Spiking Systems in Population-Infection Dynamics

Motivated by a class of models in population dynamics, we introduce the concept of spiking dynamical systems. A spiking system admits an asymptotically stable equilibrium but, under proper perturbations on the initial conditions in a compact region including the equilibrium, its output exhibits a spike of arbitrarily large magnitude before the state returns within the region. We consider a model that describes a well-documented phenomenon in caterpillar-virus dynamics: a sudden increase of the caterpillar population occurs, due to a temporary reduction of the viral population, and is then followed by a sudden decrease. We prove that the caterpillar-virus system is spiking according to our proposed mathematical definition: the model can yield arbitrarily large population densities for caterpillars, and then the original conditions are suddenly restored. When the model also takes into account environmental constraints that keep the caterpillar population bounded, the spike cannot be arbitrarily large, but the population density can get arbitrarily close to the maximal one that can be achieved in the absence of virus.