A Study on the Specialist Predator with the Allee Effect on the Prey

TL;DR

The study investigates how a tunable Allee effect on prey reshapes predator–prey dynamics under Holling type‑I and II functional responses. By deriving non‑dimensional models and locating boundary and interior fixed points, it shows that predator extinction, stable coexistence, or oscillatory cycles can arise depending on parameters, with Hopf bifurcations driving limit cycles. A novel Allee function is proposed and validated ecologically, revealing that the effect’s strength modulates stability thresholds and the onset of oscillations. The results advance understanding of specialist predation in density‑dependent prey and highlight mechanisms leading to sustained population cycles in ecological systems.

Abstract

Predator-prey models in theoretical ecology have a long and complex history, spanning decades of research. Most of the models rely upon simple reproduction and mortality rates associated with different types of functional responses. A key development in this field occurred with the introduction of a density dependent reproduction rate, originally introduced by Allee. In this manuscript, a new function representing the Allee effect is introduced and justified from the ecological point of view. This paper aims to analyze predator-prey models incorporating Holling type-I and II functional responses, influenced by this new Allee function. A rich dynamics shows up in the presence of the said function, including the emergence of the limit cycles through the Hopf bifurcation for a particular parameter domain.

Figures (5)

• Figure 1: Phase portrait containing four phase trajectories for each parameter set has been obtained. Each set has $\epsilon = 1.0$. In Figure 1(a), choice of parameter $d = 1.5, a = 2.0$ shows prey population extinct over time.Phase portrait in Figure 1(b) has parameter value $d = 0.75, a = 2.0$ showing stable interior fixed point. Figures 1(c) and 1(d) have been drawn for parameter values $d = 0.4, a = 2.0$ and $d = 0.4, a= 1.8$ reflecting the occurrence of Hopf-bifurcation for $1.8 < a < 2.0$.
• Figure 2: Phase portraits drawn in $\epsilon<\epsilon_0 = 1.56$ parameter domain. In each case other parameters were kept fixed at $a = 3.0$, $d = 0.6$ and $h= 1.6$ while choosing (a) $\epsilon = 0.5$ (b) $\epsilon = 1.0$ and (c) $\epsilon = 1.5$, for respective figures.
• Figure 3: Phase portraits drawn in $\epsilon_0 (= 1.56) < \epsilon< \epsilon_{\rm critial} (\approx 1.963)$ parameter domain. In each case other parameters were kept fixed at $a = 3.0$, $d = 0.6$ and $h= 1.6$ while choosing (a) $\epsilon = 1.6$ (b) $\epsilon = 1.7$ and (c) $\epsilon = 1.8$, for respective figures.
• Figure 4: Graphical solution for Trace $=0$ condition with $h = 1.6, d=0.6$ and $\epsilon = \epsilon_{\rm critical} - 0.15, \epsilon_{\rm critical}, \epsilon_{\rm critical} +0.15$, respectively
• Figure 5: Phase portrait for different parameter values: (a) and (b) for $\epsilon > \epsilon_{\rm critical}$ with $a>a_H$ and $a<a_H$, respectively showing the emergence of stable limit cycle via Hopf bifurcation, (c) represents a relatively larger stable limit cycle away from Hopf condition