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HIERARCHICAL HETEROGENITY OF POPULATIONS: MODELING BY THE OPEN

EIGEN HYPERCYCLE

Vasiliy Ye. Belozyorov Serge V. Chernyshenko

Faculty of Applied Mathematics Faculty of Informatics

O. Gonchara Dnipropetrovsk National University University of Koblenz-Landau

49010, Dnipropetrovsk, Ukraine D-56070, Koblenz, Germany

Email: belozvye@mail.ru Email: svc@a-teleport.com

Vsevolod S. Chernyshenko Computer Software Department

National Mining University 49600, Dnipropetrovsk, Ukraine Email: VseVsevolod@hotmail.com

KEYWORDS

System of ordinary quadratic differential equations, asymp­totic stability, biological population.

ABSTRACT

The case of a biological population, which consists of sev­eral sub-populations (different kinds of the population "so­cial" groups: families, bevies, etc.), has been considered. For description of non-trivial interactions between these groups, a model of "open the Eigen hypercycle" has been proposed. Its bifurcation analysis for 3-dimension case has been carried out. Ecological interpretation of the results has been discussed.

INTRODUCTION

A huge number of mathematical models of ecological pop­ulation structure (for example, (Allen 1974; Austin 1990; Billik and Case 1994; Chernyshenko 1995; Maurer 1999; Williamson 1990)) are designed for the description of pop­ulation dynamics while taking into account different kinds of physiological differences between specimens (of age, sex, size, etc.) At the same time it is known (Breder 1959; Urich 1938) another form of population heterogeneity, based on the "social" population structure.Contrary to sex difference, specimens can change their group membership (though this changes are not as mechanistic as change of age). The main feature of social groups is an hierarchical character of the "social structure" and dependence of the existance of higher groups on the proper functionality of the lower ones. Big populations of relatively "intellectual" animals form new levels of population organization. The most bright example is the row "specimen - family - bevy" and, correspondingly, different kind of specimens (ordinary specimen, head of family, bevy leader) (Manteyfel 1992).

In the famous hypercycle model, proposed by Eigen and Shuster (Eigen and Schuster 1979; May 1991), similar re­lationship between system elements is described. At the same time, relations between "social groups" are not cyclic, and worsening of life conditions of the populations leads to elimination of higher levels while keeping lower ones. That gives an idea to use an open modification of the hypercy-cle model, which was proposed initially for the descrip­tion of ecological successions and some social processes (Sole and Bascompte 2006; Ellner and Rees 2006).

The open hypercycle model is based on a matrix rep­resentation, which is non-linear (it is natural for a model of "social self-organization" of the population). In the ar­ticle we focus on a continuous case, although the results may be used for discrete versions also. The model shows dependence of a complexity of population "social struc­ture" from the size of its niche. Population "chooses" its complexity (or dimension) itself. We show it exactly for 3-dimension case (as well as for 2-dimension ear­lier (Chernyshenko 2005); generalization for N-dimension case is matter of further research.

1. OPEN HYPERCYCLE MODEL.

Let's consider the dynamic behavior of the heterogeneous biological population x(t) = (xi(t),xn(t))T, which describing by Eigen's model hypercycle (here and further symbol T signifies transposition)

(t) = xi(t)( Fi(t) - S_£xj(t)

x 1

So

(1)

b n(t) = xn(t)[ Fn(t) - so £ xj (t)Fj (t)

(here S0 > 0). The population consists of n subpopu-lations xj(t), i = 1, ...,n,, which represent different lev­els of "social" hierarchy. The vector of initial values is xT(0) = (xio, ...,xno).

We will consider that Fi(t) = N - xi(t),Fi(t) = ai-ixj_i(t) - xi(t); i = 2, ...,n, where N, ai,an_i are positive numbers. (The functions Fi (t), ... , Fn(t) are called as Allen's functions (Allen 1974).)

These functions determine very special interaction be­tween the sub-populations, whene each of them depends on all the previous ones. Contrary to Eigen's hypercycle, the dependence has no cyclic character, so we can call this model as 'open' hypercycle.

2. THE EQUILIBRIUM POINTS OF THE SYSTEM (1).

Let's investigate analytically main features of the model dynamics.

Let's define as ii,ik (k < n) permutations of any k symbols 1, 2,n, for which the condition

1 < ii < ... < ifc < n

2.1. Determinant of the first k equations of the system (1) is not equal to zero.

In this case, according to (2) we get that Fi1 = ... = Fik = xifc+1 = ... = xin = 0. Condition Fi1 = 0 leads to xi = 0, then xi = N, x2 = aiN, x3 = a^N, ...,xn = ai ... an-iN; in this case the condition

So = N(1 + ai + aia2 + ... + ai ... an_i)

has to be fulfilled.

Let xi =0, x2 = 0. Then x3 = ... = xn = 0. Assume

xi = 0, F2 = 0. Then xi = N, x2 = aiN, x3 = ... = xn = 0.

It is easy to find out that in this case we have n + 1 equi­libriums:

0

0

0

fc + i

1

ai

ai • ... • afc_i

0

N, .

is fulfilled.

Assume that xi1 = 0, ...,xik = 0 and xik+1 = 0, ...,xin = 0. Then the system of equations that define equilibriums has the form

1-

xi1

so

So

V So

Sxo

1   _ 2

So

so

xifc \

"So X

xifc

S

o

1

xifc

So /

Fi1

Fi2

xifc + 1 ... xin 0.

= 0,

(2)

A determinant of the first k equations of system (2) may be presented as:

1

det

xi1

so

So

Sxo

So

So

So

1

So X

xifc

So

xifc

So /

=1

xi1 + ... + xic

so

Here we may face two cases: 1)1 - xi1 + " + xik = 0

and2)1 -xi1 +::.+xik =0.

So

So

En+i

1

ai

\ ai ... an_i /

N є Rn.

2.2. Determinant of the first k equations of the system (1) is equal to zero.

Here, we have xik = So - xi1 - ... - xik1. Substituting the last formula in the system (2) we derive Fi1 = Fi2 = ... = Fic = F, where F is a nonzero function. Taking into account the last equations, system (2) may be presented as

xi1 ( ai1 _ i xi1 _ i - xi1 - F) = 0,

xifc (aifc_ixifc_i      xifc      F) 0,

xi + ... + xn = So,

xic+1 = 0,

(3)

0,

where F = Fifc+1 ,...,F = Fi„.

From this system we derive that xi1 = 0, ... , xic = 0. As k =1,2,n - 1, we get Cn + Cn + ... + Cn_i + C™ solutions. Here Cn is a number of combinations of k sets from n elements (k < n).

Taking into account the case when determinant of the rst k equations of the system (1) is not equal to zero, we get for this system (1) n + 1 + Cn + ... + C^1 + 1 = 2n + n equilibriums.

0

2.3. Jacobian matrix building.

The Jacobi matrix for arbitrary n can be evaluated as J = A + B + C, where:

A =

\

(Indefinite form of function F doesn't affect the present analysis.)

So, for the system (1) where n = 3 we got following 11 equilibriums:

Ei :

xi = 0, x2 = 0, x3 = 0.

B =

0 \

0

( -xi      0      0 •••

X201     X2       0

V    0        0        xnan_i    xn )

E2 :

E3 :

n           So (a2 + 1)So

xi = 0,x2 =- ,x3 =

a2 + 2

a2 + 2

So + N So - N

xi =---,x2 = 0,x3 =---.

C = ­

So

/ xi (N 2xi + £11X2)

V xn (N 2xi + 01x2)

xi(dn-i xn_i 2xn)

xn(an—ixn—i 2xn)

\    E4 :

So + N (ai + 1)So - N

xi = --ТГ, x2 = -—Z-, x3 = 0.

ai + 2 ai + 2

3. EIGENVALUES OF JACOBI MATRIX IN EQUI­LIBRIUM POINTS FOR n = 3.

Let n = 3 and Fi(t) = N - xi (t), F2(t) = aixi(t) -x2(t),F3(t) = a2x2(t) - x3(t), where N > 0, ai > 0, a2 > 0.

Here we come to two cases: 1)xi + x2 + x3 = So and 2)xi + x2 + x3 = So.

In the first case (xi + x2 + x3 = So) we have four equi­libriums:

(0,0, 0)T, (N, 0,0)T, (N, aiN, 0)T, (N, aiN, aia2N)T.

In the second case (xi + x2 + x3 = So) we have seven systems of equations, watch (3), to determine additional seven equilibriums:

aixi - x2 - F = 0 a2 x2 - x3 - F = 0

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V Belozyorov - Hierarchical heterogenity of populations modeling by the openeigen hypercycle