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Optimal Harvesting of Prey in Three Species Ecological Model with a Time Delay on Prey and Predator


Affiliations
1 Department of Mathematics, JNTUK, University College of Engineering,Vizianagarm-535003, India
2 Department of Mathematics, VITS, Deshmukhi, Hyderabad - 508284, India
     

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This paper deals with the stability analysis of three species Ecological model consists of a Prey (N1), a predator (N2) and a competitor (N3) .The competitor (N3) is competing with the Predator Species (N2) and neutral with the prey (N1). In this model the third species is competing with the predator for food other than the prey (N1).In addition to that, the death rates, carrying capacities of all three species are also considered , a delay in the interaction between the prey and the predator (gestation period of the predator) and harvesting effort of prey population is also considered. The model is characterized by a couple of integro- differential equations. All the eight equilibrium points of the model are identified and their local stability is discussed for interior equilibrium point. The global stability is studied by constructing a suitable Lyapunov’s function. Suitable parameter are identified for Numerical simulation which shows that this continuous time delay model exhibits rich dynamics and time delay can further stabilize or destabilize the system.

Keywords

Prey, Predator, Competitor, Equilibrium Points, Local Stability, Global Stability, Harvesting Numerical Simulation.
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  • Vidyanath T, Laxmi Narayan K and Shahnaz Bathul, A three species ecological model with a predator and two preying species, International Frontier Sciences Letters, 9; 2016: 26-32.

  • Paparao AV and Lakshmi Narayan K, Dynamics of a prey predator and competitor model with time delay. International Journal of Ecology and Development. 32(1); 2017: 75-86.


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  • Optimal Harvesting of Prey in Three Species Ecological Model with a Time Delay on Prey and Predator

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Authors

A. V. Paparao
Department of Mathematics, JNTUK, University College of Engineering,Vizianagarm-535003, India
K. Lakshmi Narayan
Department of Mathematics, VITS, Deshmukhi, Hyderabad - 508284, India

Abstract


This paper deals with the stability analysis of three species Ecological model consists of a Prey (N1), a predator (N2) and a competitor (N3) .The competitor (N3) is competing with the Predator Species (N2) and neutral with the prey (N1). In this model the third species is competing with the predator for food other than the prey (N1).In addition to that, the death rates, carrying capacities of all three species are also considered , a delay in the interaction between the prey and the predator (gestation period of the predator) and harvesting effort of prey population is also considered. The model is characterized by a couple of integro- differential equations. All the eight equilibrium points of the model are identified and their local stability is discussed for interior equilibrium point. The global stability is studied by constructing a suitable Lyapunov’s function. Suitable parameter are identified for Numerical simulation which shows that this continuous time delay model exhibits rich dynamics and time delay can further stabilize or destabilize the system.

Keywords


Prey, Predator, Competitor, Equilibrium Points, Local Stability, Global Stability, Harvesting Numerical Simulation.

References