Nonsmooth Bifurcations and the Role of Density Dependence in a Chaotic Infectious Disease Model
Hughes, Ryan Patrick
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Discrete dynamical systems can exhibit rich and interesting dynamics at lower dimensions (and co-dimensions) than that of ODE models. Classically, the minimal dimension to observe chaotic behavior in an ODE model is three; whereas it can be achieved in a one-dimensional discrete map. It is often the choice of mathematical biologists to use discrete systems as it fills many roles such as sparse data, incorporation of life cycle stages and noisy measurements. This work is analyzes a discrete time model of an infected salmon population. It provides an in-depth analysis of non-smooth bifurcations for alternate functional forms for density dependence in the growth function of a given model. These demonstrate interesting structures and chaotic behaviors with biologically feasible interpretations such as intrinsic growth rate and probability of death. The choice of density dependence function, as well as parameterization, leads to whether chaos occurs or not.
General Audience Abstract
Often times biological processes do not happen in a continuous streamlined chain of events. We observe discrete life stages, ages, and morphological differences. Similarly, data is generally collected in discrete (and often fixed) time intervals. This work focuses on the role that population density has on the behavior of these systems. We dive into a case study for a viral infection in a salmon population. We show chaotic behavior can be observed as low as a single dimension model and discuss the biological implications. Additionally, we show that the choice of density dependence in a given infectious disease model directly impacts disease dynamics and can allow or prohibit chaotic behavior.
- Masters Theses