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 C. Bifurcations in the population dynamics.

We can derive the functional response for both prey together (S = N₁ + N₂) by realizing that on the coexistence plane the prey densities are defined by: N₁ = cN₂, with c a constant (with value β₂ ∕ β₁). This means that N_i can be written as a function of S (N₁ = Sc ∕ (1 + c); N₂ = S ∕ (1 + c)). If we substitute S and c in f₁(N₁, N₂) and f₂(N₁, N₂) the result is a type II functional response (this is the case for any c):

(Eq. 7 in the main text.) For c = β₂ ∕ β₁ we find that

,

and δ = (a₁₁ - a₂₁)(a₂₂ - a₁₂).

Because this functional response is rather complex we study a simple, but illustrative scenario. We will study the case for which all handling times and attack rates only differ for predators that change prey type: T_ii = T; T_ij = T_s; a_ii = a; a_ij = a_s. Under this scenario the functional response is:

.

(C.1)

This leads to a situation where both prey species will have identical densities on the coexistence plane (η^* = 0.5). If we also assume the prey to have similar conversion rates resulting in (c = c_i), it is possible to get a complete understanding of the possible dynamics.

If there is only one prey we recover the Rosenzweig-MacArthur model for one species (either N₁ or N₂) and the predator (P). If the attack rate is sufficiently high, i.e., if:

,

(C.2)

the equilibrium is unstable and the dynamics go to a limit cycle, otherwise the system converges to a stable equilibrium. This transition from equilibrium to a limit cycle is known as a Hopf bifurcation. When

,

(C.3)

the predator will die out and the prey goes to carrying capacity, i.e., N₁ + N₂ = K. The case where the system switches from a stable equilibrium with the predator present to a stable equilibrium without the predator present is a transcritical bifurcation. Next we look at the possible invasion by the second prey species. If a > a_s β₁ and β₂ are positive and a second prey species can invade. In any other case a second prey species can not invade the system.

For the dynamics of two coexisting prey types we can get comparable results (Fig. 4b in the main text). First we only consider the attack rate and set T = T_s in Eq. C.1 . The stability boundaries are found in a similar way as on the plane N₂ = 0. The Hopf bifurcation:

,

and the transcritical bifurcation:

.

Note that these conditions correspond to Eqs. C.2 and C.3, but with the attack rate a replaced by the average attack rate over the two types ((a + a_s) ∕ 2).

We next discuss the effect of handling time on the dynamics of the system. For clarity we set a = a_s in Eq. C.1 . Again we first analyze what happens when one prey species is extinct. The system will go through a Hopf bifurcation at:

,

and a transcritical bifurcation at:

.

Note that the stability of the coexistence plane is independent from the handling time. On the coexistence plane we will see the same behaviour as when one species is extinct, except that it is dependent of the mean handling time ((T + T_s) ∕ 2). The equilibrium has a Hopf bifurcation, where a limit cycle is formed around the equilibrium at:

,

and a transcritical bifurcation at which the predator becomes extinct at:

.