Evolution of the population of Microtus Epiroticus: the Yoccoz-Birkeland model

0.93 MB·English

by  Juan José Nieto

Evolution of the population of Microtus Epiroticus: the
1
Yoccoz-Birkeland model.
1
0
2
J. J. Nieto∗ M. J. Pacifico† J. L. Vieitez‡
n
a
J January 31, 2011

8
2

]
S
Abstract
D
We study the discretized version of a dynamical system given by a model proposed by
.
h
Yoccoz and Birkeland to describe the evolution of the population of Microtus Epiroticus
t
a onSvalbardIslands,see http://zipcodezoo.com/Animals/M/Microtus epiroticus. We prove
m that this discretized version has an attractor Λ with a hyperbolic 2-periodic point p in it.
[ Forcertainvaluesoftheparametersthesystemrestrictedtotheattractorexhibitssensibility

to initial conditions. Under certain assumptions that seems to be sustained by numerical
1
simulations, the system is topologically mixing (see definition 4.1) explaining some of the
v
3 high oscillationsobservedin Nature. Moreover,we estimate its order-2Kolmogoroventropy
4 obtaining a positive value. Finally we give numerical evidence that there is a homoclinic
5 point associated with p.
5
.
1 2000 Mathematics Subject Classification: 37M05, 37N25, 92D25
0
1
Contents
1
:
v
1 Introduction 2
i
X 1.1 The discrete model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
r
a 2 The dynamical system 4
2.1 Permanence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Existence of fixed points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Existence of an attractor for the discrete model. 7
4 Study of Λ for (A ,ρ,γ)=(0.18,0.30,8.25). 9
0
4.1 Estimation of the Kolmogorov Entropyof theAttractor . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Existence of homoclinic points: numerical approach. 13
5.1 Approximated Wu (p).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
loc
5.2 Returningpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.3 Far from tangencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.4 Evidenceof homoclinic points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
∗J.J.NietoispartiallysupportedbyMinisteriodeEducacionyCiencia,Spain,andFEDER,ProjectMTM2010-
15314.
†M. J. Pacifico is partially supported byCNPq, FAPERJ and PRONEXDynamical Systems, Brazil.
‡J. L. Vieitez is partially supported byPEDECIBA and ANIICE 10065, Uruguay.
1
6 Appendices. 15
6.1 AppendixA:numerical results for the entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.2 AppendixB: description of algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.3 AppendixC: Pseudo-code of thealgorithms employed . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.4 AppendixD:pseudo-code of homclin4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.5 AppendixE: coordinates of fixedpoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.6 AppendixF: homoclinic points search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1 Introduction
We study the evolution of the population of Microtus Epiroticus (sibling vole) on Svalbard Islands in the Arctic
Ocean,usingamodelproposedbyJ.C.YoccozandH.Birkeland,see[Ar]. Itisknownthattherearenosignificant
predation of these small mammals but in spite of that, the population presents high oscillations in its number
albeitthelackoffoodisnotadeterminantfactortotheoccurrenceofthesephenomena. Thispopulationexhibits
dramatic multi-annual fluctuations, bya factor greater than 20, [YI].
The Sibling Vole (Microtus Epiroticus) is a species of vole found through much of northern Europe. First
discovered in 1960 in theGrumantbyen area, they were thought to be theCommon Vole until a genetic analysis
correctly identified them in 1990, [FJASY].
Since these rodents were introduced from Russia on Svalbard Isles between 1930 and 1960, [YI], the annual
oscillations of their number may be explained, at least in part, by a non total adaptation to the environment,
andbythepronouncedseasonalfluctuationinclimaticvariabilityatSvalbardwheretemperaturesof 30degrees
−
Celsius are common, see [YI, LBY].
Figure 1: Microtus Epiroticus.
Let usfirst sketch thetaxonomy classification of Microtus Epiroticus.
Domain: Eukaryota
•
Kingdom: Animalia
•
Phylum: Chordata
•
Class: Mammalia
•
Order: Rodentia
•
Family: Muridae
•
Subfamily: Arvicolinae
•
Genus: Microtus
•
Species: Microtus Epiroticus
•
Jean Christophe Yoccoz and H.Birkeland, see [Ar], haveproposed the following equation
A1
N(t)= N(t a)m(N(t a))m (t a)S(a)da, (1)
ρ
Z − − −
A0
2
to model theevolution in time of the population of Microtus Epiroticus. In theequation it is taken intoaccount
only the number N(t) of fertile females at certain time t. Indeed, the inclusion in a model of the number of
males is justified when there are difficulties for a female to find a male (for instance if the density of population
is too small or if the ratio male-population : female-population is far away from 1:1) which is not the case for
these rodents. In fact as has been pointed out by R. A. Ims in [Ims], “spatial clumping of sexually receptive
females inducesspacesharingamongmalevoles”whichimpliesthatitisnotdifficultforafemaletofindamale.
Moreover, thequantityof females is about thesame as those of males for theserodents, [Ims2, YI].
Let usdescribe the parameters of themodel given byequation (1):
1. t: is thetime measured in years.
2. N(t): is thepopulation of active females at time t.
3. A : is thematuration age,
0
4. A : is themaximal age expected for Microtus Epiroticus.
1
5. m(N): annualindividual reproduction rate for a population of N individuals
6. m (t): is thereproduction probability at time t of the year.
ρ
7. S(a): probability to surviveup to a years.
The model take intoaccount thefollowing facts:
(a) TheagewhenthefemalesofMicrotus Epiroticushavetheirfirstoffspringisabout50days,i.e.,A 0.14
0
≈
years (see [YIS]).
(b) The maximal age of survivalis about 2 years, i.e., A =2 (see [YI]).
1
(c) The seasonal factor m (t), that is, the reproduction probability at time t of the year, varies sharply from
ρ
0 in Winterto 1 from Springto Autumn. Thus, thedefinition of m (t) we adopt is
ρ
0 if 0 t mod (1)<ρ
mρ(t)=(cid:26) 1 if ρ≤t mod (1)<1 .
≤
(d) Theannualindividualreproductionratem(N)forapopulationofN individuals,istoohighwhenN(t)is
small. Indeed m(N) of the order of a constant m >30 individuals is realistic due to the high fertility of
0
these rodents. The value of m(N(t)) decays sharply when the population N(t) increases. Following [Ar],
for m(N) we adopt
m if N 1
m(N)= 0 ≤ ; γ >1. (2)
(cid:26) m0N−γ if N >1
We will assume that γ >1 and for some calculations we take γ =8.25. The reason for that is that there
is numerical evidence,see [Ar], that for thisvalue of the parameter we havechaotic behavior.
(e) Finally for thesurvival probability S(a), again following [Ar], we consider a linear function:
a
S(a)=1 , if 0 a A , and S(a)=0 elsewhere.
− A1 ≤ ≤ 1
Remark 1.1. Another choice of functions for S(a), for instance S(a) = exp( κa), with κ > 0, are also usual
−
in the literature. It would be interesting to test the model given by (1) replacing the linear function at (e) by a
exponential one.
Let usdescribe how the integral equation
A1
N(t)= N(t a)m(N(t a))m (t a)S(a)da arises.
ρ
Z − − −
A0
For N(t), thecontribution of females of age in between [a,a+∆a] [A ,A ] is
0 1
⊂
fem(t a) (reprod. rate(t a) ( season factor(t a) (prob. survive) ℓ([a,a+∆a])
− × − × − × ×
=N(t a) m(N(t a)) m (t a) S(a) ∆a,
ρ
− × − × − × ×
where ℓ(J) is the length of the interval J and fem(t) is the number of females at time t. Here we assume that a
female of age near A can reproduce and ∆a is small. Taking a partition a = t A ,a ,...,a = t A of
1 0 1 1 n 0
{ − − }
theinterval [t A ,t A ] we find
1 0
− −
n−1
N(t) N(t a )m(N(t a ))m (t a )S(a )∆a , where ∆a =(a a )
j j ρ j j j j j+1 j
≈ − − − −
Xj=0
Letting n we get at thelimit theintegral equation given by (1).
→∞
3
1.1 The discrete model.
There is no special reason to prefer the continuous model above to its discretization: most of the quantities
involved, as N(t) and m(N), are by nature of discrete type. Moreover, from the experimental point of view, it
is more natural to split the year on days and even in groups of days since it is very difficult to monitor N(t)
experimentally. Hence, we assume that theyear is split into p equalparts.
Since the expected value of survival is bounded by A = 2 years we will study the evolution of N(t) for
1
discrete values of t, modeling the period [0,A ] as a vector of A p+1 real entrances, from t = 0 at the initial
1 1
time of the first year, to t=2p corresponding to A =2 thefinal timeof thesecond year.
1
In this case the probability of survivalat age j is given by
p
j
S(j)=1 , j =0,1,2...,2p.
− 2p
where A p = 2p. It is also convenient to consider S(j) = 1 j . This takes into account the case where
1 − 2p+1
S(2p)>0, i.e., when these animals can reproducetill thefinal of their lives.
Given an initial vector value (N0,N1,N2,...,NA1p−1,NA1p)∈IRA1p+1, the evolution of N(t)=Nt, t∈IN,
is governed by
A1p−1
N = N m(N )m (t h)S(h)∆h (3)
t t−h t−h ρ
−
h=XA0p
2p−1
1
= N m(N )m (t h)S(h).
p t−h t−h ρ −
h=XA0p
Next we explain thechoices in equation (3).
1. We take A p = 50×p 14 which corresponds to the age at which the females have their first litter of
0 365 ≈
pups(about 50 d(cid:2)ays).(cid:3)Note that if p=100 then A0 =0.14 corresponds to 51 days.
2. Wetake∆h= 1 yearsthatcorrespondstothelengthoftheunitintervalinwhichwesplittheyear. When
p
p=100 this gives ∆h= 1 years=3.65 days.
100
Notethat thevalueof N at t dependsonly on thevaluesof N in [t A ;t A ]. Thus,theknowledge of N for
1 0 t
− −
t [ A p,0] (two years of observation) enables us to predict N for t [0,A p]. When p= 100 and A = 0.14
1 t 0 0
∈ − ∈
this means that the knowledge of (N ,N ,N ,...,N ) enables us to compute N ,...,N . Recursively we
0 1 2 200 201 214
may compute N for all j 0.
j
≥
2 The dynamical system
Equation (3) defines a discrete dynamical system in IR2p+1 as follows:
(N ,N ,...,N ) T(N ,N ,...,N )=(N ,N ,...,N ),
0 1 2p 0 1 2p p p+1 3p
7→
where we have used that A = 2 and T : IR2p+1 IR2p+1 is defined recursively by equation (3) for t =
1
→
2p+1,...,3p.
Inordertodescribetheoreticalpropertiesofasystemgivenbythediscretizedversion(3)ofYoccoz-Birkeland
equation (1), let us assume the following restrictions that weaken those given by conditions (a)–(e) described
before. Doingthisallowstoapplytheconclusionstodifferentspeciesrespectingequation(3)andthoserestrictions.
In particular, these conclusions will apply to theoriginal system modeling Microtus Epiroticus.
1. m(N) is a continuousfunction, m:IR+ IR+,
→
2. there is m IR+ such that
0
∈
m m(N) m /2 if N 1 and
(cid:26) m00N≥−γ ≥m≥(N)0≥min{m20,m0·N−γ} if N >≤1 , (4)
3. 0 m (t) 1 (so that we now allow 0<m (t)<1 for certain valuesof t),
ρ ρ
≤ ≤
4. There is ǫ 0 such that m (t)=1 for t in an interval of length 1 ρ ǫ>0,
ρ
≥ − −
4
5. 0<2A <A and A +1<A (thismeansthat inaverage each individualhasat least twoopportunities
0 1 0 1
to reproduce),
6. Definingc as c = 1 A0p+p S(h) we requirec m >2. From the definition of S(h) it follows
0 0 p(cid:16) h=A0p+(ρ+ǫ)p (cid:17) 0 0
P
1 A0p+p h (1+ρ+ǫ)+2A
c0 = ph=A0Xp+(ρ+ǫ)p(1− pA1)=(1−ρ−ǫ)(cid:18)1−(cid:18) 2A1 0(cid:19)(cid:19) . (5)
The condition c m >2 will imply, as we will see below in Proposition 2.3, that the population does not
0 0
extinguish, that is, it has thepermanence property (see Definition 2.1).
7. The exponent γ satisfies γ >1.
The following proposition shows that a dynamical system governed by equation (3) and respecting the re-
strictions 1. to 7. above is bounded.
Proposition 2.1. For all t=1,2,...,A p we have N N :=m (A1−A0)2 .
0 t ≤ max 0(cid:16) 2A1 (cid:17)
Proof. Since γ >1, inequalities (4) imply N m(N ) m for all j. Moreover from m 1 we obtain
j j 0 ρ
≤ ≤
A1p−1
N = N(t h)m(N(t h))m (t h)S(h)∆h
t ρ
− − − ≤
h=XA0p
A1p 1 m A1p m A1p h
m S(h) = 0 S(h)= 0 (1 )=
h=XA0p 0 p p h=XA0p p h=XA0p − pA1
m A p(A p+1) A p(A p+1)
p0 (cid:18)(A1−A0)p− 1 1 2A−1p0 0 (cid:19)≤
A (A +1/p) A (A +1/p)
m0(cid:18)(A1−A0)− 1 1 2−A1 0 0 (cid:19)≤
A2 A2 (A A )2
m0(cid:18)(A1−A0)− 12−A1 0(cid:19)=m0(cid:18) 12−A1 0 (cid:19) .
By induction we obtain that for all t 0, N N .
t max
≥ ≤
Remark 2.2. For the values A =2, A =0.18, m =50 we have N 41.4.
1 0 0 max
≈
2.1 Permanence.
In this section we verify that thepopulation given by equation (3) and respecting therestrictions 1. to 7. above,
in particular conditions (4) and (5),does not extinguish.
Definition 2.1. We say that a system P(t) modeling the evolution of a population is permanent, or satisfies the
permanence property, iffor any positive initialvector value P , there isǫ>0 such that the solution P(t) satisfies
0
liminfP(t) ǫ.
t≥0 ≥
If a given system is permanent then, assuming that the environmental conditions do not change in time,
the associated population will not extinguish. Thus, concerning with population dynamics this property is very
important.
The next proposition shows that thesystem understudy satisfies thepermanence property.
Proposition 2.3. If i(N)=min N , t [ pA ,0] >0 then N >0 for all t=0,1,...,A p. Moreover,
t 1 t 0
{ ∈ − }
• If i(N)≤Nm1−aγx then Nt ≥ c02m0i(N)>i(N), t∈[0,pA0].
• If i(N)≥Nm1−aγx then Nt ≥ c02m0Nm1−aγx, t∈[0,pA0].
5
Proof. If N(t h) 1 then,by (4),
− ≤
m m
N(t h)m(N(t h)) N(t h) 0 i(N) 0 .
− − ≥ − 2 ≥ 2
Otherwise N(t h)>1 and then,again by (4),
−
m m m m
N(t h)m(N(t h)) min N(t h)1−γ 0,N(t h) 0 min N1−γ 0,i(N) 0 .
− − ≥ { − 2 − 2 }≥ { max 2 2 }
Hencewe have
2p−1
N = N(t h)m(N(t h))m (t h)S(h)∆h
t ρ
− − − ≥
h=XA0p
2p−1
1 m m
min N1−γ 0,i(N) 0 m (t h)S(h)
p { max 2 2 } ρ − ≥
h=XA0p
m m 1 A0p+p h c m c m
minnNm1−aγx 20,i(N) 20o ph=A0Xp+(ρ+ǫ)p(1− pA1)=minnNm1−aγx 02 0,i(N) 02 0o .
Since, by (5), c m >2 we get by induction that N(t)>0 for all t [0,A p].
0 0 0
∈
Clearly Proposition 2.3 implies that N >0 for all t 0.
t
≥
Corollary 2.4. There is t0 > 0, depending on the initial vector value, such that we have N(t) ≥ c02m0Nm1−aγx,
t t .
0
≥
2.2 Existence of fixed points.
The following corollary is a straightforward consequence of Propositions 2.1 and 2.3.
Corollary 2.5. If Nt >0 for all t∈[−Ap,0] then there is t0 >0 such that c02m0Nm1−aγx ≤Nt ≤Nmax for t≥t0.
In particular T maps the compact set
= c0m0N1−γ,N pA1+1 into itself.
K h 2 max maxi
Now set
H := N =(N ,N ,...,N ) IR2p+1 : j =0,1,...2p: N >0 . (6)
0 1 2p j
∈ ∀
Observethat Proposition 2.(cid:8)3 together with Corollary 2.4 imply that T maps H intoitse(cid:9)lf.
Next we provethat T :H H is Lipschitz.
→
Lemma 2.6. T :H H is a Lipschitz function.
→
Proof. We put in IR2p+1 thesup norm: x = (x ,x ,...,x ) =sup x .
k k k 0 1 2p k t=0,...,2p| j|
From the definition of T we have T(N ,N ,...,N ) = (N ,N ,...,N ). Hence for all j = 0,...,p we
0 1 2p p p+1 3p
have
(T(N) T(N′)) = N N′ N N′ . (7)
| − j| | j+p− j+p|≤k − k
For j =p+1,...,2p, the difference N m(N ) N′ m(N′ ) can be estimated as follows:
| (t−h) (t−h) − (t−h) (t−h) |
(a) If N 1 and N′ 1 then by inequalities (4) we havethat
(t−h) ≤ (t−h) ≤
N m(N ) N′ m(N′ ) m N N′ .
| (t−h) (t−h) − (t−h) (t−h) |≤ 0| (t−h)− (t−h)|
(b) If N 1 and N′ 1 then,again by (4), we havethat
(t−h) ≥ (t−h) ≥
N m(N ) N′ m(N′ ) (N )1−γ (N′ )1−γ m .
| (t−h) (t−h) − (t−h) (t−h) |≤| (t−h) − (t−h) | 0
By theMean Value Theorem, there is N (N ,N′ ) such that
∈ (t−h) (t−h)
(N )1−γ (Ne′ )1−γ = 1 γ N−γ N N′ .
| (t−h) − (t−h) | | − | | (t−h)− (t−h)|
Since γ >1 and N >1 we obtain e
e m (N )1−γ (N′ )1−γ m (γ 1)N N′ .
0| (t−h) − (t−h) |≤ 0 − | (t−h)− (t−h)|
6
(c) If one of the abovequantities is greater than 1 and theother is not, say N′ >1 and N 1, then
(t−h) (t−h) ≤
N m(N ) N′ m(N′ ) =m N (N′ )1−γ .
| (t−h) (t−h) − (t−h) (t−h) | 0| (t−h)− (t−h) |
If N (N′ )1−γ then,since 0<N 1 and 1 γ <0 we get
(t−h) ≥ (t−h) (t−h) ≤ −
m N (N′ )1−γ =m (N (N′ )1−γ)
0| (t−h)− (t−h) | 0 (t−h)− (t−h) ≤
m ((N )1−γ (N′ )1−γ)=
0 (t−h) − (t−h)
m (N )1−γ (N′ )1−γ m (γ 1)N N′ .
0| (t−h) − (t−h) |≤ 0 − | (t−h)− (t−h)|
Otherwise, if N < (N′ )1−γ then, since N′ > 1 and 1 γ < 0, we have 0 > N
(t−h) (t−h) (t−h) − (t−h) −
(N′ )1−γ >N N′ and therefore
(t−h) (t−h)− (t−h)
N m(N ) N′ m(N′ ) =m N (N′ )1−γ m N N′ .
| (t−h) (t−h) − (t−h) (t−h) | 0| (t−h)− (t−h) |≤ 0| (t−h)− (t−h)|
Next, to estimate N N′ for t = p,p + 1,...,A p, we use (a), (b) and (c) above as below. Let L =
| t − t| 0
max m ,m (γ 1) . Taking into account that m (t h) and S(h) are between 0 and 1 and ∆h = 1 we
{ 0 0 − } ρ − p
obtain that:
2p−1 2p−1
|Nt−Nt′|=(cid:12)(cid:12) N(t−h)m(N(t−h))mρ(t−h)S(h)∆h− N(′t−h)m(N(′t−h))mρ(t−h)S(h)∆h(cid:12)(cid:12)=
(cid:12)(cid:12)h=XA0p h=XA0p (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
2p−1
1
(cid:12)(cid:12)p N(t−h)m(N(t−h))−N(′t−h)m(N(′t−h)) mρ(t−h)S(h)(cid:12)(cid:12)≤
(cid:12)(cid:12) h=XA0p(cid:0) (cid:1) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)1 2p−1 (cid:12)
N m(N ) N′ m(N′ ) m (t h)S(h)
p (t−h) (t−h) − (t−h) (t−h) ρ − ≤
h=XA0p(cid:12)(cid:12) (cid:12)(cid:12)
2p−1 2p−1
1 1
LN N′ Lmax N N′ (A A )L N N′ . (8)
ph=XA0p | (t−h)− (t−h)|≤ ph=XA0p h | (t−h)− (t−h)|≤ 1− 0 k − k
Taking into account that 1 (A A )L and inequalities (7) and (8) we have that for all j = 0,...,p,p+
1 0
≤ −
1,...,p+ A p, T(N) T(N′) (A A )L N N′ . By induction, since A p > 1, we obtain that
0 j 1 0 0
T(N) T(N′) | (A − A )L| N≤ N′−finishin·gkthe−prookf.
1 0
k − k≤ − ·k − k
Corollary 2.7. There is a fixed point p for T : .
K→K
Proof. ByLemma2.6themapT isLipschitzhencecontinuous. Moreover isa(2p+1)-dimensionaltopological
K
disk. Hence Brouwer Fixed Point Theorem applies, [Sp, Chapter 4, Section 7].
Remark 2.8. Since every two years (A = 2) the rodent population is renewed perhaps it is more natural to
1
search for fixed points for T2 : . So, we are interested in both, fixed points and period-two points N .
K →K ∈K
Their existence is guaranteed by Corollary 2.7.
In Appendix E we estimate the coordinates of a fixed point p of T2:H H. We find that the distance given
bythenormofthesupremumbetweenpandT2(p)isabout8.0148 10−14 an→dthel1 normisabout4.0353 10−12.
Thisestimate of p isbetter than that obtained byArlot, [Ar, Secti×on B.8],which isof order 10−4 forthe l×1 norm.
3 Existence of an attractor for the discrete model.
Proposition 3.1. Let Λ = Tn( ). Then Λ = is compact T-invariant and there is a neighborhood
n≥0 K 6 ∅
U =U(Λ) such that T(U) UT, i.e., Λ is an attractor for T.
⊂
Remark 3.2. We are not assuming that Λ is transitive in the definition of attractor.
7
Proof. SinceT( ) wehavethatC = n Tj( )isadecreasing sequenceofnonemptycompact subsetsof
IR2p+1; C CK ⊂K C . Thuns, by∩jB=a0ire TKheorem, we have that Λ= and Λ is compact.
0 1 n
⊃ ⊃···⊃ ⊃··· 6 ∅
By definition of Λ we have
T(Λ)=T( Tn( )) Tn+1( ) Tn( )=Λ,
n≥0 n≥0 n≥0
∩ K ⊂∩ K ⊂∩ K
provingthat Λ is T-invariant.
Let [ ] := N IR2p+1 : dist(N, ) ǫ and ǫ>0 beso small that
ǫ
K { ∈ K ≤ }
[ ] H = N =(N ,N ,...,N ) IR2p+1 : j =0,1,...2p: N >0 .
ǫ 0 1 2p j
K ⊂ { ∈ ∀ }
By Proposition 2.3 and Proposition 2.5, for all x [ ] there is n(x)>0 such that Tn(x)(x) . By continuity
ǫ
of T,see Lemma 2.6, there is U(x) a neighborhoo∈d oKf x contained in H such that Tn(x)(y) ∈Kfor all y U(x).
∈K ∈
By compactness of [ ] there is n >0 such that Tn([ ] ) for all n n .
ǫ 1 ǫ 1
K K ⊂K ≥
Let now U(Λ) bea neighborhood of Λ contained in [ ] .
ǫ
K
Claim 3.1. There is n >0 such that Tn( ) U(Λ) for all n n .
0 0
K ⊂ ≥
Proof. Theproofgoesbycontradiction. Ifitwerenottrue,forallj IN therewouldexistx andn >n ,
j j j−1
∈ ∈K
sWucithhotuhtatloTssnjw(xej)ma∈/yUas(sΛu)m. eStihnacet {KTnisj(cxojm)}pja∈cINt thitesreelfecxoinstvseragecsontoveargepnotinstuzbse∈quKe.nceSufcrhoma{pToinnjt(xzj)c}ajn∈nINot.
be in Λ since Tnj(xj) / U(Λ) for every j IN. But, since Tn+1( ) Tn( ) for all n IN, we obtain
TBunjt(xjn)j ∈T∩h(nh=j0)Th(Kn)j.+1∈MTohr(eo)vefro,rzal∈l j∩hn=jIN0T,∈hs(oKt)h,aotthdeisrtw(izs,e tnhj+erleThis(Kǫ)>)⊂0 ǫsufcohrKtehvaertydlist(z0,∩c∈onhn=jt0rTadhi(cKti)n)g>thǫe.
fact t∩hha=t0Tnj+Kl(x⊃j+∩l)h=0z whKen l . I∈t follows that Tn( )∩h=U0(Λ)Kfor a≥ll n n0, provi≥ngthe claim.
→ →∞ K ⊂ ≥
To conclude the proof of the proposition it is enough to verify that there is n2 > 0 such that Tn2(U(Λ))
U(Λ). This follows from the fact that Tn0( ) U(Λ) U(Λ) [ ]ǫ taking n2 = n0 +n1, thus Λ is a⊂n
K ⊂ ⊂ ⊂ K
attractor.
It is clear that the fixed point p given by Corollary 2.7 belongs to Λ. In [Ar, Section B.8] by numerical
methods it is found a candidate to be a fixed point. As we havepointed out above, in view of Corollary 2.7, the
search for such a fixedpoint has sense.
Remark 3.3. ByCorollary 2.5,the basin ofattraction of Λisthe wholeset H ofpoints withpositive coordinates
(see equation (6) ). Moreover, since isa disk, wecan choose U(Λ) simplyconnected inthe proof of Proposition
K
3.1. These facts have some theoretical implications that we discuss in section 4.
Lemma 3.4. Assume that S(2p 1) > 0 and m (1) = 1. Moreover also assume that T depends smoothly on
ρ
−
N =(N ,N ,...,N ) and that ∂(Njm(Nj)) =0, j =0,1,...,p. Then the differential D T :IR2p+1 IR2p+1 is
0 1 2p ∂Nj 6 N →
a non singular linear map.
Proof. Let us duplicate the (p + 1)-th coordinate, N , of N = (N ,...,N ,...,N ), i.e., we write N =
p 0 p 2p
(N ,...N ,N ,...N ) = (N ,N ), and consider T(N ,N ) = (N ,N ) where N = (N ,...,N ).
0 p p 2p (0) (1) (0) (1) (1) (2) (2) 2p b3p
Thus,sincethepth-coordinateequalsthe(p+1)th-coordinate,T(N ,N )issuchthatΠ (T(N ,N )=T(N)
b (0) (1) p (0) (1)
and if T is locally injective then T is locally injective too. Here Π :IR2p+2 IR2p+1 is theprojection
b p → b
b Π (x ,...,x ,x ,...,x )=(x ,...,x ,x ,...,x ).
p 0 p p+1 2p+1 0 p−1 p+1 2p+1
Taking into account that (N ,...,N ) depends on (N ,...,N ,...,N ), this artifice allows us to write
2p 3p 0 p 2p
T(N ,N )=(N ,F(N ,N )), and therefore
(0) (1) (1) (0) (1)
b A Id
|
DT = 
−−−− −−−− −−−−
∂F ∂F
b  ∂N(0) | ∂N(1) 
where A is a (p+1) (p+1) matrix of the form
×
0 0 0 1
···
 0 0 0 0 
A= ···
 ··· ··· ··· ··· 
 0 0 0 0 
 ··· 
8
and Id is the identity(p+1) (p+1) matrix.
×
To prove that T is locally injective it suffice to prove that detDT = 0. Hence, since det(A)=0 we are left
6
to provethat det ∂F =0. For this we proceed as follows. Using the expression for N given at equation (3)
(cid:16)∂bN(0)(cid:17)6 b t
and denoting ∂(Njm(Nj)) by h(N ) we compute ∂F and find
∂Nj j ∂N(0)
h(N )m (1)S(2p 1) h(N )m (2)S(2p 2) ......... h(N )m (p)S(p)
0 ρ 1 ρ p ρ
− −
1 0 h(N1)mρ(1)S(2p−1) ......... h(Np)mρ(p−1)S(p+1) 
0 0 ......... ......... .
p 
 
 ········· ········· ········· ········· 
 0 0 ......... h(N )m (1)S(2p 1) 
 p ρ − 
Since byhypothesis h(N )=0 thethesis follows.
j
6
Corollary 3.5. Under the hypothesis of Lemma 3.4 we have that T :Λ Λ is locally injective.
→
Remark 3.6. Albeit T : H H is locally injective, by Lemma 3.4, it is not globally injective. To see this
→
assume that N >1, γ >1 and that the definition of m(N) is given by equation (2). If (N ,N ,...,N )=
max 0 1 2p−1
(N ,N ,...,N ) then for t=2p, 2p+1,...,2p+A p we get
max max max 0
2p−1 2p−1
1 1
N = N m(N )m (t h)S(h) = N m(N )m (t h)S(h)
t p t−h t−h ρ − p max max ρ −
h=XA0p h=XA0p
2p−1
1
= N1−γm m (t h)S(h). (9)
p max 0 ρ −
h=XA0p
Similarly if we put (N ,N ,...,N ) = (N1−γ,N1−γ,...,N1−γ) we obtain the same values for N . By
0 1 2p−1 max max max t
induction we get that all values are the same for t 2p implying that T is not globally injective.
≥
Let uspoint out that:
1. Intheoriginalmodel,[Ar],m(N )isgivenbyequation(2)i.e.,m(N )=m ifN 1andm(N )=m N−γ
j j 0 j j 0
if N >1. HenceN m(N )=m N if N 1 and N m(N )=m N1−γ if N >≤1 implying that
j j j 0 j j ≤ j j 0 j j
∂(N m(N )) m if N 1
h(Nj)= j∂Nj j =(cid:26) m00(1−γ)jN≤j−γ if Nj >1 .
Since γ >1, we haveh(N )=0 for all N =1.
j j
6 6
2. AssumingthatT2isC1,Lemma3.4givesthatthefixedpointpfoundatCorollary2.7hasallitseigenvalues
different from zero. The numerical approximation of the eigenvalues of DT2, for the estimated value of p
p
obtained by [Ar, Section 4.2.7] and our own estimates gives that there is a single eigenvalue of modulus
greater than 1 which is negative, and there are A p eigenvalues of modulus less than 1. Hence p is a
1
codimension one hyperbolicfixed point of T2.
3. The hypothesis S(2p 1) = 0 is reasonable: otherwise one can see that for two initial vectors N =
(N ,N , ,N ) and−N′ =6 (N′,N , ,N ) with N = N′ we get T(N) = T(N′). Thus the number
0 1 ··· 2p 0 1 ··· 2p 0 6 0
N of individuals at time A p is not affected by the first set N of initial individuals. In another words
A1p 1 0
the system looses memory for a number of years less than A and so the actual dimension of the domain
1
of T would beless than A p+1.
1
4 Study of Λ for (A ,ρ,γ) = (0.18,0.30,8.25).
0
In what follows we will assume that T is smooth ( see [Ar, Section 2]) and that the calculations made for the
parametervalues(0.18,0.30,8.25)areaccurateenoughtoobtainthatifpisthefixedpointgivenbyCorollary2.7
then the eigenvalues λ ,λ ,...,λ and µ of D T satisfies λ < 1 for every j = 1,...,2p, and µ 3.335, in
1 2 2p p j
| | ≈ −
particular µ >11. Lemma 3.4 provesthat p is in fact a hyperbolicfixedpoint with Ws(p) being acodimension
| |
1Arlot in [Ar, Section 4.2.7], obtains that µ 2,29 for theparameter values (0.15,0.30,8.25).
≈−
9
one manifold and Wu(p) an arc. Moreover, since Λ is an attractor, we have that Wu(p) Λ from which the
⊂
fractal dimension of Λ is strictly greater or equal than 1. The calculations made in [Ar, Section 4.2.5] give for
this fractal dimension a valuearound 1.33 from which Arlot conjectures that locally theattractor is theproduct
of a line bya Cantor set.
Here we shall discuss if for the choice A = 0.18, ρ = 0.30 and γ = 8.25 the system given by T can be
0
transitive.2
Definition 4.1. Let f :X X be a continuous map defined in the topological space X. We say that the system
→
defined by f is (topologically) transitive if for every pair of non-empty open subsets A, B of X there is n ZZ
∈
suchthatfn(A) B= . Thedynamicalsystem defined byf istopologicallymixingifforevery pairofnon-empty
open subsets A,∩B of6 X∅there is N >0 such that fn(A) B= for all n N.
∩ 6 ∅ ≥
In [Ar, Section 5] it is pointed out the interest in studying the case where the parameters are A = 0.18,
0
ρ = 0.30, γ = 8.25: it is because the numerical simulations indicates that for this parameter choice T is
|Λ
transitive, see [Ar, Section 4.1.3, figure 12]. Moreover, in [Ar, Section 4.2.7, figures 34 and 35] the geometry of
the attractor Λ is depicted from the successive iterates of the local unstable manifold of the fixed point p. This
suggeststhatWu(p)isdenseinΛ. Thiswasconfirmedbythenumericalsimulationsdonebyus,seefigure2. The
nextpropositionshowsthatiftheorbitofapointinWu(p)isdenseinΛthenT isinfacttopologically mixing.
|Λ
Proposition 4.1. Let us assume that there exists x Wu(p) such that clos(orbit+(x ))=Λ that there exists a
0 0
∈
homoclinicpoint x forp that we dodo not have tangencies between the stable and unstable manifoldof pand that
forward iterates by T2 of an unstable segment s Wu(p) has diameter bounded away from zero. Then T :Λ Λ
⊂ →
is topologically mixing.
Proof. Observe that by hypothesis we have in particular that closWu(p) = Λ. Let A = and B = be open
subsetsofΛ,i.e.,thereareopensubsets and ofIR2p+1suchthatA= ΛandB= 6 ∅Λ. Wewi6llp∅rovethat
A B A∩ B∩
thereexistsn suchthatforalln n wehaveTn(A) B= thusprovingthatT istopologically mixing. Since
0 0
≥ ∩ 6 ∅
Wu(p) is densein Λthere is n2 >0 such that Tn2(x0) . ThusWu(p) cuts in an arc s containing Tn2(x0).
Since orbit(x0) is dense in Λ there exists n1 > n2 such∈tAhat Tn1−n2(x0) U(Ap) where U(p) is a neighborhood
of p in which we may assume that we have C1-linearizing coordinates, an∈d Tn1−n2(s) contains an arc J which
intersects transversally Ws (p), thisfollows from theassumptions we havedone. By theInclination Lemma, see
loc
[PM, Chapter 2, 7], Tn(J) C1-approaches on compact segments of Wu(p). Let ν > 0 be the radius of a ball
contained in . T§here is n0 >n1 such that Tn0(J) is ν/2-dense in Λ and hence Tn(J) is ν/2-dense in Λ for all
B
n>n . Thus Tn(J) cuts implying that Tn( ) = for n n . But since Wu(p) Λ (Λ is an attractor)
0 0
we conclude that Tn(A) BB= for n n proAvin∩gBth6at∅T is top≥ologically mixing. ⊂
0
∩ 6 ∅ ≥
Remark 4.2. Roughly speaking the above result means that for the parameter values A = 0.18, ρ = 0.30 and
0
γ =8.25, from the topological viewpoint we have that all possible states (N ,N ...,N ) Λ are visited and so a
0 1 2p
∈
chaotic behavior should beexpected. Onthe other hand, sincethere are fixed points likepinΛif(N ,N ...,N )
0 1 2p
is very near p in practice we will see the same behavior for large periods of time seeming that the population of
these rodents is in equilibria. On the other hand the hypothesis we have assumed seems to be rather strong.
Anotherconsequenceofthedensityoftheunstablemanifold ofpinΛisthefollowing (seealsoRemark6.1).
Proposition 4.3. If clos(Wu(p))=Λ then T2 :Λ Λ is injective.
|Λ →
Proof. Indeed, T2 is injective when restricted to Wu(p), for, if it were not true, there would exist x,y Wu(p)
∈
such that T2(x) = T2(y). But, since T2(p) = p it holds that Wu(p) = ∪n∈INT2(Wεu(p)) where Wεu(p) is the
ε-local-unstable manifold of p. Thus there is N > 0 such that x,y T2N(Wu(p)) and, hence, there is an arc
γ Wu(p) with end points x and y. Applying T2 to γ we find a clo∈sed loopεT2(γ) contained in Wu(p) which
co⊂ntradicts the fact that Wu(p) is homeomorphic to IR.
Assume now that there are x,y Λ such that T2(x) = T2(y). Since T2 is locally injective there is r > 0
1
such that y / B(x,r ) where T2 ∈: B(x,r ) H is a homeomorphism. There exists also r > 0 such that
∈ 1 |B(x,r1) 1 → 2
T2 : B(y,r ) H is a homeomorphism. Hence we may find V(x) B(x,r ) a neighborhood of x and
|B(y,r2) 2 → ⊂ 1
V(y) B(y,r ) a neighborhood of y such that T2(V(x))=T2(V(y)) . Since, by assumption, Wu(p) is dense in
2
Λ, the⊂re is an arc γ Wu(p) such that has its end points x′ V(x) and y′ V(y) such that T2(x′) = T2(y′)
contradicting that W⊂u(p) is homeomorphic to IR. ∈ ∈
2We thankEnriquePujals for fruitful discussions on this topic.
10