Prey Predator System

Predation is often defined as an interspecific interaction in which an individual of one animal species kills an individual of another species for dietary use. As a broader definition, predation can include an interaction between an animal and seeds or between a parasitoid and a host. However, predation rarely includes disease-causing organisms or herbivores that do not kill their food.

Predation is one of the most important interactions between species, ranking with parasitism, competition, and mutualism. Predation can affect changes in population sizes, traits, or phenotypes, and consequently promote the evolution of underlying genetic traits. These interactions are termed 'prey-predator' or 'predator-prey' systems. The existence of such interactions creates a link between the prey and predator species, termed a 'trophic link'. The assembly of trophic links within a community forms a 'food web'. Predation probably plays a major role in determining the life-history pattern of every species, and organismal complexity may increase due to predation. Here we consider the co-dynamics of prey and predator populations and coevolution of prey and predator species. We also consider the relationship between population dynamics and evolutionary change in trait values.

Hereafter we focus on a system involving one predator and one prey species. The following dynamic model describes temporal changes in the predator and prey populations:

where P and N are the population sizes (or densities) of the predator and prey; t is time; d(P) and r(N) are the intrinsic death rate of the predator and the intrinsic growth rate of the prey; and fN, P) and g(N, P) are the per capita rate of predation and the contribution of predation to the predator's per capita growth rate, respectively. These factors are often considered independent of predator abundance P. Conditions where f and g depend on the ratio N/P are analyzed (this type of predation is called 'ratio-dependent predation').

Functions f and g are characterized by the predatory interaction; they are termed the functional response and the numerical response, respectively. Three variations of fN) exist: (1) a linear relationship (fN) = aN); (2) a convex curve f(N) = aN/(1 + ahN)); and (3) a sigmoid curve (e.g., fN) = aN2/(1 + ahN2)), where h is handling time and a is the predation coefficient. For the mathematically simplest model with a type 2 functional response:

where k is the magnitude of an intraspecific density effect from growth of the prey population, and b is the conversion efficiency of ingested prey into the predator. This system can produce either a stable or unstable equilibrium (Figures 1a and 1b). Note that increases in the predator population lag one-quarter of a period behind increases in the prey population (Figure 1c). This is fairly intuitive, since the predator population decreases to the left of the predator's 'null cline' (the vertical line in Figures 1a and 1b) and increases on the right side of the line, whereas the prey population increases below the prey null cline (the parabolic curve in Figures 1a and 1b) and decreases above the curve. These null clines are obtained by the solution of the equations dP/dt = 0 and dN/dt = 0, respectively. The equilibrium of coexistence is obtained by the intersection of these null clines.

Especially in host-parasitoid dynamics, time-discrete models, such as the famous Nicholson-Bailey model, are more reasonable because the hosts reproduce seasonally, and the life cycle of a parasitoid often synchronizes with that of its host. Because of time discreteness, these models are less likely to produce a stable equilibrium.

Using the Nicholson-Bailey model, many studies have incorporated factors representing the spatial distributions and behavioral characteristics of both the host and para-sitoid. If parasitoids focus on the center of the host distribution, the risk of parasitism is low for hosts that are far from the population center. This aggregative response has a stabilizing effect on the host-parasitoid

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Figure 1 Prey-predator dynamics of model [2] and related empirical data. Parameter values are: r = 2, b = 1, k = 0.003, d = 1.6, h = 0.3; a = 0.02 in panel (a) and a = 0.01 in panels (b) and (c). (d) From Shertzer KW, Ellner SP, Fussmann GF, and Hairston NG, Jr. (2002) Predator-prey cycles in an aquatic microcosm: Testing hypotheses of mechanism. Journal of Animal Ecology 71: 802-815.

system. If parasitoids that share an individual host interfere mutually, a type 3 functional response is again possible because the potential for interference increases as the host density decreases. Another important factor in prey-predator systems is stochasticity. Many stochastic factors have been studied. For example, demographic stochasticity and genetic drift usually destabilize the equilibrium of prey-predator systems, despite a few examples to the contrary.

Hereafter we focus on time-continuous models. The Lotka-Volterra predator-prey model is an extremely simplified model of a prey-predator system, involving the case where k = 0 and h = 0 in model [2]. Even in this simple model, a time-dependent analytical solution has not been obtained. The Lotka-Volterra model has an interior equilibrium, (N, P) = (d/ba, r/a). Because this equilibrium is neutrally stable, an interior equilibrium of variants of the Lotka-Volterra model can produce either stability or instability, as shown in Figures 1a and 1b. Type 3 functional responses and the density effects from both prey and predator growth rates produce stabilizing effects on the prey-predator dynamics. Type 2 functional responses and time lags between predation and population growth have destabilizing effects.

Increasing productivity of the prey usually has a destabilizing effect on the equilibrium. This is called the 'paradox of enrichment'. This is an intuitive result because the null cline of the prey (parabolas in Figures 1a and 1b) shifts to the right; therefore, the equilibrium (the intersection of the parabola and the vertical line) point occurs to the left of the parabola's peak, as in Figure 1b.

Changes in the predator population lag behind changes in the prey population by one-quarter of a period (Figure 1c); however, few examples of such prey-predator cycles have been observed in the field. The prey population often regulates the predator population, whereas the latter less frequently regulates the former. One of the best examples is presented in Figure 1d, showing a one-half period lag between prey and predator cycles, as discussed later.

Strategy and Tactics of Prey-Predator Systems

Predators and parasites usually have traits that improve their predatory efforts, such as reduced handling time, increased search ability, or increased capture rate. However, a tradeoff can exist between predation ability and the per capita death rate of the predator population.

If the predator can process more than one species or type of prey, then prey choice becomes another important factor in the prey-predator dynamics. If the predator focuses its foraging effort on one prey species, the other prey species may experience a reduced risk of predation.

Frequency-dependent prey choice is called switching predation. Positive and negative switching refer to positive and negative relationships, respectively, between relative prey density and the per capita risk of predation for that prey species. Prey switching is a typical mechanism that enhances a type 3 functional response. Many theoretical works have shown that switching has a stabilizing effect on the relative frequency of the two prey populations, but does not always stabilize total prey density.

There are many empirical examples of the existence of switching predation; there are equally many examples of nonswitching and negative switching. Switching predation has been reported in a variety of predators and even herbivores. There are two types of indices for prey choice. A classic index is Ivlev's index (denoted here by I), which is defined as Ij = (Ij' - 1)/(I,-' + 1), where Ij' is Shorygin's index, I,' = (N, / P tN, )/(X, / p X,), and Nj and X j, respectively, represent the number of preys of type in the environment and the number of preys of type j in the predator's stomach contents. If I is positive for a particular prey species, the relative number of preys eaten by the predator is larger than the relative number of preys in the environment. However, I varies depending on which species are included as food items. In cases 1 and 2 in Table 1, a predator does not consume species 3 (X3 = 0). I for prey 1 is 0.1 if species 3 is excluded from prey items, whereas a value of 0.3 results if species 3 is included. If a predator only rarely consumes prey 3, I will change considerably. I also depends on the abundances ofrarely consumed species. In cases 3 and 4 in Table 1, I for prey 1 is highly dependent upon the abundance of prey 2. Jean Chesson's index, which is defined as (Xj/Nj)/(^2 Xj/N'), is more robust for the abundance of species that are rarely or never consumed, as shown in Table 1 .

Prey and hosts typically have traits that help them avoid predation and parasitism. Some examples of such antipredator traits are: being vigilant, seeking refuges, having hard or thorny skin, and producing toxic chemicals. Tradeoffs usually exist between escape from predation and other factors that affect fitness, for example, growth rate, fecundity, or survival rate. If such tradeoffs do not exist, antipredator efforts could evolve infinitely. If more than one species or type of predator shares a common prey species, the prey may have multiple anti-predator traits. If some antipredator efforts are effective against one predator but not against other predators, a tradeoff is likely to exist between antipredator traits that are effective against different predators.

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