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Qualitative structure of perturbations propagation process of the Fisher­Kolmogorov equation with a deviation of spatial variable
Sergey Aleshin, Sergey Glyzin P.G. Demidov Yaroslavl State University

November 17-19, 2015

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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Intro duction

In 1937 Kolmogorov, Petrovskii and Piskunov [1] proposed the logistic equation with diffusion for simulate the propagation of genetically wave u 2u + u[1 - u], = t x2 In the same year Fisher [2] published the article devoted to the analysis of a similar equation.
´ 1 Kolmogorov A., Petrovsky I., Piscounov N. Etude de l'´ equation de la diffusion avec croissance de la quantit´ de mati` et son application ` un probl` e ere a eme biologique // Moscou Univ. Bull. Math., 1 (1937). P. 1­25. 2 Fisher R. A. The Wave of Advance of Advantageous Genes // Annals of Eugenics. 1937. V. 7. P. 355­369.

(1)

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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Intro duction Logistic equation generalization for simulation of population density distribution with dependencies of spatial and time deviations was considered in [1-3]. u(t, x) = u(t, x) + u(t, x)[1 + u(t, x) - (1 + (g u)(t, x)] t and convolution has following form
t

(2)

(g u)(t, x) =
-

g (t - , x - y )u( , y )dy d ,

(3)

1 Gourley S. A., So J. W.-H., Wu J. H. Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics // Journal of Mathematical Sciences. 2004. Vol. 124, Issue 4. PP 5119­5153. 2 Britton N. F. Reaction-diffusion equations and their applications to biology / New York: Academic Press, 1986. 3 Britton N. F. Spatial structures and perio dic travelling waves in an integro-differential reaction-diffusion p opulation model // SIAM J. Appl. Math. 1990. V. 50. P. 1663­1688.
S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable YarSU


Logistic equation with a deviation of spatial variable

u 2u + u[1 - u(t, x - h)]. = t x2

(4)

u(t, x) = w(2t ± x)

s = 2t ± x (5) (6)

w - 2w + w[1 - w(s - h)] = 0, P () 2 - 2 - exp(-h).

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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Logistic equation with a deviation of spatial variable

2 - 2 - exp(-h) = 0, 2 - 2 - h exp(-h) = 0. -1.23141 Lemma (1) Quasipolynomial P () has one positive and two negative real roots at 0 < h < h and only one positive real root at h > h . 1 1 Lemma (2) h = h 1.12154 1

(7)

All roots of quasipolinom P () lie in the left half-plane for 0 < h < h , except 2 (- for one real positive root. Here h = arccos 5+2) 3.72346, The pair 2 = ±i0 of pure imaginary roots goes to the imaginary axis at h = h and 2 0 = 5 - 2 0.48587.
S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable YarSU

5-2


Logistic equation with a deviation of spatial variable

h = h + µ 2 w(s, µ) = 1 +

0<µ

1

µ z ( ) exp(i0 s) + z ( ) exp(-i0 s) + ¯
3/2

+ µw1 (s, ) + µ

w2 (s, ) + . . . ,

= µs, wj (s, )(j = 1, 2)

(8)

dz = 0 z + 1 |z |2 z , d
2 20 (-1 + i0 ) , P (i0 ) 1 1 2 2 2 1 = 20 (1 - 0 - 2i0 ) + (0 + 2i0 )2 - 2 P (i0 ) 0 + 2i 2 0 + 2i0 = . 2 2 40 + 4i0 + (0 + 2i0 )2

(9)

at 0 =

,
0

0 0.136807 - 0.20660i
S.V. Aleshin, S.D. Glyzin

1 -0.04429 - 0.03664i
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Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable


Logistic equation with a deviation of spatial variable

Lemma (3) Let h = h + µ and 0 < µ 1 then there exists µ0 > 0 such that for all 2 0 < µ < µ0 equation (5) has dichotomous cycle which one-dimensional unstable manifold and following asymptotic -Re (0 )/Re (1 ) exp is Im (0 )Re (1 )-Re (0 )Im (1 ) /Re (0 )+i and -- is an arbitrary constant, which determines the phase shift along the cycle.

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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Logistic equation with a deviation of spatial variable

u(t, x) = u(t, x + T ), v v - v (t, x - h), = t x2
2

T >0

(10) (11)

v (t, x) = v (t, x + T ).

v (t, x) = exp exp i x = - 2 - exp i h. h = 2.791544, = 0.88077. (12) (13)

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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Logistic equation with a deviation of spatial variable

T = 2 /



h = h +

u(t, x, ) = 1 + and = t,

u0 (t, , x) + u1 (t, , x) +

3/2

u2 (t, , x) + . . . ,

(14)

u0 (t, , x) = z ( ) exp i(0 t+ x) +z ( ) exp -i(0 t+ x) , ¯ dz = 0 z + 1 |z |2 z , d 0 = i exp(-i h ),


0 = sin h .

(15)

1 = 2 cos h 1 + exp(-i h ) - exp(-2i h ) + exp(i h ) w2 . 0 0.5558 - 0.6833i, 1 -0.1701 + 0.59i.

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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Logistic equation with a deviation of spatial variable

Lemma (4) Let h = h + then there exists 0 > 0 such that for all 0 < < 0 boundary value problem (4), (10) has orbitally asymptotically stable cycle with following asymptotic -Re (0 )/Re (1 ) exp it Im (0 )Re (1 ) - Re (0 )Im (1 ) /Re (0 ) + i and -- is an arbitrary constant, which determines the phase shift along the cycle.

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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Numerical analysis

uj =

u

j +1

- 2u j + u (x)2

j -1

+ 1-u

j -k

uj ,

(16)

j = 0, . . . , N - 1, k = h/x N = 1.8 · 10
5

N = 1.8 · 10

6

uj (0) =

0.1, if j [89950, 90050], 0, otherwise.

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S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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h = 1 .2

Wave propagation in logistic equation with spatial variable deviation h = 1.2 and cross-section t = 425

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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h = 2 .7

Wave propagation in logistic equation with spatial variable deviation h = 2.7 and cross-section t = 425

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

YarSU


h = 2.81

Wave propagation in logistic equation with spatial variable deviation h = 2.81 and cross-section t = 4500

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

YarSU


movie

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

YarSU


h=3

Wave propagation in logistic equation with spatial variable deviation h = 3 and cross-section t = 425

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

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Thank you for attention!

S.V. Aleshin, S.D. Glyzin Qualitative structure of p erturbations propagation pro cess of the Fisher­Kolmogorov equation with a deviation of spatial variable

YarSU