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 A. A derivation of the functional response.

We will derive the functional response of a predator preying on two prey species, incorporating the predator’s dietary history, in a similar fashion as the Holling type II functional response is usually derived (e.g., Metz and Diekmann 1986) using reaction kinetics. We follow Oaten and Murdoch(1975) and divide predators according to foraging state: some are searching for prey, others are handling one prey species and a third group is handling the second prey species. To be able to describe the change in handling time and attack rate with repeated encounters we need to keep track of the successful attack history of the predator, but for simplicity, we will only consider the last prey successfully attacked. The introduction of dietary history means that we will have to keep track of all possible states that the predator can be in. We discriminate between predators that have successfully attacked different prey last time. Therefore we have two states currently searching, one has successfully attacked prey 1 (P₀₁) the other prey 2 (P₀₂). Similarly we have two states currently handling prey 1 (P₁₂ and P₁₁) and two states handling prey 2 (P₂₁ and P₂₂). Predators in the states P₁₂ and P₂₁ are predators that changed from one prey type to another. A graphical representation of the model is shown in figure 1 of the main text.

Predators change from one state to another. When a predator handling prey is done its state will change to a predator searching for prey and, depending on which prey it just handled, change to state P₀₁ or P₀₂. The rate of change from the handling state to the searching state depends on the handling time (T_ij). The handling time for one prey can be different from the handling time for the other prey. Also it might take a predator longer to handle prey 1 if its previous prey was 2. Therefore there are four different handling times (T₁₁, T₁₂, T₂₂ and T₂₁) in our model. Similarly, predators will change from a searching to a handling state when they capture a prey. The rate of change depends on the rate of successful attacks (a_ij) and the density of the prey (N₁ or N₂). Again we have rates that depend on a predator’s last prey and on its current prey (a₁₁, a₁₂, a₂₂ and a₂₁).

The prey densities (N₁ and N₂) change over time as:

(A.1)

with g_i(N_i, N_j) the growth function for prey species i. The number of predators in their respective states (following Fig. 1 of the main text) are described by:

	= -a11N1P01 - a21N2P01 + P11 + P12 + f(b,d)	(A.2)
	= -a12N1P02 - a22N2P02 + P21 + P22 + f(b,d)	(A.3)
	= a11N1P01 -P11 + f(b,d)	(A.4)
	= a12N1P02 -P12 + f(b,d)	(A.5)
	= a21N2P01 -P21 + f(b,d)	(A.6)
	= a22N2P02 -P22 + f(b,d)	(A.7)

with f(b,d) the change in predator density due to birth and death. The total number of predators, P, is:

which can be rewritten as:

(A.8)

The number of predators in each state changes due to births and deaths, but because one can reasonably assume that prey attack and handling are fast processes compared to the births and death rates of predators, we can neglect the birth and death terms and assume that the number of predators in a certain state will converge to a quasi-steady state where the number of predators in each state is more or less constant. Therefore, we can set the left hand side of Eqs. A.2 to A.7 to zero. We can substitute Eq. (A.8 ) into Eq. (A.7 ) and then solve Eqs. A.2 to A.7 for the searching predators (P_0i) resulting in an equation were the searching predator density is dependent on the species densities N₁, N₂ and P:

(A.9)

From the prey Eqs. (A.1 ) it follows that the number of prey successfully attacked is:

Substituting P_0i with (A.9 ) results in the functional response (Eq. 2 in the main text). Note that the functional response f_i(N₁, N₂) is linear in N_i. Therefore, the population dynamical model is invariant in the positive octant: if the initial conditions are positive for the densities N₁, N₂ and P these densities remain positive ever after.

LITERATURE CITED

Metz, J. A. J., and O. Diekmann. 1986. The dynamics of physiologically structured populations, number 68 in Lecture Notes in Biomathematics. Springer-Verlag, Berlin, Germany.

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