Operational stability maps for climate-driven predator–prey dynamics: Distributed-order memory & hopf shifts

We develop a distributed-order fractional predator-prey model that incorporates climate-driven thermal forcing, fear-mediated behavioural suppression, and memory-dependent vigilance dynamics. The model is governed by the distributed-order Caputo operatorCDt(μ)x(t)=∫01μ(α)CDtαx(t)dα,which captures a continuum of ecological memory scales. Temperature-dependent demographic rates r(T), a(T), and μ(T) modulate growth, predation, and mortality, while the dynamic fear variable f(t) reduces prey reproduction and encounter rates. We derive positivity, boundedness, and equilibrium conditions, and characterize the onset of oscillations via the Hopf bifurcation threshold G*. Stability analysis shows that distributed-order memory shifts Hopf boundaries and modifies invasion thresholds compared with the classical integer-order limit. A global sensitivity analysis using partial rank correlation coefficients (PRCC) and Sobol indices identifies temperature elasticities θr, θβ, and θμ as dominant drivers of the predator invasion number Ry, while fear suppression sp and low-order memory weights strongly influence G*. A climate-driven zooplankton-phytoplankton case study illustrates how warming and behavioural feedback jointly shape predator-prey resilience, highlighting scenarios in which distributed-order memory dampens oscillations and expands the region of stable coexistence.