A logistic model of interspecific competition
The 'LotkaVolterra' model of interspecific competition (Volterra, 1926; Lotka, 1932) is an extension of the logistic equation described in Section 5.9. As such, it incorporates all of the logistic's shortcomings, but a useful model can none the less be constructed, shedding light on the factors that determine the outcome of a competitive interaction.
The logistic equation:
dt K
contains, within the brackets, a term responsible for the incorporation of intraspecific competition. The basis of the LotkaVolterra model is the replacement of this term by one which incorporates both intra and interspecific competition.
The population size of one species can be denoted by N1, and that of a second species by N2. Their carrying capacities and intrinsic rates of increase are K1, K2, r1 and r2, respectively.
Suppose that 10 individuals of species 2 have, between them, the a: the competition same competitive, inhibitory effect on coefficient species 1 as does a single individual of species 1. The total competitive effect on species 1 (intra and interspecific) will then be equivalent to the effect of (Nl + N2/10) species 1 individuals. The constant (1/10 in the present case) is called a competition coefficient and is denoted by a12 ('alphaonetwo'). It measures the per capita competitive effect on species 1 of species 2. Thus, multiplying N2 by a12 converts it to a number of 'Njequivalents'. (Note that a12 < 1 means that individuals of species 2 have less inhibitory effect on individuals of species 1 than individuals of species 1 have on others of their own species, whilst a12 > 1 means that individuals of species 2 have a greater inhibitory effect on individuals of species 1 than do the species 1 individuals themselves.)
LotkaVolterra model: a logistic model for two species
The crucial element in the model is the replacement of N1 in the bracket of the logistic equation with a term signifying 'Nj plus Nrequivalents', i.e.:
must also therefore be 'zero isoclines' for each species (lines along which there is neither an increase nor a decrease), dividing the combinations leading to increase from those leading to decrease. Moreover, if a zero isocline is drawn first, there will be combinations leading to an increase on one side of it, and combinations leading to a decrease on the other.
In order to draw a zero isocline for species 1, we can use the fact that on the zero isocline dN1/dt = 0 (by definition), that is (from Equation 8.3):
and in the case of the second species:
dN 2 dt
This is true when the intrinsic rate of increase (r1) is zero, and when the population size (N1) is zero, but  much more importantly in the present context  it is also true when:
behavior of the LotkaVolterra model is investigated using 'zero isoclines'
These two equations constitute the LotkaVolterra model.
To appreciate the properties of this model, we must ask the question: when (under what circumstances) does each species increase or decrease in abundance? In order to answer this, it is necessary to construct diagrams in which all possible combinations of species 1 and species 2 abundance can be displayed (i.e. all possible combinations of N1 and N2). These will be diagrams (Figures 8.7 and 8.9), with N1 plotted on the horizontal axis and N2 plotted on the vertical axis, such that there are low numbers of both species towards the bottom left, high numbers of both species towards the top right, and so on. Certain combinations of N1 and N2 will give rise to increases in species 1 and/or species 2, whilst other combinations will give rise to decreases in species 1 and/or species 2. Crucially, there which can be rearranged as:
In other words, everywhere along the straight line which this equation represents, dNj/d t = 0. The line is therefore the zero isocline for species j and since it is a straight line it can be drawn by finding two points on it and joining them. Thus, in Equation 8.7, when:
and when:
and joining them gives the zero isocline for species 1. Below and to the left of this, the numbers of both species are relatively low, and species 1, subjected to only weak competition, increases in abundance (the arrows in the figure, representing this increase, point from left to right, since Nl is on the horizontal axis). Above and to the right of the line, the numbers are high, competition is strong and species 1 decreases in abundance (arrows from right to left). Based on an equivalent derivation, Figure 8.7b has combinations leading to an increase and decrease in species 2, separated by a species 2 zero isocline, with arrows, like the N2 axis, running vertically.
Finally, in order to determine the outcome of competition in this model, it is necessary to fuse Figures 8.7a and b, allowing the behavior of a joint population to be predicted. In doing this, it should be noted that the arrows in Figure 8.7 are actually vectors  with a strength as well as a direction  and that to determine
Joint
N2 population
Joint
N2 population
the behavior of a joint N1, N2 population, the normal rules of vector addition should be applied (Figure 8.8).
Figure 8.9 shows that there are, in fact, four different ways in which the two zero isoclines can be arranged relative to one another, and the outcome of competition will be different in each case. The different cases can be defined and distinguished by the intercepts of the zero isoclines. For instance, in Figure 8.9a:
The first inequality (K1 > K2a12) indicates that the inhibitory intraspecific effects that species 1 can exert on itself are greater than the interspecific effects that species 2 can exert on species 1. The second inequality, however, indicates that species 1 can exert more of an effect on species 2 than species 2 can on itself. Species 1 is thus a strong interspecific competitor, whilst species 2 is a weak interspecific competitor; and as the vectors in Figure 8.9a show, species 1 drives species 2 to extinction and attains its own carrying capacity. The situation is four ways in which the two zero isoclines can be arranged strong interspecific competitors outcompete weak interspecific competitors
Figure 8.9 The outcomes of competition generated by the LotkaVolterra competition equations for the four possible arrangements of the N1 and N2 zero isoclines. Vectors, generally, refer to joint populations, and are derived as indicated in (a). The solid circles show stable equilibrium points. The open circle in (c) is an unstable equilibrium point. For further discussion, see the text.
Figure 8.9 The outcomes of competition generated by the LotkaVolterra competition equations for the four possible arrangements of the N1 and N2 zero isoclines. Vectors, generally, refer to joint populations, and are derived as indicated in (a). The solid circles show stable equilibrium points. The open circle in (c) is an unstable equilibrium point. For further discussion, see the text.
reversed in Figure 8.8b. Hence, Figures 8.8a and b describe cases in which the environment is such that one species invariably outcompetes the other.
In Figure 8.9c:
Thus, individuals of both species compete more strongly with individuals of the other species than they do amongst themselves. This will occur, for example, when each species produces a substance that is toxic to the other species but is harmless to itself, or when each species is aggressive towards or even preys upon individuals of the other species, more than individuals of its own species. The consequence, as the figure shows, is an unstable equilibrium combination of N1 and N2 (where the isoclines cross), and two stable points. At the first of these stable points, species 1 reaches its carrying capacity with species 2 extinct; whilst at the second, species 2 reaches its carrying capacity with species 1 extinct. Which of these two outcomes is actually attained is determined by the initial densities: the species which has the initial advantage will drive the other species to extinction.
Finally, in Figure 8.9d:
K1 K2
In this case, both species have less competitive effect on the other species than they have on themselves. The outcome, as Figure 8.9d shows, is a stable equilibrium combination of the two species, which all joint populations tend to approach.
Overall, therefore, the LotkaVolterra model of interspecific competition is able to generate a range of possible outcomes: the predictable exclusion of one species by another, exclusion dependent on initial densities, and stable coexistence. Each of these possibilities will be discussed in turn, alongside the results of laboratory and field investigations. We will see that the three outcomes from the model correspond to biologically reasonable circumstances. The model, therefore, in spite of its simplicity and its failure to address many of the complexities of the dynamics of competiton in the real world, serves a useful purpose.
Before we move on, however, one particular shortcoming of the Lotka Ks, as and rs Volterra model is worth noting. The outcome of competition in the model depends on the Ks and the as, but not on the rs, the intrinsic rates of increase. These determine the speed with which the outcome is achieved but not the outcome itself. This, though, seems to be a result peculiar to competition between only two species, since in models of competition between three or more species, the Ks, as and rs combine to determine the outcome (Strobeck, 1973).
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