American Journal of Applied Mathematics 2018; 6(3): 97-108 http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20180603.11 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) Modeling and Simulation Study of the Population Dynamics of Commensal-Host-Parasite System Geremew Kenassa Edessa 1 , Boka Kumsa 1 , Purnachandra Rao Koya 2 1 Department of Mathematics, Wollega University, Nekemte, Ethiopia 2 Department of Mathematics, Hawassa University, Hawassa, Ethiopia Email address: To cite this article: Geremew Kenassa Edessa, Boka Kumsa, Purnachandra Rao Koya. Modeling and Simulation Study of the Population Dynamics of Commensal-Host-Parasite System. American Journal of Applied Mathematics. Vol. 6, No. 3, 2018, pp. 97-108. doi: 10.11648/j.ajam.20180603.11 Received: April 30, 2018; Accepted: June 1, 2018; Published: July 5, 2018 Abstract: This paper deals with the modeling and simulation study of Commensal-host species system together with the inclusion of parasite population. The model comprises of three populations viz. Host, Commensal and Parasite. The Commensal population gets benefit from Host population but the former do not do any harm to the latter. The parasite population gets benefit and also do harm to Host population. However, the Commensal population only harms the parasites. The mathematical model is comprised of a system of three first order non-linear ordinary differential equations. Mathematical analysis of the model is conducted. Positivity and boundedness of the solution have been verified and thus shown that the model is physically meaningful and biologically acceptable. Scaled model is constructed so as to reduce the number of model parameters. Equilibrium points of the model are identified and stability analysis is conducted. Simulation study is conducted in order to support the mathematical analysis. In the present model the Commensal population lies higher and the parasite population lies below respectively the host population. This fact is well supported by the mathematical analysis as well as simulation study. The results of analysis and simulation are presented and discussed lucidly in the text of the paper. Keywords: Commensalism, Parasite, Stability, Eco-epidemiology, Numerical Simulation 1. Introduction It has been the general observation that populations can sometimes undergo drastic changes in their abundance. These changes are not only unexpected but also are equally difficult to explain in simple manner either theoretically or empirically [1-2]. Many species have been driven to extinct and many others are at the verge of extinction due to several reasons such as overexploitation, predation, and the like [3]. In nature populations may interact in different ways with one another during their lifetime and even they may undergo from one type of interaction to another. The species populations in an ecosystem will interact in different ways and these interactions can be regarded as positive (+), negative (-) or neutral (0). For instance the interactions such as mutualism and commensalism are positive interactions while those such as intra and inter- specific competitions, parasitism and predation are considered as negative interactions. Mathematical representations have been developed to address all these kinds of interactions. In general, in case of a two-species interaction the mathematical formulation can be expressed by a system of two equations as    r  N  1       a        (1) Here in (1) ,   1, 2 but   . The parameters   and   represent the intrinsic growth rate and carrying capacity of the species  respectively. The variables   and  ! represent the population densities of the species  and  respectively. The quantity " ! is the interaction coefficient between species  and . Some classical forms of the models addressing two interacting populations are represented by these types of system of equations depending on the signs of the interaction