Kolmogorov system with explicit hyperbolic limit cycle Текст научной статьи по специальности «Математика»

YflK 517.5

Kolmogorov System with Explicit Hyperbolic Limit Cycle

A class of Kolmogorov differential system is introduced.We show that under suitable assumptions on parameters, an algebraic hyprbolic limit cycle can occur, the explicit expression of this limit cycle is given.

Keywords: Kolmogorov differential system, Invariant curve, Singular point, Periodic solution, Algebraic limit cycle.

DOI: 10.17516/1997-1397-2017-10-2-216-222.

Introduction

The analysis of the existence, number and stability of limit cycles of the non linear differential system:

where P and Q are polynomials, has long been a topic of interests.

Many mathematical models in biology science and population dynamics, frequently involve the systems of ordinary differential equations having the form

where x(t) and y(t) represent the population density of two species at time t, and F(x,y) , G(x, y) are the capita growth rate of each specie, usually, such systems are called Kolmogorov systems.

Kolmogorov models are widely used in ecology to describe the interaction between two populations, and a limit cycle corresponds to an equilibrium state of the system.

When F(x,y) and G(x,y) are polynomials of degrees > 2, limit cycles can occur and there is an extensive literature dealing with their existence, number and stability (see for instance May [13], Lloyd, Pearson,Saez and Szanto [11,12], Huang [8], Huang, Wang, Cheng [9], Huang, Zhu [10], Boqian and Demeng [4], Cheng [5], and references therein), but to our knowledge, the

* salahben21@yahoo.com 1 bendjeddou@univ-setif.dz © Siberian Federal University. All rights reserved

Salah Benyoucef*

Department of Mathematics Faculty of sciences University Badji Mokhtar-Annaba

Algeria

Ahmed BendjeddoU

University of Setif 1, department of Mathematics, 19000

Algeria

Received 07.08.2016, received in revised form 10.01.2017, accepted 06.02.2017

(1)

(2)

exact analytic expressions of the limit cycles for a given kolmogorov system is still unknown except in simplest and specific cases.

This paper is a contribution in that direction, motivated by the recent publication of some research papers exhibiting planar polynomial systems with one or more algebraic limit cycles analytically given (see for instance Bendjeddou and Cheurfa [1,2], Benyoucef, Bendjeddou [3], Chengbin, Boqian [4], Peng Yue-hui [15] and references therein), we will prove the existence of a limit cycle of a class of Kolmogorov system, and give its explicit form.

1. Some useful notions

Let us recall some useful notions.

For U G R [x,y], the algebraic curve U = 0 is called an invariant curve of the polynomial system (2), if for some polynomial K G R [x, y] called the cofactor of the algebraic curve, we have

dU dU

xF (x,y) — + yG(x,y) dy = KU. (3)

The curve r = {(x,y) G R2 : U(x,y) = 0} is non-singular of system (2) if the equilibrium points of the system that satisfy

f xF(x,y) = °

\yG(x,y)=0 ()

are not contained on the curve r.

A limit cycle 7 = {(x (t) ,y (t)) ,t G [0,T]} , is a T-periodic solution of system (2), isolated with respect to all other possible periodic solutions of the system.

Let y (t) be periodic orbit of system(2) of period T, then 7 is an hyperbolic limit cycle if

rT

/ div(7)dt is different from zero. J 0

We construct here a multi-parameter Kolmogorov system admitting a limit cycle if some conditions on the parameters are satisfied.

2. The main result

As a main result, we have the following theorem, Theorem 1. The polynomial differential system

( x = x((axn+1 + bxn + x(cym + dym-1 + fe)) - (x + y) (mcym + d (m - 1) ym-1)) , 1 y = y ((y(axn + bxn-1 + h)+ cym+1 + dym) + (x + y) (naxn + b(n - 1)xn-1)) , where a, c are positive real, b, d are negative real, and h satisfied

(5)

< (-1)" ()""' (n)' + <-1) (^P (mr » > > 2

(6)

admits an hyperbolic limit cycle in realistic quadrant.The limit cycle is represented by the closed trajectory of the curve T.

r= {(x,y) G R2 : axn + bx"- + cy) + dy)-i + h = 0, (n > 2, m > 2)} . (7)

Proof. We will prove that r is composed of closed trajectory, it is nonsingular and an invariant

rT

curve of system (5), and / div(r)dt = 0 (see for instance Perko [14]).

J 0

i) The curve r is composed of closed trajectory. We consider U(x, y) as a function

fx(y) = cym + dym—1 + axn + bxn— 1 + h, where x is a real parameter, a, c positive real, b, d negative real, -j^ = ym—2 (dm — d + cmy)

dfx

dy

_d (mm_1)

= 0 ^ y = 0 or y =-i-, fx(0) = axn + bxn— 1 + h, and

dy

fx( )=(—ir-i(^y~ (

cm c

m—l/j \ m

— ) + axn + bxn—1 + h.

m

—d — 1)

Let p(x) = fx(0), and q(x) = fx(-). We distinguish the following cases.

1. Numbers m and n are even.

Function y ^ fx(y) is decreasing for y e

d( m 1) —---,

,

— d( m — 1)

and increasing for y e

If p(x) is positive and q(x) is negative, by applying the theorem of intermediate values we conclude that the equation fx(y) = 0 admits two solutions,

yi(x) e

0,

— d( m — 1)

and y2 (x) e

— d( m — 1)

,

dp

Function — = xn (b(n — 1) + anx), x ^ p(x) is decreasing for x e

dx

creasing for x e

—b(n — 1)

,

—b(n — 1) an

'—b(n — 1)'

and in-

, for p(x) to be positive it is sufficient that p (——— ) > 0,

an

(—^\=—1)n-ii 2—1)7 M"+h,

an a n

(—t—H) > 0 „ h > —Wn—±)n—1f i)\

an a n

Function = xn 2 (b(n — 1) + anx), x ^ q(x) is decreasing for x e dx

,

—b(n — 1)

and

increasing for x e

—b(n — 1)

If q(0) > 0 and q —b(n — 1)

—b(n — 1)

< 0, then there exist x1 e

—b(n — 1)

and x2 e

that satisfied q(x1) = q(x2) = 0, and V x e ]x1 ,x2[ q(x) < 0,

q(0) > 0«(—1)m—^ m + h> 0«h> (—1)^ (my, q(—) <0• h< (—1)m (m^)m—1 (m)m + —1 )n ()n—1 (n)n

cm

cm

cm

cm

p

p

an

0

an

an

an

We conclude that if h satisfies the relationship (6) then for x £ [x1,x2], there exist y1 (x) £

—d(m — 1)

and y2 (x) £

yi(x2) = y2(x2) =

— d( m — 1)

— d( m — 1)

that satisfied yy(xy) = y2(xy)

— d( m — 1)

and Vx £ ]xy,x2[ yy(x) = y2(x).

Then r is composed by closed trajectory in first quadrant, more precisely in domain

D = <(x,y) £ R2, xy < x < x2, yy( —--) < y < y2(—--)

j(x,i

.

x £

2. Number m is even and n is odd number. Function x ^ p(x) is increasing for x £ 0[U

—b(n — 1)

—b(n — 1)

,

0,

, for p(x) to be positive it is sufficient that p (—b(n—— )

\ an J

and decreasing for > 0, and this is

verified if H> (—l)n ( .

x

Function x ^ q(x) is increasing for x £ 0[ U

—b(n — 1)

—b(n — 1)

,

0,

and x2 £

an —b(n — 1)

—b(n — 1)

when q(0) > 0 and q ( - ) < 0, there exist xy £

and decreasing for —b(n — 1)

,

We have

q(0) > 0 ^ h> ( —1)

0,

such that q(xy) = q(x2) = 0, and V x £ ]xy,x2[ q(x) < 0. m — 1N m 1 (d

4 —bn—1:) < 0«h < (—1)^m—1Y-' (dY' + —ir(n—X (b) .

y an J y c J \m J \ a J \n J

As before we conclude that r is composed by closed trajectory in first quadrant.

3. Number m is odd and n is even number. Function y ^ fx(y) is increasing for y £ 0[ U

y£

0,

—d( m — 1)

— d( m — 1)

,

and decreasing for

If p(x) is positive and q(x) is negative, by applying the theorem of intermediate values the equation fx(y) = 0 admits two solutions,

yi(x) £

0,

—d( m — 1)

and y2(x) £

—d( m — 1)

,

As before we conclude that if h satisfied the relationship (6), the curve r represents a closed trajectory in the first quadrant.

4. Numbers m and n are odd.

Simply combine between the case 3, when m is odd number and the case 2, when n is odd number, we conclude also, if h satisfied the relationship( 6), the curve r represents a closed trajectory in the first quadrant.

ii) The curve r is nonsingular of system (5).

We recall that the curve r is nonsingular of system( 5 ) if the following system has no real solution

axn + bxn-1 + cym + ym-1 + h = 0, xn-1 (x + y) (anx + b(n — 1)) = 0, ym-1 (x + y) (cmy + d(m — 1)) = 0.

0

cm

cm

cm

an

an

an

an

an

m

cm

cm

cm

cm

Note that the closed trajectory of the curve r is in realistic quadrant then it does not intersect

i 11 • ii- i (—b(n — 1) —d(m — 1)

the axes, and does not intersect the line y = —x, the pair '

second and the third equations but not the first, because h

m1

cm

= «-"I "-^r (n)"+

cancels the

-i

(-1)r

(m-" ) (

d \ m

— ) , then the curve r is nonsingular.

m

iii) r is an invariant curve of system (5).

dU. dU.

dj~x + dyy = (xn (nax + b(- — 1)) + ym (mcy + d (m — 1))) U, the cofactor is K(x, y) = xn (nax + b(n — 1)) + ym (mcy + d (m — 1)).

iv) / div(r)dt = 0. Note that

(9)

div(r)dt

K (x(t),y(t))dt,

(10)

see for instance Giacomini & Grau [12].

[ K(x(t),y(t))dt = f xn (nax + b(n — 1)) dt + f ym (mcy + d (m — 1)) dt J 0 J 0 J 0

rT 10

= r = r

xn (nax + b(n — 1))

xy (x + y) (naxn 1 + b(n — 1)xn 2)

—^dy ——^—dx, y (x + y) x (x + y)

dy —

ym (mcy + d (m — 1))

xy (x + y) (mcym 1 + d (m — 1) ym 2)

dx =

by applying the Green formula,

r x dy — y

y (x + y) x (x + y)

where int(r) denotes the interior of r.

1 T

dx = 2

(r) (x + y)

ïdxdy,

(11)

As the factor 2

(x + y)

3. Examples

Example 1. The system

is positive then / K(x(t),y(t))dt is nonzero.

0

x = x (x4 — 2x3 — 2xy2 + 2x — 4y3 + 3y2) , y = y (3x4 + 4x3y — 4x3 — 6x2y + 2y3 — 3y2 + 2y)

(12)

admits one limit cycle represented by the curve x3 — 2x2 + 2y2 — 3y + 2 = 0, and it has six singular points, the limit cycle around a stable focus (Fig. 1).

Example 2. Let a = 3, b = —5, c =1, d = —3, h = 5,

Note that, h satisfied max The system

f 27b4 —4d3| \ 256a3 ' 27c2 J

< h <

27b4 4d3

256a3 27c2

| x = x (3x5 — 5x4 — 2xy3 + 3xy2 + 5x — 3y4 + 6y3) , \ y = y (I2x5 + 15x4y — 15x4 — 20x3y + y4 — 3y3 + 5y) admits one limit cycle represented by the curve 3x4 — 5x3 + y3 — 3y2 + 5 singular points, the limit cycle encloses a stable focus (Fig. 2).

(13)

0, and it has six

0

0

0

1

Fig. 1. Limit cycle of system (13) with singular points

Fig. 2. Limit cycle of system (13) with singular points.

Conclusion

We proposed in this paper a polynomial Kolmogorov system, where just choose the parameters satisfying the conditions of the theorem (1), we conclude directly that the system has a limit cycle in the realistic quadrant, and we give it explicitly.

References

[1] A.Bendjeddou, R.Cheurfa, On the exact limit cycle for some class of planar differential systems, Nonlinear differ. equ. appl., 14(2007), 491-498.

[2] A.Bendjeddou, R.Cheurfa, Cubic and quartic planar differential systems with exact algebraic limit cycles, Elect. J. of Diff. Equ., (2011), no. 15, 1-12.

[3] S.Benyoucef, A.Bendjeddou, A class of Kolmogorov system with exact limit cycles, International journal of pure and applied mathematics, 103(2015), no. 4, 439-451.

[4] Shen Boqian,Liu Demeng, Existence of limit cycles for a cubic Kolmogorov system with an hyperbolic solution, Northwest Math., 16(2000), no. 1, 91-95.

[5] K.S.Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12(1981), no. 4, 541-548.

[6] SI Chengbin, Shen Boqian, The existence of limit cycles for the Kolmogorov cubic system with a quartic curve solution, J. Sys. Sci. & Math. Scis., 28(2008), no. 3, 334-339.

[7] H.Giacomini, M.Grau, On the stability of limit cycles for planar differential systems, J. of Diff. Equ., 213(2005), no. 2, 368-388.

[8] X.C.Huang, Limit cycle in a Kolmogorov-type model, Internat. J. Math. & Math Sci., 13(1990), no. 3, 555-566.

[9] X.Huang, Y Wang, A Cheng, Limit cycles in a cubic predator-prey differential system, J. Korean Math. Soc., 43(2006), no. 4, 829-843.

10] X.C Huang and Lemin Zhu, Limit cycles in a general Kolmogorov model, Nonlin. Anal. Theo. Meth. and Appl. 60 (2005), 1393-1414.

11] N.G.Lloyd, J.M.Pearson, E.Saez, I.Szanto, Limit cycles of a Cubic Kolmogorov System, Appl. Math. Lett., 9(1996), no. 1, 15-18.

12] N.G.Lloyd, J.M.Pearson, E.Saez, I.Szanto, A cubic Kolmogorov system with six limit cycles, International Journal Computers and Mathematics with Applications, 44(2002), 445-455.

13] R.M. May, Limit cycles in predator-prey communities, Science, 177(1972), 900-902.

14] L.Perko, Differential equations and dynamical systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 2001.

[15] Peng Yue-hui, Limit Cycles in a Class of Kolmogorov Model with Two Positive equilibrium Points, Natural Science journal of Xiangtan University, 32(2010), no. 4, 10-15.

Система Колмогорова с явными гиперболическими предельными циклами

Салах Бенусеф

Департамент математики Факультет наук, Университет Баджи Мохтар-Аннада

Алжир

Ахмед Бенжедду

Университет Сетиф 1 Департамент математики, 19000 Алжир

Вводится класс дифференциальных систем Колмогорова. Показано, что при подходящих предположениях относительно параметров для алгебраического предельного гиперболического цикла можно получить явное выражение.

Ключевые слова: Колмогорова дифференциальная система, инвариантная кривая, особая точка, периодическое решение, предельный алгебраический цикл.