Different levels of Ammensalism in a Mathematical Model with Harvesting at Variable Rate for Enmity Species

Abstract

The purpose of this study is to explore an ecological Ammensal model with four species  of Species-1 i.e Ammensal Prey (P_A), Species-2  i.e Ammensal Predator (Q_A), Species-3 i.e Ammensal Enemy (E_A), and Species-4  i.e Enmity(N) with harvesting at variable rate(χ)  . The System mainly consists of an Ammensal Prey (P_A), a Ammensal Predator (Q_A) that survives on an Ammensal –prey(P_A), Ammensal Enemy (E_A) and Enmity(N) for which P_A, Q_A are Ammensal, i.e., E_A and N have an undesirable effect on P_A and Q_A without being affected in any way. Furthermore, E_A is enormous for N, and N is harmful to E_A. First level Ammensalism is represented by the pair P_A & E_A , the pair of second level Ammensalism  is Q_A & N and third level Ammensalism by the pair E_A & N. The system's model equations form a set of four non-linear ordinary differential coupled equations of first order. There are sixteen equilibrium points found. The interactions between the four species are explored in light of different changes in various species’ growth rate.