The model is based on logistic growth. Remember that logistic growth models intraspecific (within one species) competition. The Lotka-Volterra model of interspecific (between species) competition includes the effects of intraspecific competition, but adds the competitive effect of another species.

The model is based on two equations of population growth; one for each of two competing species. The species are referred to (unimaginatively enough) as species 1 and species 2; subscripts 1 and 2 are used on the symbols to indicate the species to which they refer. The equations follow; the symbols are defined below the equations.

Equation for population growth of species 1:

Equation for population growth of species 2:

The symbols are largely the same ones that were used in logistic growth; note that the r values refer to rm values for species 1 and 2.  The main symbol that is added here is alpha (a).

• a₁₂ refers to the impact on population growth of species 1 of an individual of species 2, relative to the impact of an individual of species 1.
• a₂₁refers to the impact on population growth of species 2 of an individual of species 1, relative to the impact of an individual of species 2.

These equations can be used to predict the outcome of competition over time.  To do this, we determine equilibria: population sizes for species 1 and 2 for which population growth of both species will be zero.  If population growth is zero, then the population sizes do not change over time, and we have an equilibrium (a situation in which conditions remain the same over time.)

To find the equilibria, we will first determine combinations of population sizes for species 1 and species 2 for which species 1 population growth is zero.  Then we will determine combinations of population sizes for species 1 and species 2 for which species 2 population growth is zero.  Finally, we will put these conditions together to find combinations of population sizes of both species for which population growth of both species is zero.

Population sizes for which species 1 growth is zero:

To find the population sizes for which species 1 growth is zero, we set the growth equation (given above) for species 1 to equal zero.  That is:

0 = r₁N₁(K₁-N₁-a₁₂N₂)

This equation will equal zero if any of the three multiplied terms is equal to zero.  The three multiplied terms are: r₁, N₁, and (K₁-N₁-a₁₂N₂).

If r₁ were zero, then the MAXIMUM population growth rate for species 1 would be zero -- such a species would never grow and would not exist, so this is not ecologically interesting, and we won't worry about this situation.

If N₁ were zero, it would mean that there are no individuals of species 1 in the population.  In other words, the population of  species 1 does not exist.  That's not very interesting either -- we want to look at situations where the species DOES exist to see what happens when it competes with species 2.  So we won't worry about this situation either.

The ecologically interesting situation occurs when:

K₁-N₁-a₁₂N₂=0

Solving this, we find:

N₁=K₁-a₁₂N₂

Now, look at the form of this equation.  If we were to plot N₁ on the Y axis and N₂ on the X axis of a graph, so that Y represents N₁ and X represents N₂, then note that this is in the form:

Y=mX + b

You should recognize this form: it is a straight line, with slope m and intercept b.  This means if we plot N₁ versus N₂, we get a line with Y intercept K₁ and slope of negative alpha₁₂.  We can also determine the X intercept of this line by setting Y=0 (that is, N1=0):

0=K₁-a₁₂N₂

N₂=K₁/a₁₂
So when we draw this graph, it looks like this:

The line is called the zero growth isocline for species 1: it represents all combinations of N₁ and N₂for which growth of N₁ is zero.

Suppose we had a combination of population sizes of species 1 and species 2 that falls below this line.  This would represent a situation with fewer individuals of the two species than the numbers required to cause growth to be zero.  In this situation, because there are few individuals of the two species, there would be plenty of resources for species 1, and the population of species 1 would increase in size.  In contrast, if we have a combination of population sizes of species 1 and species 2 that falls above the zero growth isocline for species 1, this means there are more individuals of the two species than what would cause growth to be zero -- in the presence of so many individuals, resources would be depleted and species 1 population size would decrease.  We represent these areas of increase or decrease on the graph by drawing arrows, like this:

An arrow pointing up represents growth of species 1 since moving a point up on this graph means that species 1, plotted on the vertical (Y) axis, gets bigger.  An arrow pointing down represents decrease in population size of species 1 since moving a point down on this graph means that species 1, plotted on the vertical (Y) axis, gets smaller.

Combinations of population sizes of species 1 and 2 for which growth of species 2 is zero:

Note that the growth equation for species 2 looks just like the growth equation for species 1, but with all the 1's and 2's reversed.  This means if we go through all the steps we did above for species 1, but apply them to the equation for species 2, we're going to get the same result (but with all the 1's and 2's reversed.)  This means we're going to get a zero growth isocline for species 2, with intercept on the N2 axis at K2 and the intercept on the N1 axis at K1/alpha21.  If you don't believe me, work out the equation following the steps we used above (this would be a good way to practice and make sure you understand how this model works!)

If we plot this graph on axes like the ones above where N1 is on the Y axis and N2 is on the X axis, the graph looks like this:

As we did for species 1, we can determine the regions of the graph for which species 2 increases or decreases in size.  Below the isocline, there are few individuals of species 1 and species 2, and therefore plenty of resources.  In this area of the graph, species 2 increases.  Above the isocline, there are many individuals of species 1 and 2, and resources are depleted.  In this area of the graph, species 2 decreases.  We represent increase and decrease in species 2 by arrows pointing to the right (increase) or left (decrease), as shown here:

Note that an arrow to the right represents increase in species 2 (rather than an arrow pointing up, as we used for species 1) because species 2 is plotted on the horizontal (X) axis so moving a point to the right means species 2 gets bigger and moving a point to the left means species 2 gets smaller.

Finding Equilibrium Situations for Species 1 and 2:

We now need to determine whether there are situations for which population growth for both species (1 and 2) is zero -- these will be situations for which neither population changes, so they will represent equilibria.  To do this, we are going to plot both isoclines on the same graph.  It turns out that there are four ways to plot these two lines relative to each other.  These are shown here:

We'll consider each case.

Case 1:  The species 1 isocline is above the species 2 isocline.

Consider what happens in each section of the graph.  Below both isoclines, species 1 and 2 both increase.  In the range of the graph between the two isoclines, we are above the species 2 isocline so it decreases, but below the species 1 isocline so it continues to increase.  The following graph shows these changes in species 1 and species 2 with arrows (vertical arrows for species 1 since it is on the vertical axis and horixontal arrows for species 2 since it is on the horizontal axis.)

The result is that species 2 declines to zero and species 1 increases to its carrying capacity.  In this case species 1 has competitively excluded species 2.

We can look at the K and alpha values to understand this in terms of the strengths of interspecific competition and intraspecific competition.  Note, on the N₁ axis, that the K₁ term falls above the K₂/alpha₂₁ term.  Thus:

                        K₁>K₂/a₂₁

so                    K₁a₂₁>K₂

This means that when species 1 is at its carring capacity, its impact on species 2 (measured by K₁ times a₂₁) is greater than the impact of K₂ individuals of species 2.  Thus, species 1 is affecting species 2 more negatively than species 2 affects itself.  Interspecific competition regulates species 2 more than species 2 is regulated by intraspecific competition.

From the N₂ axis, note that K₂ falls below the K₁/alpha1₁₂ term.  Thus:

                    K₂<K₁/a₁₂

so                K₂a₁₂<K₁

This means that the impact of species 2 at its carrying capacity on species 1 is less than the impact of species 1 at carrying capacity on itself.  Thus, species 1 is regulated more by intraspecific competition than by interspecific competition.

We see here that competitive exclusion is occurring if one species is more regulated by intraspecific competition, the other more by interspecific competition.  In this case, it is species 1 that is regulated more by intraspecific competition, and species 1 "wins" -- it competitively excludes species 2.  In the next case we will see something similar, but with species 1 and species 2 reversed.
 

Case 2:  The species 2 isocline is above the species 1 isocline.

Again, consider what happens in each section of the graph.  Below both isoclines, species 1 and 2 both increase.  We represent the increase as above with arrows.  In the range of the graph between the two isoclines, we are above the species 1 isocline so it decreases, but below the species 2 isocline so it continues to increase.  The result is that species 1 declines to zero and species 2 increases to its carrying capacity.  In this case species 2 has competitively excluded species 1.

As we did above, we will look at the K and alpha values to understand this in terms of the strengths of interspecific competition and intraspecific competition.  From the N₁ axis, we see that:

K₂/a₂₁>K₁

so K₂ > K₁a₂₁

So, using the same reasoning as above, intraspecific competiton impacts species 1 more than does interspecific competiton with species 1.

And from the N2 axis, we see that:

                   K₁/a₁₂ < K₂

so                K₁ < K₂a₁₂

indicating that species 1 is more impacted by interspecific competition with species 1 than it is by intraspecific competition.

From cases 1 and 2, we see that that competitive exclusion is occurring if one species is more regulated by intraspecific competition, the other more by interspecific competition.

Case 3: Isoclines for the two species cross; the K values on each axis are lower than the K/a values

This time, there are four sections of the graph to consider.  Below both isoclines, both species increase.  Above the isoclines, both decrease. In the triangular sections of the graph combinations of species are above the isocline for one species but not the other.  In the upper left, the combination is above the species 1 isocline so species 1 declines, but above the species 2 isocline so species 2 increases, and the populations move toward the point where the lines cross.  In the lower right, the combination is above the species 2 isocline so species 2 declines, but below the species 1 isocline so species 1 increases.  The populations move toward the point where the lines cross.

In this case, there is a stable equilibrium where both species coexist.

Examining K and K/alpha values from the axes, we find from the N1 axis that:

K₂/a₂₁>K₁

so
K₂>K₁a₂₁

Indicating that species 2 is regulated more by intraspecific competition than by interspecific competition

And from the N2 axis we see that:

K₁/a₁₂>K₂

So

K₁ > K₂a₁₂

Indicating that species 1 is regulated more by intraspecific competition than by interspecific competition.

So we can see that when each species is regulated more by intraspecific competition than by competition with the other species, the two species coexist.

Case 4: Isoclines for the two species cross; the K values on each axis are higher than the K/a values

As with case 3, we need to consider all four sections of the graph.  When both populations are small, both increase.  When both are large, both decrease.  In the upper left, species 1 is below its isocline and species 2 is above its isocline, so species 1 increases and species 2 decreases; this leads to competitive exclusion of species 2 by species 1.  In the lower right, species 2 is below its isocline and species 1 is above its isocline, so species 2 increases and species 2 decreases; this leads to competitive exclusion of species 1 by species 2.  The point where the lines cross is not a stable equilibrium because populations tend to move away from it, so it will not be observed in natural populations.

This situation is unlike the others in that there are two possible outcomes.  If the populations grow into the upper left triangle, species 1 "wins."  If the populations grow into the lower right triangle, species 2 "wins."  Situations in which a population of a species is large give that species the edge -- make it more likely to "win."  This could occur if one species established a population in an area before the other species, or if one species had a higher per capita growth rate than the other.

Once again we can examine the K/alpha and K terms to understand this situation in terms of inter and intraspecific competition.  In this situation, from the N₁ axis:

K₁>K₂/a₂₁

so

K₁a₂₁>K₂

Indicating higher impact of interspecific competition than intraspecific competition on species 2

From the N₂ axis:

K₂>K₁/a₁₂

so K₂a₁₂>K₁

Indicating higher impact of interspecific competition than intraspecific competition on species 1.

Thus, the situation in which either species could exclude the other occurs when, for each species, interspecific competition is stronger than is intraspecific competition.  This situation is termed mutual antagonism.