Dual fear phenomenon in an eco-epidemiological model with prey aggregation

TL;DR

This work develops an eco-epidemiological SIP model with prey aggregation ($S^r$, $0<r<1$) and dual fear effects mediated by predators, and analyzes the resulting dynamical system. It establishes nonnegativity and boundedness, derives equilibria including a coexistence state, and performs comprehensive co-dimension-one and -two bifurcation analyses (saddle-node, Hopf, transcritical, zero-Hopf, and SNTC) using Sotomayor’s theorem and numerical continuation. The study reveals finite-time extinction of susceptible prey due to aggregation, and demonstrates fear-based and selective predation strategies as viable disease-control mechanisms. These results illuminate how predator-induced fear and prey behavior modulate disease transmission and population outcomes, with implications for ecological management and conservation planning.

Abstract

This study presents a thorough analysis of an eco-epidemiological model that integrates infectious diseases in prey, prey aggregation, and the dual fear effect induced by predators. We establish criteria for determining the existence of equilibrium points, which carry substantial biological significance. We establish the conditions for the occurrence of Hopf, saddle-node, and transcritical bifurcations by employing fear parameters as key bifurcation parameters. Furthermore, through numerical simulations, we demonstrate the occurrence of multiple zero-Hopf (ZH) and saddle-node transcritical (SNTC) bifurcations around the endemic steady states by varying specific key parameters across the two-parametric plane. We demonstrate that the introduction of predator-induced fear, which hinders the growth rate of susceptible prey, can lead to the finite time extinction of an initially stable susceptible prey population. Finally, we discuss management strategies aimed at regulating disease transmission, focusing on fear-based interventions and selective predation via predator attack rate on infectious prey.

Figures (7)

• Figure 1: Saddle-node (SN) bifurcation of model \ref{['Mainsystem']} as fear parameters are varied (a) $k_2=1$ is fixed and $k_1^*=0.4181$ around $E_4=(0.4615,1.0565,0.8523)$ (b) $k_1=0.1$ is fixed and $k_2^*=0.4417$ around $E_4=(0.6418,0.9591,1.5854)$. All other parameters are fixed and given as $b_0=8,~K=4,~a_0=0.5,~d_0=0.7,~r=0.5,~e_0=4,~a_1=0.4,~d_1=0.7,~a_2=0.8,~d_2=0.4,~d_3=0.5$. Solid line denotes stable and dotted line denotes unstable.
• Figure 2: Hopf and transcritical bifurcations of model \ref{['Mainsystem']} as the fear parameters are varied. (a) Transcritical point (TC) at $k_1^{TC}=0.4219$ around $E_3=(4.0600,0,0.9978)$. Also, the Hopf point (H) $k_1^H=2.5075$ around $E_4=(1.6184,0.7596,0.3308)$ with Lyapunov coefficient of $L_{k_1}=-8.827\times 10^{-3}$, (b) Hopf point $k_2^H=0.1536$ around $E_4=(1.6694,0.7411,0.5311)$ with Lyapunov coefficient of $L_{k_2}=-9.9094\times 10^{-3}$. Also, the transcritical point at $k_2^{TC}=1.7885$ around $E_3=(4.0600,0,0.7081)$. All other parameters are fixed and given as $b_0=2,k_1=0.99,~k_2=0.85,~K=8,~a_0=0.3,~d_0=0.6,~r=0.7,~e_0=0.5,~a_1=0.4,~d_1=0.7,~a_2=0.8,~d_2=0.3,~d_3=0.5$. Solid line denotes stable and dotted line denotes unstable.
• Figure 3: Two-parameter bifurcation diagrams for model \ref{['Mainsystem']} as key parameters are varied. (a) 3-D bifurcation surface of the saddle-node and zero-Hopf curves. (b) Zero-Hopf ($ZH$) point at $(k_2,d_0)=(0.9917,1.4225)$. (c) 3-D bifurcation surface of the saddle-node, saddle-node transcritical, and zero-Hopf curves. (d) Multiple zero-Hopf points at $(k_2,K)=(0.2868,5.0261)$ and $(k_2,K)=(0.2585,5.5590)$. Multiple saddle-node transcritical ($SNTC$) points at $(k_2,K)=(0.4508,3.9784)$ and $(k_2,K)=(0.2271,5.7454)$. All other parameters are fixed and given as $b_0=8,~K=4,~a_0=0.5,~d_0=0.7,~r=0.5,~e_0=4,~a_1=0.4,~d_1=0.7,~a_2=0.8,~k_1=0.1,~d_2=0.4,~d_3=0.5$.
• Figure 4: Time evolution of population densities with varying $k_1$. (a) Stable dynamics observed at $k_1=0$ for $E_4=(2.8194,0.5925,4.5959)$ (b) Finte time extinction of $S$ when $k_1=0.2$. All other parameters are fixed and given as $b_0=10,~K=5,~a_0=0.5,~d_0=0.7,~r=0.5,~e_0=6,~a_1=0.4,~d_1=0.7,~a_2=0.8,~k_2=0.8,~d_2=0.3,~d_3=0.5$ and initial condition is given as $(3,2,4)$.
• Figure 5: Figures depicting time evolution of population densities and phase portrait with IC$=(0.8,0.9,1.1)$. ((a) & (d)) Stable $E_3=(4.0600,0,1.7382)$ at $k_1=0$, ((b) & (e)) stable $E_4=(2.5030,0.4596,0.6076)$ at $k_1=1.2$, ((c) & (f)) oscillatory coexistence at $k_1=4$ where $E_4=(1.2898,0.8830,0.2102)$. All other parameters are fixed and given as $b_0=2,~K=8,~a_0=0.3,~d_0=0.6,~r=0.7,~e_0=0.5,~k_2=0.85,~a_1=0.4,~d_1=0.7,~a_2=0.8,~d_2=0.3,~d_3=0.5$.

Theorems & Definitions (37)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 4.1
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3