PREDATOR - PREY OR PREDATOR POPULATION – RESOURCE POPULATION Theoretical Models The first models to predict the changes in the population number of a predator population and its resource population were developed by A.J. Lotka and V. Volterra in the 1920s. In their equations, they accounted for inverse effect of population numbers of one species on the other in a predatorresource system by assuming that the growth rates of a predator and its resource could be modeled using a modification of the exponential growth equation. The basic assumptions of this approach are 1) that the birth rate of the predator is a function of the number of the resource population and 2) that the death rate of the resource population is a function of the number of predators. We will designate
P and 2) the number in the resource population as R. 1) the number in the predator population as
First, let us consider the growth rate of the resource population. When the predator is not present, then it is assumed that the resource increases exponentially, or
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dR = rR dt where r is the intrinsic rate of increase for the resource population. However, when the predator is present, then the resource growth rate is reduced by the number of predators and their ability to obtain the resource. This reduction can be incorporated into the model as
dR = (r – aP)R dt where a is a constant that indicates the successful individual attack rate by the predators. If P were constant, then R would grow (or decline) exponentially. Because the predator is dependent upon the resource population as a food resource, when the resource is absent the predator population declines exponentially. This rate of decrease can be symbolize as
dP = - dP dt where d is the death rate of the predator in the absence of the resource population. However, in the presence of the resource, the predator population will increase at a rate dependent upon 1) the number of the resource population and
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2) the ability of the predator to utilize the resource population. The growth rate in predator population is then dP = (- d + bR) P dt where b is a constant that indicates the individual conversion rate by the predator of the resource population into new predators. If R were constant, then P would follow the exponential growth Notice that the last terms in these two equations when they are multiplied out,
- aPR and bPR, involve the products of the numbers of the two species. In other words, either the reduction in the growth rate of the resource or the increase in the growth rate of the predator due to predation may change very quickly because the terms resulting in these changes are a function of the joint densities of the two species. Let us solve these two equations for their equilibria. The equation for the growth rate of the resource population can only be zero if
r – aP = 0 In other words, if
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P=r a
then
dR = 0 dt
and there is no change in the numbers of the resource population. As we did for two competing species, let us plot this on a graph that gives the number of predators on one axis and the number in resource population on the other axis.
In general, the number in the predator population will be larger than the number in the resource population, so the scales on these two axes will be different. The horizontal line in Figure 1 indicates this equilibrium, the zero isocline for the resource population. Above this line, the number resource population decreases because of the large number predators, and below the line it increases because of the small number of predators.
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Figure 1: The change in population numbers for the resource species
Likewise, the equilibrium for the predator population occurs when -d + bR = 0 because the growth rate pf the predator population is zero. Therefore, if
R=d b then dP = 0 dt and there is no change in the number of predators. This equilibrium is called the predator zero isocline and is plotted as a vertical line in Figure 2. To the right of this line, the number of predators increases because of the excess of the resource, and to the left of it
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the number of predators decreases because there are few individuals in the resource population.
Figure 2: The change in population numbers for the predator species
At first glance, it would seem that the point where these two lines cross – that is, where both
dR = 0 and dt
dP = 0 dt
would be stable equilibrium. However if the two parts of Figure 1 and 2 are combined, as in Figure 3, then it is obvious that the numbers of the two species do not approach these equilibrium densities when they are initially different from these values.
For example, in upper right quadrant, the number of predators is increasing while the number of prey is decreasing. Likewise, in the lower left quadrant, the number of predators is decreasing and the number of prey increasing. As a result, the
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numbers never converge to the equilibrium but oscillate around it. The size of the oscilllation is dependent upon the starting numbers and does not increase or decrease in size with time. However, random variation in the environment affecting the species numbers may move them to other predator-resource combinations. The numbers will then cycle from this point, indicating that the cycles do not converge to a single stable cycle. However, the equilibrium numbers do indicate the average expected number of each species over an extended period of time.
Figure 3: The joint change in numbers for the resource and predator populations
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The numbers of the two species can also be plotted over time. Figure 4 gives the population numbers for both species for two cycles, beginning at a point in the middle of the lower section of Figure 3.
Notice that the predator numbers follow the resource numbers, reaching maximums and minimums a quarter of a cycle later. The differences in the number of the two species results from a time lag effect resource numbers on predator numbers.
For example, in lower left quadrant in Figure 3, resource numbers are increasing and predator numbers are still declining, and the predator numbers do not begin to increase until the lower right quadrant.
Let us illustrate these changes with a numerical example. Assume that r = 0.2 and a = 0.01, so that the number of predators at which dR/dt = 0, the resource zero isocline, is
P = r/a = 0.2/0.01 = 20. Likewise, if d = 0.2 and b = 0.001, the resource number at which dP/dt = 0, the predator isocline, is
R = d/b = 0.2/0.001 = 200.
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If we let P = 10 and R = 100, a point in the lower left quadrant, then the change in resource numbers is dR/dt = (r –aP)R = [0.2 – (0.01)(10)]100 = 10. The change in predator numbers is dP/dt = (-d + bR)P = [-0.2 + (0.001)(100)]10 = -1 Therefore, the number in the resource population is increased to 110 and the number of the predator population decreased to 9. Similar change can be calculated for points in the other three quadrants. One prediction of this simple model is that predator and prey numbers should be oscillate.
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