[Paper Review] An ecosystem with Holling type II response and predators' genetic variability

Clara Viberti, Ezio Venturino|arXiv (Cornell University)|Mar 18, 2014

Mathematical and Theoretical Epidemiology and Ecology Models10 references3 citations

TL;DR

This paper proposes a predator-prey model with genetically distinct predator genotypes (y and z) and a Holling type II functional response to model feeding satiation. Using Routh-Hurwitz stability analysis, it demonstrates that persistent limit cycles can emerge under specific parameter conditions, showing that genetic variability in predators can drive sustained oscillations—contrasting with models using bilinear responses, which lack such oscillations.

ABSTRACT

A new model to investigate environmental effects of genetically distinguishable predators is presented. The Holling type II response function, modelling feeding satiation, leads to persistent system's oscillations, as in classical population models. An almost complete classification of the cases arising in the Routh-Hurwitz stability conditions mathematically characterizes the paper. It is instrumental as a guideline in the numerical experiments leading to the findings on the limit cycles. This result extends what found in an earlier parallel investigation containing a standard bilinear response function.

Motivation & Objective

• To investigate the ecological consequences of genetically distinguishable predators in a predator-prey system.
• To determine whether the inclusion of a Holling type II functional response—modeling feeding satiation—can lead to persistent oscillations in a system with genetically variable predators.
• To classify the stability conditions of the coexistence equilibrium using Routh-Hurwitz criteria, providing a mathematical framework for identifying parameter regimes with limit cycles.
• To compare the dynamics of this model with previous models using bilinear interaction terms, where no oscillations were observed.
• To explore how genetic diversity in predators influences ecosystem persistence and coexistence, particularly through mutation-like transitions between genotypes.

Proposed method

• Formulates a three-species model with prey (x), and two genetically distinct predator genotypes (y, z), using a Holling type II functional response to model satiation in predation.
• Incorporates genotype-specific hunting efficiency (h for y, g for z) and a shared conversion factor e for predator reproduction from consumed prey.
• Uses a shared reproduction term in both predator equations, allowing each genotype to produce offspring of both types (via p and q, with p+q=1), modeling genetic transitions.
• Applies nondimensionalization to reduce the number of parameters and simplify analysis.
• Performs a comprehensive Routh-Hurwitz stability analysis on the coexistence equilibrium to classify all possible sign combinations of coefficients in the characteristic equation.
• Conducts numerical simulations guided by the analytical classification to validate the existence of limit cycles under specific parameter regimes.

Experimental results

Research questions

• RQ1 Can a predator-prey system with genetically variable predators and a Holling type II functional response exhibit persistent limit cycles?
• RQ2 How do the Routh-Hurwitz conditions classify the stability of the coexistence equilibrium in this model?
• RQ3 What role does the parameter B (representing predator invasion threshold) play in determining system stability and persistence?
• RQ4 Why does the model prevent the existence of equilibria with only one predator genotype, despite potential extinction of one population?
• RQ5 How does the inclusion of genetic variability in predators alter ecosystem dynamics compared to models with bilinear predation responses?

Key findings

• The system exhibits persistent limit cycles when the parameter A falls below a critical threshold, as demonstrated in numerical examples with A=0.2 (oscillations) and A=0.6 (stable equilibrium).
• For Example 1, the critical value of A is approximately 0.433, beyond which the coexistence equilibrium becomes stable, confirming the presence of a Hopf bifurcation.
• In Example 2, the threshold for oscillations is A ≈ 0.638; when A=0.25, limit cycles emerge, while A=0.85 leads to a stable equilibrium.
• In Example 3, the critical A value is approximately 0.496, and oscillations are observed at A=0.248 (half the critical value), while stability is restored at A=0.744 (1.5× critical value).
• The coexistence equilibrium is always stable when A exceeds the critical value in each case, indicating a clear transition from oscillatory to stable dynamics.
• The model ensures that no equilibrium with only one predator genotype can exist, due to the mutual production of offspring across genotypes, which prevents permanent extinction of either genotype.