Analysis of the impact of fear in the presence of additional food and prey refuge with nonlocal predator-prey models

TL;DR

The paper addresses how fear of predation modulates prey growth in a predator–prey system that includes prey refuge and predator supplementary food. It develops both local and nonlocal spatio-temporal models with a Beddington–DeAngelis functional response and analyzes equilibria, stability, transcritical and Hopf bifurcations, and Turing instabilities. The key contributions show that fear can stabilize or destabilize coexistence, prey refuge and added food shift bifurcation thresholds, and nonlocal interactions expand the Turing domain and foster spatial heterogeneity, producing patterns such as hot spots and labyrinths. These findings highlight the intricate coupling of ecological behavior, nonlinear functional responses, and space in shaping predator–prey dynamics with potential implications for management of ecosystems exhibiting fear and refuge effects. The work advances the understanding of how nonlocal perception of predation risk affects pattern formation and persistence in ecological communities.

Abstract

There are many positive and negative factors present in the predator-prey interaction which affect the net growth of the species. Fear of predation is one such factor that creates psychological stress in a prey species, which causes a negative impact on their overall growth. This work considers a predator-prey model where the prey species faces a reduction in their growth out of fear, and the predator has an alternative food source that helps the prey to hide in a safer place. As an extension into the nonlocal spatio-temporal model, a nonlocal term is considered in the prey growth to incorporate a fear-effect range around their spatial location. Linear stability analysis helps to analyze the temporal model and produces a wide range of interesting results, including the presence of a certain amount of fear or even prey refuge, which helps in population coexistence. Furthermore, the numerical simulations of the local and nonlocal spatio-temporal models show different types of spatial-temporal patterns, such as Turing and non-Turing patterns. Nevertheless, an increase in fear level reduces the range of the Turing domain for the local model, whereas the opposite happens when the range of nonlocal interaction is increased.

Figures (16)

• Figure 1: Impact of predator interference on (a) prey (u) and (b) predator (v) population in the presence and absence of the fear effect.
• Figure 2: Comparison of the components of $E^{*}$ for $\alpha=0.1,\ 0.5,\ 0.9$ while varying $\omega.$
• Figure 3: Change of dynamical behaviour of temporal system (\ref{['eq:det1']}) with increasing (a) $c$, (b) $\eta$, (c) $\mu_{2}$, and (d) $\omega$.
• Figure 4: (a) Change of dynamical behaviour of the temporal system with increasing $m$. (b) Impact of prey refuge $(m)$ on the predator population $(v)$ in the presence of alternative food sources.
• Figure 5: Change of dynamical behaviour of the temporal system with increasing $A$. The parametric values are chosen from Table \ref{['Table:1']} and $\omega=5$.

Theorems & Definitions (5)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5