Stoichiometric knife-edge model on discrete time scale

Ming Chen,Lale Asik,Angela Peace

doi:10.1186/s13662-019-2468-7

Ming Chen, Lale Asik + Show 1 more

Open Access

https://doi.org/10.1186/s13662-019-2468-7

Copy DOI

Abstract

Ecological stoichiometry is the study of the balance of multiple elements in ecological interactions and processes (Sterner and Elser in Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, 2002). Modeling under this framework enables us to investigate the effect nutrient content on organisms whether the imbalance involves insufficient or excess nutrient content. This phenomenon is called the “stoichiometric knife-edge”. In this paper, a discrete-time predator–prey model that captures this phenomenon is established and qualitatively analyzed. We systematically expound the similarities and differences between our discrete model and the corresponding continuous analog. Theoretical and numerical analyses show that while the discrete and continuous models share many properties, differences also exist. Under certain parameter sets, the models exhibit qualitatively different dynamics. While the continuous model shows limit cycle, Hopf bifurcation, and saddle-node bifurcation, the discrete-time model exhibits richer dynamical behaviors, such as chaos. By comparing the dynamics of the continuous and discrete model, we can conclude that stoichiometric effects of low food quality on predators are robust to the discretization of time. This study can possibly serve as an example for pointing to the importance of time scale in ecological modeling.

Highlights

Ecological stoichiometry is the study of the balance of energy and essential chemical elements throughout ecological systems [23]
These stoichiometric models are defined on the continuous time scale; empirical data in ecological systems are collected on discrete time intervals
We focus on whether the discrete model can retain the most important dynamical features exhibited in the continuous stoichiometric knife-edge model, like deterministic extinction of grazer when faced with excess food-nutrient content

Summary

Introduction

Ecological stoichiometry is the study of the balance of energy and essential chemical elements throughout ecological systems [23]. Elser 2012 [10] and Peace 2013 [21] proposed a stoichiometric model in order to investigate the growth response of grazer to producer of varying P:C ratios capturing the mechanism of the knife-edge phenomenon These stoichiometric models are defined on the continuous time scale; empirical data in ecological systems are collected on discrete time intervals. We discretize the model (2.1) applying the method developed by Cook, Busenberg, Wiener, and Shah [5, 8] and used in many papers [7, 12, 16, 22, 24, 28] This method employs the differential equations with piecewise constant arguments (EPCA) by assuming that on a given time interval [t, t + 1] the per capita growth rate stays constant. Letting t → n + 1, we obtain the following discrete-time analog of system (2.1): bx(n) x(n + 1) = x(n) exp b – min{K, (PT – θ y(n))/q}.

Internal equilibria

Conclusion and discussion