Prey Switching with a Linear Preference Trade-Off

Abstract

In ecology, prey switching refers to a predator's adaptive change of habitat or diet in response to prey abundance. In this paper, we study piecewise-smooth models of predator-prey interactions with a linear trade-off in a predator's prey preference. We consider optimally foraging predators and derive a model for a 1 predator-2 prey interaction with a tilted switching manifold between the two sides of discontinuous vector fields. We show that the 1 predator-2 prey system undergoes a novel adding-sliding-like (center to two-part periodic orbit; "C2PO") bifurcation in which the prey ratio transitions from constant to time-dependent. Further away from the bifurcation point, the period of the oscillating prey ratio period doubles, suggesting a possible cascade to chaos. We compare our model predictions with data and demonstrate that we successfully capture the periodicity in the ratio between the predator's preferred and alternative prey types in data on freshwater plankton. Our study suggests that it is useful to investigate prey ratio as a possible indicator of how population dynamics can be influenced by ecosystem diversity. 1 We have chosen to follow this generally accepted convention instead of planktic, which is the correct adjective.A standard Lotka-Volterra model for one predator and two prey that allows diversity in the prey community predicts extinction of the prey type that has a smaller capacity to survive [22]. Therefore, such a framework is inappropriate for investigations of ciliate-phytoplankton dynamics, because several predators and prey coexist, and there is a known preference towards one of the phytoplankton prey [32]. One way to resolve this discrepancy is to examine adaptive predator behavior in response to changes in prey densities. There have been many investigations of prey switching [33], in which predators express preference for more abundant prey. For example, prey switching has been demonstrated to promote coexistence of competing prey species [42] and to decrease prey competition due to a shared predator [1]. As a result, prey switching has been suggested as a candidate mechanism for coexistence in communities with diverse prey [2].In smooth differential-equation models for population dynamics, prey switching can be represented using a Holling type-III functional response [17], in which a sigmoid function gives the increased degree of predation with increasing densities of a principal prey. In a 1 predator-1 prey system, a Holling type-III functional response corresponds to a situation in which predation is low at low prey densities but saturates quickly at a high value when prey is abundant. Such a functional response was observed in a system of protist predators and their yeast prey [15]. A low abundance of yeast resulted in increased sedimentation of yeast at the bottom of a glass tube and the walls, so prey were safe from predation, which resulted in a decrease of predator concentration when the prey concentration was below a threshold. Predator preference towards more abundant prey ...

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

“…Early piecewise-smooth models arose in electronics and mechanics, but are increasingly a feature of the life sciences and an array of other physical problems, from superconductors [4] to predator-prey strategies [5,26]. For example take the three systems (2) x i = B (z 1 , z 2 , ..., z n ) − γ i x i , z i = H(x i − v i ) , (3)…”

Inferring parameters of prey switching in a 1 predator–2 prey plankton system with a linear preference tradeoff

“…Although these nonlinearities are standard, it is convenient for us to use non-standard notation for the model coefficients that describe them. This notation allows us both to derive the switching condition introduced previously in [35] and to compare the two smooth models that we develop in the present paper to this piecewise-smooth system.…”

“…Consequently, we can conclude that it is justified to use a piecewise-smooth dynamical system, which has fewer parameters than the associated smooth models introduced in this paper, as a simplifying approximation of a smooth dynamical system. We also show that the piecewise-smooth model in [35] is both biologically and mathematically consistent as the limit of two smooth systems, which we construct by (1) using a hyperbolic tangent as a transition function from one diet choice to another and (2) incorporating a parameter that changes abruptly across the discontinuity in the model in [35] as a system variable with dynamics on a time scale comparable to that of the population dynamics of a predator and its two prey. In the second construction, we examine a system with one more dimension than the corresponding piecewise-smooth system.…”

“…From these results, we see that when a q is large, which corresponds to a predator with a steep tradeoff in its prey preference (i.e., a small increase in specialization towards the preferred prey comes at a large cost in growth from feeding on the alternative prey), the coexistence steady state (2.7) is stable for all k > k 0 . When the prey switching is steep (i.e., when k → ∞), the coexistence steady state (2.7) is equal to the steady state of the piecewise-smooth system that lies on the switching manifold (i.e., it is a pseudoequilibrium), and it has a complex-conjugate pair of eigenvalues with negative real part when a q > q 2 /q 1 [35]. However, in contrast to the piecewise-smooth system, in which the coexistence steady state is repelling for shallow or flat prey preference tradeoffs (i.e., when a q < q 2 /q 1 ), the smooth system (2.2) has an interval of intermediate prey-switching slopes k ∈ (k 0 , k 1 ) (see Equation (A.9) for the expression for k 1 ) for which the coexistence state is also stable for a q < q 2 /q 1 .…”

Dynamics at a Switching Intersection: Hierarchy, Isonomy, and Multiple Sliding

“…Such generalizations are of interest beyond control, however, for example in applications to switching behaviour in biology and mechanics (see e.g. [7,18,20]). …”