<p>Understanding the spatio-temporal dynamics of interacting populations is crucial for studying ecological systems. In this work, we develop an eco-epidemic system of susceptible and infected preys and predators, incorporating memory-driven delays due to a carryover effect <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(f_1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in susceptible prey and a predator-induced fear <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(f_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, along with a recovery process of infected preys parametrized by a constant recovery rate (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>). We prove the existence and boundedness of solutions and establish Hopf bifurcation conditions for four cases of time delays, which are also verified numerically. Without delays, the temporal system exhibits saddle–node and Hopf bifurcations with respect to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, where higher carryover stabilizes and higher fear destabilizes the dynamics, as shown numerically, while variations in the recovery rate significantly influence population densities by increasing susceptible prey and suppressing predator persistence under different transmission rates. In the presence of delays and in the absence of recovery (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>), delays do not affect the stability of an initially stable temporal system; however, in unstable regimes, carryover and fear delays lead to chaotic oscillations, confirmed by the computation of Lyapunov exponents, and bursting dynamics, respectively. When the recovery rate is nonzero and exceeds a threshold value, temporal stability becomes independent of the delays. PRCC-based global sensitivity analysis identifies key parameters that significantly influence coexistence and system stability. Beyond temporal dynamics, small delays induce Turing instability and generate diverse spatial patterns in a reaction-diffusion framework, where increasing fear-induced delay <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(\tau _2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> enhances aggregation by transforming micro-spirals into dense clusters, carryover delay <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(\tau _1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> stabilizes larger spirals, and their combined effects produce four-headed spirals at high prey diffusion <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(D_S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mi>S</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> that become denser at lower diffusion; increasing recovery shifts large spirals to micro-spirals, confirming the existence of a critical recovery rate beyond which the destabilizing effects of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\tau _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>τ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\tau _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>τ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are suppressed. Overall, this study shows that time delays and recovery jointly govern ecosystem stability, driving transitions between regular, chaotic, and patterned dynamics, and offering insights for ecological management and disease control.</p>

Emergence of Bursting and Delay-Induced Spiral Patterns in Eco-Epidemiological Systems

期刊论文
发表日期：

2026年3月24日

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Namrata Mani Tripathi,
Ranjit Kumar Upadhyay,
Dipesh Barman,
Anotida Madzvamuse

摘要

Understanding the spatio-temporal dynamics of interacting populations is crucial for studying ecological systems. In this work, we develop an eco-epidemic system of susceptible and infected preys and predators, incorporating memory-driven delays due to a carryover effect $(f_1)$ $(f_{1})$ in susceptible prey and a predator-induced fear $(f_2)$ $(f_{2})$ , along with a recovery process of infected preys parametrized by a constant recovery rate ( $\gamma $ $γ$ ). We prove the existence and boundedness of solutions and establish Hopf bifurcation conditions for four cases of time delays, which are also verified numerically. Without delays, the temporal system exhibits saddle–node and Hopf bifurcations with respect to $f_1$ $f_{1}$ and $f_2$ $f_{2}$ , where higher carryover stabilizes and higher fear destabilizes the dynamics, as shown numerically, while variations in the recovery rate significantly influence population densities by increasing susceptible prey and suppressing predator persistence under different transmission rates. In the presence of delays and in the absence of recovery ( $\gamma =0$ $γ = 0$ ), delays do not affect the stability of an initially stable temporal system; however, in unstable regimes, carryover and fear delays lead to chaotic oscillations, confirmed by the computation of Lyapunov exponents, and bursting dynamics, respectively. When the recovery rate is nonzero and exceeds a threshold value, temporal stability becomes independent of the delays. PRCC-based global sensitivity analysis identifies key parameters that significantly influence coexistence and system stability. Beyond temporal dynamics, small delays induce Turing instability and generate diverse spatial patterns in a reaction-diffusion framework, where increasing fear-induced delay $(\tau _2)$ $(τ_{2})$ enhances aggregation by transforming micro-spirals into dense clusters, carryover delay $(\tau _1)$ $(τ_{1})$ stabilizes larger spirals, and their combined effects produce four-headed spirals at high prey diffusion $(D_S)$ $(D_{S})$ that become denser at lower diffusion; increasing recovery shifts large spirals to micro-spirals, confirming the existence of a critical recovery rate beyond which the destabilizing effects of $\tau _1$ $τ_{1}$ and $\tau _2$ $τ_{2}$ are suppressed. Overall, this study shows that time delays and recovery jointly govern ecosystem stability, driving transitions between regular, chaotic, and patterned dynamics, and offering insights for ecological management and disease control.