Here

dx1/dt = r1x1 + a11x12 + a12x1x2 = P (1a)

dx2/dt = r2x2 + a21x1x2+ a22x22 = Q (1b)

with r being the intrinsic growth rate of the living thing (group of living things) within those examined, and a being an interaction term.

The singular point is

P = Q = 0

P is nullcline

x1 = 0

that is, the vertical axis, and

x2 = (-r1 – a11x1)/ a12 (2)

The image of this is a straight in the phase plane. The section point of the vertical axis is given by the value of

-r1 / a12, (2a)

that of the straight direction-tangent, by the value of

– a11/ a12 (2b)

Figures 1a-1d show the possible cases.

Fig. 1a

Fig. 1b

Fig. 1c

Fig. 1d

Q is nullcline

x2 = 0

that is, the horizontal axis, and

x1 = (-r2 – a22x2)/ a21 (3)

This is a straight too. Here, the section point of the horizontal axis is given by the value of

-r2 / a21 (3a)

and that of the straight direction-tangent, by the value of

– a22/ a21 (3b)

Figures 2a-2d show the possible variations belonging to this.

Fig. 2a

Fig. 2b

Fig. 2c

Fig. 2d

P, Q – nullclines and the direction vectors.

The number of cases is not sixteen, as it could follow from the pairing of nullclines P and Q , but more. The reason for this is that even within a pair there can be cases where the null lines section each other, and ones where they do not. Besides, it can be the case that at the section that ‘P’ comes from above or ‘Q’ does.

There are however fewer cases than expected on the basis of this, as in the cases of

r1 = a11 > 0

and

r2 = a22 > 0

There is no such doubling.

Thus, there are thirty-four cases altogether. These are shown in figures 1a2a-1d2d.

Fig. 1a2a

Fig. 1a2a2

Fig. 1a2b

Fig. 1a2b2

Fig. 1a2c

Fig. 1a2c2

Fig. 1a2c2b

Fig. 1a2c3

Fig. 1a2d

Fig. 1b2a

Fig. 1b2a2

Fig. 1b2b

Fig. 1b2b2

Fig. 1b2b2B

Fig. 1b2bB

Fig. 1b2c

Fig. 1b2c2

Fig. 1b2c2B

Fig. 1b2cB

Fig. 1b2d

Fig. 1c2a

Fig. 1c2a2

Fig. 1c2a2B

Fig. 1c2aB

Fig. 1c2b

Fig. 1c2b2

Fig. 1c2b2B

Fig. 1c2c

Fig. 1c2c2

Fig. 1c2d

Fig. 1d2a

Fig. 1d2b

Fig. 1d2c

Fig. 1d2d

The thirty-four cases can be categorized according to different aspects.

On the basis of singular points, there are 81 singular points altogether.

According to types, of these 28 are stable singular points and 53 are unstable singular points.

From the point of view of the existence of singular points, there are singular points in each case.

According to the number of singular points existing in one case, there is at least one and there are maximum four singular points.

From the point of view of the stability of the singular point, in 28 cases there are, and in 6 cases there are not (1c2aB, 1c2B, 1d2a, 1d2b, 1d2c, 1d2d) stable singular points. Where there are, there is always only one. Of these, 8 are globally stable. Of the 28 cases, 20 are extinction. Each of the remaining cases belong in group 1c, and there is only one case (1c2a2) where both remain, while in all the rest, of which one is globally stable (1c2d), only one does (1c2a2B, 1c2b2, 1c2b2B, 1c2c, 1c2c2).

The singular case where both remain is where

a11 = -1

r1 = 1

r2 = a22 = -1

Literature: EVOLUTION, CONSTRAINT, COOPERATION, AND COMMUNITY STRUCTURE IN SIMPLE MODELS, Lee Worden A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY, IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE PROGRAM IN APPLIED AND COMPUTATIONAL MATHEMATICS, November 2003

Dr. Endre Simonyi