A Linear State-Space Analysis of the Migration Model in an Island Biogeography System

Biogeography deals with the study of the distribution of biodiversity over space and time and has been well studied by naturists and biologists for over the last five decades. Recently, the theory of biogeography has been applied to solve difficult engineering optimization problems in the form of a nature-inspired metaheuristic, known as biogeography-based optimization (BBO) algorithm. In this correspondence paper, we present an in-depth analysis of the linear time-invariant (LTI) system model of immigration and emigration of organisms in an island biogeography system that forms the basis of BBO. We find the bound of the eigenvalues of the general LTI system matrix using the Perron-Frobenius theorem from linear algebra. Based on the bounds of the eigenvalues, we further investigate four important properties of the LTI biogeography system, including the system reasonability with probability distribution vectors, stability, convergence, and nature of the equilibrium state. Our analysis gives a better insight into the dynamics of migration in actual biogeography systems and also helps in the understanding of the search mechanism of BBO on multimodal fitness landscapes.