persistence and extinction of stochastic kolmogorov systems · 2 0 the population ispersistentand...

Persistence and Extinction of Stochastic KolmogorovSystems

Dang Nguyen

(This talk is based on joint papers withAlex Hening, Du Nguyen and George Yin)

Wayne State University

IMA - Workshop on Ecological and Biological Systems

Outline

1 Introduction

2 Persistence and Extinction of Stochastic Kolmogorov Systems

3 Applications2D SystemsComments on Higher Dimensional SystemsDegenerate Systems

4 Further Remarks

Introduction

Real populations do not evolve in isolation and as a result much ofecology is concerned with understanding the characteristics thatallow two or more species to coexist, or one species to take overthe habitat of another.

The dynamics of an ecological system is traditionally modeled byODE.The fluctuations of the environment make the dynamics ofpopulations inherently stochastic.The combined effects of biotic interactions and environmentalfluctuations are key when trying to determine species richness.

Introduction

Real populations do not evolve in isolation and as a result much ofecology is concerned with understanding the characteristics thatallow two or more species to coexist, or one species to take overthe habitat of another.The dynamics of an ecological system is traditionally modeled byODE.The fluctuations of the environment make the dynamics ofpopulations inherently stochastic.

The combined effects of biotic interactions and environmentalfluctuations are key when trying to determine species richness.

Introduction

Real populations do not evolve in isolation and as a result much ofecology is concerned with understanding the characteristics thatallow two or more species to coexist, or one species to take overthe habitat of another.The dynamics of an ecological system is traditionally modeled byODE.The fluctuations of the environment make the dynamics ofpopulations inherently stochastic.The combined effects of biotic interactions and environmentalfluctuations are key when trying to determine species richness.

There are some interesting situations (see e.g. [Benaim-Lobry(AAP16)] and [Du-Nguyen (JDE11)])

Sometimes biotic effects can result in species going extinct.However, if one adds the effects of a random environmentextinction might be reversed into coexistence.In other instances deterministic systems that coexist becomeextinct once one takes into account environmental fluctuations.

There are some interesting situations (see e.g. [Benaim-Lobry(AAP16)] and [Du-Nguyen (JDE11)])Sometimes biotic effects can result in species going extinct.However, if one adds the effects of a random environmentextinction might be reversed into coexistence.

In other instances deterministic systems that coexist becomeextinct once one takes into account environmental fluctuations.

There are some interesting situations (see e.g. [Benaim-Lobry(AAP16)] and [Du-Nguyen (JDE11)])Sometimes biotic effects can result in species going extinct.However, if one adds the effects of a random environmentextinction might be reversed into coexistence.In other instances deterministic systems that coexist becomeextinct once one takes into account environmental fluctuations.

A Simple 1D- Equation

The well-known logistic equation:

dXt = Xt (r −kXt )dt , (1.1)

where r is the growth rate and k is the intrinsic competition rate.

Adding a stochastic term:

dXt = Xt (r −kXt )dt + σXtdWt . (1.2)

The key factor is the so-called stochastic growth rate r − σ2

2.

If r − σ2

2 < 0 the population goes extinct almost surely.

If r − σ2

2 > 0 the population is persistent and converges to itsunique invariant probability measure on (0,∞).If r − σ2

2 = 0 the process is null-recurrent. This is the critical casewhere the population does not go extinct but also does not havean ‘equilibrium’ (invariant probability measure).



dXt = Xt (r −kXt )dt , (1.1)

where r is the growth rate and k is the intrinsic competition rate.Adding a stochastic term:



2.

If r − σ2


If r − σ2





dXt = Xt (r −kXt )dt , (1.1)




2.

If r − σ2


If r − σ2

2 > 0 the population is persistent and converges to itsunique invariant probability measure on (0,∞).

If r − σ2




dXt = Xt (r −kXt )dt , (1.1)




2.

If r − σ2


If r − σ2



The dynamics of n interacting populationsX(t) = (X1(t), . . . ,Xn(t))t≥0 is given by

dXi(t) = Xi(t)fi(X(t))dt , i = 1, . . . ,n (1.3)

This kind of systems is usually named after Kolmogorov.

If we add stochastic effects we get

dXi(t) = Xi(t)fi(X(t))dt + Xi(t)gi(X(t))dEi(t), i = 1, . . . ,n (1.4)

We assume E(t) = (E1(t), . . . ,En(t))T = Γ>B(t) where Γ is a n×nmatrix such that Γ>Γ = Σ = (σij)n×n and B(t) = (B1(t), . . . ,Bn(t)) isa vector of independent standard Brownian motions.

The dynamics of n interacting populationsX(t) = (X1(t), . . . ,Xn(t))t≥0 is given by

dXi(t) = Xi(t)fi(X(t))dt , i = 1, . . . ,n (1.3)

This kind of systems is usually named after Kolmogorov.If we add stochastic effects we get


We assume E(t) = (E1(t), . . . ,En(t))T = Γ>B(t) where Γ is a n×nmatrix such that Γ>Γ = Σ = (σij)n×n and B(t) = (B1(t), . . . ,Bn(t)) isa vector of independent standard Brownian motions.

Fundamental questions

Under what conditions is the system persistent (in other words:when do n-species coexist?)

If the system is not persistent, which species persist, whichspecies go extinct?Many papers have attempted to answer these questions.However, nearly all of them cannot give sharp conditions even forspecific 2D systems.A common approach is to choose a function and to imposeconditions such that the function has some Lyapunov-typeproperties. The choice of Lyapunov function is usually artificial.The results one gets are therefore limited as the particularLyapunov function does not reflect the true nature of thedynamical system.The papers [Schreiber-Benaim-Atchade (JMB11)] [Benaim 14]give good conditions for persistence (in the case of compactspaces). Conditions for extinction are quite restrictive.


Under what conditions is the system persistent (in other words:when do n-species coexist?)If the system is not persistent, which species persist, whichspecies go extinct?

Many papers have attempted to answer these questions.However, nearly all of them cannot give sharp conditions even forspecific 2D systems.A common approach is to choose a function and to imposeconditions such that the function has some Lyapunov-typeproperties. The choice of Lyapunov function is usually artificial.The results one gets are therefore limited as the particularLyapunov function does not reflect the true nature of thedynamical system.The papers [Schreiber-Benaim-Atchade (JMB11)] [Benaim 14]give good conditions for persistence (in the case of compactspaces). Conditions for extinction are quite restrictive.


Under what conditions is the system persistent (in other words:when do n-species coexist?)If the system is not persistent, which species persist, whichspecies go extinct?Many papers have attempted to answer these questions.However, nearly all of them cannot give sharp conditions even forspecific 2D systems.A common approach is to choose a function and to imposeconditions such that the function has some Lyapunov-typeproperties. The choice of Lyapunov function is usually artificial.The results one gets are therefore limited as the particularLyapunov function does not reflect the true nature of thedynamical system.

The papers [Schreiber-Benaim-Atchade (JMB11)] [Benaim 14]give good conditions for persistence (in the case of compactspaces). Conditions for extinction are quite restrictive.


Under what conditions is the system persistent (in other words:when do n-species coexist?)If the system is not persistent, which species persist, whichspecies go extinct?Many papers have attempted to answer these questions.However, nearly all of them cannot give sharp conditions even forspecific 2D systems.A common approach is to choose a function and to imposeconditions such that the function has some Lyapunov-typeproperties. The choice of Lyapunov function is usually artificial.The results one gets are therefore limited as the particularLyapunov function does not reflect the true nature of thedynamical system.The papers [Schreiber-Benaim-Atchade (JMB11)] [Benaim 14]give good conditions for persistence (in the case of compactspaces). Conditions for extinction are quite restrictive.

General Stochastic Kolmogorov Systems [Hening-Nguyen(AAP18)]

We set Rn+ = [0,∞)n and Rn,◦

+ = (0,∞)n.Consider the stochastic Kolmogorov system


Standing Assumption

Assumption 2.11 fi(·),gi(·) : Rn

+→ R are locally Lipschitz functions for anyi = 1, . . . ,n.

2 There exist c = (c1, . . . ,cn) ∈ Rn,◦+ and γb > 0 such that

limsup‖x‖→∞

[∑i cixi fi(x)

1 + ∑i cixi− 1

2∑i ,j σijcicjxixjgi(x)gj(x)

(1 + ∑i cixi)2

+ γb

(1 +∑

i(|fi(x)|+ g2

i (x))

)]< 0.

3 (gi(x)gj(x)σij)n×n is a positive definite matrix for any x ∈ Rn,◦+ .

Conditions (1) and (2) are satisfied for most of well-known Kolmogorovsystems.

Some Definitions

Xi is said to (stochastically) persist if for any ε > 0, there existsδ > 0 satisfying

liminft→∞

Px{Xi(t) > δ}> 1− ε, x ∈ Rn,◦+ .

The system is said to be (stochastically) persistent if allXi , i = 1, . . . ,n persist stochastically.If X(0) = x ∈ Rn,◦

+ we say the population Xi goes extinct withprobability px > 0 if

Px

{limt→∞

Xi(t) = 0}

= px.

If px = 1, Xi goes extinct almost surely.

Some Definitions


liminft→∞

Px{Xi(t) > δ}> 1− ε, x ∈ Rn,◦+ .

The system is said to be (stochastically) persistent if allXi , i = 1, . . . ,n persist stochastically.

If X(0) = x ∈ Rn,◦+ we say the population Xi goes extinct with

probability px > 0 if

Px

{limt→∞

Xi(t) = 0}

= px.


Some Definitions


liminft→∞

Px{Xi(t) > δ}> 1− ε, x ∈ Rn,◦+ .

The system is said to be (stochastically) persistent if allXi , i = 1, . . . ,n persist stochastically.If X(0) = x ∈ Rn,◦

+ we say the population Xi goes extinct withprobability px > 0 if

Px

{limt→∞

Xi(t) = 0}

= px.


Main Ideas of Our Approach

In the deterministic setting one usually characterizes theasymptotics of the system by first looking at the equilibrium points(or rest points). The stability of an equilibrium is quantified by theLyapunov exponents of the linearized systemWe want to do something similar in the stochastic setting. We lookat the behaviors of the systems near the boundary to determinewhether or not the system is persistent.

lnXi(t)t

=lnXi(0)

t+

1t

∫ t

0gi(X(s))dEi(s)

+1t

∫ t

0

[fi(X(s))−

g2i (X(s))σii

2

]ds (2.2)

If X is close to the support of an ergodic measure µ supported on∂Rn

+ for a long time, then

1t

∫ t

0

[fi(X(s))−

g2i (X(s))σii

2

]ds

can be approximated by the average with respect to µ

λi(µ) =∫

∂Rn+

(fi(x)−

g2i (x)σii

2

)µ(dx), i = 1, . . . ,n

As t → ∞ the termlnXi(0)

t+

1t

∫ t

0gi(X(s))dEi(s)

is negligible. This implies that

λi(µ) =∫

∂Rn+

(fi(x)−

g2i (x)σii

2

)µ(dx), i = 1, . . . ,n

are the Lyapunov exponents of µ.It can also be seen that λi(µ) gives the long-term growth rate ofXi(t) if X is close to the support of µ.

Let M be the set of ergodic probability measures of X supportedon the boundary ∂Rn

+ := Rn+ \R

n,◦+ .

The set of invariant probability measures (i.p.m) on the boundaryis the convex hull of M , denoted by Conv(M).Consider µ ∈M . Since the diffusion X is nondegenerate in eachsubspace, there exist 0 < n1 < · · ·< nk ≤ n (if k = 0, there are non1, . . . ,nk ) such that supp(µ) = Rµ

+ where

Rµ

+ := {(x1, . . . ,xn) ∈ Rn+ : xi = 0 if i ∈ Eµ}

for Iµ := {n1, . . . ,nk} and Eµ := {1, . . . ,n}\{n1, . . . ,nk}.

We can call λi(µ), i ∈ Iµ internal Lyapunov exponents,λi(µ), i ∈ Eµ external Lyapunov exponents.We have

λi(µ) = 0, i ∈ Iµ

Intuitively this is expected because µ is in a way an ‘equilibrium’so the process should not tend to grow or decay when it evolves inRµ,◦+ .

Let M be the set of ergodic probability measures of X supportedon the boundary ∂Rn

+ := Rn+ \R

n,◦+ .

The set of invariant probability measures (i.p.m) on the boundaryis the convex hull of M , denoted by Conv(M).Consider µ ∈M . Since the diffusion X is nondegenerate in eachsubspace, there exist 0 < n1 < · · ·< nk ≤ n (if k = 0, there are non1, . . . ,nk ) such that supp(µ) = Rµ

+ where

Rµ

+ := {(x1, . . . ,xn) ∈ Rn+ : xi = 0 if i ∈ Eµ}

for Iµ := {n1, . . . ,nk} and Eµ := {1, . . . ,n}\{n1, . . . ,nk}.We can call λi(µ), i ∈ Iµ internal Lyapunov exponents,λi(µ), i ∈ Eµ external Lyapunov exponents.We have

λi(µ) = 0, i ∈ Iµ

Intuitively this is expected because µ is in a way an ‘equilibrium’so the process should not tend to grow or decay when it evolves inRµ,◦+ .

If all i.p.m on ∂Rn,◦+ are repellers, the system must be persistent.

If there is one ergodic i.p.m. on ∂Rn,◦+ that is an attractor, the

extinction happens.

Because the diffusion is nondegenerate, there are only 2possibilities for solutions in the interior Rn,◦

+ :I The solution does not tend to the boundary with probability 1 for

any initial value x ∈ Rn,◦+ .

I The solution tends to the boundary with probability 1 for any initialvalue x ∈ Rn,◦

+ .

No Bifurcation!

If all i.p.m on ∂Rn,◦+ are repellers, the system must be persistent.

If there is one ergodic i.p.m. on ∂Rn,◦+ that is an attractor, the

extinction happens.Because the diffusion is nondegenerate, there are only 2possibilities for solutions in the interior Rn,◦

+ :I The solution does not tend to the boundary with probability 1 for

any initial value x ∈ Rn,◦+ .

I The solution tends to the boundary with probability 1 for any initialvalue x ∈ Rn,◦

+ .

No Bifurcation!

Theorem 2.1 (Condition for persistence)Suppose that for any µ ∈ Conv(M )

max{i=1,...,n}

{λi(µ)}> 0 (2.3)

(This says that any invariant probability measure on the boundary is arepeller). We can show that X is stochastically persistent and ittransition probability converges exponentially fast in total variation to itsunique invariant probability measure π∗ on Rn,◦

+ .

Theorem 2.2 (Condition for extinction)

There exists µ ∈M such that maxi∈Eµ

{λi(µ)}< 0. (2.4)

and maxi∈Iµ{λi(ν)}> 0,∀ν whose support is contained in ∂Rµ

+ (2.5)

then the system is not persistent.

We need the condition

maxi∈Iµ{λi(ν)}> 0

to ensure that µ is a “sink” in Rµ,◦+ , that is, if X is close to Rµ,◦

+ , it isnot pulled away to the boundary ∂Rµ,◦

+ of Rµ,◦+ .

This condition is not restrictive because it is a sharp condition forthe existence of i.p.m of µ on Rµ,◦

+ .

Let M 1 be the set of ergodic measures on the boundary satisfying(2.4), that is, the set of attractors.In the case when there is extinction one may want to know whichspecies go extinct and which persist.We need an additional assumption

Assumption 2.2Suppose that one of the following is true

I M \M 1 = /0

I For any ν ∈M \M 1, max{i=1,...,n} {λi (ν)}> 0.

The assumption basically means that there are only two types:“attractors” and “repellers”,there are no ergodic measures that we cannot determine whetherthey are attractors or repellers.

Define the random normalized occupation measures

Π̃t (·) :=1t

∫ t

01{X(s)∈·}ds, t > 0

Theorem 2.3Suppose M 1 6= /0 (that is the conditions for extinction are satisfied). LetAssumption 2.2 holds. Then for any x ∈ Rn,◦

+

∑µ∈M 1

Pµ

x = 1 (2.6)

where for x ∈ Rn,◦+ ,µ ∈M 1

Pµ

x := Px

{Π̃t (·)

weakly⇒ µ and limt→∞

lnXi(t)t

= λi(µ) < 0, i ∈ Eµ

}> 0.

2D SystemsConsider the two dimensional system{

dX (t) = X (t)f1(X (t),Y (t))dt + X (t)g1(X (t),Y (t))dE1(t)dY (t) = Y (t)f2(X (t),Y (t))dt + Y (t)g2(X (t),Y (t))dE2(t)

(3.1)

Let δ be the Dirac measure concentrated at the origin. ItsLyapunov exponents given by

λi(δ ) = fi(0)− 12

g2i (0)σii .

If λ1(δ ) < 0, there is no i.p.m on R◦1+ = (0,∞)×{0}.If λ1(δ ) > 0, there is an i.p.m µ1 on R◦1+.Similarly, If λ2(δ ) < 0, there is no i.p.m on R◦2+. If λ2(δ ) < 0, thereis an i.p.m µ2 on R◦2+.δ ,µ1,µ2 are ergodic measures. The density pi(·) of µi can befound explicitly (in terms of integrals) by solving the Fokker-Plankequation. Then λj(µi), i , j = 1,2, i 6= j can be computed in terms ofintegrals.



(3.1)


λi(δ ) = fi(0)− 12

g2i (0)σii .

If λ1(δ ) < 0, there is no i.p.m on R◦1+ = (0,∞)×{0}.If λ1(δ ) > 0, there is an i.p.m µ1 on R◦1+.Similarly, If λ2(δ ) < 0, there is no i.p.m on R◦2+. If λ2(δ ) < 0, thereis an i.p.m µ2 on R◦2+.

δ ,µ1,µ2 are ergodic measures. The density pi(·) of µi can befound explicitly (in terms of integrals) by solving the Fokker-Plankequation. Then λj(µi), i , j = 1,2, i 6= j can be computed in terms ofintegrals.



(3.1)


λi(δ ) = fi(0)− 12

g2i (0)σii .

If λ1(δ ) < 0, there is no i.p.m on R◦1+ = (0,∞)×{0}.If λ1(δ ) > 0, there is an i.p.m µ1 on R◦1+.Similarly, If λ2(δ ) < 0, there is no i.p.m on R◦2+. If λ2(δ ) < 0, thereis an i.p.m µ2 on R◦2+.δ ,µ1,µ2 are ergodic measures. The density pi(·) of µi can befound explicitly (in terms of integrals) by solving the Fokker-Plankequation. Then λj(µi), i , j = 1,2, i 6= j can be computed in terms ofintegrals.

Classification

If λi(δ ) < 0, i = 1,2 then X (t),Y (t) converge to 0 almost surely at theexponential rate λ1(δ ),λ2(δ ) respectively.

λ1(δ ) > 0,λ2(δ ) < 0 and λ2(µ1) < 0 then Y (t) converges to 0almost surely at the exponential rate λ2(µ1) (left)λ1(δ ) < 0,λ2(δ ) > 0 and λ1(µ2) < 0 then X (t) converges to 0almost surely at the exponential rate λ1(µ2) (right)

X and Y coexist if either of the following holds.λ1(δ ) > 0,λ2(δ ) < 0 and λ2(µ1) > 0 (left)λ1(δ ) < 0,λ2(δ ) > 0 and λ1(µ2) < 0 (right)

Suppose λ1(δ ) > 0,λ2(δ ) > 0.If λ2(µ1) < 0 and λ1(µ2) > 0 then Y (t) converges to 0 almostsurely at the exponential rate λ2(µ1) (right)If λ1(µ2) < 0 and λ2(µ1) > 0 then X (t) converges to 0 almostsurely at the exponential rate λ1(µ2) (left)

X and Y coexist if λ1(δ ) > 0,λ2(δ ) > 0, λ2(µ1) > 0 and λ1(µ2) > 0.

If λ1(δ ) > 0,λ2(δ ) > 0, λ2(µ1) < 0 and λ1(µ2) < 0,then px ,y

i > 0, i = 1,2 and px ,y1 + px ,y

2 = 1 where

px ,y1 = Px ,y

{limt→∞

lnX (t)t

= λ1(µ2)

},

px ,y2 = Px ,y

{limt→∞

lnY (t)t

= λ2(µ1)

}.

Higher Dimensional Systems

In general it is difficult to compute the invariant probabilitymeasures on the boundary and their corresponding Lyapunovexponents.

However, we have proved that the rates of convergence areexponential, so the approximations are usually very good.In certain cases, we can have explicit computations. For instance,the stochastic Lotka-Volterra systems with linear diffusion parts:

dXi(t) = Xi(t)

(bi −

n

∑j=1

ajiXj(t)

)dt + σiXi(t)dEi(t).

(see e.g. [Hening-Nguyen (JMB17)])


In general it is difficult to compute the invariant probabilitymeasures on the boundary and their corresponding Lyapunovexponents.However, we have proved that the rates of convergence areexponential, so the approximations are usually very good.

In certain cases, we can have explicit computations. For instance,the stochastic Lotka-Volterra systems with linear diffusion parts:

dXi(t) = Xi(t)

(bi −

n

∑j=1

ajiXj(t)




In general it is difficult to compute the invariant probabilitymeasures on the boundary and their corresponding Lyapunovexponents.However, we have proved that the rates of convergence areexponential, so the approximations are usually very good.In certain cases, we can have explicit computations. For instance,the stochastic Lotka-Volterra systems with linear diffusion parts:

dXi(t) = Xi(t)

(bi −

n

∑j=1

ajiXj(t)



How to Treat Degenerate Systems

The system is degenerate if (gi(x)gj(x)σij)n×n is NOT a positivedefinite matrix for some x ∈ Rn,◦

+

Our previous results basically work for degenerate systems.I The condition for persistence is the same.I To analyzing the extinction, we need some extra conditions which

can be verified for specific models.

Basically, we need to verify Hormander’s condition and investigatethe corresponding control systems

I to determine the number of i.p.m in each subspace and the supportof i.p.m

I and to show whether the boundary can be approachable from aninitial value in the interior Rn,◦

+ .

A Stochastic Predator Prey Model [Du-Dang-Yin (JAP16)]

dx(t) =x(t)(a1−b1x(t)− c1y(t)

m1 + m2x(t) + m3y(t))dt + αx(t)dB(t),

dy(t) =y(t)(−a2−b2y(t) +

c2x(t)m1 + m2x(t) + m3y(t)

)dt + βy(t)dB(t),

(3.2)

The term x(t)y(t)m1+m2x(t)+m3y(t) is called the Beddington-Deanglis

functional response.

By a detailed analysis, we show thatThe conditions for extinction and persistence are the same as thenon-degenerate case.The invariant probability measure in R2,◦

+ is unique if it exists.We can describe the support of the invariant probability measure.If the system is nondegenerate, the support of an ergodicmeasure is the whole subspace supporting it




dy(t) =y(t)(−a2−b2y(t) +

c2x(t)m1 + m2x(t) + m3y(t)

)dt + βy(t)dB(t),

(3.2)


functional response.By a detailed analysis, we show thatThe conditions for extinction and persistence are the same as thenon-degenerate case.

The invariant probability measure in R2,◦+ is unique if it exists.

We can describe the support of the invariant probability measure.If the system is nondegenerate, the support of an ergodicmeasure is the whole subspace supporting it




dy(t) =y(t)(−a2−b2y(t) +

c2x(t)m1 + m2x(t) + m3y(t)

)dt + βy(t)dB(t),

(3.2)


functional response.By a detailed analysis, we show thatThe conditions for extinction and persistence are the same as thenon-degenerate case.The invariant probability measure in R2,◦

+ is unique if it exists.We can describe the support of the invariant probability measure.If the system is nondegenerate, the support of an ergodicmeasure is the whole subspace supporting it

Figure: The phase portrait of (3.2) with a set of parameters.The support of the invariant measure is the domain below the blue curveOur theoretical results show the explicit equation of the blue curve

Figure: The empirical measure of the process, which approximates theinvariant probability measure.

Further Remarks

Our methods work for other types of random noise.Some random factors are given by switching process. (see[Mao-Yuan 06], [Yin-Zhu 10], [Nguyen-Yin 16,17]).We can use our methods to treat Kolmogorov systems underregime-switching (with or without diffusion parts). (see e.g.[Du-Nguyen (JDE11)], [Du-Nguyen-Yin (JDE14)])

The equation for Kolmogorov systems under regime-switching isgiven by

dXi(t) = Xi(t)fi(Xi(t),ξ (t))dt , i = 1, . . . ,m (4.1)

where ξ (t) is a cadlag process taking values in a finite state spaceS = {1, . . . ,m}.Suppose that the switching intensity of r(t) depends on the stateof X(t)

P{ξ (t + ∆) = j | ξ (t) = i ,X(s),ξ (s),s ≤ t}= qij(X(s))∆ + o(∆) if i 6= j andP{ξ (t + ∆) = i | r(t) = i ,X(s),ξ (s),s ≤ t}= 1 + qii(X(t))∆ + o(∆).

(4.2)

Further Remarks

Our methods work for other types of random noise.Some random factors are given by switching process. (see[Mao-Yuan 06], [Yin-Zhu 10], [Nguyen-Yin 16,17]).We can use our methods to treat Kolmogorov systems underregime-switching (with or without diffusion parts). (see e.g.[Du-Nguyen (JDE11)], [Du-Nguyen-Yin (JDE14)])The equation for Kolmogorov systems under regime-switching isgiven by

dXi(t) = Xi(t)fi(Xi(t),ξ (t))dt , i = 1, . . . ,m (4.1)

where ξ (t) is a cadlag process taking values in a finite state spaceS = {1, . . . ,m}.Suppose that the switching intensity of r(t) depends on the stateof X(t)

P{ξ (t + ∆) = j | ξ (t) = i ,X(s),ξ (s),s ≤ t}= qij(X(s))∆ + o(∆) if i 6= j andP{ξ (t + ∆) = i | r(t) = i ,X(s),ξ (s),s ≤ t}= 1 + qii(X(t))∆ + o(∆).

(4.2)

Further Remarks

Ignoring critical cases, there is still a gap in our results forextinction and persistence.Main condition for persistence: maxi∈Eµ

{λi(µ)}> 0 for any i .p.mon the boundary.Main condition for extinction: maxi∈Eµ

{λi(µ)}< 0 for someERGODIC i .p.m on the boundary.

The gap does not appear for n = 2. But it does for n ≥ 3.Question: How to close the gap?The idea is to use the Morse decomposition (from Schreiber(JDE2000)) to break the boundary into separate suitable parts tohandle the behavior near the boundary.

Further Remarks



{λi(µ)}< 0 for someERGODIC i .p.m on the boundary.The gap does not appear for n = 2. But it does for n ≥ 3.Question: How to close the gap?

The idea is to use the Morse decomposition (from Schreiber(JDE2000)) to break the boundary into separate suitable parts tohandle the behavior near the boundary.

Further Remarks



{λi(µ)}< 0 for someERGODIC i .p.m on the boundary.The gap does not appear for n = 2. But it does for n ≥ 3.Question: How to close the gap?The idea is to use the Morse decomposition (from Schreiber(JDE2000)) to break the boundary into separate suitable parts tohandle the behavior near the boundary.

Further Remarks

The exponential rates of convergence relies on the assumption:

limsup‖x‖→∞

[∑i cixi fi(x)

1 + ∑i cixi− 1

2∑i ,j σijcicjxixjgi(x)gj(x)

(1 + ∑i cixi)2

+ γb

(1 +∑

i(|fi(x)|+ g2

i (x))

)]< 0.

Question: Can we still prove extinction and persitence if thatcondition is relaxed?The answer should be YES, although we are likely to lose theexponential rates of convergence.

Thank you

persistence and extinction of stochastic kolmogorov systems · 2 0 the population ispersistentand...

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