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Spatially inhomogenious modes of logistic equation with delay and small diusion in a at area
Vladimir Goryunov

Novemb er 17-19, 2015

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with17-19, and small diusion November delay 2015 1 / 18


We consider the boundary value problem from the population dynamics

N = d N + r

(1 -

N t -1 )N

,

N

= 0,


where N = N (t , x ) R population density; Nt -1 = N (t - 1, x ); x R2; Laplace operator; D diusion coecient; r Malthusian coecient of linear growth; the direction of the outer normal to the border of bounded at area . The objective is to detect and study the periodic and nonperiodic complex modes at t 1.

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with17-19, and small diusion November delay 2015 2 / 18


Separately we consider the equation without diusion:

N=r

(1 -

Nt -1)N

.

In this case, it is well known that a unit equilibrium state becomes unstable when r = . 2 If

N

1

r> 2 N (t + T

then N 1 is unstable and it is a ) N (t ).

T

-periodic mode

It is proving analytically when r > + . 2

r

is close to , and numerically when 2

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with17-19, and small diusion November delay 2015 3 / 18


Now we assume that with diusion:

r

>



2

or about , and we consider the problem 2

N = d N + r

(1 -

N t -1 )N

,

N

= 0,


(1)

Fix a parameter r . Then we have certainly a N 1 for r < , and 2 N (t , x ) = N (t ) periodic spatially homogeneous solution otherwise. If the diusion d is large enough (d 1) then the spatially homogeneous solutions are certainly stable. There is a critical diusion

d

crit

for which this stability is lost.

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with17-19, and small diusion November delay 2015 4 / 18


In a numerical experiment we consider the area

= {x R2 | 0

x1

1, 0

x

2

1}.

The area is covered with a uniform grid with a step h = 0.01. The values in the appropriate squares of area are considered identical and are denoted as Ni ,j (i , j [1, M ], where M = 100). Then the Laplace operator is replaced by its dierence analogue

n

N i ,j

=

Ni -1,j

- 2Ni ,j +

h

2

Ni +1,j

+

N i ,j

-1

-2

N i ,j + N i , j h2

+1

,

and the boundary value problem (1) is replaced by a system of dierential-dierence equations with the following boundary conditions:

N i ,0 = N i ,1 , N i ,M N0,j = N1,j , NM ,j
Vladimir Goryunov

= =

Ni ,M +1, i NM +1,j , j

[1, [1,

M M

], ].

Spatially inhomogenious modes of logistic equation with17-19, and small diusion November delay 2015 5 / 18


Thus, given the step h = 0.01, we consider the system of 10000 equations with delay. In the process of computing the value of d is varied. As a numerical method for solving the system with initial conditions Ni ,j (s ) = i ,j (s ), s [-1, 0], where i ,j (s ) are continuous by s functions, it was chosen the DormandPrince method of the fth order with variable length of the integration step. The calculations were performed on a computing cluster of YSU (МНИЛ ?Дискретная и вычислительная геометрия? им. Б.Н. Делоне).

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with17-19, and small diusion November delay 2015 6 / 18


Spiral waves

Spiral wave generation by r = 2 and t = 10, 48, 99, 753.

d

= 10-4 . Times

Wandering of the spiral wave. Times
Vladimir Goryunov

t

= 4713, 9807.

Spatially inhomogenious modes of logistic equation with17-19, and small diusion November delay 2015 7 / 18


Distribution of the average value of

Ni ,j

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with17-19, and small diusion November delay 2015 8 / 18


Spiral wave generation by r = 3 and t = 5, 25, 83.

d

= 5 10-5 . Times

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with17-19, and small diusion November delay 2015 9 / 18


Spiral wave generation through spontaneous self-organization by r = 2 and d = 3 10-5. Times t = 120, 551, 2337, 5415, 6138, 8006, 8146, 8515, 8768, 9952, 10010, 11200.
Vladimir Goryunov Spatially inhomogenious modes of logistic equation with delay2015 small10 / 18 November 17-19, and diusion


Distribution of the average value of

N i ,j

in bubble structure

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with delay2015 small11 / 18 November 17-19, and diusion


t

Double spiral wave generation by r = 2 and d = 1 10-4 . Times = 40, 57, 145, 600, 1600, 2200, 3400, 4880, 5442, 5771, 8173, 15000.
Vladimir Goryunov Spatially inhomogenious modes of logistic equation with delay2015 small12 / 18 November 17-19, and diusion


Chaotic structure by r = 2 and d = 1 10-5 . Times t = 23, 51, 146, 374, 466, 750, 2161, 15000.
Vladimir Goryunov Spatially inhomogenious modes of logistic equation with delay2015 small13 / 18 November 17-19, and diusion


Distribution of the average value of

N i ,j

in chaotic structure

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with delay2015 small14 / 18 November 17-19, and diusion


Coexistense of spiral wave and chaotic structure for a long time by r = 2 and d = 1 10-5. Times t = 410, 544, 2370, 4630, 7460, 8302, 9280, 9960.
Vladimir Goryunov Spatially inhomogenious modes of logistic equation with delay2015 small15 / 18 November 17-19, and diusion


Numerical artifacts by r = 2 and d = 5 10-6 . Times t = 202, 598, 917, 3691.

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with delay2015 small16 / 18 November 17-19, and diusion


Halo generation by

r

= 2 and

d

= 5 10-5 . Times

t

= 2400, 4360.

Distribution of the average value of
Vladimir Goryunov

Ni ,j

in halo structure

Spatially inhomogenious modes of logistic equation with delay2015 small17 / 18 November 17-19, and diusion


Thank you for attention!

Vladimir Goryunov

Spatially inhomogenious modes of logistic equation with delay2015 small18 / 18 November 17-19, and diusion