Nonlocal generalized models of predator-prey systems

Abstract

The method of generalized modeling has been applied successfully in many different contexts, particularly in ecology and systems biology. It can be used to analyze the stability and bifurcations of steady-state solutions. Although many dynamical systems in mathematical biology exhibit steady-state behaviour one also wants to understand nonlocal dynamics beyond equilibrium points. In this paper we analyze predatorprey dynamical systems and extend the method of generalized models to periodic solutions. First, we adapt the equilibrium generalized modeling approach and compute the unique Floquet multiplier of the periodic solution which depends upon so-called generalized elasticity and scale functions. We prove that these functions also have to satisfy a flow on parameter (or moduli) space. Then we use Fourier analysis to provide computable conditions for stability and the moduli space flow. The final stability analysis reduces to two discrete convolutions which can be interpreted to understand when the predator-prey system is stable and what factors enhance or prohibit stable oscillatory behaviour. Finally, we provide a sampling algorithm for parameter space based on nonlinear optimization and the Fast Fourier Transform which enables us to gain a statistical understanding of the stability properties of periodic predator-prey dynamics.

Structural sensitivity and resilience in a predator–prey model with density-dependent mortality

“…Bazykin's model is equivalent to Rosenzweig and MacArthur's model (1963) with density-dependent mortality for the predator. The predator has no density-dependent mortality in previous studies on structural sensitivity and generalized predator-prey models (Kuehn and Gross, 2011;Yeakel et al, 2011). However, density-dependent mortality represents the effects of diseases and/ or competition and can be relevant for a wide range of predator species (Loreau, 2010).…”

Predicting critical transitions in a model of systemic inflammation

“…Although circadian rhythms are often suppressed in critically ill patients (Lowry, 2009; Mundigler et al, 2002), there is still a need to very accurately distinguish between movement that is part of a healthy circadian rhythm and a pathological transition between steady states. Generalized models of periodic systems have been studied (Kuehn and Gross, 2011), but not yet applied in the prediction of critical transitions. Random noise is another confounding factor which makes the estimation of partial derivatives from limited data more difficult, although appropriate smoothing of the data may be sufficient to retain predictive power (Lade and Gross, 2012).…”

Dynamical analysis of evolution equations in generalized models

“…The examples have also demonstrated that known results about bifurcations and unfoldings carry over easily to generalized models. The same calculations can be carried out for all other local bifurcations which will yield conditions analogous to (42).…”

“…However, this approach cannot capture global bifurcations such as saddle-nodes of periodic orbits. Recent work of Kuehn and Gross [42] shows how to extend generalized modeling to periodic orbits for the predator-prey system (1). The main problem is that the generalized parameter β i,k and f i,k,x j become time-dependent functions β i,k (t) and f i,k,x j (t).…”

“…This changes the algebraic structure of the generalized model and introduces a so-called moduli flow constraint on the scale functions β i,k (t). The detailed description of this approach is beyond the scope of this paper and we refer the interested reader to [42].…”