Continuous-time Models of Plankton Interactions and a Discrete System of Larch Budmoth Population
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Motivated by the increasing incidences of harmful algal blooms worldwide, we propose deterministic models of interactions between non-toxic producing phytoplankton (NTP), toxin producing phytoplankton (TPP), and zooplankton to explore the role of TPP on population interactions. In these models, the mechanisms of mutual interference between predator zooplankton and the avoidance of TPP by zooplankton are incorporated. The interaction between NTP and TPP is modeled by the classic Lotka-Volterra competition equations. The toxin produced by the TPP has no negative effect on the NTP but the zooplankton suffer an extra mortality due to the ingestion of TPP. One of the models assumes that the toxin production by TPP is a constant over time and the resulting model is an autonomous system of ordinary differential equations. To simulate day/night, tidal or seasonal cycles occurring in natural systems, we extend the autonomous model to a periodic system. For both of these models, we investigate existence of steady states/trivial periodic solutions and their stability. Sufficient conditions based on the model parameters are derived for which the three interacting populations can coexist. From these conditions, we show that increasing the avoidance of TPP by zooplankton can promote coexistence of the populations in both of these systems. The deterministic systems proposed do not take demographic stochasticity into consideration. To account for the random birth and death embedded in the populations, we formulate continuous-time Markov chain models and stochastic models of Ito differential equations based on the deterministic systems. We numerically simulate the stochastic models and explore the differences between these two types of models. On a separate topic, the population of larch budmoth in the Swiss Alps is well known for its periodic outbreaks and is one of the best examples of a complex population system. In this thesis, we investigate a discrete consumer resource system based on a model originally proposed for studying the cyclic dynamics of the larch budmoth population in the Swiss Alps. We study existence of steady states and their stability. It is shown that the moth population can persist indefinitely for all of the biologically feasible parameter values. We also discuss local bifurcations of the model. Sufficient conditions based on the model parameters are deduced for which either a period-doubling bifurcation or a Neimark-Sacker bifurcation occurs. In the latter case, the moth population will be quasi-periodic over time, which may be qualitatively similar to the observed moth population data.