mathematical models of ecological niches search · mathematical models of ecological niches search...

Applied Mathematical Sciences, Vol. 10, 2016, no. 38, 1907 - 1921

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2016.64139

Mathematical Models of Ecological Niches Search

Eugeny Petrovich Kolpak

Saint Petersburg State University

Faculty of Applied Mathematics and Control Process

199034, Saint Petersburg, Universitetskaya nab., 7/9, Russian Federation

Ekaterina Valerievna Gorynya

Saint Petersburg State University

Faculty of Applied Mathematics and Control Process

199034, Saint Petersburg, Universitetskaya nab., 7/9, Russian Federation

Copyright © 2016 Eugeny Petrovich Kolpak and Ekaterina Valerievna Gorynya. This article is

distributed under the Creative Commons Attribution License, which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This work develops mathematical models of two communities escaping the

competition. Models are represented by the Cauchy problem for the system of

ordinary differential equations and boundary value problems for systems of partial

differential equations.

Keywords: population, modeling, differential equations, stability, competition

1 Introduction

First mathematical models of competition designed for biological populations

belong apparently to Volterra [1]. Within the framework of these models in a

competitive environment, only that species “survives” which, ceteris paribus,

breeds faster than others. However, in an environment of biological populations,

there are many examples [2] of “peacefully” co-existence of species with different

reproduction speeds in the same trophic resource. Constant competitive

suppression of one community by another would significantly reduce the number

of species. However, the fact of existence in the nature of a variety of animals and

plants does not comply in general with this hypothesis [2]. The abundance of

species can be explained by the fact that competition was in the past, and

eventually species found its environmental niche in the form of other trophic re-

1908 Eugeny P. Kolpak and Ekaterina V. Gorynya

sources and habitats [2]. In today's system of economic relations between the vast

majority of small businesses compared to the number of large enterprises also

requires an explanation [3-5]. Modern mathematical models describing the

individual elements of a competitive relationship in the economic environment

offer a variety of options for reducing the level of competition [6-8].

It is very difficult to compare the theoretical results with the experimental

ones. Dynamics of population growth depends on many factors: weather

conditions, the emergence of new trophic resources, accidents and many other

reasons. The change of the habitat of the population leads to a change in the

existing population system [2, 9-13]. The formation of spatial structure of

population leads to the similar effect [14-16]. Under the influence of

anthropogenous factors a redistribution of the territory takes place, which happens

between populations and between groups of the same population. Emergence of

toxins in the territory creates prerequisites for the changes in the metabolism of

individuals, as well as for birth and death rates variations [17, 18].

Anthropogenous mechanical pressure on habitat [19, 20] and the human

competition for resources and territories [2] lead to physical destruction of trophic

resources and shelters [6, 21-25]. All these factors do not allow individuals of

population and population as a whole to adapt to new conditions of life. Therefore

theoretical results do not coincide well with the experimental data on a long-term

investigation [2].

Volterra's followers developed models admitting simultaneous stable existence

of two competing species in a variety of biological communities [26-30]

(medicine), [31, 32] (plant), [33, 34] (microbiology), [35, 36] (sport). In other

words, they proposed the description options for “survival of the weakest species”

among “strong,” which from the point of view of Volterra’s competition model

would have to perish. The paper offers a model of avoiding competition or, in

other words, the population’s escape to another ecological niche. The models take

into account different strategies of population’s survival [2]. The models are

represented by the boundary value problem for a system of nonlinear differential

equations in partial derivatives. Homogeneous solutions were analyzed for

stability. To build solutions of nonlinear equations, numerical methods are used,

grid method and Bubnov — Galerkin method.

2 Mathematical model of competition (Volterra—Bigon)

For a description of the population dynamics of two competing species, the

“modified” Volterra mathematical model is used that takes into account both

interspecific and intraspecific competition

11 1 1 2

22 2 2 1

(1 ),

(1 ).

duu u u

dt

duu u u

dt

(1)

Mathematical models of ecological niches search 1909

In these equations, 1u and

2u are the quantities of two competing populations, ,

1 , 2 are positive constants.

The system of equations (1) has four stationary points

1. 1 0u ,

2 0u .

2. 1 1u ,

2 0u .

3. 1 0u ,

2 1u .

4. 1 1 1 21 / 1u , 2 2 1 21 / 1u if 1 1 and

2 1 , or

1 1 and 2 1 .

The first stationary point is an unstable, the second will be stable if 1 1 and

2 1 , and the third one - if 1 1 and

2 1 . The forth stationary point is

implemented and is stable if 1 1 and

2 1 .

Thus, the model of competition (1) allows the simultaneous stable existence of

two competing populations.

3 “Active” escape from the competition

Evolution of the species occurred in such a way that under the influence of

external factors and internal changes in the organisms of individual species the

internal metabolism was changing, and they gradually began to use the new

trophic resources [2]. At that, the species continued to exist together. This can be

accommodated by assuming that in the model (1) 1 and

2 are time functions.

Then the model of the “escape” from the competition takes the form of

11 1 1 2

22 2 2 1

11 1 1 1 2

22 2 2 1 2

(1 ),

(1 ),

( , , ),

( , , ).

duu u u

dt

duu u u

dt

du u t

dt

du u t

dt

(2)

In this model, it is assumed that 1 1 2( , , ) 0u u t and

2 1 2( , , ) 0u u t . That is, it is

considered that the functions 1( )t and

2( )t are not increasing time function.

This enables a gradual escape from the competitive relationship. Adoption as

functions of 1( )t and

2( )t

1( ) 0t if 1 0du

dt , and

1( ) 1t if 1 0du

dt ,

2( ) 0t if 2 0du

dt , and

2( ) 1t if 2 0du

dt


means that the populations “feel” reducing their numbers and immediately “react”

to it. This model assumes that each population is trying to reduce the impact on

them by a competing population.

4 “Passive” avoidance of competition

This model assumes that the avoidance of competition occurs “naturally.”

1 2

2 1

11 1 1 2

22 2 2 1

(1 ),

(1 ).

u

u

duu u e u

dt

duu u e u

dt

(3)

Here it is assumed, that the functions 1 2( )f u and

2 1( )f u are positive and

decreasing functions its arguments, and 2

1 2lim ( ) 0u

f u

,1

2 1lim ( ) 0u

f u

. For

instance, such conditions satisfy the function 1 2

1 2( )uf u e

and 2 1

2 1( )uf u e

.

A non-trivial stationary point of the system of equations (3), wherein

10 1u and 20 1u , is the solution of the system of equations

1 2

2 1

1 1 2

2 2 1

1 0,

1 0.

u

u

u e u

u e u

(4)

The function ( ) 1 zf z e z in the point 0z is positive ( (0) 1f ) and

decreasing ( (0)f ); and in the point 1z , it will take positive values if the

inequality is 1e . When the inequality is 1 in the interval of [0,1] , ( )f z

will be monotonically decreasing function. If both of the inequalities are 1

1 1e , 2

2 1e ,

1 1 , and 2 1 , the system of equations (4) has a unique

solution satisfying 10 1u and

20 1u . When the inequality is 1 , the

function ( )f z will be minimal [0,1] in the point 1/z . Therefore, when the

inequality is 1 1 or

2 1 , the system of equations (4), depending on the

values of the constants in the equation (3) can have two solutions satisfying the

conditions 10 1u and

20 1u . Thus, this model may have three stationary

points.

The eigenvalues of the Jacobi matrix of the right side of the equations (3) are

the roots of the polynomial

1 2 2 12

1 2 1 2 1 2 1 2 2 11 (1 )(1 ) 0u uu u u u e u u .

If both of the inequalities are 1

1 1e , 2

2 1e ,

1 1 , and 2 1 , the

constant term of this polynomial is positive, its roots have negative real parts, and

the non-trivial stationary point is therefore stable.


5 Change of the habitat

In competition, one of the species can gradually avoid the competitive

relationship by moving to another trophic resource. Suppose there are two

competing populations in a common area or a trophic resource, and one of them

can move to a trophic resource not available to the other populations. Let 1v be

the number of the first population in a new trophic resource. Then the model (1)

goes into the model

11 1 1 2 1

22 2 2 1

11 1 1

(1 ) ,

(1 ),

(1 ) ,v

duu u u bu

dt

duu u u

dt

dvv v bu

dt

(5)

where v is the constant, and b is the rate of transition of the first population

species to a new trophic resource.

As follows from the analysis of equations (1), when 1 1 and

2 1 , the first

population in the model (1) dies. With 1 and

2 satisfying these inequalities, the

stationary point of the system (5)

1 0u , 1 1v ,

2 1u

will be stable since all the eigenvalues

1 1 1b , 2 ,

3 v

of the Jacobi matrix of the right hand side of equation (5) in the fixed point will be

negative. This corresponds to a complete transition of the first population to a new

trophic resource.

If both of the inequalities are 1 1 and

2 1 , the system of equations may

have a non-trivial stationary point

1 21 2 1 1

1 2 1 2

1 1 (1 ) 1, , 1 1 4 / ,

1 1 2v

b bu u v bu

if the inequality is 1 1b . At that, the eigenvalues of the Jacobi matrix of the

right side of the equation (5) being the roots of the equation

2

1 1 2 2 1 1 21 4 / ( ) (1 ) 0v vbu u u u u ,

will have negative real parts. That is, in case of implementation, this stationary

point is stable.


6 Heterogeneous areal

Populations live in areas with different trophic functions in different parts of it.

Moving species within areas occurs randomly. The intensity of the competitive

relationship may depend on what part of the area it takes place. When building

models of interacting populations in the territory, the mathematical apparatus of

continuum mechanics is used [28, 37-40]. The competition model in the interval

[0, ]l in this case is a system of differential equations in in partial derivatives

2

1 11 1 1 1 22

2

2 22 2 2 2 12

(1 ( ) ),

(1 ( ) ).

u uD u u x u

t x

u uD u u x u

t x

(6)

In these the equations parameters 1D and

2D characterize mobility of

individuals of populations. Functions 1( )x and

2( )x determine the intensity of

the competitive relationship.

The system of equations is added by the boundary conditions

at 0x : 1 0u ,

2 0u (7)

at x l : 1 0u

x

, 2 0

u

x

. (8)

If the functions 1( )x and

2( )x in some points of the interval are greater than

one and in some points are less than one, the system of equations (6) may have

heterogeneous solutions.

The number of populations 1( )M t and

2( )M t on the time cell t are calculated

by the following formula

1 1

0

( , )

l

M u t x dx , 2 2

0

( , )

l

M u t x dx .

To assess the influence of parameters 1D and

2D , we assume that the solution

of the equations (6) with boundary conditions (7)-(8) as a first approximation is

described by the functions

1( , ) ( )sin2

xu t x A t

l

, 2 ( , ) ( )sin

2

xu t x B t

l

.

Then with the application of the variational method Bubnov-Galerkin

equations (10)-(11) [38] followed by the equations for the coefficients ( )A t and

( )B t

2

1 1

2

2 2

8,

4 3

8.

4 3

dAD A A A A B

dt

dBD B B B A B

dt

(9)


The system of equations (9) has four stationary points.

1. 0A , 0B .

2. 0A , 2

2

31

8 4B D

.

3. 2

1

31

8 4A D

, 0B .

4. 2 2

1 1 2

1 2

1 31 ,

1 8 4 4A D D

2 2

2 2 1

1 2

1 31 .

1 8 4 4B D D

The number of populations on the time cell are calculated by the formulae

1 2 /M A and 2 2 /M B . Therefore, only positive values of A and B have

the physical meaning. Unlike model (1), the coordinates of stationary points

depend not only on the parameters 1 and

2 , and parameters 1D and

2D . Taking

values of 2

2 4 /D the second stationary point doesn't make sense, and taking

values of 2

1 4 /D the third doesn't make sense.

In the performance of inequalities 1 1 and

2 1 unlike model (2) the

existence region of positive values A and B for the fourth fixed point in the

coordinate system (1D ,

2D ) is determined by a system of inequalities.

2

1 14

D

, 2

2 14

D

, 1 1 22

4 10 1 D D

, 2 2 12

4 10 1 D D

.

In performance of inequality 1 2 1 the second and third stationary points

will be unstable. Eigenvalues of the Jacobian matrix in the fourth stationary point

are the roots of the quadratic equation

2

2

1 2

8 8(1 ) 0

3 3A B AB

,

that has complex conjugated roots. Therefore the fourth stationary point will be

steady if it is realized. Thus in model (6)-(8) the escape from the competitive

relationship could be executed due to the change of mobility of individuals of one

of the populations.

7 Numerical experiment

Assume that the two populations do not compete throughout their areal, but

only in one part and 1 and

2 depend on coordinate x


1

1

0, ,

, .

if x l b

c f x l b

, 2

2

0, ,

, .

if x l b

c if x l b

The solution for the system of equations (6) for these functions are is obtained

by applying numerical methods.

These algorithms have been developed to solution for non-linear equations in

partial derivatives. In one case the discretization of differential operators by finite

differences has been done. The obtaining system of non-linear algebraic equations

has been solved by simple iteration method. Numerical solution of equations (6)

for the case1 0.001D ,

2 0.01D , 1 1.2c and

2 0.9c , 1l , / 2b l on the

time cell 100t is given at Figure 1 - 2. Figure 1 shows the dependence

1 1( )u u x and 2 2( )u u x on the time cell 100t . In Figure 2 shows the

dependence of the stationary values1M ,

2M and the parameter b .

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

u 1, u 2

u1, b=0.5

u1, b=0

u2, b=0.5

u2, b=0

Figure 1: the dependence of the functions

1 1( )u u x and 2 2( )u u x in the time of

t .

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

b

M1, M

2

M1

M2

Figure 2: the dependence

1( )M t and 2( )M t on b .


8 Travelling wave

The solution of equations (6) are looking for in the form of a travelling wave

propagating with v the velocity from to .

That is, we believe that the solution depends only on one argument z x vt .

Under this assumption equation (6) is converted to the form

11

11 1 1 1 1 2

12

22 2 2 2 2 1

,

(1 ),

,

(1 ).

duw

dz

dwD vw u u u

dz

duw

dz

dwD vw u u u

dz

(10)

These equations are added conditions imposed on the functions 1u and

2u [1],

at x : 1 1 1 21 / 1u , 2 2 1 21 / 1u ;

at x : 1 1u ,

2 0u .

These conditions correspond to the case where the area on which there is only

one population with a linear density of population 1 1u , the second population

moves into. In this case, under the coexistence equilibration is observed at the

point x . Supposed that 1 1 and

2 1 .

Eigenvalues of the Jacobian matrix of the right part of the equations (10)

1 1 2 1 1

1 1 1

22 2 2 1

2 2 2

0 1 0 0

1 1(1 2 ) 0

0 0 0 1

0 (1 2 )

vu u u

D D DJ

vu u u

D D D

at 1 1 1 21 / 1u , 2 2 1 21 / 1u are roots of the equation

2 2

1 2

1 1 2 2

10

v vu u

D D D D

. 11)

At the point x the function1 1( )u u z must be increasing, but the function

2 2( )u u z -decreasing. Equation (11) has two positive and two negative roots.

Therefore, we can get a decision on which the function 1 1( )u u z increases and

the function 2 2( )u u z decreases.

Under 1 1u and

2 0u eigenvalues of the Jacobian matrix satisfy the

equations


2 1

1 1

0uv

D D , 2 2

2 2

0uv

D D .

The first equation has roots of opposite signs, and the second may have

complex-conjugate roots. In this case, the function 2 2( )u u z can take negative

values. Under such condition 2 2 1 22 (1 ) /(1 )v D the second equation

will have negative roots. In this case we can obtain a such solution that at x

the function is decreasing and 1 1( )u u z - increasing. This decision will represent

a travelling wave for 2 2( )u u x tv , propagating from x to x , and for

the function 1 1( )u u x tv is "running" at a velocity no less than

2 2 1 22 (1 ) /(1 )v D .

The solution of equations (6) with boundary conditions

at 0x and at x l

1 2 0u u

x x

and initial conditions

1 1u , 0

2 2 (0)u u ,

where ( )x is delta-function, and 0

2u is a small positive value. This was

calculated by applying numerical methods. Over time, in the neighborhood of

0x the values 1( )u x tended to value 1 1 21 / 1 , and

2( )u x - to

2 1 21 / 1 . Figure 3 shows the dependence 1 1( )u u x and

2 2( )u u x at

=1 , 1 =0.8 ,

2 =0.5 , 1 D =0.00001 ,

2 D =0.00001 on the time cell 100t and

t=200 . The velocity of wave propagation was close to the value

2 2 1 22 (1 ) /(1 )v D .

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

u 1, u 2

u1, t=200

u2, t=200

u1, t=100

v

u2, t=100

Figure 3 The shape of the travelling wave in the time t=100 and t=200 .


9 Conclusion

The above models of two competing populations allow us to describe the

mechanisms of search of ecological niches by different populations in the course

of their evolution.

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