Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise

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 f

J Differential Equations ••• (••••) •••–•••

www.elsevier.com/locate/jde

N.H Danga,1, N.H Dub,2, G Yina,∗,1

aDepartment of Mathematics, Wayne State University, Detroit, MI 48202, USA

bDepartment 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

ThisworkfocusesonpopulationdynamicsoftwospeciesdescribedbyKolmogorovsystemsofitivetypeundertelegraphnoisethatisformulatedasacontinuous-timeMarkovchainwithtwostates.Ourmaineffortisonestablishingtheexistenceofaninvariant(orastationary)probabilitymeasure.Inaddition,theconvergenceintotalvariationoftheinstantaneousmeasuretothestationarymeasureisdemonstratedundersuitableconditions.Moreover,theΩ-limitsetofamodelinwhicheachspeciesisdominantinastate

compet-ofthetelegraphnoiseisexaminedindetail

©2014ElsevierInc.All rights reserved

1 This research was supported in part by the National Science Foundation DMS-1207667.

2 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.

1 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 ecosys-tem [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 environ-ments 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 ing values in M = {1, 2} Individual equations corresponding to the states 1 and 2 are different

tak-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 trate 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

concen-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 tion 2 Then Section3 takes up the issue of the existence of an invariant probability measure Section4continues the investigation by focusing on Ω-limit sets and properties of the invariant

Sec-measure Section5deals with dynamics of the system when each species dominates in one state Finally, Section6concludes the paper with further remarks

2 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

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

Such a two-state Markov chain is commonly referred to as telegraph noise because of the graph

of its sample paths Let F t be the σ -algebra generated by ξ(·) up to time t and P-null sets The

filtration {Ft}t≥0satisfies the usual condition That is, it is increasing and right continuous while

F0contains all P-null sets Then ( Ω, F, {F t }, P) is a complete filtered probability space.

In this paper, we focus on a Kolmogorov system under telegraph noise given by

are continuously differentiable in (x, y) ∈ R2

+= {(x, y) : x ≥ 0, y ≥ 0} Note that in the above and henceforth, we write a i (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

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

the initial value (x(0), y(0)) = (x0, y0) ∈ R2,◦

+ , where R2,◦

+ = {(x, y) : x > 0, y > 0} denotes the

interior of the set R2

+ Moreover, it is noted in Assumption 2.1that 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)aξ(t), u(t),0

˙v(t) = v(t)bξ(t ), 0, v(t)

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

and for each g i (u) defined on (0, ∞) and continuously differentiable in u By Assumption 2.1,

there is a unique pair (u1, u2) satisfying a1(u1, 0) = 0 and a2(u2, 0) = 0 In case u1 2, 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∗μ i (u) = 0 for i ∈ M (L∗denotes the adjoint of L)

Likewise, under Assumption 2.1, for (2.6), there exist v1, v2such that for v1< v2, the stationary

density of (ξ(t), v(t)), say (ν1, ν2), is given by

In case u1= u2(resp v1= v2), (μ1(u), μ2(u)) = (δ u −u1, δ u −u1) (resp (ν1(u), ν2(u) = (δ v −v1,

δ v −v1)) ) is a generalized density given by δ u −u1 (resp δ v −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 values of the Jacobian matrix

∂x ( 0, v1) < 0 Therefore, a sufficient condition for (0, v1)to repel positive

so-lutions is that it is a saddle point or equivalently a1( 0, v1) >0 In the same manner, we need

b1(u1, 0) > 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 λ1and λ2, which play a crucial role to determine the dynamical behaviors of (2.2)

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 λ1and

λ2implies 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

open The purpose of this paper is to address these questions In Section3, we prove the

exis-tence of an invariant probability measure, provided that λ1> 0 and λ2>0, which is assumed throughout this paper Section4 is an improvement of the results in [3] In particular, we de-

scribed 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 Section5, we consider (2.2)in the case where each species is dominant in one state

3 The existence of an invariant probability measure

The trajectory of ξ(t) is piecewise-constant and cadlag (right continuous having left limits)

functions Let

0= τ0< τ1< τ2< · · · < τ n <· · ·

be its jump times Put

σ1= τ1− τ0, σ2= τ2− τ1, , σ n = τ n − τ n−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=1are conditionally independent given the sequence

{ξ τ k}∞

k=1 Note that if ξ(0) is given, then ξ(τ n ) is known because the process ξ(t) takes only two

values Hence, {σk}∞

n=1is a sequence of independent random variables taking values in [0, ∞)

Moreover, if ξ(0) = 1, then σ 2n+1has the exponential density α1 [0,∞) (t) exp(−αt) and σ 2nhas

the density β1 [0,∞) (t) exp(−βt) Conversely, if ξ(0) = 2, then σ 2n has the exponential density

α1 [0,∞) (t) exp(−αt) and σ 2n+1 has the density β1 [0,∞) (t) exp(−βt) (see [4, vol 2, p 217])

Here 1[0,∞) = 1 for t ≥ 0 (= 0 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

0 < 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 ≥ 0 Note that δ, M are chosen such that a i (δ, 0), b i ( 0, δ) > 0 and a i (M, 0),

b i ( 0, M) < 0 for all i ∈ M.

We now state the main theorem of this section

Theorem 3.1 If λ1and λ2are positive, the Markov process (ξ(t), x(t), y(t)) has an invariant

probability measure π∗on the state space M × R 2,◦

+ .

To prove this theorem, we need to estimate the average time that (x(t), y(t)) spends on some

compact subset of M × R 2,◦

+ 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 formly 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

uni-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

pro-cess Therefore, there exists a semigroup of P-measure preserving transformations θt satisfying

ξ(t + s, ω) = ξ(t, θ s ω) ∀t, s > 0 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.

Let m be a positive number such that ∂a i (x,0)

∂x ≤ −m and ∂a i (0,y)

ln u∗(t) − ln u(t)≤ −mu∗(t) − u(t)≤ −mδln u∗(t) − ln u(t).

In view of the comparison theorem, we obtain

Letting t→ ∞ obtains the desired result The proof is complete 2

Lemma 3.2 For any ε > 0, there exists a T2= T2(ε) > 0 and a subset A ∈ F∞with P(A) > 1 −ε

such that ∀t > T and ω ∈ A,

Proof Since (ξ(t), v(t)) has a unique invariant distribution (whose density is (ν1(v), ν2(v))),

it follows from the Birkhoff Ergodic theorem that limt→∞1t

t

0a(ξ(s), 0, v(s))ds = λ1almost

surely given that v(0) admits (ν1(v), ν2(v)) as its density function Moreover, it is not

diffi-cult to show that for any given initial value v(0) > 0, and any (˜i, ˜v) ∈ M × (v1, v2), we have

P{(ξ(t), v(t)) = (˜i, ˜v) for some t ≥ 0} = 1 As a result, we obtain the strong law of large bers, that is, for any initial value v(0) > 0, P{lim t→∞1t

t

0b(ξ(s), u∗(s), 0)ds = λ1} = 1 The rest of this proof is now forward 2

straight-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 > T3and δ ≤ u0, v0≤ M,

for almost all ω ∈ A, where A is the set mentioned in Lemma 3.2

Let L = max{| ∂a i (x,0)

∂x |; |∂b i (0,y)

∂y | : 0 ≤ 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\

Proof For any sufficiently small ε1> 0, let u ε1(t, ω, u0)be the solution to

mδ ∀0 ≤ s ≤ t Choosing suitable ε1= ε1(ε) and γ such that γ+Mε1

mδ < L ε, we have the claim The proof of Lemma 3.3is complete 2

Now, we are in a position of proving Theorem 3.1

Proof of Theorem 2.1 To simplify the notation, denote

Fix a T > T3such that T1( ln M − ln δ) < ε Put χ n (ω) = 1A (θ nT ω) , where A is as in Lemma 3.2

and 1A ( ·) is the indicator function {χ n } is obviously a stationary process since θ T is a preserving transformation (see [9, Section 16.4]) We now prove that θ T is ergodic with re-

measure-spect to (  Ω, F, {F t}t≥0, P) Since F∞⊂ F and A ∈ F∞, where F∞= {F t : t ≥ 0}, χ n 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 0 < P(B ∩ θ −T B) = P(B) < 1 Suppose that there is a set

B ∈ F∞satisfying this almost invariant property Since F∞= σ ({F t : t ≥ 0}), for any ε >0, we

can find a set B ∈ F t for some t such that P(BB) < ε where BB = (B \ B ) ∪ (B \ B)

Let M t be the space of functions from [0, t] to M and B t the cylindrical σ -algebra on M t Since F t 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 = {ξ t ( ·) ∈ C } for some C ∈ Bt where ξ t (h + ·) denotes the trajectory

of ξ(·) in [h, h + t ] for each h ≥ 0, that is ξ t (h + t) = ξ(h + t) ∀t ∈ [0, t ] Let n0 be so

large that n0T > t and that |P{ξ(n0T − t ) = i | ξ(0) = j} − p i | < ε ∀i, j ∈ M We have P(B ∩ θ −n0T B ) = P{ξ t ( ·) ∈ C , ξ t (n0T + ·) ∈ C} Using the Markov property, we deduce that

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])

It follows from Remark 3.1that χ n = χ1+ χ2+ χ3 For convenience, put χ4= 1 − χ n By (3.6),

if χ n1= 1 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, χ1