[Paper Review] Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions

Yun Kang, Dieter Armbruster|arXiv (Cornell University)|Feb 11, 2011

Animal Ecology and Behavior Studies33 references3 citations

TL;DR

This paper investigates plant-herbivore dynamics using monotone plant growth functions, showing they generate a unique interior equilibrium and uniform persistence under certain parameters. It identifies noise-induced bursting as a key mechanism for almost periodic herbivore outbreaks, contrasting monotone models with non-monotone ones in terms of bistability and chaotic crises.

ABSTRACT

We formulate general plant-herbivore interaction models with monotone plant growth functions (rates). We study the impact of monotone plant growth functions in general plant-herbivore models on their dynamics. Our study shows that all monotone plant growth models generate a unique interior equilibrium and they are uniform persistent under certain range of {parameters} values. However, if the attacking rate of herbivore is too small or the quantity of plant is not enough, then herbivore goes extinct. Moreover, these models lead to noise sensitive bursting which can be identified as a dynamical mechanism for almost periodic outbreaks of the herbivore infestation. Monotone and non-monotone plant growth models are contrasted with respect to bistability and crises of chaotic attractors.

Motivation & Objective

• To analyze the impact of monotone plant growth functions on plant-herbivore interaction dynamics.
• To determine conditions under which herbivores persist or go extinct in response to plant growth and attack rates.
• To investigate the role of noise in generating almost periodic herbivore outbreaks via bursting dynamics.
• To compare monotone plant growth models with unimodal and multimodal models in terms of bistability and chaotic attractor crises.
• To establish uniform persistence and global stability properties for general monotone plant-herbivore systems.

Proposed method

• Formulates general discrete-time plant-herbivore models using monotone plant growth functions, including Beverton-Holt and Holling-Type III models.
• Applies dynamical systems theory to analyze equilibrium existence, stability, and uniform persistence under conditions H1 and H3.
• Uses numerical simulations to explore noise-induced bursting behavior in herbivore populations under stochastic perturbations.
• Employs bifurcation analysis to identify Neimark-Sacker and heteroclinic bifurcations in specific model instances.
• Compares model outcomes across monotone (e.g., Beverton-Holt) and non-monotone (e.g., Ricker) growth functions.
• Quantifies burst periodicity via resident time ratios and average period as functions of noise amplitude.

Experimental results

Research questions

• RQ1 Do monotone plant growth functions lead to a unique interior equilibrium in plant-herbivore models?
• RQ2 Under what parameter conditions is the herbivore population uniformly persistent or driven to extinction?
• RQ3 How does noise induce almost periodic outbreaks in herbivore populations within monotone plant-herbivore systems?
• RQ4 What dynamical mechanisms—such as Neimark-Sacker or heteroclinic bifurcations—underlie periodic bursting in these models?
• RQ5 How do monotone plant growth models differ from non-monotone models in terms of bistability and crisis of chaotic attractors?

Key findings

• All monotone plant growth models generate a unique interior equilibrium under standard assumptions.
• Herbivore extinction occurs when the attack rate is too low or plant biomass is insufficient, even with monotonic growth.
• Noise-induced bursting is a dominant dynamical mechanism for almost periodic herbivore outbreaks, with burst periods ranging from 8 to 19 years depending on noise amplitude.
• The Beverton-Holt model exhibits Neimark-Sacker bifurcations leading to a unique stable periodic orbit, while the Holling-Type III model undergoes heteroclinic bifurcations.
• Monotone models like Beverton-Holt are uniformly persistent under H1 and H3, whereas Holling-Type III models may not satisfy uniform persistence.
• Numerical results show that noise amplitude controls the apparent periodicity of outbreaks, with periods of 8–12 years achievable at moderate noise levels, matching gypsy moth outbreak data.