Existence of positive solution, stability, and long-time dynamics of a biologically motivated fractional-order delay integro-differential equation with integral-type condition

In this paper, we present a novel qualitative framework for analyzing two-dimensional fractional integro-differential equations of the following form: [Formula: see text] subject to the integral-type condition [Formula: see text] This formulation captures both memory effects and nonlocal interactions, which are essential features in modeling biological systems, where past states and spatial heterogeneity drive current dynamics. Our key contributions include establishing the existence, positivity, Ulam–Hyers stability, boundedness, and long-term behavior of solutions under general nonlinear and nonlocal coupling structures. Under the setting of minimal assumptions-regularity of kernel functions, growth, and Lipschitz continuity, all results are derived. Using fixed-point theory and fractional integral inequalities, we derive Ulam–Hyers stability estimates that guarantee robustness under small perturbations. The use of integral initial conditions reflects distributed biological measurements, such as tissue-level concentrations, more realistically than classical pointwise data. Applications to biologically motivated scenarios, including nutrient transport in neural tissues and epidemic spread with spatial mobility, demonstrate the framework’s ability to model delayed responses and long-range interactions. The proposed approach advances the current theory on multidimensional FIDEs and provides a mathematically rigorous tool for simulating complex biological phenomena with memory and spatial feedback.