Some models and their nonstandard discretizations for honeybee (Apis mellifera L) colony population dynamics

Abstract

This dissertation focuses on two mathematical models to study biologically, theoretically and computationally the alarming declines of the colony population of honey bees, specifically the Colony Collapse Disorder (CCD) and capensis calamity (cc). A comprehensive description of honey bees as model organisms is provided with the aim of understanding and elaborating the assumptions that are made in order to formulate the two models. The first model is due to Khoury, Meyerscough and Barron [43]. Assuming that the rate at which the maximum eclosion is approached is sufficiently large, we have established the following result: There exists a critical value, mc; of the foragers mortality rate, m; which is a transcritical bifurcation. More precisely, the CCD occurs for m > mc in the sense that the trivial equilibrium point, (0,0), is globally asymptotically stable (GAS) for such large values of the mortality rate. If m < mc; the colony is healthy in the sense that a new interior equilibrium point which is GAS is born , while the trivial equilibrium point is unstable. In the second step, we propose a social parasite (SP) model which is characterized by a low recruitment rate of the host population. We prove that the cc occurs in the sense that the total population of the host decays to zero. We design nonstandard finite difference (NSFD) schemes that preserve the stability properties of the two continuous models including the CCD and cc phenomena. The faster decline in the SP setting is demonstrated theoretically for the NSFD scheme. Numerical simulations are provided to support the theory.