A Three Species Ecological Model with A Prey, Predator and Competitor to the Predator and Optimal Harvesting of the Prey

The present paper is devoted to an analytical investigation of three species ecological model with a Prey ( 1 N ), a predator ( 2 N ) and a competitor ) ( 3 N to the Predator without effecting the prey ( 1 N ). in addition to that, the species are provided with alternative food. The model is characterized by a set of first order non-linear ordinary differential equations. All the eight equilibrium points of the model are identified and local and global stability criteria for the equilibrium states except fully washed out and single species existence are discussed. Further exact solutions of perturbed equations have been derived. The analytical stability criteria are supported by numerical simulations using mat lab .Further we discussed the effect of optimal harvesting on the stability.


Abstract:
The present paper is devoted to an analytical investigation of three species ecological model with a Prey ( 1 N ), a predator ( 2 N ) and a competitor ) ( 3 N to the Predator without effecting the prey ( 1 N ). in addition to that, the species are provided with alternative food. The model is characterized by a set of first order non-linear ordinary differential equations. All the eight equilibrium points of the model are identified and local and global stability criteria for the equilibrium states except fully washed out and single species existence are discussed. Further exact solutions of perturbed equations have been derived. The analytical stability criteria are supported by numerical simulations using mat lab .Further we discussed the effect of optimal harvesting on the stability.

Introduction:
Ecology relates to the study of living beings in relation to their living styles. Research in the area of theoretical ecology was initiated by Lotka [15] and by Volterra [16].Since then many mathematicians and ecologists contributed to the growth of this area of knowledge as reported in the treatises of Paul Colinvaux [14], Freedman [2], Kapur [3,4] etc. Recently Archana Reddy [1] discussed the stability analysis of two interacting species with harvesting of both species. Lakshmi Narayan and Pattabhiramacharyulu [5,6] and Shiva Reddy [7,8] et al., Ravidra Reddy [9,10,11] et al. Shanker [12]et al and Papa Rao [13] have discussed different prey-predator models in detail. T.K.kar [17] studied the stability of several species models by incorporating the harvesting term. Inspired from that, we discussed a more general three species model. The model is characterized by a set of first order ordinary differential equations. All the eight equilibrium points of the model are identified and stability criteria for some equilibrium states are discussed. Further we discussed the effect of optimal harvesting on the stability.

Basic Equations:
The model equations for a three species Prey -Predator and competitor to the predator system is given by the following system of first order ordinary differential equations employing the following notation: N are the populations of the prey and predator and a competitor to the predator with the natural growth rates 1 a 2 a and 3 a respectively, 11  is rate of decrease of the prey due to insufficient food 12  is rate of decrease of the prey due to inhibition by the predator, 21  is rate of increase of the predator due to successful attacks on the prey, 22  is rate of decrease of the predator due to insufficient food other than the prey, 23  is rate of decrease of the predator due to the competition with the third species 33  is rate of decrease of the Competitor to the predator due to insufficient food 32  is rate of decrease of the competitor to the predator due to the competition with the predator q Catchability coefficient of prey (N1), E: effort applied to the harvest of the prey Throughout the analysis, we assume that (a 1 -qE) > 0.
The characteristic equation for the system is 3) The equilibrium state is stable, if three roots of the equation (4.3) are negative in case they are real or the roots have negative real parts in case they are complex. The local and global stability of the equilibrium states E 1, E 3, and E 4 are found to be unstable. Reaming is stable. We restricted our study to the equilibrium states E 5, E 6, E 7 and E 8.

Theorem
Let the Lapnouv function for the case E 5 is: dV dt  Therefore , 12 5 ( , ) E N N is globally asymptotically stable 26 BSMaSS Volume 1

Theorem
Let the Lapnouv function for the case E 6 is: Therefore , 13 6 ( , ) E N N is globally asymptotically stable

Theorem
Let the Lapnouv function for the case E 7 is: Differentiating V w.r.to 't' we get 3 Therefore , 23 7 ( , ) E N N is globally asymptotically stable

Conclusion:
In the analysis of the considered prey, predator and a competitor to the predator and optimal harvesting of the prey model, we discussed the local, global stability of the model and exact solutions of perturbed equations have been derived for stable cases. Two set of Numerical examples are studied for which first example with complex roots and second example with real roots. And also study the stability of the system (2.1) with harvesting (qE#0) and without harvesting (qE=0). From the graphs shown fig 6.1.5 and 6.2.5 it is evident that the harvesting of the prey does not have any influence on the stability.