Here we show a new interaction mechanism of colliding population waves. It provides a stable coexistence of two similar but different species competing for the same limiting resource during their asexual propagation in a limited homogeneous environment under constant conditions. The revealed mechanism opens new opportunities in conservation biology

Lev V. Kalmykov

Vyacheslav L. Kalmykov

English

Nature Precedings

 1 
Inter-Tunneling Mechanism of Colliding Population Waves 
Lev V. Kalmykov
1,2
, Vyacheslav L. Kalmykov
2,3
* 
1
Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Moscow 
Region, 142290 Russia. 
2
Pushchino State University, Pushchino, 142290, Moscow Region, Russia. 
3
Institute of Cell Biophysics of the Russian Academy of Sciences, Pushchino, 142290, Moscow Region 
 
*To whom correspondence should be addressed. E-mail: vyacheslav.l.kalmykov [at] gmail.com 
 
Abstract  
Here we show a new interaction mechanism of colliding population waves. It 
provides a stable coexistence of two similar, but different species competing for the 
same limiting resource during their asexual propagation in a limited homogeneous 
environment under constant conditions. The revealed mechanism opens new 
opportunities in conservation biology.  
 
Population waves are autowaves, i.e. they are self-sustaining waves, which use resources 
of media in which they propagate. The importance of autowaves based on the fact that 
they have universal properties that are independent of specific implementation (1). One 
such property is that these waves annihilate each other after collision. This annihilation 
occurs, for example, in result of collision of two fronts of the fire, which spread in a field 
with dry grass. Previously a paradoxical phenomenon of behavior of population waves 
was revealed - the colliding waves were not annihilated, but they looked as if they are 
penetrating through / reflecting from each other (2). The problem of interpretation of this 
phenomenon has been in the difficulty of understanding what specific mechanism is 
implemented - reflection, inter-penetration, or both. It is so because in both experiments 
N
at
ur
e 
Pr
ec
ed
in
gs
 : 
hd
l:1
01
01
/n
pr
e.
20
12
.6
99
0.
1 
: P
os
te
d 
14
 M
ar
 2
01
2
 2 
in vitro and in silico individuals of different species were not discerned because they 
were not marked. The mathematical modeling by partial differential equations also has 
not helped to understand what mechanism really happens (3) because it shows what 
happens with population waves but do not show how and why. In this paper, for the first 
time, we directly model the mechanistic mechanism of inter-tunneling of colliding 
population waves of two similar but different competing species, which are identical 
consumers (Fig 1; Movie S1). The species differ in their fitness - individuals of the first 
species are dominant, i.e. have the primary ability to use resources of an environment for 
propagation in situation of direct competition for a free microhabitat. The model 
reproduces propagation of two species of loose-bunch turf grasses in a small 
homogeneous ecosystem. Our cellular automaton model of interspecific competition is 
individual-based and its rules are based on deterministic logical "if-then" statements, 
which define direct processes of competition. It makes the model mechanistic. Earlier the 
phenomenon of tunneling of chemical autowaves through gaps in polymer membranes 
was demonstrated in experiment and in computer simulation (4). The existence of gaps in 
fronts of autowaves was experimentally demonstrated as dash waves based on Belousov-
Zhabotinsky reaction (5). We simulate here gaps in population waves. An important 
feature of our model is that all individuals have moderate propagation. This is reflected in 
the fact that every individual can use for propagation no more than one third of resources 
of its nearest environment. It leads to formation of gaps with possibility of their 
occupation by individuals of another species. Individuals of colliding population waves 
freely penetrate the perimeters of hexagonal fronts due to arising gaps (Fig. 1A and 
Movie S1). As result, both species have the same numbers of individuals on all iterations 
N
at
ur
e 
Pr
ec
ed
in
gs
 : 
hd
l:1
01
01
/n
pr
e.
20
12
.6
99
0.
1 
: P
os
te
d 
14
 M
ar
 2
01
2
 3 
(Fig. 1B). The discovered in silico new mechanism of inter-tunneling of colliding 
population waves is the direct mechanistic answer to the old question of ecology. It 
shows how two similar but different species can coexist in homogeneous ecosystem 
under constant environmental conditions in spite of that fact that they need the same 
limiting resource. The revealed mechanism opens new opportunities in conservation 
biology and is the step towards a solution of the mystery of biodiversity.  
 
 
Fig. 1. The inter-tunneling of colliding population waves during interspecific 
competition. (A) Outline of the inter-tunneling of colliding fronts of hexagonal 
population waves on the third iteration of the cellular automaton. ‘1’ - an individual of 
the first (dominant) species; ‘2’ - an individual of the second (recessive) species. 
Hexagons indicate the positions of the fronts. The dotted front line represents the 
collision line of the fronts of population waves. (B) The graph with the number of 
individuals of both species.  
 
N
at
ur
e 
Pr
ec
ed
in
gs
 : 
hd
l:1
01
01
/n
pr
e.
20
12
.6
99
0.
1 
: P
os
te
d 
14
 M
ar
 2
01
2
 4 
References and Notes 
1. V. I. Krinsky, in Self-Organization: Autowaves and Structures Far from Equilibrium V. I. 
Krinsky, Ed. (Springer-Verlag, Berlin, 1984), pp. 9-19. 
2. M. A. Tsyganov, I. B. Kresteva, A. B. Medvinskii, G. R. Ivanitskii, A Novel Mode of Bacterial 
Population Wave Interaction. Doklady Akademii Nauk 333, 532 (1993). 
3. M. A. Tsyganov, V. N. Biktashev, J. Brindley, A. V. Holden, G. R. Ivanitsky, Waves in systems 
with cross-diffusion as a new class of nonlinear waves. Physics-Uspekhi 50, 263 (2007). 
4. K. Agladze, C. Dupont, V. Krinsky, Tunneling chemical waves. Nuovo Cimento Della Societa 
Italiana Di Fisica D-Condensed Matter Atomic Molecular and Chemical Physics Fluids 
Plasmas Biophysics 20, 103 (1998). 
5. V. K. Vanag, I. R. Epstein, Dash waves in a reaction-diffusion system. Physical Review 
Letters 90, (2003). 
6. Author Contributions: Vyacheslav L. Kalmykov designed the research. Lev V. Kalmykov 
created the program, investigated the model and made the figures and the movie of the 
experiments. Both authors discussed the results, interpreted them and wrote the manuscript. 
7. This paper is a format-modified version of a manuscript which was submitted to Science on 
June 01, 2011 (it was rejected without in-depth review).  
 
N
at
ur
e 
Pr
ec
ed
in
gs
 : 
hd
l:1
01
01
/n
pr
e.
20
12
.6
99
0.
1 
: P
os
te
d 
14
 M
ar
 2
01
2
 5 
Supporting Material for 
Inter-Tunneling Mechanism of Population Waves  
 
 
This Supporting Material includes: 
 
Materials and Methods 
Fig. S1 
Legend for Supporting Online Movie S1 - 
http://www.youtube.com/watch?v=gZa2QIAdRBA   
 
 Materials and Methods 
Deterministic cellular automaton was used for modeling the spatio-temporal 
dynamics of simultaneous colonization of a free small ecosystem by two similar but 
different competing species. An integral ecosystem (macrohabitat) is modeled by the 
whole cellular automaton. The lattice is closed to a torus by periodic conditions for 
avoiding boundary effects. The hexagonal lattice was used, because it most naturally 
corresponds to the principle of densest packing of round projections of microhabitats. 
Resources of ecosystem are homogeneously distributed over all microhabitats. Each site 
models a microhabitat, which in the free state contains resource for existence of any one 
individual. A life cycle of an individual lasts one iteration of the automaton. All states of 
microhabitats have the same duration. All microhabitats are identical, i.e. every 
individual of both species consumes identical quantity of identical resources. The lattice 
consists of 26x26 sites. The initiate individuals of the two competing species are placed 
in the coordinates (10, 10) for the first and (15, 15) for the second species. One of the 
species is dominant and the other one is recessive. The dominance is defined as the 
primary ability of an individual to occupy a free microhabitat in a direct conflict of 
interests with an individual of the other species. Propagation of individuals is defined by 
N
at
ur
e 
Pr
ec
ed
in
gs
 : 
hd
l:1
01
01
/n
pr
e.
20
12
.6
99
0.
1 
: P
os
te
d 
14
 M
ar
 2
01
2
 6 
the neighborhood of the cellular automaton (Fig. S1A). Each site (microhabitat) can be in 
one of the five states. Specification of states and possible transitions between the states of 
a site are represented in Fig. S1B.  
 
 
 
 
Figure S1. Characteristics of the model. (A) Cellular automaton’s neighborhood (i, j are 
integer). A parental individual has coordinates (i, j) and marked by the grey color. 
Possible offsprings have coordinates (i, j-2), (i-2, j), (i-2, j+2), (i, j+2), (i+2, j), (i+2, j-2) 
on the next iteration and marked by orange color. (B) Directed graph of transitions 
between states of a lattice site (microhabitat).  
‘0’ – a free microhabitat that can be occupied by offspring of any species.  
‘1’ – a living individual of the first (dominant) species occupies a microhabitat.  
‘2’ – the regeneration state of a microhabitat after living of an individual of the first 
species. In Movie S1 this state is represented as symbol ‘.’ 
‘3’ – a living individual of the second (recessive) species occupies a microhabitat. In 
Movie S1 this state is represented as symbol ‘2’.  
‘4’ - the regeneration state of a microhabitat after living of an individual of the second 
species. In Movie S1 this state is represented as symbol ‘*’.  
N
at
ur
e 
Pr
ec
ed
in
gs
 : 
hd
l:1
01
01
/n
pr
e.
20
12
.6
99
0.
1 
: P
os
te
d 
14
 M
ar
 2
01
2
 7 
Movie S1 is available on YouTube - 
http://www.youtube.com/watch?v=gZa2QIAdRBA 
Movie S1. Inter-tunneling mechanism of colliding population waves. This video consist 
of three dynamical parts of the one model: cellular automaton’s lattice with all processes 
on it up to the 20
th
 iteration; the graph of the number of both species; statistics with the 
number of individuals and the number of iteration. After collision of population waves’ 
fronts on the 3
rd
 iteration, the mechanistic demonstration of their free inter-penetration 
through each other is appeared. States of a lattice site (microhabitat):  
‘0’ – a free site that can be occupied by offspring of any species; 
‘1’ – a site is occupied by a living individual of the first (dominant) species; 
‘.’ – the regeneration state of a microhabitat after living of an individual of the first 
species; 
‘2’ – a site is occupied by a living individual of the second (recessive) species; 
‘*’ - the regeneration state of a microhabitat after living of an individual of the second 
species.   
 
 
N
at
ur
e 
Pr
ec
ed
in
gs
 : 
hd
l:1
01
01
/n
pr
e.
20
12
.6
99
0.
1 
: P
os
te
d 
14
 M
ar
 2
01
2


A Novel Mode of Bacterial Population Wave Interaction. Doklady Akademii Nauk 333,

Dash waves in a reaction-diffusion system. Physical Review Letters 90,

in Self-Organization: Autowaves and Structures Far from Equilibrium

This paper is a format-modified version of a manuscript which was submitted to Science on

Tunneling chemical waves. Nuovo Cimento Della Societa Italiana Di Fisica D-Condensed Matter Atomic Molecular and Chemical Physics Fluids Plasmas Biophysics 20,

Waves in systems with cross-diffusion as a new class of nonlinear waves.

Inter-Tunneling Mechanism of Colliding Population Waves

http://precedings.nature.com/documents/6990/version/1

Inter-Tunneling Mechanism of Colliding Population Waves

Abstract

Similar works

Full text

Available Versions

Nature Precedings