Local stability analysis of two density-dependent semelparous species in two age classes

Abstract

It is crucial to take into account the dynamics of the species while investigating how a species may survive in an environment. A species can be classified as either semelparous or iteroparous depending on how it reproduces. In this article, we present a model, which consists of two semelparous species by considering two age classes. We specifically discuss the effects of density-dependent in the interaction between the two semelparaous species and examine the equilibria of the system in the absence and presence of harvesting in the system. Then, the local stability of the equilibria is also investigated. A modified Leslie matrix population model with the addition of density-dependent in the equation is used. The model is analyzed in the presence and absence of competition between these species. We assume that density-dependent only occurred in the first age class of both species and that harvesting only occurred in the second age class of both species. Then, we assume that competition only occurs in the first age class in both species in the form of interspecific and intraspecific competition. This assumption is intended to simplify the complexity of the problem in the model. Our results show that there are three equilibria in the model without competition and four equilibria in the model with the competition. Hence, the presence of competition has influenced the number of equilibria. We also investigate the relation between the stability of the equilibria with the net reproduction rate of the system. Furthermore, we found the condition for the local stability of the co-existence equilibrium point, which is related to the degree of interspecific and intraspecific competition. This theory may be applied to investigate the dynamics of natural resources, whether in the absence of human exploitation and in the presence of various strategies in managing the exploitation of the resources, such as in fisheries industries.