Lecture 19: Competition Models

Reading: Economy of Nature, pp. 430-432.

Logistic Model of Interspecific Competition

Lotka-Volterra Model
Lotka (1932) and Volterra (1926)

Logistic Equation

Let:

= number of individuals of species 1
= number of individuals of species 2
and = respective carrying capacities
and = respective intrinsic rates of increase

If:

Ten individuals of species 2 have collectively the same competitive-inhibitory effect on species 1 as one individual of species 1

Then:

The total effect on species 1 of intra- and inter-specific competition will be equal to the effect of species 1 individuals.

The one-tenth factor = competition coefficient =

"The competitive effect on species 1 by species 2"

When < 1, each individual of species 2 has less inhibitory effect on each individual of species 1 than each individual of species 1 has on other species 1 individuals. In other words, when < 1, intra-specific competition in species 1 is more intense than inter- specific competition.

For species 1: equivalents

so:

For species 2:

How does population growth of one species respond to the presence of another species?

When a population is stable: = 0 (zero growth condition)

If = 0 then

This is true if: or (trivial examples)

or if: = 0 then (a straight line, zero growth isocline)

In species 1 terms:

when then and when then

Species 1 responses to the numbers of species 1 and numbers of competing species 2 (after Begon, Harper and Townsend, 1996, p 275, Fig. 7.6).

 Similarly for species 2:

If = 0 then

and = 0 then

In species 2 terms:

when then and when then

Species 2 responses to the numbers of species 2 and numbers of competing species 1 (after Begon, Harper and Townsend, 1996, p 275, Fig. 7.6).

The outcomes of competition can be modeled by combining the graphs for the two competing species.

The behavior of the two species populations can be determined by vector addition.

There are four different zero isocline arrangements possible for competition between species 1 and 2.

When the inhibitory effects on species 1 by itself (intraspecific) are greater than the inhibitory effects on species 1 by species 2 (interspecific), and species 1 has a greater inhibitory effect on species 2 than species 2 has on itself: Then species 1 drives species 2 to extinction, species 1 wins [see graph (a), below].

and so and

Outcome: Competitive Exclusion

Species 1 goes to carrying capacity and species 2 is excluded.

The opposite outcome occurs when the inhibitory effects on species 2 by itself (intraspecific) are greater than the inhibitory effects on species 2 by species 1 (interspecific), and species 2 has a greater inhibitory effect on species 1 than species 1 has on itself: Then species 2 drives species 1 to extinction, species 2 wins [see graph (b), below].

and so and

Outcome: Competitive Exclusion
Species 2 goes to carrying capacity and species 1 is excluded.

An unstable "equilibrium" occurs when the inhibitory effect on species 1 by species 2 is greater than the inhibitory effect on species 1 by itself, and the inhibitory effect on species 2 by species 1 is greater than the inhibitory effect on species 2 by itself [see graph (c), below].

and so and

Outcome depends on the starting densities.
        The species with the initial advantage wins.

Outcome: Competitive Exclusion (ultimately)
        Either: Species 1 goes to and species 2 is excluded,
                   or species 2 goes to and species 1 is excluded.

A stable equilibrium will occur when the inhibitory effect on species 1 by itself is greater than the inhibitory effect on species 2 by species 1, and the inhibitory effect on species 2 by itself is greater than the inhibitory effect on species 1 by species 2 [see graph (d), p 83].

and so and

Outcome: Stable Coexistence

Both species will persist (ultimately) at the equilibrium densities where their zero growth isoclines cross.

Outcomes of competition between two species (after Begon, Harper and Townsend, 1996, p 277, Fig. 7.8).

Competitive exclusion occurs when interspecific competition is greater than intraspecific competition. When addition of one more individual causes more harm to the other species than to conspecifics.

Competitive exclusion occurs because complete competitors cannot coexist (Gause).

Coexistence occurs when intraspecific competition is greater than interspecific competition. When addition of one more individual causes more harm to conspecifics than to the other species.

Demonstrating Competition:

Interaction reduces: individual or population growth rates population density
Interaction changes: habitat utilization

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