One-dimensional Lottka – Volterra

“These encounters have a positive or negative effect on population i described by the sign and magnitude of their interaction term aij, and an effect on population j described by the corresponding term aji. If aij is negative, encounters with j are detrimental to i, while if it is positive, these encounters are beneficial to i. When both aji and aij are negative, these populations are said to have a competitive interaction; when both are positive, it is a mutualistic interaction, and a pair with mixed signs is a victim-exploiter interaction, which may represent a predatorprey, host-parasite or similar relationship. Relationships with at least one negative term in the pair are antagonistic. Self-interactions aii are usually required to be competitive, representing an overcrowding or other negative density dependence effect which helps to prevent populations from growing without bound. In the absence of interactions, each population grows or declines at an intrinsic rate ri.”

Here: dx / dt = rx + ax2

(1)
with r being the intrinsic growth rate of the living thing (group of living things) within the examinations, and a being an interaction term.

On the basis of this, a singular point is
xs = 0
and
xs = – r / a

with s referring to the singular point.

“The most commonly considered case is r >0 and a <0, since it is well behaved and biologically nontrivial. In this case, the growth rate r + aX is positive when X is small and negative when X is large, so the population X is driven to a positive equilibrium value (which is easily seen to be -_r/a). This is the standard logistic growth equation, often written dX/dt=_rX(_1- _X/_K), using the equilibrium value K= -r/a.

If r< _0 and a< _0, the growth rate r+ aX is strictly negative for all X> _0, so all positive-sized populations decline to 0. If r > 0 and a > 0, the growth rate is positive for all positive X, so the population grows to infinity.

Finally, if r <_ 0 but a >_ 0, the equilibrium dX/dt1 = -_r/a is positive, but the growth rate is negative below the equilibrium and positive above, making it an unstable equilibrium that repels nearby trajectories either to zero or to infinity, depending on which side they fall.

There are several degenerate cases as well, when one or both of the parameters is zero, but in general I intend to avoid degenerate cases as much as possible.”

As x is the population of the living thing, which cannot be negative, the second root is realistic only if
SIG(r) = – SIG(a)

(2)
The so-called phase velocity is:
v = xdx /dt = rx2 + ax3

In the case of a primary system it is a necessary and sufficient precondition of the asymptotic stability of a singular point that the phase straight have a non-infinitely small section which contains the singular point, and each point of which with an x value higher than at the singular point has a negative v value, while in the case of the points with an x value lower than at the singular point, the v value is positive, and there are not any points within the section from which the moving point can get into the singular point. This latter precondition rules out the existence of a point within the section in which the phase velocity approaches the infinite. Put in another way, the phase velocity is limited everywhere.

To express the above with formulas, if
x > x2
then
v < 0
and if
x < x2
then
v > 0
and also,
there is a finitely high v1 number regarding which
ABS(v) < ABS(v1).

Phase velocity, that is, the speed of the increase or decrease of the population of the living thing, is always limited. This precondition is therefore satisfied in reality. Let us examine whether or not it holds true for this model too.

Cases:
1 There is one singular point

On the basis of this, if (2) does not come about, that is,
SIG(r) = SIG(a)
then the origin, that is, extinction, is stable, in case
SIG(v) = -1

(3)
As the precondition x > 0 is always there, another form of (3) is:
SIG(r + ax) = -1

(3a)
In case
SIG(r) = SIG(a) = 1
that is, both are positive, this is not come about anywhere, the origin is unstable, the system approaches infinity.
However, in case
SIG(r) = SIG(a) = -1
that is, both are negative, this will come about everywhere, and thus the origin is a globally stable singular point.
Thus, it can be stated that if the signs of the two constants are identical, there is only one singular point, which is extinction, or the origin. Extinction takes place if both constants are negative; however, if both are positive the number signifying the population approaches infinity.

2 There are two singular points

As mentioned earlier, in this case the signs of the two constants are different.

If
r > 0
a < 0
at the other singular point
x < xs
that is, in the case of
x < -r/ a
v > 0
and in the case of
x > -r/ a
v < 0

So, this singular point is stable, and globally too.

In a reverse case however if
x < xs
v < 0
and is positive in the case of the higher value, so here the second is unstable. The origin is stable, but not globally, as in the case of an x value higher than the second the population approaches infinity.

Thus, it can be stated that in the case of two singular points the origin is unstable, that is, the system does not approach infinity if the value of r is positive, while the other singular point is globally stable, that is, the system approaches it from everywhere. However, if the value of r is negative, there is not one globally stable point, and the population approaches infinity if its number was lower than the r/a value even at the beginning, and its number increases infinitely if it was higher than the r/a value even at the beginning.

To sum up: A precondition of the steady existence, leading neither to extinction nor to limitless increase, of the living thing (group of living things) free of other living things’ effect influencing this living thing (group of living things) is the satisfying of the
r > 0
and
a < 0
preconditions.

This statement corresponds to that written down in the literature.

Time function

Equation (1) arranged and integrated,

LN(x/ (r + ax)) = rt
of which
x = -r / (a-exp(-rt))

(4)
Zero point, if the denominator approaches infinity, that is, time approaches infinity and
r < 0

Point of discontinuity, where
a = exp(-rT)
that is,
T = -LN(a)/r
and
T > 0

Cases:

1 The two constants bear the same sign

1.1 Both are positive

There is no zero point.

Point of discontinuity
As only
x > 0
is to be examined, which comes to be the case if
LN(a) < 0
This comes about if
a < 1

In this case it approaches infinity, which can be seen in Figure 1b, but in the case of
a > 1 it approaches the value
x0 – (r /a) (Figure 1a).
Both Figures show left-to-right motion.

Figure 1b

Thus, here the origin is always unstable; the other is either stable or unstable, depending on the value of a.

This result differs both from the one written down in the literature and the one obtained through my method.

1.2 Both are negative

Here, there is decrease only (Figure 2), so the Figure shows right-to-left motion.

Figure 2

The second singular point is unstable here. The constants influence the velocity of moving away only. Motion is towards the origin.

2 The two constants bear different signs.

2.1 r is positive

It always comes about and approaches the non- extinction value (Figure 3).

Figure 3

2.2 r is negative

In case
a < 1
it approaches the origin, that is, in Figure 4a motion is also from right to left.

Here the precondition of discontinuity is satisfied if

a > 1 but in this case it has discontinuity in finite time (Figure 4b).

Thus, in this respect the model does not square with reality.

This result too differs both from the one written down in the literature and the one obtained through my method.

Figure 4a

Figure 4b

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