Prey-Predator-Parasite: an Ecosystem Model With Fragile Persistence

abstract: Using a simple $SI$ infection model, I uncover the overall dynamics of the system and how they depend on the incidence function. I consider both an epidemic and endemic perspective of the model, but in both cases, three classes of incidence functions are identified. In the epidemic form, power incidences, where the infective portion $I^p$ has $p\in(0,1)$, cause unconditional host extinction, homogeneous incidences have host extinction for certain parameter constellations and host survival for others, and upper density-dependent incidences never cause host extinction. The case of non-extinction in upper density-dependent incidences extends to the case where a latent period is included. Using data from experiments with rhanavirus and salamanders, maximum likelihood estimates are applied to the data. With these estimates, I generate the corrected Akaike information criteria, which reward a low likelihood and punish the use of more parameters. This generates the Akaike weight, which is used to fit parameters to the data, and determine which incidence functions fit the data the best. From an endemic perspective, I observe that power incidences cause initial condition dependent host extinction for some parameter constellations and global stability for others, homogeneous incidences have host extinction for certain parameter constellations and host survival for others, and upper density-dependent incidences never cause host extinction. The dynamics when the incidence function is homogeneous are deeply explored. I expand the endemic considerations in the homogeneous case by adding a predator into the model. Using persistence theory, I show the conditions for the persistence of each of the predator, prey, and parasite species. Potential dynamics of the system include parasite mediated persistence of the predator, survival of the ecosystem at high initial predator levels and ecosystem collapse at low initial predator levels, persistence of all three species, and much more.Dissertation/ThesisDoctoral Dissertation Mathematics 201