Abstract
The paper presents the study of three species ecological model with Prey N 1 , predator N 2 and competitor to the Predator N 3 and neutral with the predator N 2 with imprecise biological parameters. The model is characterized by a set of first order nonlinear ordinary differential equations. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. Equilibrium points of the model are identified, the local stability is discussed using Routh - Hurwitz criteria and global stability by Liapunov function. The existence of bionomic equilibrium of the system has been discussed and optimal harvesting policy is given using Pontryagin’s maximum principle. The stability analysis is supported by Numerical simulation using Mat lab. Keywords : Prey; Predator; Competitor to the predator; Equilibrium points; interval number, Stability of the equilibrium points; Bionomic Equilibrium; Optimal harvesting policy; Pontryagin’s maximum principle; Numerical simulation using mat lab. DOI : 10.7176/JAAS/52-02
Highlights
By assuming that the predator and competitor to the predator have alternative food in addition to prey population, the model for one Prey and two Predator and harvesting on the both species is given by the following system of first order ordinary differential equations employing the following notation: Let N1 denotes the size of the prey population, N2 denotes the size of the predator population and N3 denotes the size of the competitor to the predator population, lets assuming that there is demand for all species in the market so the harvesting of both species are carried out
By the construction of the prey–predator model the parameters such as prey population growth rate r, predator population growth rate s, competitor to the predator growth rate l and predation coefficients 1, 1O, 1T, P, PO, PT and H- are positive in nature and are considered precise
Most of the researchers have developed the prey, predator and competitor to the predator harvesting model based on the assumption that the biological parameters are precisely known but the scenario is different in real life situation
Summary
1. Background Mathematical modeling of ecosystems is a field of study which helps us to understand the interactions between different species and the mechanisms that influence the growth of species and their existence and stability. Mathematical models have been used to study the dynamics of prey-predator systems since Lotka (1925) and Volterra (1927). They proposed the simple mathematical model which describes the interaction between prey and the predator. A mathematical model to study the ecological dynamics of prey and predator system is proposed and analyzed. As an example some of the prey and predator system in some areas be studied
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