Population Dynamics in Predation PDF

## 7.5 Population Dynamics in Predation - What role do predators play in the dynamics of their prey, or prey in the dynamics of their predators? Are there any common patterns of dynamics that emerge? - Previous sections showed that there are no simple answers to these questions. It depends on the...

## 7.5 Population Dynamics in Predation - What role do predators play in the dynamics of their prey, or prey in the dynamics of their predators? Are there any common patterns of dynamics that emerge? - Previous sections showed that there are no simple answers to these questions. It depends on the details of individual predator and prey behavior, on the possible compensatory responses at the individual and population levels, and so on. - However, rather than becoming discouraged by its complexity, we can develop a logical thinking process on these dynamics by starting "simply" and then adding new aspects one at a time to build a more realistic picture. ### 7.5.1 The Bases of Predator-Prey Dynamics: The Tendency for Cycles - We start by making a conscious oversimplification, ignoring everything except the predator and prey, and asking what basic trend we would see in the dynamics of their interactions. - It turns out that the basic trend is to show joint oscillations—cycles—in abundance. With this established, we can turn to many other important factors that could modify or abolish this basic trend. However, rather than exploring all of them individually, Sections 7.5.4 and 7.5.5 examine just two of the most important: spatial density and spatial fragmentation. - Of course, these two factors alone cannot tell the whole story; they illustrate how differences in the dynamics of the prey and predator can be explained by the different impacts of different factors, each potentially impacting those dynamics. - Starting simply, then, assume that there is a large prey population. Predators should do well in this situation; they should consume a lot of prey and therefore increase their own abundance. - A large prey population, then, allows the existence of a large predator population. However, a large predator population causes a small prey population. Predators then face a problem: a high number of them, and very little food. Their abundance declines. This, however, puts less pressure on the prey. A small predator population allows a large prey population to build—and the two populations come back around again. - In short, there is a basic tendency for predators and prey to experience joint oscillations in abundance—population cycles (Figure 7.15)—essentially due to the time lag in the predator’s abundance response relative to the prey's abundance response, and vice versa. - A "time lag" in the response means, for example, that a high predator abundance reflects a high prey abundance in the past, but it coincides with a decline in prey abundance, and so on. A simple mathematical model—the Lotka-Volterra model—essentially conveys this same message as described in Box 7.2. ### 7.2 Quantitative Aspects #### The Predator-Prey Model of Lotka-Volterra Like the models in Boxes 5.4 and 6.1, one of the cornerstones of mathematical models in ecology is described and explained here. The model is known (like the interspecific competition model in Box 6.1) by the names of its creators: Lotka and Volterra (Volterra, 1926; Lotka, 1932). It has two components: P, the number present in a predator population (or consumer), and N, the number or biomass present in a prey population or plant population. It is assumed that, in the absence of consumers, prey populations increase exponentially (Box 5.4): - $$dN/dt = rN$$ However, we also need a term to indicate that prey individuals are removed from the population by predators. They will do this at a rate that depends on the frequency of encounters between prey and predator, which will increase with increasing numbers of predators (P) and prey (N). The exact number of encounters and consumption, however, will also increase with the efficiency of the predator’s search and attack, given as a. The consumption rate of prey will then be aPN, and in total: - $$dN/dt = rN - aPN$$ (1) Turning to the number of predators, in the absence of food, it is assumed that they decline exponentially over the period of starvation: - $$dP/dt = -qP$$, where q is their mortality rate. This, however, is countered by the birth of predators, a rate that is assumed to be dependent on (i) the rate at which food is consumed, aPN, and (ii) the efficiency of the predator, f, in turning that food into offspring of the predator. In total: - $$dP/dt = faPn - qP$$ (2) Equations 1 and 2 comprise the Lotka-Volterra model. The properties of this model can be investigated by obtaining zero isoclines (see Box 6.1). There are separate zero isoclines for predators and prey, both plotted on a graph as prey density (x axis) versus predator density (y axis) (Figure 7.16). The prey zero isocline represents combinations of prey and predator densities that lead to an unchanged prey population, dN/dt = 0, while the predator zero isocline represents combinations of prey and predator densities that lead to an unchanged predator population, dP/dt = 0. In the case of the prey, we "solve" by substituting dN/dt = 0 into Equation 1, obtaining the equation for the isocline as: - $$P = r/a$$ Thus, given that r and a are constants, the zero isocline for the prey is a line where P itself is constant (Figure 7.16a): the prey increases when predator abundance is low (P < r/a), but it decreases when it is high (P> r/a). Similarly, for the predators, we solve by substituting dP/dt = 0 into Equation 2, giving the equation for the isocline as: - $$N = q/fa$$ The predator zero isocline is therefore a line along which N is constant (Figure 7.16b): predators diminish when prey abundance is low (N < q/fa), but increase when it is high (N > q/fa). Plotting the two isoclines (and their corresponding sets of arrows) together in Figure 7.17 shows the joint behavior of the populations. The various combinations of increases and decreases listed above mean that the populations undergo "joint oscillations" or "joint cycles" in abundance; "joint" in the sense that the numerical ups and downs of predators and prey are linked, with predator abundance following the prey's abundance (discussed biologically in the main body of the text). It is important to note that the model does not "predict" the exact patterns of abundance that it generates. The real world is obviously more complex than that imagined by this model. Therefore, the model does not capture every detail, but it does contain the essential tendencies of the joint cycles found in predator-prey interactions. ### 7.5.2 Predator-prey Cycles in Practice From the basic tendency of a predator-prey interaction to produce a joint oscillation in abundance, we might expect to see these cycles in real populations. However, several aspects of predator and prey ecology have been ignored in order to show this basic trend, and these can modify the expectation significantly. It is not a surprise, then, that there are relatively few clear examples of predator-prey cycles, though some have received much attention from biologists and demonstrate the basic tendency for cycles. They do occur sometimes. In several cases, for example, joint oscillations of prey and predator can be generated over several generations in the laboratory (Figure 7.18a; see also Figure 7.22c). Among populations in the wild, there are several examples where cycles in prey and predator abundances can be discerned. Cycles in populations of hares, in particular, have been discussed by ecologists since the 1920s, and they have been recognized by fur trappers for more than 100 years. The most famous of these is the snowshoe hare (Lepus americanus), which exhibits a "10-year" cycle in the boreal forests of North America (it actually varies from 8 to 11 years; see Figure 7.18b). The snowshoe hare is the dominant herbivore in the region, feeding on the terminal branches of a variety of shrubs and small trees. It has several predators, including the Canada lynx (Lynx canadensis), which also shows cycles of similar length. The hare cycles often involve changes in abundance of 10 to 30 times, and in some habitats, changes on the order of 100 times can occur These changes are remarkable in that they are virtually synchronous across a broad area from Alaska to Newfoundland. However, are the hare and lynx participants in a predator-prey cycle? This might seem less likely once we consider the number of other species with which both interact. Their food web (see Section 9.5) is shown in Figure 7.19. In fact, both experimental studies (Krebs et al., 2001) and more sophisticated statistical analyses of population dynamic data (Stenseth et al., 1997) suggest that, while the hare’s dynamics are determined by interactions with both their food and their predators (particularly the lynx), lynx dynamics are largely determined by their interaction with their snowshoe hare prey, consistent with the food web. Both the plant-hare and the predator-hare interactions have a tendency to cycle—but, in practice, the cycle appears to be driven by the interaction between them. This reminds us that even when we have a predator-prey pair, we may not simply be observing predator-prey oscillations. ### 7.5.3 Disease Dynamics and Cycles - Cycles are also apparent in the dynamics of many parasites, especially microparasites (bacteria, viruses, etc.). To understand the dynamics of any parasite, a good starting point is its basic reproduction rate, R0. - In relation to microparasites, R0 is the mean number of new infected hosts that would result from a single infected host in a population of susceptible hosts. An infection will eventually die out for R0 < 1 (each infection in the present leads to less than one infection in the future), but an infection will expand for R0 > 1. Therefore, there is a “threshold of transmission,” where R0 = 1, which must be crossed for the disease to spread. A derivation of R0 for microparasites with direct transmission (see Figure 7.12c) is provided in Box 7.4.