Dynamics of Diseased-Impacted Prey Populations: Defense and Allee Effect Mechanisms

TL;DR

This work develops a three-variable eco-epidemiological model with susceptible $S$, infected $I$, and predator $P$, incorporating prey aggregation via $S^r$ ($r\in(0,1)$) and a density-dependent Allee threshold $L$. Through stability and bifurcation analyses, the authors identify multiple equilibria and rich co-dimension one and two bifurcations (transcritical, Hopf, saddle-node, cusp, zero-Hopf, generalized Hopf, Bogdanov–Takens), outlining how $L$ and $r$ shape disease persistence and population persistence. They establish conditions for disease eradication in the infected prey or finite-time extinction of susceptible prey, revealing a threshold $r^*$ under which $I(t)\to 0$ even in coexistence regimes. The results offer practical insights for wildlife disease management via behavioral aggregation and have broad implications for conservation and ecosystem resilience, supported by analytical results and numerical continuations.

Abstract

This study introduces an innovative framework for merging ecological and epidemiological modeling via the formulation of a sophisticated predator-prey model that addresses the intricacies of disease dynamics, the Allee effect, and defensive mechanisms through prey aggregation. Employing rigorous stability and bifurcation analyses, we identify multiple feasible equilibria and establish critical thresholds that influence population survival and extinction. Our mathematical model reveals that the intensity of the Allee effect plays a crucial role in shaping population recovery and disease persistence, offering pivotal insights into finite time extinction mechanisms. We further illustrate, through extensive numerical simulations, that adjusting susceptible prey aggregation strategically can substantially reduce disease transmission, emphasizing the applicability of our findings for practical conservation interventions. The combined modulation of the aggregation constant and Allee effect determined three primary ecological outcomes: stable coexistence, elimination of infected prey, and complete population extinction. Moreover, these results have significant implications for wildlife management and ecosystem resilience, providing a solid theoretical framework for interdisciplinary strategies aimed at protecting endangered species.

Figures (6)

• Figure 2: Intersection of nullclines for varying values of $L$. (a) 3D plot of the nullclines when $L=-0.5$, showing one interior equilibrium point. (b) 2D projection of the S-P nullclines corresponding to (a). (c) 3D plot of the nullclines when $L=-0.1$. (d) 2D projection of the S-P nullclines corresponding to (c). All other parameters are fixed and given as $a_0=3,~a_1=0.4,~a_2=0.8,~r=0.5,~d_0=0.4,~d_1=0.7,~d_2=0.3,~d_3=0.4,~e_0=0.9,~K=4$
• Figure 3: One parameter bifurcation plots illustrating the effects of varying the Allee threshold, and also the aggregation constant. (a) $L$ as a bifurcation parameter (b) $r$ as a bifurcation parameter. All other parameters are fixed and given as $a_0=3,~a_1=0.4,~a_2=0.8,~r=0.5,~d_0=0.4,~d_1=0.7,~d_2=0.3,~d_3=0.4,~e_0=0.9,~K=4,~L=-0.5$. For clarity, the following notation is used in the figures: SN denotes a saddle-node bifurcation, TC denotes a transcritical bifurcation, and H denotes a Hopf bifurcation. Stable equilibria are represented by solid lines, and unstable equilibria are represented by dashed lines
• Figure 4: Two parameter bifurcation plots. (a) $L-a_0$ parametric space depicting $GH$ and $ZH$ in the weak Allee regime (b) $L-e_0$ parametric space depicting $CP_1,~CP_2,~BT_1,~BT_2,$ and $ZH$ in the strong Allee regime. All other parameters are fixed and given as $a_0=3,~a_1=0.4,~a_2=0.8,~r=0.5,~d_0=0.4,~d_1=0.7,~d_2=0.3,~d_3=0.4,~e_0=0.9,~K=4,~L=-0.5$
• Figure 5: Bifurcation diagram illustrating the effects of varying $L$ and $r$ on the population dynamics in model \ref{['Mainsystem']}. Stable coexistence region (green), infectious prey free region (yellow), and finite time extinction of susceptible prey region or total collapse (orange). The parameters are fixed as $a_0=3,~a_1=0.4,~a_2=0.8,~d_0=0.4,~d_1=0.7,~d_2=0.3,~d_3=0.4,~e_0=0.9,~K=4$ with initial condition $(S(0),I(0),P(0))=(2,1,3)$
• Figure 6: Time series and phase portrait plots illustrating the transition from a stable coexistence state to an infected prey free state as the aggregation constant is altered. (a) stable coexistence equilibrium point $E_4=(2.61341,0.787546,2.78867)$, here $r=0.5$ (b) extinction of the infected population $E_3=(3.4077,0,5.54573)$, here $r=0.8$. (c) 3D phase portrait of the SIP model for $r=0.5$ (d) 3D phase portrait of the SIP model for $r=0.8$. All other parameters are fixed and given as $a_0=3,~a_1=0.4,~a_2=0.8,~d_0=0.4,~d_1=0.7,~d_2=0.3,~d_3=0.4,~e_0=0.9,~K=4,~L=-0.5$. with initial condition $(S(0),I(0),P(0))=(2,1,3)$

Theorems & Definitions (26)

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