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 Abstract—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. I. K´evin Perrot LIP (UMR 5668 School of Applied Mathematics, CNRS-ENS de Lyon-UCBL1), and Informatics, 46 alle dItalie 69364 Hanoi University of Science Lyon Cedex 07 - France. and Technology, Email: kevin.perrot@ens-lyon.fr Hanoi, Vietnam. Email: doanh.nguyenngoc@hust.vn I NTRODUCTION An important issue in ecology is to understand the effects of the tactics that individuals may adopt at the population and community levels. Individuals migrate because the food is limited, they compete 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 relationship between migration of species individuals among multizone environment and species distribution. One of the most common and simple theoretical 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 factor 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. II. zone 1 zone 2 nB1 nA2 nB2 nA3 nC1 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. P RESENTATION OF THE MODEL nA1 the densities of the two other species in this zone. Definition 3. A stable configuration is a configuration such that its density matrix does not change over the time. nC2 III. nB3 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 nC3 zone 3 Fig. 1. Three species S = {A, B, C} and three zones Z = {1, 2, 3}. nXi is the density of species X in zone i, X ∈ 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. Initial configuration A B (0.2) (0.5) The system evolves in discrete time and continuous 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 distribution 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 : i∈Z nXi (t) = 1. Formally, a configuration is determined by its density matrix: nA1 nB1 nC1 nA2 nB2 nC2 nA3 nB3 nC3 (t) (1) If there is no ambiguity, we will usually omit the dependence on the time t and simply refer to a notation 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 dominates a zone if its density is strictly greater than ⇒ Stable configuration (0.8) C n(t) = S IMULATION zone 1 A(0.1) B (0.6) C (0.1) zone 2 A(0.1) B (0.45) C A B C (1) (1) (1) zone 1 zone 2 zone 3 (0.4) zone 3 A(0.2) A (0.5) A (0.3) B (0.8) B A (0.6) (1) A B (0.1) C (0.4) B (0.2) C (0.3) (0.3) (1/3) (1/3) (1/3) zone 1 zone 2 zone 3 zone 1 zone 2 zone 3 A(0.1) B (1) C C C (0.4) C C A (0.3) A (0.6) B (0.5) A B (0.55) (0.4) B (0.45) B (0.1) C C (0.3) (0.4) C (0.3) (1/3) (1/3) (1/3) zone 1 zone 2 zone 3 zone 1 zone 2 zone 3 C C 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. IV. M AIN RESULTS 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 follows: Definition 4. Let c and c be two configurations with density matrices n and n , respectively. The distance between the two configurations is defined by d(c, c ) = max {|nXi − nXi |} . X∈S i∈Z Definition 5. Starting from a configuration c(t0 ), we say that the system converges to a stable configuration s if ∀ > 0, ∃ k( ), ∀ k > k( ) : d(c(t0 +k∆t), s) < . Moreover, if k( ) in O log2 1 we say that the systems exponentially converges to s. 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 restriction 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 reference: 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 consider 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 0 so A dominates zone 1. for Xi ∈ {A1, B2, C3} 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∆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 + 1)∆t), s) = d(c(t0 + k∆t), s) < ⇐⇒ k > log2 d(c, s) which concludes the proof. Lemma 2 (One-two). If, during two consecutive 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 = i mY i = 0, and mY j = nY j + n2Y i ,j = i  m = 1 for Z ∈ / {X, Y }, ∀j. Zj 3 where n 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)    0 +∆t) ≥ 12 + 1−nA1 (t 2 nB1 (t0 + 2∆t) = 0    o +∆t) nC1 (t0 + 2∆t) = 1−nC1 (t ≤ 12 . 2 Consider zone 2, after two steps we have:  nA2 (t0 + 2∆t) = nA3 (t0 +∆t) ≤ 1    n (t + 2∆t) = n (t2 + ∆t) 2 B2 0 B2 0 nB1 (t0 +∆t) ≥ 21 +   2  0 +∆t) nC2 (t0 + 2∆t) = 1−nC2 (t ≤ 12 . 2 so B dominates zone 2. • Consider zone 3, after one step, the density of the three species are the following:  nA2 (t0 )   nA3 (t0 + ∆t) = 2 nB3 (t0 + ∆t) = nB3 (t0 ) + nB12(t0 )   n (t + ∆t) = 1−nC3 (t0 ) . C3 0 2 By hypothesis, we know that B dominates zone 3 after one step, then nB3 (t0 ) + nB12(t0 ) is greater that nA22(t0 ) (t0 ) and 1−nC3 . 2 Let us consider now the situation after two steps:   nA3 (t0 + 2∆t) = nA2 (t20 +∆t)     = nA3 (t0 ) < nB3 (t0 )    n (t + 2∆t) = n 4 (t + ∆t) B3 0 B3 0 = n (t0 ) + nB12(t0 )  B3   1−n  C3 (t0 +∆t)  nC3 (t0 + 2∆t) =  2   (t0 ) = 1+nC3 . 4 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 t0 and t0 + ∆t, we have: nB3 (t0 ) > nC3 (t0 ) nB3 (t0 ) + nB12(t0 ) > 1−nC3 (t0 ) , 2 this implies that 4nB3 (t0 ) + nB1 (t0 ) 1 + nC3 (t0 ), then nB3 (t0 + 2∆t) nB3 (t0 ) + nB12(t0 ) > nB3 (t0 ) + nB14(t0 ) 1+nC3 (t0 ) = nC3 (t0 + 2∆t). 4 We can conclude that after two steps, dominates zone 3. > = > B We now prove that the system exponentially converges to the stable configuration s. For the species B, after one step, their individ- nA1 nB1 nC1 zone 1 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) − 13 for i ∈ {1, 2, 3}. The density of C in zone 1 after one step is: nC2 (t0 ) + nC3 (t0 ) 2 1 d2 (t0 ) + d3 (t0 ) = + 3 2 1 d1 (t0 ) . = − 3 2 nC1 (t0 + ∆t) = = −1 2 di (t0 ), (−1)k di (t0 ). 2k It means that di (t0 + ∆t) and more This fact generally di (t0 + k∆t) = implies the exponential convergence for species C. nA2 nB2 nC2 zone 2 nA3 nB3 nC3 zone 3 The first disjunction goes according to the dominant species in zone 2: max{nX2 (t0 )} = X∈S (case 1) nA2 (case 2) nB2 (case 3) nC2 (case 1) We know all the dominancies, therefore we can perform one time step. We picture c(t0 + ∆t) below. nA1 + nA3 2 nB2 2 nC2 +nC3 2 zone 1 nA2 + nA3 2 nB1 2 nC1 +nC3 2 zone 2 0 nB1 +nB2 2 nC1 +nC2 2 nB3 + zone 3 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 a cOE or a cOT . Proof: Let c = c(t0 ) be any configuration. We show that after k steps with k ≥ 2, the configuration 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{nXi } X∈S i∈Z and nB3 = max{nX3 }. X∈S The initial picture, where dominant densities are boxed, is pictured below. Species A dominates zone 1 because nA1 is the maximal density, and nA1 is greater than nA2 which dominated over nB2 at time t0 . The comparison with C uses similar arguments. Analogously, species B dominates zone 3. At this stage, we perform again a case disjunction, according to the dominant species in zone 2: (case 1.1) If A dominates zone 2, i.e. max{nX2 (t0 + ∆t)} = nA2 + nA3 /2. Then we X∈S apply Lemma 2 and deduce that the system converges exponentially to a cOT . (case 1.2) If B dominate zone 2, i.e. max{nX2 (t0 +∆t)} = nB1 /2. This case is imposX∈S sible. At time t0 , nC2 < nA2 and at time t0 + ∆t, nC1 +nC3 < nB1 2 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{nX2 (t0 + ∆t)} = (nC1 + nC3 )/2We apply X∈S 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{nXi } and nB3 = max{nX3 }. X∈S X∈S i∈Z Then the system exponentially converges to a cOT if c satisfies one of the following conditions, otherwise the system exponentially converges to a cOE . 1) nB2 = max{nX2 } 2) nC1 +nC3 and nB2 + nB1 2 > 2 nB1 C2 and nB3 + 2 > nC1 +n 2 nA2 = max{nX2 } and nA2 + V. nA3 2 > ACKNOWLEDGMENT This work was done while the authors were at Vietnam Institute of Advanced Study in Mathematics (VIASM). This work was also partially supported by the project VAST.DLT.01/12-13. R EFERENCES [1] X∈S X∈S (density dependent) migration tactics and distribution 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. [2] nC1 +nC3 2 D ISCUSSION AND C ONCLUSION We have presented a discrete model for migration of individuals of three competing species for territory over three zones. As a first results, from a mathematical point of view, we have distinguished 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 uniform distribution depending on initial conditions. Top of Fig. 2 shows a typical stable configuration where species individuals form a clumped distribution. Below, species A and B form a clumped distribution while species C forms an uniform distribution. 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. [3] [4] [5] [6] [7] [8] [9] [10] The main conclusion that emerges from this study is the existence of a relationship between E. Abdllaoui, P. Auger, B. W. Kooi, R. Bravo de la Parra and R. Mchich. Effects of density-dependent migrations on stability of a two-patch predator-prey model. Mathematical Biosciences, 210(1):335-354, 2007. P. Auger, R. Bravo de la Parra, C. Poggiale, E. Sanchez and T. Nguyen-Huu. 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Behavioral Ecology and Sociobiology. 23 (1), 51-53, 1988.  ... first stable configuration each zone contains only one species (top of Fig 2), in the second one species is of density in one zone, another species stays and dominates in the two other zones, and... exponentially converges to s Definition We call a one-each configuration, denoted by cOE , a configuration such that each zone contains only one species Definition We call a one-two configuration,... of each species are equal to in one zone and are equal to 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,