemma 3.3 For ε > 0, we can find a γ = γ (ε) ∈ (0, δ] (not depend on (x0 , y0 ) ∈ [0, M]2 \ [0, δ]2 ) such that • If y(s, ω, x0 , y0 ) < γ ∀0 ≤ s ≤ t , then y(s, ω, x0 , y0 ) + u(s, ω, x0 ) − x(s, ω, x0 , y0 ) < ε L ∀0 ≤ s ≤ t ε L ∀0 ≤ s ≤ t • Similarly, if x(s, ω, x0 , y0 ) < γ ∀0 ≤ s ≤ t , then x(s, ω, x0 , y0 ) + v(s, ω, y0 ) − y(s, ω, x0 , y0 ) < JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.9 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• Proof For any sufficiently small ε1 > 0, let uε1 (t, ω, u0 ) be the solution to u˙ ε1 (t) = uε1 (t) a ξ(t), uε1 (t), − ε1 starting from u0 We prove that for almost all ω ∈ Ω, ≤ u(t, ω, u0 ) − uε1 (t, ω, u0 ) ≤ Mε mδ ∀u0 ∈ [δ, M], t ≥ First, using the comparison theorem, it is clear that ≤ u(t, ω, u0 ) − uε1 (t, ω, u0 ) for all t ≥ Moreover, d ln u(t, ω, u0 ) − ln uε1 (t, ω, u0 ) dt = a ξ(t, ω), u(t, ω, u0 ), − a ξ(t, ω), uε1 (t, ω, u0 , 0) + ε1 ≤ −m u(t, ω, u0 ) − uε1 (t, ω, u0 ) + ε1 ≤ −mδ ln u(t, ω, u0 ) − ln uε1 (t, ω, u0 ) + ε1 (3.4) Hence, in view of the comparison theorem, u(t, ω, u0 ) − uε1 (t, ω, u0 ) ≤ M ln u(t, ω, u0 ) − ln uε1 (t, ω, u0 ) Mε1 − e−mδt mδ Mε1 ≤ ∀t ≥ mδ ≤ By the continuity of (x, y), we can find a γ = γ (ε1 ) such that whenever ≤ y(s, ω) ≤ γ and ≤ x(s, ω) ≤ M, then a ξ(s, ω), x(s, ω), − ε1 ≤ x(s, ˙ ω) = a ξ(s, ω), x(s, ω), y(s, ω) ≤ a ξ(s, ω), x(s, ω), + ε1 By the comparison theorem, x(s, ω, x0 ) ≥ uε1 (s, ω, x0 ), ∀0 ≤ s ≤ t , which derives that u(s, ω, x0 ) − x(s, ω, x0 ) ≤ Mε mδ ∀0 ≤ s ≤ t Analogously, we have x(s, ω, x0 ) − u(s, ω, x0 ) ≤ Mε1 Mε1 ε mδ ∀0 ≤ s ≤ t Choosing suitable ε1 = ε1 (ε) and γ such that γ + mδ < L , we have the claim The proof of Lemma 3.3 is complete ✷ Now, we are in a position of proving Theorem 3.1 Proof of Theorem 2.1 To simplify the notation, denote z0 = (x0 , y0 ) and z(t, ω, z0 ) = x(t, ω, x0 , y0 ), y(t, ω, x0 , y0 ) (3.5) Since ξ(t + s, ω) = ξ(t, θ s ω) ∀t, s > 0, we have z(t + s, ω, z0 ) = z t, θ s ω, z(s, ω, z0 ) ∀t, s > (3.6) JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.10 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 10 Fix a T > T3 such that T1 (ln M − ln δ) < ε Put χn (ω) = 1A (θ nT ω), where A is as in Lemma 3.2 and 1A (·) is the indicator function {χn } is obviously a stationary process since θ T is a measurepreserving transformation (see [9, Section 16.4]) We now prove that θ T is ergodic with respect to (Ω, F, {Ft }t≥0 , P) Since F∞ ⊂ F and A ∈ F∞ , where F∞ = {Ft : t ≥ 0}, χn is F∞ -measurable It thus suffices to prove that θ T is ergodic with respect to (Ω, F∞ , P), that is, there is no set B ∈ F∞ satisfying < P(B ∩ θ −T B) = P(B) < Suppose that there is a set B ∈ F∞ satisfying this almost invariant property Since F∞ = σ ({Ft : t ≥ 0}), for any ε > 0, we can find a set B ∈ Ft for some t such that P(B B ) < ε where B B = (B \ B ) ∪ (B \ B) Let Mt be the space of functions from [0, t ] to M and Bt the cylindrical σ -algebra on Mt Since Ft is the σ -algebra generated by ξ(t), t ∈ [0, t ] and P-null sets, we can choose B to be of the form B = {ξt (·) ∈ C } for some C ∈ Bt where ξt (h + ·) denotes the trajectory of ξ(·) in [h, h + t ] for each h ≥ 0, that is ξt (h + t) = ξ(h + t) ∀t ∈ [0, t ] Let n0 be so large that n0 T > t and that |P{ξ(n0 T − t ) = i | ξ(0) = j } − pi | < ε ∀i, j ∈ M We have P(B ∩ θ −n0 T B ) = P{ξt (·) ∈ C , ξt (n0 T + ·) ∈ C } Using the Markov property, we deduce that P ξt (·) ∈ C , ξt (n0 T + ·) ∈ C = P ξt (·) ∈ C P ξt (n0 T + ·) ∈ C ξ(n0 T ) = i P ξ(n0 T ) = i ξ t = j i,j ∈M × P ξ t = j ξt ∈ C ≤ P ξt (·) ∈ C P ξt (n0 T + ·) ∈ C ξ(n0 T ) = i pi + ε i∈M ≤ P ξt (·) ∈ C P ξt (·) ∈ C ξ(0) = i pi + 2ε i∈M ≤ P ξt (·) ∈ C ≤ P B 2 + 2ε since P ξ(0) = i = pi + 2ε ≤ P(B) + ε + 2ε Since θ t preserves measures and P(B B ) < ε , it is easy to see that P B ∩ θ −n0 T B ≤ P B ∩ θ −n0 T B + 2ε ≤ P(B) + ε + 4ε < P(B) for sufficiently small ε On the other hand, it follows from the property P(B ∩ θ −T B) = P(B) that P(B ∩ θ −nT B) = P(B) ∀n ∈ N This contradiction means that the transformation θ T is ergodic In view of the Birkhoff Ergodic theorem, (see [9, Theorem 16.14]) k→∞ k k−1 χn = P(A) lim n=0 a.s (3.7) JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.11 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 11 Let χn1 (ω) = if χn (ω) = and y(t, ω, z0 ) < γ ∀nT ≤ t ≤ (n + 1)T otherwise, χn2 (ω) = if χn (ω) = and x(t, ω, z0 ) < γ ∀nT ≤ t ≤ (n + 1)T otherwise, χn3 (ω) = if χn (ω) = and ∃nT ≤ t ≤ (n + 1)T : x(t, ω, z0 ), y(t, ω, z0 ) > γ otherwise It follows from Remark 3.1 that χn = χn1 + χn2 + χn3 For convenience, put χn4 = − χn By (3.6), if χn1 = then y(t + nT , ω, z0 ) < γ ∀0 ≤ t ≤ T (or y(t, θ nT ω, z(nT , ω)) < γ ∀0 ≤ t ≤ T ), which is combined with Remark 3.1 to obtain that δ ≤ x(nT , ω, z0 ) ≤ M Moreover, χn1 = implies χn (ω) = 1, i.e., θ nT ω ∈ A Thus, it follows from Proposition 3.1 and Lemma 3.3 that T T b ξ t, θ nT ω , u t, θ nT ω, x(nT , ω, z0 ) , dt − λ1 < ε, (3.8) and that T T y t, θ nT ω, z(nT , ω, z0 ) dt T + T u t, θ nT ω, x(nT , ω, z0 ) − x t, θ nT ω, z(nT , ω, z0 ) ds < ε L (3.9) Using (3.6) again and the inequality |bi (x, y) − bi (u, 0)| ≤ L(|x − u| + y), for all x, y, u ∈ (0, M], we have ln y (n + 1)T , ω, z0 − ln y(nT , ω, z0 ) T (n+1)T = T T b ξ(t, ω), z(t, ω, z0 ) dt = T nT b ξ(t + nT , ω), z(t + nT , ω, z0 ) dt T = T b ξ t, θ nT ω , z t, θ nT ω, z(nT , ω, z0 ) dt T = T b ξ t, θ nT ω , u t, θ nT ω, x(nT , ω, z0 ) , dt T + T b ξ t, θ nT ω , z t, θ nT ω, z(nT , ω, z0 ) JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.12 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 12 − b ξ t, θ nT ω , u t, θ nT ω, x(nT , ω, z0 ) , dt T ≥ T T b ξ t, θ nT ω , u t, θ nT L ω, x(nT , ω) , )dt − T y t, θ nT ω, z(nT , ω, z0 ) dt T L − T u t, θ nT ω, x(nT , ω, z0 ) − x t, θ nT ω, z(nT , ω, z0 ) dt (3.10) Applying inequalities (3.8) and (3.9) to (3.10), we claim that if χn1 (ω) = then ln y(nT + T , ω, z0 ) − ln y(nT , ω, z0 ) ≥ λ1 − 2ε T (3.11) In case χn2 (ω) = 1, we have x(t, ω, x0 , y0 ) < γ ∀t ∈ [nT , (n + 1)T ] Therefore, by Remark 3.1, we derive δ ≤ y(nT , ω, x0 , y0 ), y(nT + T , ω, x0 , y0 ) ≤ M As a result, 1 ln y(nT + T , ω, x0 , y0 ) − ln y(nT , ω, x0 , y0 ) ≥ (ln δ − ln M) > −ε T T (3.12) Let H = max{|ai (x, y)|, |bi (x, y)| : ≤ x, y ≤ M, i ∈ M} If χn3 (ω) = or χn4 (ω) = 1, then ln y(nT + T , ω, z0 ) − ln y(nT , ω, z0 ) T = T nT +T b ξ(t, ω), z(t, ω, z0 ) dt ≥ −H (3.13) nT Applying (3.10), (3.12), and (3.13) to the equation ln y(nT + T , ω, x0 , y0 ) − ln y(nT , ω, x0 , y0 ) T = T ln y(nT + T , ω, x0 , y0 ) − ln y(nT , ω, x0 , y0 ) χni , i=1 we have ln y(nT + T , ω, z0 ) − ln y(nT , ω, z0 ) T ≥ (λ1 − 2ε)χn1 (ω) − εχn2 (ω) − H χn3 (ω) + χn4 (ω) summing over from n = to k − 1, and dividing by k, we obtain (3.14) JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.13 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 13 ln y(kT , ω, z0 ) − ln y0 ) kT ≥ (λ1 − 2ε) k k−1 k−1 χn1 (ω) − ε n=0 k−1 χn2 (ω) − H n=0 χn3 (ω) + χn4 (ω) (3.15) n=0 Consequently, lim sup k→∞ (λ1 − 2ε) k k−1 k−1 χn1 (ω) − ε n=0 k−1 χn2 (ω) − H n=0 χn3 (ω) + χn4 (ω) ≤0 a.s (3.16) χn3 (ω) + χn4 (ω) ≤0 a.s (3.17) n=0 Similarly, we can obtain lim sup k→∞ (λ2 − 2ε) k k−1 k−1 χn2 (ω) − ε n=0 k−1 χn1 (ω) − H n=0 n=0 Let λ = min{λ1 , λ2 }, adding side by side (3.16) and (3.17) yields lim sup k→∞ (λ − 3ε) k k−1 k−1 χn1 (ω) + χn2 (ω) − H n=0 χn3 (ω) + χn4 (ω) ≤0 a.s (3.18) n=0 Moreover, using (3.7), we have − lim k→∞ k k−1 χn1 (ω) + χn2 (ω) + χn3 (ω) n=0 = − lim k→∞ k k−1 χn (ω) n=0 = −P(A) ≤ −(1 − ε), (3.19) and k→∞ k k−1 χn4 (ω) = − P(A) ≤ ε lim (3.20) n=0 Multiplying both sides of (3.19) by (λ − 3ε) and multiplying both sides of (3.20) by H , then adding to (3.18), we have lim sup k→∞ −(H + λ − 3ε) k k−1 χn3 (ω) ≤ −(λ − 3ε)(1 − ε) + H ε a.s., n=0 or for ε sufficiently small, lim inf k→∞ k k−1 χn3 (ω) ≥ n=0 (λ − 3ε)(1 − ε) − H ε := m > H + λ − 3ε a.s (3.21) JID:YJDEQ AID:7505 /FLA 14 [m1+; v 1.193; Prn:5/06/2014; 13:46] P.14 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• We can find a γ = γ (γ , T ) > such that z(t, ω, z) ∈ [γ , M]2 ∀t ∈ [0, T ], z ∈ (0, M]2 , provided that there is an s ∈ [0, T ] such that z(s, ω, z) ∈ [γ , M]2 As a result, if χn3 = 1, then nT +T 1{z(t,ω,z0 )∈[γ ,M]2 } dt = T (3.22) nT It follows from (3.21) and (3.22) that t lim inf t→∞ t 1{z(s,ω,z0 )∈[γ ,M]2 } ds = m > a.s In view of Fatou’s lemma, t lim inf t→∞ t P z(s, ω, z0 ) ∈ [γ , M]2 ds = m > (3.23) Since the process (ξ(t), z(t)) = (ξ(t), x(t), y(t)) has the Feller property, (3.23) guarantees the 2,◦ ; see [12] or [16] ✷ existence of an invariant probability measure on M × R+ Ω-limit set and stability We denote by πt1 (u, v) = (x1 (t, u, v), y1 (t, u, v)), (resp πt2 (u, v) = (x2 (t, u, v), y2 (t, u, v)) the solution of Eq (2.3) (resp (2.4)) with initial value (u, v) The Ω-limit set of the trajectory starting from an initial value (x0 , y0 ) is defined by Ω(x0 , y0 , ω) = x(t, ω, x0 , y0 ), y(t, ω, x0 , y0 ) T >0 t>T We use the notation “Ω-limit set” in lieu of the usual one “ω-limit set” in dynamic systems to avoid notational conflict We still use ω as an element of the probability space For simplicity, we set Xn = x(τn , x0 , y0 ); F0n = σ (τk : k ≤ n); Yn = y(τn , x0 , y0 ); Fn∞ = σ (τk − τn : k > n) It is clear that (Xn , Yn ) is F0n -measurable and if ξ(0) is given then F0n is independent of Fn∞ The following lemma can be proved using arguments similar to that of [3, Theorem 2.2] by noting that the Lebesgue measure on R+ is absolutely continuous w.r.t any exponential distribution Lemma 4.1 Let {ηn }∞ be a sequence of strictly increasing finite stopping times with respect to filtration Fn Suppose that A is a Borel subset of R+ with positive Lebesgue measure Then, the events An = {σηn +1 ∈ A} occur infinitely often a.s., that is, JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.15 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 15 ∞ ∞ P Ai = k=1 i=k Lemma 4.2 If λ1 and λ2 are positive, we can find an h¯ > such that with probability the events {h¯ ≤ X2k , Y2k ≤ M} as well as {h¯ ≤ X2k+1 , Y2k+1 ≤ M} occur infinitely often Proof It is clear that (ξ(t), x(t), y(t)) is a Feller–Markov process with respect to the filtration {Ft } hence it is also a strong Markov process For a stopping time ζ , the σ -algebra at ζ is Fζ = {A ∈ F∞ : A ∩ {ζ ≤ t} ∈ Ft ∀t ∈ R+ } Fix T4 > 0, by [3, Theorem 2.1], we can define almost surely finite stopping times η1 = inf t > : x(t) ≥ δ, y(t) ≥ δ , η2 = inf t > η1 + T4 : x(t) ≥ δ, y(t) ≥ δ , ηn = inf t > ηn−1 + T4 : x(t) ≥ δ, y(t) ≥ δ For a stopping time ζ , we write τ (ζ ) as the first jump time of ξ(t) after ζ , i.e., τ (ζ ) = inf{t > ζ : ξ(t) = ξ(ζ )} Let σˆ (ζ ) = τ (ζ ) − ζ and Ak = {σˆ (ηk ) < T4 }, k ∈ N Then Ak+1 is in the σ -algebra generated by {ξ(ηn+1 + s) : s ≥ 0} while Ak ∈ Fηk+1 Therefore, in view of [4, Theorem 5, p 59] and an analogue of [4, Theorem d), p 36] for strong Markov processes, we obtain for Ack := Ω \ Ak , k ∈ N that P Ack ξ(ηk ) = i = P σˆ (0) > T4 ξ(0) = i ≤ p := max P σˆ (0) > T4 ξ(0) = i i∈M 0, λ2 > 0, and that system (2.3) has a globally stable positive equilibrium (x1∗ , y1∗ ) Let JID:YJDEQ AID:7505 /FLA 16 [m1+; v 1.193; Prn:5/06/2014; 13:46] P.16 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• (n) S = (x, y) = πtn ◦ · · · ◦ πt1 (1) x1∗ , y1∗ : < t1 , t2 , , tn ; n ∈ N , (4.3) where (k) = if k is even, otherwise (k) = Then a) With S denoting the closure of S, S is a subset of the Ω-limit set Ω(x0 , y0 , ω) with probability b) If there exists a t0 > such that the point (x , y ) = πt20 (x1∗ , y1∗ ) satisfying the following condition det a1 (x , y ) b1 (x , y ) a2 (x , y ) b2 (x , y ) = (4.4) 2,◦ Then, S absorbs all positive solutions in the sense that for any initial value (x0 , y0 ) ∈ R+ , the value γ˜ (ω) = inf{t > : (x(s, ω, x0 , y0 ), y(s, ω, x0 , y0 )) ∈ S ∀s > t} is finite outside 2,◦ with a P-null set Consequently, S is the Ω-limit set Ω(x0 , y0 , ω) for any (x0 , y0 ) in R+ probability Moreover, we can characterize the invariant probability measure by the following theorem Theorem 4.2 Suppose that Assumptions 2.1 and 2.2 are satisfied, that λ1 > 0, λ2 > 0, and that system (2.3) has a globally stable positive equilibrium (x1∗ , y1∗ ) Assume further that there exists a t0 > such that the point (x , y ) = πt20 (x1∗ , y1∗ ) satisfying (4.4) Then the stationary distribution 2,◦ and for any initial π ∗ has a density f ∗ with respect to the Lebesgue measure m on M × R+ distribution, the distribution of (ξ(t), x(t), y(t)) converges to π ∗ in total variation Before giving the proof, we recall some concepts in [11] to be used to prove the convergence in total variation Let X be a locally compact, separable metric space, B(X) the Borel σ -algebra on X Let Φ = {Φt : t ≥ 0} be a homogeneous Markov process with state space X, B(X) and transition semigroup P (t, x, ·) We can consider the process Φ on a probability space (Ω, F, {Px }x∈X ) where the measure Px satisfies Px (Φt ∈ A) = P (t, x, A) for all x ∈ X, t ≥ 0, A ∈ B(X) Suppose further that Φ is a Borel right process Note that if Φ is a Feller process then it is a Borel right process; see [15] for the definition of a Borel right process and the aforesaid implication For B ∈ B(X), define ∞ τB = inf{t ≥ : Φt ∈ B}, ηB = 1{Φt ∈B} dt Φ is said to be Harris recurrent if either of the following equivalent conditions is satisfied (H1) There is a nontrivial σ -finite measure ϕ1 such that Px (τB < ∞) = ∀x whenever ϕ1 (B) > (H2) There is a nontrivial σ -finite measure ϕ2 such that Px (ηB = ∞) = ∀x whenever ϕ2 (B) > Φ is positive Harris recurrent if it is Harris recurrent and it has an invariant probability measure For a probability measure a on R+ , we define a sampled Markov transition function Ka of Φ by JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.17 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 17 ∞ Ka (x, B) = P (t, x, B)a(dt) Ka is said to possess a nowhere-trivial continuous component if there is a kernel T : (X, B(X)) → R+ satisfying • For each B ∈ B(X), the function T (·, B) is lower semi-continuous • For any x ∈ X, T (x, ·) is a non-trivial measure satisfying Ka (x, B) ≥ T (x, B) ∀B ∈ B(X) Φ is called a T -process if for some probability measure a, the corresponding transition function Ka admits a nowhere-trivial continuous component Proof of Theorem 4.2 It suffices to prove the convergence in total variation since the other assertions of this theorem follow from Theorem 3.1 and [3, Theorem 3.1] To proceed, we will prove that our process is a positive Harris recurrent T -process Let P (t, i, z, E) be the transition 2,◦ and E belongs to the probability function of (ξ(t), x(t), y(t)), where t ∈ R+ , (i, z) ∈ M × R+ 2,◦ ∗ ∗ ∗ Borel σ -algebra B(M × R+ ) Denote z1 = (x1 , y1 ) and z0 = (x , y ) By the existence and continuous dependence of solutions on initial conditions, there exists a small positive number c such that ϕ(s, t) = πs1 ◦ πt2 (z0 ) is defined and continuously differentiable in (−3c, 3c)2 Since det ∂ϕ ∂ϕ , ∂s ∂t = x y det (0,0) a1 (x , y ) b1 (x , y ) a2 (x , y ) b2 (x , y ) =0 2,◦ ∂ϕ we can suppose that det( ∂ϕ ∂s , ∂t ) = ∀(s, t) ∈ (−3c, 3c) Let K = t0 + 2c For z ∈ R+ , ◦ πs22 ◦ πs11 (z) in the domain W := {(s1 , s2 ) : we consider a function ψz (s1 , s2 ) = πK−s −s2 ◦ πs22 (z1∗ ) = < s1 , t0 − c < s2 , s1 + s2 < K} In particular, we have ψz1∗ (s1 , s2 ) = πK−s −s2 ϕ(K − s1 − s2 , s2 − t0 ) Therefore, det( ∂ψz∗ ∂ψz∗ ∂s1 , ∂s2 )|(c,t0 ) = Thus, there exists an ε > such ∂ψz ∂ψz ∗ that det( ∂s1 , ∂s2 )|(c,t0 ) = for all z ∈ Uε For each z ∈ Uε∗ , there exists a neighborhood Wz of (c, t0 ) such that ψz is a diffeomorphism between Wz and ψz (Wz ) Since ψz is also continuously 1 differentiable with respect to z, by modifying slightly the inverse function theorem, when ε is sufficiently small, we can choose Wz ⊂ W such that W = ψz (Wz ) is the same for all z ∈ Uε∗ and that d1 := inf z∈Uε∗ ,w∈W Jz (w) > where Jz (w) is the determinant of the Jacobian matrix of ψz−1 at w Let f (s, t) is the density function of (σ1 , σ2 ) provided that ξ(0) = Recall that given ξ(0) = 1, σ1 , σ2 are independent exponential variables, so f (s, t) is a smooth function and d2 := inf(s,t)∈W f (s, t) > Let d3 := P{σ3 > K | ξ(0) = 1} > It is easy to see that for any z ∈ Uε∗ and any Borel set B ⊂ W , P K, i, z, {1} × B ≥ P (σ1 , σ2 ) ∈ ψz−1 (B), σ3 > K ξ(0) = ≥ d3 Wz f (s, t)1ψ −1 (B) (s, t)dsdt z JID:YJDEQ AID:7505 /FLA 18 [m1+; v 1.193; Prn:5/06/2014; 13:46] P.18 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• ≥ d2 d3 1ψ −1 (B) (s, t)dsdt z Wz ≥ d2 d3 1B (w1 , w2 ) Jz (w1 , w2 ) dw1 dw2 W ≥ d1 d2 d3 m(B) ˜ = d4 m {1} × B , (4.5) 2,◦ where m(·) ˜ is the Lebesgue measure on R+ and d4 is some positive constant In the proof of [3, 2,◦ , there is with Theorem 2.2], we claim that for any initial value (i, z0 ) = (i, x0 , y0 ) ∈ M × R+ ∗ probability a sequence sn (ω) ↑ ∞ such that ξ(sn ) = 1, (x(sn ), y(sn )) ∈ Uε Combining this property and (4.5), it follows from the strong Markov property of (ξ(t), x(t), y(t)) that if B ⊂ W and m({1} × B) > 0, then (ξ(t), x(t), y(t)) will enter {1} × B at some moment t = t (ω) almost surely It means that the condition (H1) is satisfied with the measure φ1 (E) = m(E ∩ ({1} × B)) 2,◦ for E ∈ B(M × R+ ) Thus, (ξ(t), x(t), y(t)) is positive Harris recurrent Since z1∗ is a globally asymptotically stable equilibrium of (2.3), for each k ∈ N, there is some nk ∈ N such that πn1k K (z) ∈ Uε∗ and πn1k K−s ◦ πs2 (z) ∈ Uε∗ for every z ∈ [k −1 , k]2 , ≤ s ≤ K We can choose {nk } to be an increasing sequence Putting pk = min{P{σ1 > nk K | ξ(0) = 1}, P{σ1 < K, σ1 + σ2 ≥ nk K | ξ(0) = 2}}, we have P nk K, i, z, {1} × Uε∗ ≥ pk ∀i ∈ M, z ∈ k −1 , k (4.6) Exploiting the Kolmogorov–Chapman equation, it follows from (4.5) and (4.6) that P (nk + 1)K, i, z, {1} × B ≥ pk d4 m {1} × B ∀i ∈ M, for any Borel set B ⊂ W (4.7) Let a is a probability measure on R+ given by a(nK) = 2−n , n ∈ N Consider the following −n Markov transition function Ka (i, z, E) = ∞ n=1 P (nK, i, z, E) Define the kernel T : (M × 2,◦ 2,◦ R+ , B(M × R+ )) → R+ by T (i, z, E) = 2−nk+1 −1 pk+1 d4 m E ∩ {1} × W if z ∈ (k + 1)−1 , k + \ k −1 , k , k ∈ N 2,◦ Hence, it follows from (4.7) that Ka (i, z, E) ≥ T (i, z, E) ∀E ∈ B(M × R+ ) Moreover, 2,◦ for each E ∈ B(M × R+ ), T (i, z, E) is a lower-semi continuous function As a result, (ξ(t), x(t), y(t)) is a T -process Applying [11, Theorems 3.2(ii) and 8.1(ii)], we conclude that 2,◦ P (t, i, z, ·) − π ∗ (·) → ∀(i, z) ∈ M × R+ where · is the total variation norm (Note that for a positive Harris recurrent process, the invariant transition function Π(x, ·) in [11, Theorem 8.1] turns out to be a unique invariant probability measure For our process, it is π ∗ ) ✷ Dynamics of the system when each species dominates one state This section considers the case that one of the two species dominates a (discrete) state (e.g., in dry season), but it becomes weaker and loses the dominance when the environment changes (e.g., in rainy season) In the mathematical modeling, we can suppose that the first species tends to be vanished in one state meanwhile the second species eventually dies out in the other state JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.19 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 19 The interesting question is that what is the behavior of the system if the states switch randomly from one to the other We describe this situation by the following assumption Assumption 5.1 (0, v1 ) (reps (u2 , 0)) is a saddle point of system (2.3) (resp (2.4)) while (u1 , 0) (resp (0, v2 )) is stable Moreover, all positive solutions to (2.3) (resp (2.4)) converge to the stable equilibrium (u1 , 0) (resp (0, v2 )) Remark 5.1 (0, v1 ) (resp (u2 , 0)) is a saddle point of system (2.3) (resp (2.4)) if and only if a1 (0, v1 ) > (resp b2 (u2 , 0) > 0) By the center manifold theorem and the attractivity of (u1 , 0) and (0, v2 ), there exists a (x1 , y1 ) such that the solution starting at (x1 , y1 ) of (2.3) can expand to the whole real line and lim π t→∞ t x1 , y1 = (u1 , 0) and lim π t→−∞ t x1 , y1 = (0, v1 ) (5.1) lim π t→−∞ t x2 , y2 = (u2 , 0) (5.2) similarly, there exists a point (x2 , y2 ) such that lim π t→∞ t x2 , y2 = (0, v2 ) and Denote by Γ1 and Γ2 their orbits, respectively Let a, b (a > 0, b < 0) be the eigenvalues of the Jacobian matrix of (xa1 (x, y), yb1 (x, y))T at the saddle point (0, v1 ), it follows from the Hartman–Grobman theorem and a suitable linear transformation of coordinates that for some sufficient small δ1 > 0, there exists a homeomorphism φ from Bδ1 = {(x, y) : x + y < δ1 } onto an open neighborhood of (0, v1 ) such that φ(0, 0) = (0, v1 ), φ(0, ϑ) ∈ y-axis ∀|ϑ| < δ1 ; φ(ς, 0) ∈ Γ1 ∀0 < ς < δ1 2,◦ φ(ς, ϑ) ∈ R+ ∀ς ≥ 0, (ς, ϑ) ∈ Bδ1 If π(t, ς, ϑ) is a local solution to the system ς˙ = aς, ϑ˙ = bϑ, ς ≥ satisfying π(t, ς, ϑ) ∈ Bδ1 , then φ(π(t, ς, ϑ)) is a solution to (2.3) Since limt→−∞ πt1 (x , y ) = (0, v1 ) ∀(x , y ) ∈ Γ1 and φ(ς, 0) ∈ Γ1 ∀0 < ς < δ1 , for any neighborhood V of (x , y ) ∈ Γ1 , there are < ε < σ0 < δ1 such that πt1 (x, y) will eventually go through V if (x, y) ∈ φ(Bε (ς0 , 0)) where Bε (ς0 , 0) = { (ς, ϑ) − (ς0 , 0) < ε} It is clear that for any ς ∈ (0, δ1 ) and < ε < δ1 − ς0 , the solution (ς(t), ϑ(t)) starting at (ς, ϑ) ∈ (0, ς0 ) × (−ε, ε) must visit Bε (ς0 , 0) = { (ς, ϑ) − (ς0 , 0) < ε} at some t > Hence, for (x, y) ∈ Vς0 ,ε = φ((0, ς0 ) × (−ε, ε)), we have {πt2 (x, y) : t ≥ 0} ∩ φ(Bε (ς0 , 0)) = ∅ Moreover, since πt1 (0, v2 ) → (0, v1 ) as t → ∞, we can exploit the continuous dependence of so2,◦ lutions on initial values to show that {πt1 (x, y) : t ≥ 0} ∩ Vς0 ,ε = ∅ when (x, y) ∈ R+ is sufficiently close to (0, v2 ) Combining the above comments yields that there is a δ2 > such that {πt2 (x, y) : t ≥ 0} ∩ V = ∅ if < x < δ2 and |y − v2 | ≤ δ2 This fact is demonstrated in Fig Furthermore, if < δ3 ≤ x ≤ δ2 and |y − v2 | ≤ δ2 , we can estimate an upper bound for the time πt1 (x, y) enters V using the compactness of {(x, y) : δ3 ≤ x ≤ δ2 , |y − v2 | ≤ δ2 } and the continuous dependence of solutions on initial values again To sum up, we can obtain the following lemma JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.20 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 20 Fig Saddle equilibrium for (2.3) and its linearized form Lemma 5.1 For any neighborhood V of (x , y ) ∈ Γ1 , there exists a δ2 > such that • For any (x, y) satisfying < x < δ2 , v2 − δ2 < y < v2 + δ2 , we have {πt1 (x, y) : t ≥ 0} ∩ V = ∅ • For any < δ3 < δ2 , there is a t3 > such that πt1 (x, y) ∈ V for some t ∈ [0, t3 ], provided that δ3 < x < δ2 , v2 − δ2 < y < v2 + δ2 For simplicity, in this section, we suppose that ξ(0) = Lemma 5.2 For any δ2 > 0, there is a < δ3 < δ2 and infinitely many integers k = k(ω) such that δ3 ≤ X2k ≤ δ2 and v2 − δ2 < Y2k < v2 + δ2 almost surely Proof In view of Lemma 4.2, with probability 1, there are infinitely many positive integers k such that h¯ ≤ X2k+1 , Y2k+1 ≤ M Since (0, v2 ) is a stable equilibrium of the system (2.4) and it attracts all positive solutions, we can find a t4 > satisfying that πt2 (x, y) − (0, v2 ) < δ2 for all t > t4 Obviously, δ3 := inf{x2 (t, x, y) : h¯ ≤ x, y ≤ M, t4 ≤ t ≤ 2t4 } > It follows from Lemmas 4.1 and 4.2 that the events {h¯ ≤ X2k+1 , Y2k+1 ≤ M, t4 ≤ σ2k+2 < 2t4 } occur infinitely often with probability Note that whenever these events occur, we have δ3 ≤ X2k+2 ≤ δ2 , v2 − δ2 ≤ Y2k+2 ≤ v2 + δ2 The lemma is therefore proved ✷ Lemma 5.3 Γ1 , Γ2 ⊂ Ω(x0 , y0 , ω) a.s Proof Let (x , x ) be arbitrarily in Γ1 and δ2 , δ3 be determined as in Lemma 5.1 and Lemma 5.2, respectively Define almost surely finite stopping times ζ1 = inf{2k : δ3 ≤ X2k ≤ δ2 , v2 − δ2 ≤ Y2k ≤ v2 + δ2 }, ζ2 = inf{2k > ζ1 : δ3 ≤ X2k ≤ δ2 , v2 − δ2 ≤ Y2k ≤ v2 + δ2 }, ζn = inf{2k > ζn−1 : δ3 ≤ X2k ≤ δ2 , v2 − δ2 ≤ Y2k ≤ v2 + δ2 } JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.21 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 21 Let t3 be defined as in Lemma 5.1 Put Ck = {σζk +1 > t3 } Applying Lemma 4.1 again, we conclude that there are infinitely many k ∈ N such that σζ k +1 > t3 It follows from Lemma 5.1 that if Ck occurs, there is an sk ∈ [τζk , τζk +1 ] such that (x(sk ), y(sk )) = πs1k (Xζk , Yζk ) ∈ V Since limk→∞ τζk = ∞, sk → ∞ as k → ∞ As a result, (x , y ) ∈ Ω(x0 , y0 , ω) a.s ✷ Moreover, we can prove a stronger result that enable us to describe completely the Ω-limit set Lemma 5.4 For any open neighborhood V of (x , y ) ∈ Γ1 , there are infinitely many k’s such that (X2k+1 , Y2k+1 ) ∈ V ✷ Proof It is analogous to that of Lemma 4.2 Similar to the notation in (4.3), we define (n) S1 = (x, y) = πtn ◦ · · · ◦ πt2 (2) ◦ πt1 (1) (u, v) : (u, v) ∈ Γ1 , ≤ t0 , t1 , t2 , , tn ; n ∈ N , (5.3) where (k) = if k is even, otherwise (k) = Theorem 5.1 Suppose that λ1 , λ2 > and Assumption 5.1 holds Then S1 is contained in the Ω-limit set Ω(x0 , y0 , ω) a.s Further, if there is a (x , y ) ∈ Γ1 such that det a1 (x , y ) b1 (x , y ) a2 (x , y ) b2 (x , y ) = 0, (5.4) 2,◦ then, S1 absorbs all positive solution in the sense that for any initial value (x0, y0 ) ∈ R+ , the value γ˜ (ω) = inf{t > : (x(s, x0 , y0 , ω), y(s, x0 , y0 , ω)) ∈ S1 (resp S2 ) ∀s > t} is finite outside a P-null set Consequently, the closure S of S1 is the Ω-limit set Ω(x0 , y0 , ω) for any (x0 , y0 ) 2,◦ a.s in R+ Proof It is proved using Lemma 5.4 and the arguments as in the proof of [3, Theorem 2.2] ✷ Remark 5.2 If we define (n+1) S2 = (x, y) = πtn ◦ · · · ◦ πt2 (3) ◦ πt1 (2) (u, v) : (u, v) ∈ Γ2 , ≤ t0 , t1 , t2 , , tn ; n ∈ N , then we have the same conclusion that the closure S of S2 is the Ω-limit set Ω(x0 , y0 , ω) for 2,◦ with probability As a result S = S any (x0 , y0 ) in R+ We now consider the existence and properties of an invariant probability measure In order to that we need an additional assumption on the Lie algebra of vector fields 2,◦ is said to satisfy Condition H if vectors w1 (z) − Definition 5.2 A point z = (x, y) ∈ R+ w2 (z), [w1 , w2 ](z), [w1 , [w1 , w2 ]](z), span R2 where wi (x, y) = (xai (x, y), ybi (x, y)) and [·, ·] is the Lie bracket JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.22 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 22 Theorem 5.3 Suppose that λ1 > 0, λ2 > and Assumption 5.1 holds Suppose further that there is a (x , y ) ∈ Γ1 (resp Γ2 ) such that (5.4) holds Moreover, there exists a (x , y ) ∈ S1 (resp S2 ) satisfying condition H Then the invariant probability measure π ∗ of (ξ(t), x(t), y(t)) 2,◦ has a density f ∗ with respect to the Lebesgue measure on M × R+ and is concentrated on M × S1 , (resp M × S2 ) Moreover, for any initial value, the distribution of (ξ(t), x(t), y(t)) converges in total variation to π ∗ Proof Suppose that there is a point (x , y ) ∈ Γ1 assuming (5.4) and that there exists a (x , y ) ∈ S1 satisfying condition H Theorems 3.1 and 5.1 declare the existence of an invariant probability measure concentrated on M × S1 In view of the two claims given below, the absolute continuity of π ∗ with respect to Lebesgue measure is proved using arguments in the proof of [3, Proposition 3.1] Finally, similar to that of Theorem 4.2, the convergence in total variation of (ξ(t), x(t), y(t)) is proved ✷ Claim Any point of M × S1 is approachable from all other points In other word, for any (i, z1 ), (j, z2 ) ∈ M × S1 , and any neighborhood {j } × Vz2 of (j, z2 ) there exist s1 , s2 , , sn ≥ such that πslnn ◦ · · · ◦ πsl22 ◦ πsl11 (z1 ) ∈ Vz2 where lk ∈ M, l1 = i, ln = j Proof By the definition of S1 , we only need to prove that if (j, z2 ) ∈ Γ1 × M, (j, z2 ) is approachable from any (i, z1 ) ∈ M × S1 Indeed, for an open neighborhood Vz2 of z2 ∈ Γ1 , it follows from Lemma 5.1 that there is a δ2 > such that if z ∈ (0, δ2 ) × (v2 − δ2 , v2 + δ2 ), there is a t (z) such that πt11 (z) (z) ∈ Vz2 Moreover, since (0, v2 ) attracts all positive solutions to Eq (2.4), for all z ∈ R12,◦ , there is t (z) > such that πt22 (z) (z) ∈ (0, δ2 ) × (v2 − δ2 , v2 + δ2 ) As a result, π 11 ◦ π 21 (z1 ) ∈ Vz2 and π 12 ◦ π 22 ◦ π 12 (z1 ) ∈ Vz2 where s11 = t (z1 ), s21 = t (π 21 (z1 )) s2 s1 s3 s2 s1 s1 and s12 > 0, s22 = t (π 12 (z1 )), s32 = t (π 22 ◦ π 12 (z1 )) which means that (1, z2 ) is approachable s1 s2 s1 Note that if z ∈ Vz2 , then πs2 (z) ∈ Vz2 when s is sufficiently small Consequently, (2, z2 ) is also approachable ✷ m Claim There exists positive numbers K, sˆ1 , , sˆm such that i=1 sˆi < K and that eiρ(m+1) ρ(m) ρ(1) ρ(m+2) ther ϕ (s1 , , sm ) = πK− m s ◦ πsm ◦ · · · ◦ πs1 (x , y ) or ϕ (s1 , , sm ) = πK− m s ◦ ρ(m+1) πs m ρ(2) i=1 i i=1 i ◦ · · · ◦ πs1 (x , y ) has the Jacobian matrix at (ˆs1 , , sˆm ) of rank Proof Note that condition H is normally called the hypo-ellipticity condition in Hormander’s theory and Claim is a classical result from the geometrical control theory which can be found on [8, Chapter 3] ✷ Example 5.4 We illustrate our results by plotting a sample orbit of the system x(t) ˙ = x a ξ(t) − b ξ(t) x − c ξ(t) y y(t) ˙ = y d ξ(t) − e ξ(t) x − f ξ(t) y , (5.5) where a1 = 6, b1 = 3, c1 = 2, d1 = 12, e1 = 4, f1 = 3, a2 = 12, b2 = 3.6, c2 = 2.2, d2 = 9.6, e2 = 4, f2 = 2, x(0) = 2, y(0) = 1, α = 3, β = By calculation, λ1 ≈ 0.13, λ2 ≈ 0.963 The JID:YJDEQ AID:7505 /FLA [m1+; v 1.193; Prn:5/06/2014; 13:46] P.23 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• 23 Fig Phase portraits of system (5.5) under the first set of parameters Fig Phase portraits of system (5.5) under another set of parameters results of Theorems 5.1 and 5.3 hold for this case since their hypothesis can be verified easily The sample orbit eventually provides us with the image of the Ω-limit set; see Fig Example 5.5 In this example, we still consider (5.5) with different parameters We use a1 = 4, b1 = 1, c1 = 2, d1 = 3, e1 = 1, f1 = 3, a2 = 3, b2 = 3, c2 = 1, d2 = 4, e2 = 2, f2 = 1, x(0) = 1.5, y(0) = 2, α = 1, β = Then λ1 and λ2 are calculated with λ1 = 0.181 and λ2 = 0.178, respectively Fig provides a sample orbit of the solution Conclusion Under simple conditions, we have established the existence of stationary distributions for Kolmogorov type systems under telegraph noise Such ergodicity results will be of essential JID:YJDEQ AID:7505 /FLA 24 [m1+; v 1.193; Prn:5/06/2014; 13:46] P.24 (1-24) N.H Dang et al / J Differential Equations ••• (••••) •••–••• utility in many applications of two-dimensional systems under random environment, especially for ecology systems Further asymptotic results have also obtained for the associated Ω-limit sets Although only a two-state Markov chain (the telegraph noise) is considered, it appears that these results can be generalized to systems perturbed by a Markov chain of more than two states without much difficulty Our results are based on the sufficient condition that λ1 > and λ2 > For future study, interesting and important questions to be addressed include: What are necessary conditions for the existence of the stationary distribution? 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Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998 [3] N.H Du, N.H Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J Differential