Eco-Epidemiological Dynamics Under State-Dependent Delays: A New Delay Differential Approach

In much of the traditional literature on eco‐epidemiology, one finds that species interactions and the spread of infectious agents are cast almost reflexively as ordinary differential equations (ODEs) or, at best, delay differential equations (DDEs) endowed with fixed time lags. Yet anyone who has looked closely at empirical data knows that living populations do not respond on a rigid clock reaction times often stretch or shrink depending on how crowded the environment is, how intense an outbreak becomes, or how scarce resources turn out to be. In light of this, we introduce here a framework that moves beyond constant‐lag assumptions by embedding state‐dependent delays into a predator–prey model where the prey itself can carry an infection. By letting the delay in disease transmission flex in direct proportion to prey density a feature all too familiar in natüre our model seeks to mirror how, say, a brief food shortage might slow immune responses or, conversely, how an epidemic surge might force populations to react more quickly. Concretely, the prey population is partitioned into susceptible and infected classes, while predators consume prey without distinction. Crucially, the transmission term is no longer governed by “one size fits all” delay; instead, the time lag varies with the prey density at each instant. Once the equations are in place, we prove under mild regularity and boundedness hypotheses the existence and uniqueness of solutions. From there, a qualitative tour begins: we examine when steady states lose or gain stability, and we hunt for periodic orbits that might correspond to recurrent outbreaks or population cycles. To show how these state‐tied delays reshape dynamics, numerical experiments reveal patterns that simply do not appear if one insists on fixed delays. For instance, what might look like a benign equilibrium under a constant lag can, with a state‐dependent delay, turn into a sequence of oscillatory flares or, interestingly, become stabilized precisely because the delay adapts to low prey densities. Such phenomena oscillatory outbreaks that dampen themselves or, in other regimes, delay induced stabilization underscore the point that variable lags can drive behaviors unseen in classic constant‐delay formulations. In sum, weaving state‐dependent delays into eco‐epidemiological equations not only aligns the mathematics with biological realism but also opens a more nuanced window onto the possible trajectories of interacting populations under disease pressure.