34 Ecology and Applied Environmental Science
predators (predator organisms or parasites) which, by attacking selectively
the more abundant species, can prevent it from increasing enough to be able
to eliminate its competitor.
3.4.2.1 Competition Models of Two Populations
Competition can be described by the typical Lotka-Volterra equations,
according to the exponential model:
dN dt r N
N N
dN dt r N
N N
1
1 1
2 1 1 2
2
2 2
1 2 1 2
D
D
The corresponding equations according to the logistic model are:
dN dt r N
N N q N
dN dt r N
N N q N
1
1 1
2 1 1 2
1 1
2
2
2 2
1 2 1 2
2 2
2
D
D
In the first case, one population’s rate of increase decreases due to the
other population’s presence. In the latter case, each individual from both
populations is subjected to a double competition, interspecific and intraspecific. The result is decreased growth rates of both populations, and sizes
which cannot reach the environmental carrying capacity that each population would achieve without the impact of the competition. However, if one
of the populations disappears, the other will increase more rapidly, enjoying the whole of the environmental carrying capacity.
Coefficients α 21 and α 12 regulate the negative effect of one of the populations to the increase rate of the other; this effect depends on the value of the
product N 1 N 2 . The presence of this product is necessary so that each population gets rid of interspecific competition if its competitor disappears, whereas
if its own size becomes zero, the rate of its change becomes zero, too.
The above equations cannot be solved analytically. In order to explore
the system’s equilibrium points, in the case of the logistic model, we write
them as follows:
dN dt r N
N
N K
dN dt r N
N
N K
1
1 1
1
21 2
1
2
2 2
2
12 1
2
1
1
D
D
*
*
where K 1 = r 1 /q 1 , K 2 = r 2 /q 2 , α *
12 = α 12 K 1 /r 1 , α *
21 = α 21 K 2 /r 2 .
We observe that: dN 1 /dt = 0, if K 1 = N 1 + α *
21 N 2 , and dN 2 /dt = 0, if
K 2 = N 2 + α *
12 N 1 . Placing the two lines, K 1 = N 1 + α *
21 N 2 and K 2 = N 2 + α *
12 N 1
predators (predator organisms or parasites) which, by attacking selectively
the more abundant species, can prevent it from increasing enough to be able
to eliminate its competitor.
3.4.2.1 Competition Models of Two Populations
Competition can be described by the typical Lotka-Volterra equations,
according to the exponential model:
dN dt r N
N N
dN dt r N
N N
1
1 1
2 1 1 2
2
2 2
1 2 1 2
D
D
The corresponding equations according to the logistic model are:
dN dt r N
N N q N
dN dt r N
N N q N
1
1 1
2 1 1 2
1 1
2
2
2 2
1 2 1 2
2 2
2
D
D
In the first case, one population’s rate of increase decreases due to the
other population’s presence. In the latter case, each individual from both
populations is subjected to a double competition, interspecific and intraspecific. The result is decreased growth rates of both populations, and sizes
which cannot reach the environmental carrying capacity that each population would achieve without the impact of the competition. However, if one
of the populations disappears, the other will increase more rapidly, enjoying the whole of the environmental carrying capacity.
Coefficients α 21 and α 12 regulate the negative effect of one of the populations to the increase rate of the other; this effect depends on the value of the
product N 1 N 2 . The presence of this product is necessary so that each population gets rid of interspecific competition if its competitor disappears, whereas
if its own size becomes zero, the rate of its change becomes zero, too.
The above equations cannot be solved analytically. In order to explore
the system’s equilibrium points, in the case of the logistic model, we write
them as follows:
dN dt r N
N
N K
dN dt r N
N
N K
1
1 1
1
21 2
1
2
2 2
2
12 1
2
1
1
D
D
*
*
where K 1 = r 1 /q 1 , K 2 = r 2 /q 2 , α *
12 = α 12 K 1 /r 1 , α *
21 = α 21 K 2 /r 2 .
We observe that: dN 1 /dt = 0, if K 1 = N 1 + α *
21 N 2 , and dN 2 /dt = 0, if
K 2 = N 2 + α *
12 N 1 . Placing the two lines, K 1 = N 1 + α *
21 N 2 and K 2 = N 2 + α *
12 N 1