Sensitivity of the dynamics of the general Rosenzweig–MacArthur model to the mathematical form of the functional response: a bifurcation theory approach

Abstract

The equations in the Rosenzweig-MacArthur predator-prey model have been shown to be sensitive to the mathematical form used to model the predator response function even if the forms used have the same basic shape: zero at zero, monotone increasing, concave down, and saturating. Here, we revisit this model to help explain this sensitivity in the case of three response functions of Holling type II form: Monod, Ivlev, and Hyperbolic tangent. We consider both the local and global dynamics and determine the possible bifurcations with respect to variation of the carrying capacity of the prey, a measure of the enrichment of the environment. We give an analytic expression that determines the criticality of the Hopf bifurcation, and prove that although all three forms can give rise to supercritical Hopf bifurcations, only the Trigonometric form can also give rise to subcritical Hopf bifurcation and has a saddle node bifurcation of periodic orbits giving rise to two coexisting limit cycles, providing a counterexample to a conjecture of Kooji and Zegeling. We also revisit the ranking of the functional responses, according to their potential to destabilize the dynamics of the model and show that given data, not only the choice of the functional form, but the choice of the number and/or position of the data points can influence the dynamics predicted.

Community dynamics and sensitivity to model structure: towards a probabilistic view of process-based model predictions

“…The underlying idea behind our approach is that the cost of choosing one model to make a prediction increases if (i) there is a high probability that another model also fits available data well and (ii) the two models make different predictions. Thus, this approach extends earlier studies on structural sensitivity ( [8,9], among others) by considering the fact that, even if alternative models predict different dynamics, one model may outperform others in fitting available data. Again with the Rosenzweig-MacArthur model, we illustrate this possibility in figure 5 by using data on the ingestion rate (i.e.…”

“…Empirical data were not considered in previous structural sensitivity analyses in Rozensweig-MacArthur model [8,9]. As our baseline, we therefore use three examples of functional-response data to provide an overview of our approach and its practical use.…”

“…All these assumptions define the Rozensweig-MacArthur model, which is often refered as a predator-prey model but is generic enough to also describe a variety of consumer-resource systems. Previous studies on structural sensitivity compared a deterministic analysis of model predictions based on best-fitted functions ( [8,9,11,18,20], among others). Fitting a function to data implies uncertainty in the parameters being inferred, which means that there is additional insight to be gained by considering both parameter and model uncertainty.…”

Pest control by generalist parasitoids: A bifurcation theory approach

“…When K 1 ≤ 1/b, the graph of F (N ) is a monotonically decreasing function for N ≥ 0. Seo and Wolkowicz [18] demonstrate that for the prey nullcline, F (N,…”

Global stability of the predator-prey model with a sigmoid functional response

“…It is interesting to mention that the above properties do not hold for the model (1) with the simplified Monod-Haldane functional response (6). It has been shown in [15,22] that, with the simplified Monod-Haldane functional response (6), the model can exhibit complicated dynamics, such as the subcritical Hopf bifurcation that leads to the co-existence of a stable positive equilibrium and multiple limit cycles, a phenomenon also observed in [17,18] for certain types of functional response.…”