Mathematical Model Describes the Interaction of Predator and Prey Populations.
In the 1920s, Alfred Lotka and Vittora Volterra turned their attention from competition (see Section 13.2) to the effects of predation on population growth. Independently, they proposed mathematical statements to express the relationship between predator and prey populations. They provided one equation for the prey population and another for the predator population.
The population growth equation for the prey population consists of two components: the exponential model of population growth (dN/dt = rN; see Chapter 9) and a term that represents mortality of prey from predation. Mortality resulting from predation is expressed as the per capita rate at which predators consume prey (number of prey consumed per predator per unit time). The per capita consumption rate by predators is assumed to increase linearly with the size of the prey population (Figure 14.1a) and can therefore be represented as cNprey, where c represents the capture efficiency of the predator, defined by the slope of the relationship shown in Figure 14.1a. (Note that the greater the value of c, the greater the number of prey captured and consumed for a given prey population size, which means that the predator is more efficient at capturing prey.) The total rate of predation (total number of prey captured per unit time) is the product of the per capita rate of consumption (cNprey) and the number of predators (Npred), or (cNprey)Npred. This value represents a source of mortality for the prey population and must be subtracted from the rate of population increase represented by the exponential model of growth. The resulting equation representing the rate of change in the prey population (dNprey/dt) is:
dNprey/dt=rNprey−(cNprey)NpreddNprey/dt=rNprey−(cNprey)Npred
The equation for the predator population likewise consists of two components: one representing birth and the other death of predators. The predator mortality rate is assumed to be a constant proportion of the predator population and is therefore represented as dNpred, where d is the per capita death rate (this value is equivalent to the per capita death rate in the exponential model of population growth developed in Chapter 9). The per capita birthrate is assumed to be a function of the amount of food consumed by the predator, the per capita rate of consumption (cNprey), and increases linearly with the per capita rate at which prey are consumed (Figure 14.1b). The per capita birthrate is therefore the product of b, the efficiency with which food is converted into population growth (reproduction), which is defined by the slope of the relationship shown in Figure 14.1b, and the rate of predation (cNprey), or b(cNprey). The total birthrate for the predator population is then the product of the per capita birthrate, b(cNprey), and the number of predators, Npred: b(cNprey)Npred. The resulting equation representing the rate of change in the predator population is:
dNpred/dt=b(cNprey)Npred−dNpreddNpred/dt=b(cNprey)Npred−dNpred
The Lotka–Volterra equations for predator and prey population growth therefore explicitly link the two populations, each functioning as a density-dependent regulator on the other. Predators regulate the growth of the prey population by functioning as a source of density-dependent mortality. The prey population functions as a source of density-dependent regulation on the birthrate of the predator population. To understand how these two populations interact, we can use the same graphical approach used to examine the outcomes of interspecific competition (Chapter 13, Section 13.2).
In the absence of predators (or at low predator density), the prey population grows exponentially (dNprey/dt = rNprey). As the predator population increases, prey mortality increases until eventually the mortality rate resulting from predation, (cNprey)Npred, is equal to the inherent growth rate of the prey population, rNprey, and the net population growth for the prey species is zero (dNprey/dt = 0). We can solve for the size of the predator population (Npred) at which this occurs:
cNpreyNpred=rNpreycNpred=rNpred=rccNpreyNpred=rNprey cNpred=r Npred=rc
Simply put, the growth rate of the prey population is zero when the number of predators is equal to the per capita growth rate of the prey population (r) divided by the efficiency of predation (c).
This value therefore defines the zero-growth isocline for the prey population (Figure 14.2a). As with the construction of the zero-growth isoclines in the analysis of the Lotka–Volterra competition equations (see Section 13.2, Figure 13.1), the two axes of the graph represent the two interacting populations. The x-axis represents the size of the prey population (Nprey), and the y-axis represents the predator population (Npred). The prey zero-growth isocline is independent of the prey population size (Nprey) and is represented by a line parallel to the x-axis at a point along the y-axis represented by the value Npred = r/c. For values of Npred below the zero-growth isocline, mortality resulting from predation, (cNprey)Npred, is less than the inherent growth rate of the prey population (rNprey), so population growth is positive and the prey population increases, as represented by the green horizontal arrow pointing to the right. If the predator population exceeds this value, mortality resulting from predation, (cNprey)Npred, is greater than the inherent growth rate of the prey population (rNprey) and the growth rate of the prey becomes negative. The corresponding decline in the size of the prey population is represented by the green arrow pointing to the left.
Likewise, we can define the zero-growth isocline for the predator population by examining the influence of prey population size on the growth rate of the predator population. The growth rate of the predator population is zero (dNpred/dt = 0) when the rate of predator increase (resulting from the consumption of prey) is equal to the rate of mortality:
b(cNprey)Npred=dNpredbcNprey=dNprey=dbcb(cNprey)Npred=dNpred bcNprey=d Nprey=dbc
The growth rate of the predator population is zero when the size of the prey population (Nprey) equals the per capita mortality rate of the predator (d) divided by the product of the efficiency of predation (c) and the ability of predators to convert the prey consumed into offspring (b). Note that these are the two factors that determine the per capita predator birthrate for a given prey population (Nprey). As with the prey population, we can now use this value to define the zero-growth isocline for the predator population (Figure 14.2b). The predator zero-growth isocline is independent of the predator population size (Npred) and is represented by a line parallel to the y-axis at a point along the x-axis (represented by the value Nprey = d/bc). For values of Nprey to the left of the zero-growth isocline (toward the origin) the rate of birth in the predator population, b(cNprey)Npred, is less than the rate of mortality, dNpred, and the growth rate of the predator population is negative. The corresponding decline in population size is represented by the red arrow pointing downward. For values of Nprey to the right of the predator zero-growth isocline, the population birthrate is greater than the mortality rate and the population growth rate is positive. The increase in population size is represented by the vertical red arrow pointing up.
As we did in the graphical analysis of competitive interactions (see Section 13.3, Figure 13.2), the two zero-growth isoclines representing the predator and prey populations can be combined to examine changes in the growth rates of two interacting populations for any combination of population sizes (Figure 14.2c). When plotted on the same set of axes, the zero-growth isoclines for the predator and prey populations divide the graph into four regions. In the lower right-hand region, the combined values of Nprey and Npred are below the prey zero-isocline (green dashed line), so the prey population increases, as represented by the green arrow pointing to the right. Likewise, the combined values lie above the zero-growth isocline for the predator population so the predator population increases, as represented by the red arrow pointing upward. The next value of (Nprey, Npred) will therefore be within the region defined by the green and red arrows represented by the black arrow. The combined dynamics indicated by the black arrow point toward the upper right region of the graph. For the upper right-hand region, combined values of Nprey and Npred are above the prey isocline, so the prey population declines as indicated by the green horizontal arrow pointing left. The combined values are to the right of the predator isocline, so the predator population increases as indicated by the vertical red arrow pointing up. The black arrow indicating the combined dynamics points toward the upper left-hand region of the graph. In the upper left-hand region of the graph, the combined values of Nprey and Npred are above the prey isocline and to the left of the predator isocline so both populations decline. In this case, the combined dynamics (black arrow) point toward the origin. In the last region of the graph, the lower left, the combined values of Nprey and Npred are below the prey isocline and to the left of the predator isocline. In this case, the prey population increases and the predator population declines. The combined dynamics point in the direction of the lower left-hand region of the graph, completing a circular, or cyclical, pattern, where the combined dynamics of the predator and prey populations move in a counterclockwise pattern through the four regions defined by the population isoclines.
