Stochastic modelling of nutrient and predator-prey populations

Student thesis: Doctoral Thesis

Abstract

This thesis first constructs a stochastic differential equation (SDE) model of a fjord nutrient, based on the hydrographic and chemical data collected from the 1991 field campaign implemented in Loch Linnhe. Stochastic modelling approach is able to account for the process noise in the nutrient data. The SDE model is first extended from a deterministic nutrient model by the parameter perturbation scheme.To capture the annual variations in the sea-loch nutrient, the SDE model is refined by considering the complex physical and biological processes that make big effects on the nutrient dynamics. The model is parameterised using the least squares approach. The goodness of fit of the SDE model is assessed by comparing the distribution graphs and by performing statistical tests. The existence of the environmental-type process noise in the nutrient data is illustrated by a residual analysis for the data. Finally a simulation study is carried out to identify the accuracy of the parameter estimation frameworks.This thesis also studies the stochastic versions of the foraging arena predator-prey system. The impacts of different types of environmental noise on the population dynamics are deduced. First of all, the SDE predator-prey model is formulated by incorporating white noise into the deterministic foraging arena system using the parameter perturbation technique. We then prove that the SDE has a unique global positive solution. We also study the asymptotic moment estimate of the model solution and produce the conditions for the system to be extinct.Furthermore the existence of a stationary distribution is pointed out under certain parametric restrictions. Secondly of all, we take a further step of incorporating telegraph noise and time delay to the stochastic foraging arena system. The stochastically ultimate boundedness, extinction and the pathwise estimation of the population system are studied. Thirdly, we introduce white noise to more system parameters since all of them can be inuenced by the complex variability.Namely, not only the growth rate of prey and the density-dependent mortality rate of predator, but also the quadratic mortality rates of the two species and the capturing rate of predator are perturbed by the stochastic noise. Then we study how the correlations between the Brownian motions affect the long-time properties of the system. The parametric conditions for the system to have a stationary distribution are deduced. Numerical simulations are carried out to substantiate the analytical results.
Date of Award1 Oct 2018
LanguageEnglish
Awarding Institution
  • University Of Strathclyde
SponsorsUniversity of Strathclyde
SupervisorXuerong Mao (Supervisor) & Michael Heath (Supervisor)

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