Effects of Migration of Three Competing Species on Their Distributions in Multizone Environment

—In this paper, we investigate the relationship between migration and species distribution in multizone environment. We present a discrete model for migration of three competing species over three zones. We prove that the migration tactics of species leads to the fact that the system exponentially converges to one of two typical configurations: the first one is a case where each zone contains only one species, the second one is a case where one species is of density 1 in one zone, another species stays and dominates in the two other zones, and the last species is evenly split into the 3 zones with a density one third in each. We also show a characterization of the initial conditions under which the system converges to one of the two configurations.

Effects of Migration of Three Competing

Species on Their Distributions in

Multizone Environment

Phan Thi Ha Duong

Institute of Mathematics,

Vietnam Academy of Science

and Technology,

Hanoi, Vietnam

Email: phanhaduong@math.ac.vn

Doanh Nguyen-Ngoc School of Applied Mathematics, and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam

Email: doanh.nguyenngoc@hust.vn

K´evin Perrot LIP (UMR 5668 CNRS-ENS de Lyon-UCBL1),

46 alle dItalie 69364 Lyon Cedex 07 - France Email: kevin.perrot@ens-lyon.fr

Abstract—In this paper, we investigate the

rela-tionship between migration and species distribution

in multizone environment We present a discrete

model for migration of three competing species over

three zones We prove that the migration tactics of

species leads to the fact that the system exponentially

converges to one of two typical configurations: the

first one is a case where each zone contains only one

species, the second one is a case where one species is

of density 1 in one zone, another species stays and

dominates in the two other zones, and the last species

is evenly split into the 3 zones with a density one third

in each We also show a characterization of the initial

conditions under which the system converges to one

of the two configurations

An important issue in ecology is to understand

the effects of the tactics that individuals may adopt

at the population and community levels

Individu-als migrate because the food is limited, they

com-pete with others, environmental conditions are not

good for them (weather, natural calamity, ) and so

on This leads to various portraits of distribution of

species in environment

There was also a lot of interest in the

rela-tionship between migration of species individuals

among multizone environment and species

distribu-tion One of the most common and simple

theoreti-cal explanation for effects of individuals’ migration

on the species distribution is ideal free distribution (IFD) theory The theory states that the number

of individuals that will aggregate (or else clump)

in various zones is proportional to the amount of resources available in each For example, if zone

1 contains twice as many resources as zone 2, there will be twice as many individuals foraging

in zone 1 as in zone 2 The IFD theory predicts that the distribution of individuals among zones will minimize resource competition and maximize fitness ( [4], [5], [6], [10], [7])

Some recent investigations studied another fac-tor leading to individuals’ migration and also showed the link between the migration and the species distribution over multizone environment (see for examples in [2], [8], [2], [9] The authors showed that interaction between species leads to migration of individuals and therefore to species distribution These investigations, however, take into account only two species and two zones The main reason is that a model with more than two species and two zones is much more complex and less tractable The aim of this work is to follow this approach by taking into account three competing species for territory among three zones The raised question is “what is the stable distribution of the three species among three zones?” There is

no simple answer to this question We show in this paper that it depends on migration tactics of

individuals as well as the initial distribution of

species

The paper is organized as follows Section

II is dedicated to the presentation of the model

Section III shows simulations of the model: typical

examples and remarks Thereafter, in section IV,

we present the main results Finally, section V is

about discussion and conclusion

n C1

n B1

n A1

zone 1

n C2

n B2

n A2

zone 2

n C3

n B3

n A3

zone 3

S, i ∈ Z In the figure: density of species A in black, density

of species B in grey while density of species C in white over

three zones.

The system evolves in discrete time and

con-tinuous space We consider the case of 3 species

S = {A, B, C} and 3 zones Z = {1, 2, 3} (see

Fig 1) We call configuration at time t a

distribu-tion of the individuals of each species into the 3

zones, composed of a density nXi(t) of individuals

of species X in zone i at time t, for example

nA1(t) is the density of individuals of species A

is zone 1 at time t, such that for every species

X : P

i∈ZnXi(t) = 1 Formally, a configuration

is determined by its density matrix:

n(t) =

nA1 nA2 nA3

nB1 nB2 nB3

nC1 nC2 nC3

!

If there is no ambiguity, we will usually omit the

dependence on the time t and simply refer to a

no-tation n instead of n(t) The set of configurations

is denoted by C

To describe the dynamics of the system, we are

going to introduce some definitions as follows:

Definition 1 In a configuration, a species

domi-nates a zone if its density is strictly greater than

the densities of the two other species in this zone Definition 2 The evolution rule is that if a species dominates a zone at time t then those individuals stay into this zone in the next time step (t + ∆t), and if a species does not dominate a zone then in the next time step half of them move into each of the two other zones

For a configuration c(t) at time t, we denote

by c(t + k∆t) the configuration obtained from c(t) after k time steps

Definition 3 A stable configuration is a configu-ration such that its density matrix does not change over the time

A lot of simulations were done We present here three of them showing the typical stable configurations that appear Top of Fig 2 shows the

Initial configuration ⇒ Stable configuration

A

(0.8)

B (0.2)

C

(0.5) zone 1

A (0.1)

B

(0.6)

C (0.1) zone 2

A (0.1)

B (0.45)

C

(0.4) zone 3 zone 1

A

(1)

zone 2

B

(1)

zone 3

C

(1)

A

(0.5)

B (0.1)

C

(0.4) zone 1

A

(0.3)

B (0.2)

C

(0.3) zone 2

A (0.2)

B

(0.8)

C

(0.3) zone 3

A

(0.6)

C

(1/3) zone 1

A

(0.4)

C

(1/3) zone 2

B

(1)

C

(1/3) zone 3

A

(0.6)

B (0.1)

C

(0.3) zone 1

A

(0.3)

B

(0.5)

C

(0.4) zone 2

A (0.1)

B

(0.4)

C

(0.3) zone 3

A

(1)

C

(1/3) zone 1

B

(0.55)

C

(1/3) zone 2

B

(0.45)

C

(1/3) zone 3

Fig 2 Three typical stable configurations Left panel is about initial configurations Right panel is about the corresponding stable configurations.

case where densities of each species are equal to 1

in one zone and are equal to 0 in the others zones

In the middle of Fig 2, species A is in two zones and dominates both, species B is only in one zone where it dominates, while species C is evenly split into the three zones At bottom of Fig 2, species

A is only in one zone where it dominates, species

B is in the two others zones and dominates both, while species C is in evenly split into the three

zones We have the following remarks from the

above simulations:

Remark 1 There are three remarks as follows:

(1) there are two typical stable configurations: in

the first stable configuration each zone contains

only one species (top of Fig 2), in the second

one species is of density 1 in one zone, another

species stays and dominates in the two other zones,

and the last species is evenly split into the 3

zones with a density 1/3 in each (bottom of Fig

2); (2) the system converges rapidly to the stable

configurations;(3) it is not easy to figure out under

which conditions the system converges to one of the

two above typical configurations

The next section is a formal analysis of these

remarks

We begin in subsection IV-A by explaining that

we can discard the cases of equality in our study,

without changing the results we obtain about the

dynamic of the system, by proving that cases of

equality almost never happen Then we describe the

two typical dynamics of the system in subsection

IV-B Finally, subsection IV-C is devoted to the

study of the dynamics of the system according to

the initial configuration

Firstly, we introduce some definitions as

fol-lows:

Definition 4 Let c and c0 be two configurations

with density matrices n and n0, respectively The

distance between the two configurations is defined

by

d(c, c0) = max

X∈S

i∈Z

{|nXi− n0Xi|}

Definition 5 Starting from a configuration c(t0),

we say that the system converges to a stable

configurations if

∀  > 0, ∃ k(), ∀ k > k() : d(c(t0+k∆t), s) < 

Moreover, ifk() in O log2 1

we say that the

systems exponentially converges tos

Definition 6 We call a one-each configuration,

denoted by cOE, a configuration such that each

zone contains only one species

Definition 7 We call a one-two configuration, denoted by cOT, a configuration such that one species is of density 1 in one zone, another species stays and dominates in the two other zones, and the last species is evenly split into the 3 zones with a density 1/3 in each

A Ignoring cases of equality

We denote C∗the set of configurations such that there is a case of equality between the densities of two species competing for dominancy in a zone Formally,

C∗= {c ∈ C | ∃ X, Y, i : nXi= nY i} where n is the density matrix of c

Intuitively, if we consider the set C which

is uncountable (continuous space) then a case of equality in C∗somehow corresponds to the restric-tion of an uncountably large degree of liberty to a countable one, hence the following result holds Theorem 1 |C|C|∗| = 0

We will apply this result without explicit refer-ence: when comparing densities of two competing species it allows to convert an inequality into a strict inequality

B Two typical behaviors Now, we are going to show the two lemmas about cOE and cOT

Lemma 1 (One each) From a configuration c = c(t0) such that each species X dominates exactly one zone i, the system exponentially converges

to the stable configuration where the density of species X in zone i is 1

Proof: Without loss of generality, let us con-sider a configuration such that A dominates zone

1, B dominates zone 2 and C dominates zone 3 First of all, we can notice that the repartition of dominancy will never change since nXi(t + ∆t) ≥

1

2 if and only if species X dominates zone i at time

t (recall Theorem 1)

We now prove that the system exponentially converges to the stable configuration s of density

matrix m such that

mXi=1 for Xi ∈ {A1, B2, C3}

0 otherwise

We can notice that

d(c(t0+k∆t), s) = 1− min

Xi∈{A1,B2,C3}nXi(t0+ k∆t) since the difference is at least as important for

species A (resp B, C) in zone 1 (resp 2, 3) than

in other zones According to the repartition of

dominancy, we have

d(c(t0+ (k + 1)∆t), s) = d(c(t0+ k∆t), s)

2 because half of the individuals in a zone where

they are not dominant move to their dominant zone

Consequently, for all  > 0, we have

d(c(t0+ k∆t), s) <  ⇐⇒ k > log2 d(c, s)





which concludes the proof

Lemma 2 (One-two) If, during two

consecu-tive configurations c = c(t0) and c(t0 + ∆t), a

species X dominates zone i and another species

Y dominates the two other zones, then the system

exponentially converges to the stable configuration

s of density matrix m, defined as follows:







mXi= 1, and mXj = 0,j 6= i

mY i= 0, and mY j= nY j+nY i

2 ,j 6= i

mZj= 13 forZ /∈ {X, Y }, ∀j

wheren is the density matrix of c

Proof:Without loss of generality, we consider

that A dominates zone 1 and B dominates zone 2

and zone 3 and that nB2> nB3 (let us denote this

property by (*)) We first prove that the property

(*) keeps satisfied during the evolution For that,

it is sufficient to prove that at time t0+ 2∆t, (*)

is still satisfied, which means that if (*) is true for

two consecutive steps then it is true for the third

step, so it is true for all steps

• Consider zone 1, after two steps we have:







nA1(t0+ 2∆t) = nA1(t0+ ∆t)

+1−nA1 (t 0 +∆t)

2

nB1(t0+ 2∆t) = 0

nC1(t0+ 2∆t) = 1−nC1 (t o +∆t)

≤1

so A dominates zone 1

• Consider zone 2, after two steps we have:







nA2(t0+ 2∆t) = nA3 (t 0 +∆t)

2

nB2(t0+ 2∆t) = nB2(t0+ ∆t)

+nB1 (t 0 +∆t)

2

nC2(t0+ 2∆t) = 1−nC2 (t 0 +∆t)

2

so B dominates zone 2

• Consider zone 3, after one step, the density

of the three species are the following:







nA3(t0+ ∆t) = nA2 (t 0 )

2

nB3(t0+ ∆t) = nB3(t0) +nB1 (t 0 )

2

nC3(t0+ ∆t) = 1−nC3 (t 0 )

By hypothesis, we know that B dom-inates zone 3 after one step, then

nB3(t0) + nB1 (t 0 )

2 is greater that nA2 (t 0 )

2

and 1−nC3 (t 0 )

Let us consider now the situation after two steps:















nA3(t0+ 2∆t) = nA2 (t 0 +∆t)

2

= nA3 (t 0 )

4 < nB3(t0)

nB3(t0+ 2∆t) = nB3(t0+ ∆t)

= nB3(t0) +nB1 (t 0 )

2

nC3(t0+ 2∆t) = 1−nC3 (t 0 +∆t)

2

= 1+nC3 (t 0 )

We will now prove that B still dominates zone 3 at this step, that means nB3(t0+ 2∆t) > nC3(t0+ 2∆t) In fact, from the hypothesis that B dominates zone 3 at time

t0and t0+ ∆t, we have:

 nB3(t0) > nC3(t0)

nB3(t0) +nB1 (t 0 )

2 >1−nC3 (t 0 )

this implies that 4nB3(t0) + nB1(t0) >

1 + nC3(t0), then nB3(t0 + 2∆t) =

nB3(t0) +nB1 (t 0 )

2 > nB3(t0) +nB1 (t 0 )

4 >

1+n C3 (t 0 )

4 = nC3(t0+ 2∆t)

We can conclude that after two steps, B dominates zone 3

We now prove that the system exponentially converges to the stable configuration s

For the species B, after one step, their

individ-uals do not move any more For the species A, we

can apply the same argument as in Lemma 1 to

prove the exponential convergence

We will now prove that the density of species

C in each zone exponentially converges to 13 Let

us denote by di(t) the different nCi(t) − 1

3 for

i ∈ {1, 2, 3} The density of C in zone 1 after one

step is:

nC1(t0+ ∆t) = nC2(t0) + nC3(t0)

2

= 1

3 +

d2(t0) + d3(t0) 2

= 1

3 −d1(t0)

2 .

It means that di(t0+ ∆t) = −12 di(t0), and more

generally di(t0+ k∆t) = (−1)2kkdi(t0) This fact

implies the exponential convergence for species C

The proof of this Lemma is then completed

C Dynamics of the system

The following theorem is about the portrait of

the dynamics The theorem proves the first and

second remark of the previous section

Theorem 2 Beginning from any configuration, the

system always converges exponentially to acOEor

acOT

Proof: Let c = c(t0) be any configuration

We show that after k steps with k ≥ 2, the

con-figuration c(t0+ k∆t) will satisfies the condition

of Lemma 1 or Lemma 2, then applying those

Lemmas, one can deduce the statement of this

theorem

To do that, we will check every possible case,

in many cases the proofs are similar We perform a

case disjunction according to the dominant species

in each zone The density of one species in a zone

has to be greater than any other one Furthermore,

no species can dominate all of the three zones

Without loss of generality, we consider that

nA1= max

X∈S

i∈Z

{nXi} and nB3= max

X∈S{nX3}

The initial picture, where dominant densities are

boxed, is pictured below

zone 1 zone 2 zone 3 The first disjunction goes according to the dominant species in zone 2:

(case 1) (case 2) (case 3) max

X∈S{nX2(t0)} = nA2 nB2 nC2 (case 1) We know all the dominancies, therefore we can perform one time step We picture c(t0+ ∆t) below

nA1+nA3

n A3

n B2

2

n B1

2 nB3+nB1 +n B2

2

n C2 +n C3

2

n C1 +n C3

2

n C1 +n C2

2

Species A dominates zone 1 because nA1 is the maximal density, and nA1 is greater than nA2 which dominated over nB2at time t0 The compar-ison with C uses similar arguments Analogously, species B dominates zone 3

At this stage, we perform again a case disjunc-tion, according to the dominant species in zone 2: (case 1.1) If A dominates zone 2, i.e max

X∈S{nX2(t0+ ∆t)} = nA2+ nA3/2 Then we apply Lemma 2 and deduce that the system con-verges exponentially to a cOT

(case 1.2) If B dominate zone 2, i.e max

X∈S{nX2(t0+∆t)} = nB1/2 This case is impos-sible At time t0, nC2 < nA2 and at time t0+ ∆t,

n C1 +n C3

2 < nB1

2 , then 1 = nC1+ nC2+ nC3 <

nB1 + nA2 which implies that nB1 > nA1, a contradiction with the maximality of nA1 (case 1.3) If C dominates zone 3, i.e max

X∈S{nX2(t0+ ∆t)} = (nC1+ nC3)/2We apply Lemma 1 and deduce that the system exponentially converges to a cOE

(case 2) Analogously, in this case, the system always converges exponentially to either a cOE or

a cOT

(case 3) We apply Lemma 1 and deduce

that the system exponentially converges to a cOE

The following theorem shows characterization

of the cases when the system converges to a cOE

(resp cOT)

Theorem 3 Let c be a configuration Without loss

of generality, one can suppose that

nA1= max

X∈S

i∈Z

{nXi} and nB3= max

X∈S{nX3}

Then the system exponentially converges to acOT

if c satisfies one of the following conditions,

other-wise the system exponentially converges to acOE

1) nB2 = max

X∈S{nX2} and nB2+nB1

2 >nC1 +n C3

2

and nB3+nB1

2 >nC1 +n C2

2

2) nA2= max

X∈S{nX2} and nA2+nA3

2 >nC1 +n C3

2

We have presented a discrete model for

mi-gration of individuals of three competing species

for territory over three zones As a first results,

from a mathematical point of view, we have

distin-guished two typical stable configurations: cOE and

cOT From an ecological point of view, we could

take into account two possibilities concerning the

species distribution: clumped distribution and

uni-form distribution depending on initial conditions

Top of Fig 2 shows a typical stable

configura-tion where species individuals form a clumped

dis-tribution Below, species A and B form a clumped

distribution while species C forms an uniform

dis-tribution However, there are differences between

the two cases In the middle of Fig 2, species A

forms a clumped distribution over two zones while

species B forms a clumped distribution only in

the other At bottom of Fig 2, species A forms

a clumped distribution only in one zone while

species B forms a clumped distribution over the

two other zones

The main conclusion that emerges from this

study is the existence of a relationship between

(density dependent) migration tactics and distribu-tion of species over the three zones In this study,

we just consider three species and three zones It would also be very interesting to take into account

of four (or in general n) species and four (or in general n) zones (n > 4) This would lead to

a more complicated model and less tractable that would be interesting to investigate in future work

This work was done while the authors were

at Vietnam Institute of Advanced Study in Math-ematics (VIASM) This work was also partially supported by the project VAST.DLT.01/12-13

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