Bifurcation, chaos, multistability, and organized structures in a predator–prey model with vigilance

Abstract

There is not a single species that does not strive for survival. Every species has crafted specialized techniques to avoid possible dangers that mostly come from the side of their predators. Survival instincts in nature led prey populations to develop many anti-predator strategies. Vigilance is a well-observed effective antipredator strategy that influences predator–prey dynamics significantly. We consider a simple discrete-time predator-prey model assuming that vigilance affects the predation rate and the growth rate of the prey. We investigate the system dynamics by constructing isoperiodic and Lyapunov exponent diagrams with the simultaneous variation of the prey’s growth rate and the strength of vigilance. We observe a series of different types of organized periodic structures with different kinds of period-adding phenomena. The usual period-bubbling phenomenon is shown near a shrimp-shaped periodic structure. We observe the presence of double and triple heterogeneous attractors. We also notice Wada basin boundaries in the system, which is quite rare in ecological systems. The complex dynamics of the system in biparameter space are explored through extensive numerical simulations.

Population dynamic study of interaction between two predators and one prey

“…Numerous researchers [17,18] have found that population models with two species cannot correctly reflect reality, and only models with three or more species can represent a large number of important behaviors. The development of mathematics also illustrate that threespecies food chain models are much more complex features than two-species models [19][20][21][22].…”

Dynamical Complexities of a Discrete Ivlev-Type Predator-Prey System

“…Examples include ecosystems and chemical reactions [19][20][21]. More signifcantly, discrete-time models are comparably simpler and less computationally intensive [22]. Because of this, compared to continuous forms, discrete-time model formulation and simulation are typically more straightforward, practical, and accurate.…”

Codimension-Two Bifurcations of a Simplified Discrete-Time SIR Model with Nonlinear Incidence and Recovery Rates

“…Most discrete-time epidemic models not only are simple analogies of continuous epidemic models, but also show complex dynamic behaviors that the corresponding continuous models cannot show. The discrete-time model is capable of producing extremely complicated dynamics, even in a one-dimensional instance [14]. Many interesting and important research works can be found in [15][16][17][18][19] and the references cited therein.…”

“…Some researchers have explored the use of computers to model the dynamics within the local parameter variation. Therefore, the numerical methods are helpful not only to validate our analytical results but also to better demonstrate and understand the dynamics of the model [14]. In this paper, we analyze a discrete-time model with the help of various numerical tools and techniques.…”