Lotka Volterra in �uctuating environment or

"how good can be bad"

Michel Benaïm∗and Claude Lobry†

December 15, 2014

Abstract

We consider two dimensional Lotka-Volterra systems in �uctuat-

ing environment. Relying on recent results on stochastic persistence

and piecewise deterministic Markov processes, we show that random

switching between two environments both favorable to the same species

can lead to the extinction of this species or coexistence of the species.

MSC: 60J99; 34A60

Keywords: Population dynamics, Persistence, Piecewise deterministic pro-cesses

Contents

1 Introduction 3

2 Preliminaries 5

2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Average growth rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Jointly favorable environments . . . . . . . . . . . . . . . . . . . . . 7

∗Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand, Neuchâtel,Suisse-2000. (michel.benaim@unine.ch).†EPI Modemic Inria and Université de Nice Sophia-Antipolis

1

3 Main results 10

3.1 Good may be good (extinction of species 2) . . . . . . . . . . . . . . 10

3.2 Good may be bad (extinction of species 1) . . . . . . . . . . . . . . . 13

3.3 Good may be fair (Persistence) . . . . . . . . . . . . . . . . . . . . . 14

4 Illustrations 18

5 Proofs of Propositions 2.1 and 2.2 22

5.1 Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 24

2

1 Introduction

It has been known for a long time that periodic variations of the environmentin Lotka-Volterra models can modify the outcome of the competition (seee.g [7, 9, 8, 16] as well as the discussion and references following Corollary5.30 in [11]). In the present paper we show that similar phenomena andparadoxical situations can also occur in the case of stochastic �uctuationsexpressed by random switching between two di�erent environments and weinvestigate thoroughly the behavior of such processes.

Let E = (A,B),

A =

(a bc d

), B =

(αβ

), (1)

where a, b, c, d, α, β are positive numbers. Such a pair is called an environ-ment. The Lotka-Volterra vector �eld associated to E is the map FE : R2 7→ R2

de�ned by

FE(x, y) =

{αx(1− ax− by)βy(1− cx− dy)

(2)

It induces a dynamical system on R+ × R+ given by the ODE

(x, y) = FE(x, y). (3)

Here x (respectively y) represents the abundance of some species 1 (respec-tively 2) and (3) describes the interaction of the two species in environmentE .

Environment E is said to be favorable to species 1 if

a < c and b < d.

We let Env1 denote the set of such environments. The following result easilyfollows from an isocline analysis (see e.g [14], Chapter 3.3).

Proposition 1.1 Suppose E = (A,B) ∈ Env1. Then, for every (x, y) ∈ R∗+×

R+ the solution to (3) with initial condition (x, y) converges to (1

a, 0) as t→

∞.

Let now E0 and E1 be two environments. Consider the process obtained byrandom switching between E0 and E1 de�ned as follows:

(X, Y ) = FEIt (X, Y ) (4)

3

where It ∈ {0, 1} is a continuous time jump process with jump rates λ0, λ1 ≥0. That is

P(It+s = 1− i|It = i,Ft) = λis+ o(s)

where Ft is the sigma �eld generated by {Iu, u ≤ t}. In other words, as-suming that I0 = i and (X0, Y0) = (x, y) the process (Xt, Yt) follows thesolution trajectory to FEi with initial condition (x, y) for an exponentiallydistributed random time, with intensity λi. Then, (Xt, Yt) follows the the so-lution trajectory to FE1−i

for another exponentially distributed random time,with intensity λ1−i and so on.

The purpose of this paper is to describe the behavior of (Xt, Yt) when E0

and E1 are both favorable to 1 (i.e E0, E1 ∈ Env1) and (X0, Y0) ∈ R∗+ × R∗+.Our main result can be roughly described as follows. Let

R = {(λ0, λ1) : λ0 ≥ 0, λ1 ≥ 0}

be the set of all possible jump rates. Let Env⊗21 ⊂ Env1 × Env1 be the set of

environments (E0, E1) such that for all (λ0, λ1) ∈ R the random Lotka-Volterramodel (4) remains favorable to 1 (i.e Yt → 0 almost surely). Then,

(i) Env⊗21 is characterized as a proper semi-algebraic subset of Env1 × Env1;

(ii) If (E0, E1) ∈ Env1 × Env1 \ Env⊗21 , R can be written as a disjoint union

R = R+,− ∪R+,+ ∪R−,+ ∪R−,−

where each Ru,v is open, R+,− and R \R+,− are nonempty, and

(a) (λ0, λ1) ∈ R+,− ⇒ Extinction of species 2;

(b) (λ0, λ1) ∈ R+,+ ⇒ Persistence of the species: the empirical occu-pation measure of {Xt, Yt} converges almost surely to a probabilitymeasure on the positive quadrant R∗+ × R∗+ absolutely continuouswith respect to the Lebesgue measure.

(c) Under some supplementary condition (that we conjecture to bealways satis�ed) (λ0, λ1) ∈ R−,+ ⇒ Extinction of species 1;

(d) (λ0, λ1) ∈ R−,− ⇒ Extinction of one species.

The proofs rely on recent results on stochastic persistence given in [3] builtupon previous results obtained for deterministic systems in [12, 17, 10, 13](see also [20] for a comprehensive introduction to the deterministic theory),

4

stochastic di�erential equations with small di�usion term in [4], stochasticdi�erential equations and random di�erence equations in [19, 18]. We alsomake a crucial use of some recent results on Piecewise deterministic Markovprocesses obtained in [1, 2] and [6].

The paper is organized as follows. In Section 2 we explicitly compute theaverage growth rates of the species. These are the growth rates averaged overthe invariant measures of the process (4) restricted to the extinction faces. Wethen relate their signs to the nature of the environment. Section 3 contains themain results and their proofs. Section 4 presents some illustrations obtainedby numerical simulation and Section 5 contains the proofs of the propositionsstated in section 2.

2 Preliminaries

2.1 Notation

For i = 0, 1, environment Ei is de�ned by (1) with (ai, bi, . . .) instead of(a, b, . . .). Throughout all the paper we assume that E0 and E1 are favorableto species 1. We suppose that λ0 and λ1 are nonzero (the case where one ofthem is zero being handled by proposition 1.1).

For η > 0 small enough the set

Kη = {(x, y) ∈ R2+ : η ≤ x+ y ≤ 1/η}

is positively invariant under the dynamics induced by FE0 and FE1 and attractsevery solution to (4) with initial condition (x, y) ∈ R2

+ \ {0, 0}. Fix such an ηand let

M = Kη × {0, 1}.Set Zt = (Xt, Yt, It). Since Zt eventually lies inM (whenever (X0, Y0) 6= (0, 0))we may assume without loss of generality that Z0 ∈M and we see M as thestate space of the process {Zt}t≥0.

The extinction set of species 2 is the set

M20 = {(x, y, i) ∈M : y = 0} ' [η, 1/η]× {0, 1}.

Extinction set of species 1, denoted M10 , is de�ned similarly (with x = 0

instead of y = 0) and the extinction set is de�ned as

M0 = M10 ∪M2

0 .

5

2.2 Average growth rates

Given a probability measure µ supported by M20 ' [η, 1/η] × {0, 1}, set

µ(A, i) = µ(A×{i}) and de�ne the average intrinsic growth rate of species 2with respect to µ as

Λ2(µ) =

∫β0(1− c0x)µ(dx, 0) +

∫β1(1− c1x)µ(dx, 1)

Note that the quantity βi(1− cix) can be seen as the intrinsic growth rate ofspecies 2 in environment Ei when the abundance of species 1 is x.

Set

pi =1

ai, γi =

λiαi,

and for all x ∈]p0, p1[

θ(x) =|x− p0|γ0−1|p1 − x|γ1−1

x1+γ0+γ1(5)

and

P (x) = [β1

α1

(1− c1x)(1− a0x)− β0

α0

(1− c0x)(1− a1x)]a1 − a0

|a1 − a0|(6)

The following proposition is proved in Section 5.

Proposition 2.1 The process {Xt, Yt, It} restricted to M20 has a unique in-

variant probability µ given as follows:

(i) If p0 = p1 = pµ = δp ⊗ ν

where ν = λ0λ1+λ0

δ1 + λ1λ1+λ0

δ0.

(ii) If p0 6= p1

µ(dx, 1) = h1(x)1[p1,p0](dx)dx,

µ(dx, 0) = h0(x)1[p1,p0](dx)dx

where

h1(x) = Cp1|x− p1|γ1−1|p0 − x|γ0

α1x1+γ0+γ1,

h0(x) = Cp0|x− p1|γ1|p0 − x|γ0−1

α0x1+γ0+γ1

6

and C (depending on p1, p0, γ1, γ0) is de�ned by the normalization con-dition ∫ p0

p1

(h1(x) + h0(x))dx = 1.

Moreover, Λ2(µ) equals the number

Λ2 =

λ1β0(1− c0p) + λ0β1(1− c1p) if p0 = p1 = p,

p0p1C

∫]p0,p1[

P (x)θ(x)dx if p0 6= p1(7)

For further reference we simply call Λ2 the average growth rate of species 2.We may write Λ2 = Λ2(λ0, λ1) if we need to emphasize the dependency onthe jump rates.

Similarly, the average growth rate of species 1, denoted Λ1, is de�ned likeΛ2 by permuting αi and βi, and replacing (ai, ci) by (di, bi).

2.3 Jointly favorable environments

The signs of the average growth rates Λ1 and Λ2 will prove to be crucial forcharacterizing the long term behavior of {Zt} in the next sections. Here weshow how they can be related to the nature of E0 and E1.

For all 0 ≤ s ≤ 1, we let Es = (As, Bs) be the environment de�ned by

sFE1 + (1− s)FE0 = FEs (8)

Then, with the notation of section 1,

Bs =

(αsβs

)=

(sα1 + (1− s)α0

sβ1 + (1− s)β0

)and

As =

as bs

cs ds

=

sα1a1+(1−s)α0a0

αs

sα1b1+(1−s)α0b0αs

sβ1c1+(1−s)β0c0βs

sβ1d1+(1−s)β0d0βs

.

SetI = {0 < s < 1 : as > cs} (9)

andJ = {0 < s < 1 : bs > ds}. (10)

7

It is easily checked that I (respectively J) is either empty or is an openinterval which closure is contained in ]0, 1[.

We shall say that E0 and E1 are jointly favorable to species 1 if, in additionof being favorable to 1, E0 and E1 are such that I and J are empty. We letEnv⊗2

1 ⊂ Env1 × Env1 denote the set of jointly favorable environments.

Remark 1 Set R = β0α1

α0β1and u = sα1

αs. Then a direct computation shows that

sβ1βs

= uu(1−R)+R

. Thus,

cs − as = u(c11

u(1−R) +R− a1) + (1− u)(c0

R

u(1−R) +R)− a0)

=Au2 +Bu+ C

u(1−R) +R

withA = (a1 − a0)(R− 1),

B = (2a0 − c0 − a1)R + (c1 − a0),

andC = (c0 − a0)R.

Then

I 6= ∅ ⇔

A 6= 0

∆ = B2 − 4AC > 0

0 < −B−√

∆2A

< 1

(11)

The condition for J 6= ∅ is obtained by replacing ai by bi and ci by di in thede�nitions of A,B,C above, R being unchanged.

Remark 2 The characterization given in Remark 1 shows that Env⊗21 is a

semi algebraic subset of Env1 × Env1.

The following proposition is proved in section 5.

Proposition 2.2 The map

Λ2 : R∗+ × R∗+ 7→ R,

λ0, λ1 7→ Λ2(λ0, λ1)

satis�es the following properties:

8

(i) If I = ∅, then for all λ0, λ1 Λ2(λ0, λ1) < 0

(ii) For all s ∈]0, 1[

limt→∞

Λ2(ts, t(1− s)) = βs(1−csas

)

> 0 if s ∈ I,= 0 if s ∈ ∂I,< 0 if s ∈]0, 1[\I

limt→0

Λ2(ts, t(1− s)) = (1− s)β0(1− c0

a0

) + sβ1(1− c1

a1

) < 0;

Remark 3 The same properties hold true if one replace I by J and Λ2 by−Λ1.

The next result follows directly from Proposition 2.2.

Corollary 2.3 For u, v ∈ {+,−}, let

Ru,v = {λ0 > 0, λ1 > 0 : Sign(Λ1(λ0, λ1)) = u, Sign(Λ2(λ0, λ1)) = v}.

Then

(i) R+− 6= ∅,

(ii) I ∩ J c 6= ∅ ⇒ R+,+ 6= ∅,

(iii) J ∩ Ic 6= ∅ ⇒ R−,− 6= ∅,

(iv) I ∩ J 6= ∅ ⇒ R−,+ 6= ∅.

In view of Proposition 2.2, the following conjecture sounds reasonable.

Conjecture 2.4 If I =]s0, s1[ 6= ∅ the set

{(s, t) ∈]0, 1[×R∗+ : Λ2(ts, t(1− s)) = 0}

is the graph of a smooth function

I 7→ R∗+, s 7→ t(s)

with lims→s0 t(s) = lims→s1 t(s) =∞. In particular, implications (ii), (iii) and(iv) in Corollary 2.3 are equivalences.

9

3 Main results

The empirical occupation measure of the process {Zt} = {Xt, Yt, It} is the(random) measure given by

Πt =1

t

∫ t

0

δZsds.

Hence, for every Borel set A ⊂ M,Πt(A) is the proportion of time spent by{Zs} in A up to time t.

Recall that a sequence of probability {µn} on a metric space E (such asM,M i

0 or R2+ ) is said to converge weakly to µ (another probability on E) if∫

fdµn →∫fdµ for every continuous bounded function f : E 7→ R.

3.1 Good may be good (extinction of species 2)

Recall that pi = 1ai.

Theorem 3.1 Assume that Λ2 < 0,Λ1 > 0 and Z0 = z ∈M \M0. Then, thefollowing properties hold with probability one:

(a) lim supt→∞log(Yt)

t≤ Λ2,

(b) The limit set of {Xt, Yt} equals [p0, p1]× {0},

(c) {Πt} converges weakly to µ, where µ is the probability on M20 de�ned in

Proposition 2.1

Remark 4 It follows from Theorem 3.1 that the marginal empirical occupa-tion measure of {Xt, Yt} converge to the marginal

µ(dx, 0) + µ(dx, 1) =

{δp if p0 = p1 = pCθ(x)[ p1

α1|x− p0|+ p0

α0|p1 − x|]dx, if p0 6= p1

with θ given by (5) and C a normalization constant.

Corollary 3.2 Suppose that E0 and E1 are jointly favorable to 1. Then con-clusions of Theorem 3.1 hold for all positive jump rates λ0, λ1.

Proof: Follows from Theorem 3.1 and Proposition 2.2. 2

10

Proof of Theorem 3.1

Record that Zt = (Xt, Yt, It). For all z ∈M we let Pz denote the law of {Zt}t≥0

given that Z0 = z and we let Ez denote the corresponding expectation.If E is one of the setsM,M \M0 orM \M i

0, and h : E 7→ R is a measurablefunction which is either bounded from below or above, we let, for all t ≥ 0and z ∈ E,

Pth(z) = Ez(h(Zt)).

For 1 > ε > 0 su�ciently small we let

M10,ε = {z = (x, y, i) ∈M : x < ε}

andM2

0,ε = {z = (x, y, i) ∈M : y < ε}.

Let V i : M \M i0 7→ R, i = 1, 2 be the maps de�ned by

V 1((x, y, i)) = − log(x) and V 2((x, y, i)) = log(y).

The assumptions Λ1 > 0,Λ2 < 0 and compactness of M0 imply the followingLemma:

Lemma 3.3 Let α1 > Λ1 > 0 and α2 > −Λ2 > 0. Then, there exist T >0, θ > 0, ε > 0 and 0 ≤ ρ < 1 such that for all z ∈M i

0,ε \M i0

(i) PTVi(z)−V i(z)T

≤ −αi,

(ii) PT (eθVi)(z) ≤ ρeθV

i(z)

Proof: Follows from Propositions 6.1 and 6.2 (proved in a more generalcontext) in [3]. 2

De�ne the stopping times

τ i,Outε = min{k ∈ N : ZkT ∈M \M i0,ε}

andτ i,Inε = min{k ∈ N : ZkT ∈M i

0,ε}.

Step 1. We �rst prove that there exists some constant c > 0 such that for allz ∈M \M1

0

Pz(τ 2,Inε/2 <∞) ≥ c (12)

11

Set V 1k = V 1(ZkT ) + kα1T, k ∈ N. It follows from Lemma 3.3 (i) that

{V 1k∧τ1,Out

ε} is a nonnegative supermartingale. Thus, for all z ∈M1

0,ε \M10

α1TEz(k ∧ τ 1,Outε ) ≤ Ez(V 1

k∧τ1,Outε

) ≤ V 10 = V 1(z).

That is

Ez(τ 1,Outε ) ≤ V 1(z)

α1T<∞. (13)

Now, because E0 ∈ Env1, (1/a0, 0) is a linearly stable equilibrium for FE0 whichbasin of attraction contains R∗+×R+. Therefore, there exists k0 ∈ N such thatfor all z = (x, y, i) ∈M \M1

0,ε and k ≥ k0

ΦE0kT (x, y) ∈M20,ε/2.

Here ΦE0 stands for the �ow induced by FE0 . It easily follows (one can writea direct proof or use the support Theorem ( Theorem 3.4 in [6]) for piecewisedeterministic Markov processes) that for some c > 0

Pz(Zk0T ∈M20,ε/2) ≥ c (14)

for all z ∈M \M10,ε Combining (13) and (15) concludes the proof of the �rst

step.Step 2. Let A be the event de�ned as

A = {lim supt→∞

V 2(Zt)

t≤ −α2}.

We claim that there exists c1 > 0 such that for all z ∈M20,ε/2

Pz(A) ≥ c1. (15)

Set W 2k = eθV

2(ZkT ). By Lemma 3.3 (ii), {W 2k∧τ2,Out

ε} is a nonnegative super-

martingale. Thus, for all z ∈M20,ε/2

Ez(W 2k∧τ2,Out

ε1τ2,Out

ε <∞) ≤ W 20 = eθV

2(z) ≤ (ε

2)θ.

Hence, letting k →∞ and using dominated convergence, leads to

εθPz(τ 2,Outε <∞) ≤ Ez(W 2

τ2,Outε

1τ2,Outε <∞) ≤ W 2

0 ≤ (ε

2)θ.

12

Thus

Pz(τ 2,Outε =∞) ≥ 1− 1

2θ= c1 > 0. (16)

LetM2n =

∑nk=1(V 2(ZkT )−PTV 2(Z(k−1)T ). By the strong law of large numbers

for martingales applied to {Mk} and Lemma 3.3 (i), it comes that

lim supk→∞

V 2(ZkT )

kT≤ −α2

on the event {τ 2,Outε =∞}. Let C = sup{βi|1− cix− diy| : (x, y, i) ∈M}. It

is easy to check that V 2(ZkT+t)− V 2(ZkT ) ≤ Ct. Thus,

lim supt→∞

V 2(Zt)

t≤ −α2

almost surely on on the event {τ 2,Outε = ∞}. This later inequality, together

with (16) concludes the proof of step 2.Step 3. From (12) and (15) we deduce that

Pz(A) ≥ cc1 (17)

for all z ∈M \M10 . Thus for all z ∈M \M1

0 ,

1A = limt→∞

Pz(A|Ft)) = limt→∞

PZt(A) ≥ cc1

where the �rst equality follows from Doob's Martingale convergence theorem,and the second from the Markov property. This concludes the proof.

3.2 Good may be bad (extinction of species 1)

Theorem 3.4 Assume that Λ1 < 0,Λ2 > 0, I ∩ J 6= ∅ and Z0 = z ∈M \M0.Then, the following properties hold with probability one:

(a) lim supt→∞log(Xt)

t≤ Λ1,

(b) The limit set of {Xt, Yt} equals {0} × [p0, p1],

(c) {Πt} converges weakly to µ, where pi = 1di

and µ is the probability on M10

de�ned analogously to µ (by permuting αi and βi, and replacing (ai, ci)by (di, bi)).

13

Proof: By permuting the roles of species 1 and 2, the proof amounts to(re)proving Theorem 3.1 under the assumption that E0 and E1 are favorableto species 2. All the arguments given in the proof of Theorem 3.1 go throughbut for the proof of (15) where we have explicitly used the fact that E0 was fa-vorable to species 1. Here, it su�ces to replace FE0 by FEs for some s ∈ I∩J 2

Remark 5 If Conjecture 2.4 is true, the assumption that I ∩ J 6= ∅ in The-orem 3.4 is unnecessary.

Theorem 3.5 Assume that Λ1 < 0,Λ2 < 0 and Z0 = z ∈ M \M0. Let A1

(respectively A2) be the event de�ned by assertions (a), (b) and (c) in Theorem3.1 (respectively Theorem 3.4). Then P(A1) + P(A2) = 1,P(A2) > 0, andP(A1) > 0 for z su�ciently close to M1

0 . If we furthermore assume thatJ ∩ Ic 6= ∅ then P(A1) > 0 for all z ∈M \M0.

Proof: The proof is similar to the proof of Theorem 3.1, so we only givea sketch of it. Reasoning like in Theorem 3.1, we show that there existsc, c1 > 0 such that for all z ∈ M i

0,ε,Pz(Ai) ≥ c1 and for all z ∈ M \M10,ε

Pz({Zt} enters M20,ε/2) ≥ c.

Thus, for all z ∈ M \ M0,Pz(A1) + Pz(A2) ≥ c1 + cc1. Hence, by theMartingale argument used in the last step of the proof of Theorem 3.1, weget that Pz(A1) + Pz(A2) = 1. Since (1/a0, 0) is a linearly stable equilibriumfor ΦE0 which basin contains R∗+ × R∗+, Pz({Zt} enters M2

0,ε/2) > 0 for all

z ∈ M \ M0 and, consequently, Pz(A2) > 0. If furthermore there is somes ∈ I ∩ J c (0, 1/as) is a linearly stable equilibrium for ΦEs which basin con-tains R∗+ × R∗+ and, by the same argument, Pz(A1) > 0. 2

3.3 Good may be fair (Persistence)

Theorem 3.6 Suppose that Λ1 > 0,Λ2 > 0 Then, there exists a uniqueinvariant probability (for the process {Zt}) Π on M \M0 i.e Π(M \M0) = 1.Furthermore,

(i) Π is absolutely continuous with respect to the Lebesgue measure dxdy ⊗(δ0 + δ1);

14

(ii) There exists θ > 0 such that∫(

1

xθ+

1

yθ)dΠ <∞;

(iii) For every initial condition z = (x, y, i) ∈M \M0

limt→∞

Πt = Π

weakly, with probability one.

Proof: By Feller continuity of {Zt} and compactness of M the sequence{Πt} is relatively compact (for the weak convergence) and every limit pointof {Πt} is an invariant probability (see e.g [6], Proposition 2.4 and Lemma2.5).

Now, the assumption that Λ1 and Λ2 are positive, ensure that the persis-tence condition given in ([3] sections 5 and 5.2) is satis�ed. Then by the Per-sistence Theorem 5.1 in [3] (generalizing previous results in [4] and [19]), everylimit point of {Πt} is a probability over M \M0 provided Z0 = z ∈M \M0.By Lemma 3.3 (ii) every such limit point satis�es the integrability condition(ii).

To conclude, it then su�ces to show that {Zt} has a unique invariantprobability on M \M0, Π and that Π is absolutely continuous with respectto dxdy ⊗ (δ0 + δ1).

We rely on Theorem 1 in [1] (see also [6], Theorem 4.4 and the discussionfollowing Theorem 4.5). According to this theorem, a su�cient conditionensuring both uniqueness and absolute continuity of Π is that

(i) There exists a point m ∈ R∗+ × R∗+ which is accessible from every point(x, y) ∈ R∗+ × R∗+,

(ii) The Lie algebra generated by (FE0 , FE1) has full rank at point m.

There are several equivalent formulations of accessibility (called D− ap-proachability in ([1]). One of them, see section 3 in [6], is that for everyneighborhood U of m and every (x, y) ∈ R∗+×R∗+ there is a solution η to thedi�erential inclusion

η ∈ conv(FE0 , FE1)(η),

η(0) = (x, y)

15

which meet U (i.e η(t) ∈ U for some t > 0). Here conv(FE0 , FE1) stands forthe convex hull of FE0 and FE1 .

For any environment E , let (ΦEt ) denote the �ow induced by FE and let

Γ+E (m) = {ΦEt (m) : t ≥ 0},Γ−E (m) = {ΦEt (m) : t ≤ 0},

Since Λ2 > 0, I 6= ∅ by Proposition 2.2. Choose s ∈ I. Then, point ms =(1/as, 0) is a hyperbolic saddle equilibrium for FEs (as de�ned by equation (8))which stable manifold is the x− axis and which unstable manifold, denotedW ums

(FEs), is transverse to the x−axis at ms.Now, choose an arbitrary point m∗ ∈ W u

ms(FEs) ∩ R∗+ × R∗+. We claim

that every point in Γ+E0(m

∗) is accessible. By de�nition of Γ+E0(m

∗) and ac-cessibility, it su�ces to show that m∗ is accessible. A standard Poincarésection argument shows that there exists an arc L transverse to W u

ms(FEs) at

m∗ and a continuous maps P :]p0 − η0, p0 + η0[×]0, η0[ 7→ L such that for all(x, y) ∈]p0 − η0, p0 + η0[×]0, η0[

Γ+Es(x, y) ∩ L = {P (x, y)}

and limy→0 P (x, y) = m∗. On the other hand, since E0 ∈ Env1,

Γ+E0(x, y)∩]p0 − η0, p0 + η0[×]0, η0[ 6= ∅.

This proves the claim.Now there must be a pointm∗ ∈ Γ+

0 at which FE0 and FE1 are non collinear.Suppose the contrary. Then there exists ε ∈ {−,+} such that for all m ∈ Γ0

FE0(m)

‖FE0(m)‖= ε

FE1(m)

‖FE1(m)‖.

Thus Γ+0 = Γε1. But, since E0 and E1 are favorable to 1, limt→∞ΦE0t (m∗) =

(p0, 0), limt→∞ΦE1t (m∗) = (p1, 0) and limt→−∞ΦE1t (m∗) = (0, 1/d1). Alsop0 6= p1 for otherwise I would be empty. A contradiction. 2

We conclude this section with a theorem describing certain properties ofthe topological support of Π. Consider again the di�erential inclusion inducedby FE0 , FE1 :

η(t) ∈ conv(FE0 , FE1)(η(t)) (18)

16

A solution to (18) with initial condition (x, y) is an absolutely continuousfunction η : R 7→ R2 such that η(0) = (x, y) and (18) holds for almost everyt ∈ R.

Di�erential inclusion (18) induces a set valued dynamical system Ψ = {Ψt}de�ned by

Ψt(x, y) = {η(t) : η is solution to (18) with initial condition η(0) = (x, y)}

A set A ⊂ R2 is called strongly positively invariant under (18) if Ψt(A) ⊂ Afor all t ≥ 0. It is called invariant if for every point (x, y) ∈ A there exists asolution η to (18) with initial condition (x, y) such that η(R) ⊂ A.

The omega limit set of (x, y) under Ψ is the set

ωΨ(x, y) =⋂t≥0

Ψ[t,∞[(x, y)

As shown in ([6], Lemma 3.9) ωΨ(x, y) is compact, connected, invariant andstrongly positively invariant under Ψ.

Theorem 3.7 Under the assumptions of Theorem 3.6, the topological supportof Π writes supp(Π) = Γ× {0, 1} where

(i) Γ = ωΨ(x, y) for all (x, y) ∈ R∗+ × R∗+. In particular, Γ is compact con-nected strongly positively invariant and invariant under Ψ;

(ii) Γ equates the closure of its interior;

(iii) Γ ∩ R+ × {0} = [p0, p1]× {0};

(iv) If I ∩ J 6= ∅ then Γ ∩ {0} × R+ = {0} × [p0, p1].

(v) Γ \ {0} × [p0, p1] is contractible (hence simply connected).

Proof: (i) Let (m, i) ∈ supp(Π). By Theorem 3.6, for every neighborhood Uof m and every initial condition z = (x, y, i) ∈M \M0 lim inft→∞Πt(U) > 0.This implies that m ∈ ωΨ(x, y) (compare to Proposition 3.17 (iii) in [6]).Conversely, let m ∈ ωΨ(x, y) for some (x, y) ∈ R∗+ × R∗+ and let U be aneighborhood of m. Then

Π(U × {i}) =

∫Pz(Zz ∈ U × {i})Π(dz) =

∫Qz(U × {i})Π(dz)

17

where Qz(·) =∫∞

0Pz(Zt ∈ ·)e−tdt. Suppose Π(U × {i}) = 0. Then for some

z0 ∈ supp(Π) \M0 (recall that Π(M0) = 0) Qz0(U × {i}) = 0. Thus Pz0(Zt ∈U×{i}) = 0 for almost all t ≥ 0. On the other hand, because z0 ∈ supp(Π) ⊂ωΨ(x, y) there exists a solution η to (18) with initial condition (x, y) and somesome nonempty interval ]t1, t2[ such that for all t ∈]t1, t2[ η(t) ∈ U. This laterproperty combined with the support theorem (Theorem 3.4 and Lemma 3.2in [6]) implies that Pz0(Zt ∈ U × {i}) > 0 for all t ∈]t1, t2[. A contradiction.

(ii) By Proposition 3.11 in [6] (or more precisely the proof of this propo-sition), either Γ has empty interior or it equates the closure of its interior. Inthe proof of Theorem 3.6, we have shown that there exists a point m in theinterior of Γ.

(iii) Point (pi, 0) lies in Γ as a linearly stable equilibrium of FEi . By stronginvariance, [p0, p1]×{0} ⊂ Γ. On the other hand, by invariance, Γ∩R+×{0}is compact and invariant but every compact invariant set for Ψ contained inR+ × {0} either equals [p0, p1] × {0} or contains the origin (0, 0). Since theorigin is an hyperbolic linearly unstable equilibrium for FE0 and FE1 it cannotbelong to Γ.

(iv) If I ∩ J 6= ∅ then for any s ∈ I ∩ J FEs has a linearly stable equi-librium ms ∈ {0} × [p0, p1] which basin of attraction contains R∗+ × R∗+.Thus ms ∈ Γ proving that Γ ∩ {0} × R+ is non empty. The proof thatΓ ∩ {0} × R+ = {0} × [p0, p1] is similar to the proof of assertion (iii). (v).Since Γ is positively invariant under ΦE0 and (p0, 0) is a linearly stable equilib-rium which basin contains R∗+×R+, Γ\({0}×R+) is contractible to (p0, 0). 2

4 Illustrations

We present some numerical simulations illustrating the results of the preced-ing sections. We consider the environments

A0 =

(1 12 2

), B0 =

(15

), (19)

and

A1 =

(3 34 4 + ρ

), B1 =

(51

). (20)

The simulations below are obtained with

λ0 = st, λ1 = (1− s)t

18

,

Figure 1: Phase portraits of FE0 and FE1

for di�erent values of s ∈]0, 1[, t > 0 and ρ ∈ {0, 1, 3}. Let S(u) = u5(1−u)+u

.Using, Remark 1, it is easy to check that

(a) I = S(]34− 1

2√

6, 3

4+ 1

2√

6[),

(b) J = I for ρ = 0,

(c) J = S(]7196−√

24196

, 7196

+√

24196

[⊂ I for ρ = 1,

(d) J = ∅ for ρ = 3.

The phase portraits of FE0 and FE1 are given Figure 1 with ρ = 3.

Figure 2 and 3 are obtained with ρ = 3 (so that J = ∅). Figure 2 withs 6∈ I and t "large" illustrates Theorems 3.1 (extinction of species 2). Figure3 with s ∈ I illustrates Theorems 3.6 and 3.7 (persistence).

Figures 4 and 5 are obtained with ρ = 1. Figure 4 with s ∈ I ∩ J, t = 10illustrates Theorems 3.6 and 3.7 (persistence) in case I∩J 6= ∅. Figure 5 withs ∈ I ∩ J and "large" t illustrates Theorem 3.4.

Figures 6 is obtained with ρ = 0 s ∈ I ∩ J and t conveniently chosen. Itillustrates Theorem 3.5.

Remark 6 The transitions from extinction of species 2 to extinction of species1 when the jump rate parameter t increases is reminiscent of the transitionoccurring with linear systems analyzed in [5] and [15].

19

Figure 2: ρ = 3, u = 0.4, t = 100 (extinction of 2)

Figure 3: ρ = 3, u = 0.75, t = 12 (persistence)

20

Figure 4: ρ = 1, u = 0.75, t = 10 (persistence)

Figure 5: ρ = 1, u = 0.75, t = 100 (extinction of 1)

21

Figure 6: ρ = 0, u = 0.75, t = 1/0.15 (extinction of 1 or 2)

5 Proofs of Propositions 2.1 and 2.2

5.1 Proof of Proposition 2.1

The process {Xt, Yt, It} restricted to M20 is de�ned by Yt = 0 and the one

dimensional dynamicsX = αItX(1− aItX) (21)

The invariant probability of the chain (It) is given by

ν =λ0

λ1 + λ0

δ1 +λ1

λ1 + λ0

δ0.

If a0 = a1 = a,Xt → 1/a = p. Thus (Xt, It) converges weakly to δp ⊗ ν andthe result is proved.

Suppose now that 0 < a0 < a1.By Proposition 3.17 in [6] and Theorem 1 in [1] ( or Theorem 4.4 in [6]),

there exists a unique invariant probability µ on R∗+×{0, 1} for (Xt, It) whichfurthermore is supported by [p1, p0]. A recent result by [2] also proves thatsuch a measure has a smooth density (in the x− variable) on ]p1, p0[.

Let Ψ : R × {0, 1} 7→ R, (x, i) 7→ Ψ(x, i) be smooth in the x variable.

Set Ψ′(x, i) = ∂Ψ(x,i)∂x

, and fi(x) = αix(1− xpi

). The in�nitesimal generator of

22

(x(t), It) acts on Ψ as follows

LΨ(x, 1) = 〈f1(x),Ψ′(x, 1)〉+ λ1(Ψ(x, 0)−Ψ(x, 1))

LΨ(x, 0) = 〈f0(x),Ψ′(x, 0)〉+ λ0(Ψ(x, 1)−Ψ(x, 0))

Write µ(dx, 1) = h1(x)dx and µ(dx, 0) = h0(x)dx. Then∑i=0,1

∫LΨ(x, i)hi(x)dx = 0.

Choose Ψ(x, i) = g(x) + c and Ψ(x, 1 − i) = 0 where g is an arbitrary com-pactly supported smooth function and c an arbitrary constant. Then, an easyintegration by part leads to the di�erential equation{

λ0h0(x)− λ1h1(x) = −(f0h0)′(x)λ0h0(x)− λ1h1(x) = (f1h1)′(x)

(22)

and the condition ∫ p0

p1

λ0h0(x)− λ1h1(x)dx = 0. (23)

The maps

h1(x) = Cp1(x− p1)γ1−1(p0 − x)γ0

α1x1+γ1+γ0, (24)

h0(x) = Cp0(x− p1)γ1(p0 − x)γ0−1

α0x1+γ1+γ0(25)

are solutions, where C is a normalization constant given by∫ p0

p1

h0(x) + h1(x)dx = 1.

Note that h1 and h0 verify the equalities:∫ p0

p1

h0(x)dx =λ1

λ0 + λ1∫ p0

p1

h1(x)dx =λ0

λ0 + λ1

.

This concludes the proof of Proposition 2.1.

23

5.2 Proof of Proposition 2.2

(i). We assume that I = ∅. If p0 = p1 then Λ2 < 0. Suppose that a0 < a1

(i.e p0 > p1) (the proof is similar for p0 < p1). Let ps = 1as

with as being givenin the de�nition of As. The function s 7→ ps maps ]0, 1[ homeomorphicallyonto ]p0, p1[ and by de�nition of Es

sα1(1− a1ps) + (1− s)α0(1− a0ps) = 0.

Thus (1− a1ps) = − (1−s)α0

sα1(1− a0ps). Hence

P (ps) =(1− a0ps)

α1sβs(1− csps) = − βs

α1s(1− a0/as)(1− cs/as).

This proves that P (x) ≤ 0 for all x ∈]p0, p1[. Since P is a nonzero polynomialof degree 2, P (x) < 0 for all, but possibly two,points in ]p0, p1[. Thus Λ2 < 0.

(ii). If a0 = a1 the result is obvious. Thus, we can assume (withoutloss of generality) that a0 < a1. Fix s ∈]0, 1[ and let for all t > 0 νt1(respectively νt0) be the probability de�ned as νt1(dx) = 1

sht1(x)1]p1,p0[(x)dx

(νt0(dx) = 11−sh

t0(x)1]p1,p0[(x)dx) where ht1 (respectively ht0) is the map de-

�ned by equation (24) (respectively (25)) with λ0 = st and λ1 = (1− s)t. Weshall prove that

νti ⇒ δps as t→∞ (26)

andνti ⇒ δpi as t→ 0, (27)

where ⇒ denotes the weak convergence. The result to be proved follows.Let us prove (26). For all x ∈]p0, p1[, νti (dx) = Ct

ietW (x)[x|x−pi|]−11]p1,p0[(x)dx

where Cti is a normalization constant and

W (x) =s

α0

log(p0 − x) +1− sα1

log(x− p1)− αsα0α1

log(x).

We claim that

argmax]p0,p1[W = ps =1

as(28)

Indeed, set Q(x) = W ′(x)(α0α1x((x− p0)(p1 − x)). It is easy to verify that

Q(x) = sα1(p1 − x)x− (1− s)α0(x− p0)x− αs(p0 − x)(x− p1).

24

Thus Q(p0) < 0, Q(p1) > 0 and since Q is a second degree polynomial, itsu�ces to show that Q(ps) = 0 to conclude that ps is the global minimum ofW. By de�nition of ps,

sα1(1− a1ps) + (1− s)α0(1− a0ps) = 0.

Thus(1− s)α0(ps − p0) =

sα1a1

a0

(p1 − ps).

Plugging this equality in the expression of Q(ps) leads to Q(ps) = 0. Thisproves the claim. Now, from equation (28) and Laplace principle we deduce(26)

We now pass to the proof of (27). It su�ces to show that νti converges inprobability to pi, meaning that νti{x : |x− pi| ≥ ε} → 0 as t→ 0. This easilyfollows from the shape of hti and elementary estimates. 2

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Acknowledgments

This work was supported by the SNF grant FN 200020-149871/1. We thankMireille Tissot-Daguette for her help with Scilab.

27