untitled

dN 1 /dtis positive so that N 1 increases, and to the right it is negative and N 1 decreases

as indicated by the arrows. At all points on the line (called an isocline) the popula-

tion is stationary. Exactly similar reasoning produces the equivalent diagram for species

2 (Fig. 9.1b). Below the line (isocline) N 2 increases, and above it N 2 decreases.

With these two diagrams describing the competitive abilities of the two species

independently we can now predict the outcome of competition between them. If we

put the two diagrams in Fig. 9.1 together, as in Fig. 9.2a, we see that K 1 is larger

than K 2 /a 21. The latter term is the number of species 1 required to drive species 2 to

extinction, and since it is possible for species 1 to exist at higher numbers than this

level (i.e. at K 1 ), species 1 will drive species 2 down. On the other axis we see that

COMPETITION AND FACILITATION BETWEEN SPECIES 137

(a) (b)

N 1

K 1

N 1

K 2

N

2

N

2

K 1 /α 12

K 2 /α 21

Fig. 9.1Isoclines for the
Lotka–Volterra equations. (a)
At any point on the isocline
dN 1 /dt=0. This indicates
where the number of species 1
is held constant for different
population sizes of species 2.
Species 1 increases to the left
of the isocline, but decreases
right of it. (b) The isocline
where dN 2 /dt=0. This shows
where the population of species
2 is held constant at different
values of species 1. Species 2
increases below the line, but
decreases above it.

(a) (b)

N 1

K 1

N 1

K 2

N

2

N

2

K 1 /α 12

K 2 /α 21

(c) (d)

K 2 /α 21

K 2

K 1 /α 12

K 1

K 1

K 2

K 1 /α 12

K 2 /α 21 K 2 /α 21

K 2

K 1 /α 12

K 1

Fig. 9.2The
relationship of the
two species’ isoclines
determines the outcome
of competition.
(a) Species 1 increases
at all values of species 2
so that species 1 wins.
(b) The converse of
(a) such that species 2
wins. (c) In the region
where the isocline of
species 1 is outside that
of species 2, species 1
wins and vice versa, so
that either can win.
(d) A stable equilibrium
occurs because all
combinations tend
towards the intersection
of the isoclines.