E. van Leeuwen, V. A. A. Jansen, and P. W. Bright. 2007. How population dynamics shape the functional response in a one-predator–two-prey system. Ecology 88:1571–1581.

Appendix B. Predator mediated coexistence.

In this section we determine the behaviour of the proportional density. Because we are mainly interested in the effects of switching here we set the growth functions of both prey species the same (g_i(N₁,N₂) = g(N₁, N₂)). This leads to the following two differential equations for the prey densities:

We are interested in the change of the proportion of prey 1 (η) over time, given by:

(B.1)

It follows from Eq. B.1 that dη∕dt = 0 at P = 0 or η = φ(η). In the main text we show that η = φ(η) for η = 0; η = 1; η^* = β₂∕(β₁ + β₂). For the last solution both prey coexist and the proportion of both prey is stable although the actual population densities can still show sustained oscillations. As in the main text, we will refer to this situation where the proportion of prey 1 becomes constant at η^*, and which corresponds to a two-dimensional plane in the three-dimensional state space, as the coexistence plane. This equilibrium only exists if β₁ and β₂ are both positive or are both negative, because otherwise η^* is below 0 or above 1. The second part of the equation is always zero or positive and represents the predation pressure on the prey, with P(f₁(N₁, N₂) + f₂(N₁, N₂)) the number of prey eaten and (N₁ + N₂) the total prey density.

To determine the stability of the coexistence plane (η^* ) we will define the function: V = (η -η^*)². We know that this function is zero at η = η^* and always positive otherwise. The derivative of this function is:

Now replace dη∕dt with Eq. B.1 and we get:

Whether this solution is positive or negative depends on the left part of the equation (-2(η - η^* )(φ(η) - η)). By substituting the functional responses and rewriting the results it can be shown that this has the same sign as -2(η - η^*)²η(1 - η)(β ₁ + β₂). The derivative of V over time is therefore always zero or negative if β₁ + β₂ > 0 and the function V is a Lyapunov function. It follows that if β₁ + β₂ > 0 and 0 < η^* < 1 the coexistence plane η^* is globally stable. In that case the fraction of prey 1 always converges to η^*. If β₁ + β₂ < 0 and 0 < η^* < 1 the opposite is true and the fraction of prey 1 in the population converges to 0 or 1. Note that although η^* is stable the prey and predator densities themselves do not have to be stable, e.g., they might show a limit cycle. The only case where the Lyapunov function need not hold is if prey densities become infinite in size. We therefore assume prey growth is bounded. This is biologically a reasonable assumption, because eventually growth will be limited by resource availability. This is always the case in a model with a finite carrying capacity.

The speed with which the system converges to the coexistence equilibrium is determined by the value of the derivative and, therefore, defined by three factors. The first one (∣η - η^* ∣) is how far we are from the coexistence plane. The second one (∣φ(η) - η∣) is a measurement of the strength of switching; the higher the preference of the predator for the more common prey species the bigger this value will be. The third one is the predator pressure on the prey. The bigger the derivative the faster the system will converge to the coexistence plane and the more robust the system is against disturbances.

Note that if we exclude dietary history (a_ij = a_ii = a_i) we get a predator-two prey system with a type II functional response (Eq. 3 in the main text). In this case β_i = a_i(a_i - a_j) and as a result β₂ = -β₁a₂∕a₁. Thus, when one β_i is positive the other will always be negative. In such a system the coexistence plane does not exist and one prey will go extinct. Also note that if the growth rates of both prey species differ and are of the form r_ig(N₁, N₂) we can still define the Lyapunov function by redefining η as:

Oaten and Murdoch (1975) analysed a similar functional response. Their results are similar to ours with one exception: they concluded that a₁₁a₂₂ > a₂₁a₁₂ must hold for switching to occur, implying that for these values prey coexistence occurs. Our analysis shows that a₁₁a₂₂ > a₁₂a₂₁ does lead to a disproportionally increasing proportion of successful attacks at increasing prey proportions, but under these parameter values the predator can still prefer the prey type over other prey types even at low prey densities and, therefore, cause prey extinction due to apparent competition. Our result does not classify this case as switching.

LITERATURE CITED

Oaten, A., and W. W. Murdoch. 1975. Switching, functional response and stability in predator-prey systems. American Naturalist 109:299–318.