A Study on Syn Ecology Consisting of Two Hosts and One Commensal with Mortality Rate for the First and Second Species

B. Hari Prasad^*

Department of Mathematics, Chaitanya Degree College (Autonomous), Hanamkonda, Telangana State, India

Abstract

The present investigation is a study on syn ecology with mortality rate for the first and second species. The system comprises of two hosts S₁, S₂ and one commensal S₃ i.e. S₁ and S₂ both benefit S₃, without getting themselves affected either positively or adversely. Further S₁ and S₂ are neutral. The model equations constitute a set of three first order non-linear differential equations. Criteria for the asymptotic stability of all the eight equilibrium states are established. Trajectories of the perturbations over the equilibrium states are illustrated. Further, the global stability of the system is established with the aid of suitably constructed Liapunov’s function and the numerical examples for the growth rate equations are computed using Runge-Kutta fourth order method.

Keywords

Commensal, Host, Liapunov’s Function, Stable, Trajectories, Unstable

Received: April 4, 2015 / Accepted: April 25, 2015 / Published online: May 19, 2015

@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/

1. Introduction

Ecology, a branch evolutionary biology, deals with living species that coexist in a physical environment & sustain themselves on common resources. It is a common observation that the species of same nature can­­ not flourish is isolation without any interaction with species of different kinds. Significant researches in the area of theoretical ecology have been discussed by Kot [1] and by [2]. Several ecologists and mathematicians contributed to the growth of this area of knowledge. Mathematical ecology can be broadly divided into two main sub-divisions, Autecology and Synecology, which are described in the treatises of Arumugam [3], Sharma [4]. Syn-ecology is an ecosystem comprised of two or more distinct species. Species interact with each other in one way or other. The Ecological interactions can be classified as Ammensalism, Competition, Commensalism, Neutralism, Mutualism, Predation, Parasitism and so on. Mathematical Modeling plays a vital role in providing insight into the mutual relationships (positive, negative) between the interacting species. The general concepts of Modeling in Biological Science have been initiated by several authors Ma [5], Murray [6], and Sze-Bi Hsu [7]. Recently the authors Papa Rao et al. [8], Shivareddy et al. [9], Srinivas [10] and Kumar et al. [11] discussed three species ecological models such as predation, completion and commensalism. Srinivas [12] studied the competitive ecosystem of two species and three species with limited and unlimited resources.  Later, Narayan  et al. [13] studied prey-predator ecological models with partial cover for the prey and alternate food for the predator.  Acharyulu et al. [14,15] derived some productive results on various mathematical models of ecological Ammensalism with multifarious resources in the manifold directions. Further, Kumar [16] studied some mathematical models of ecological commensalism. The present author Prasad [17-20] investigated continuous and discrete models on the three species syn-ecosystems.

In this paper we study on the three species syn-eco system with mortality rate for the first and second species. The system comprises of two hosts S₁, S₂ and one commensal S₃ i.e. S₁ and S₂ both benefit S₃, without getting themselves affected either positively or adversely. Further, S₁ and S₂ are neutral. Commensalism is a symbiotic interaction between two populations where one population (S₁) gets benefit from (S₂) while the other (S₂) is neither harmed nor benefited due to the interaction with (S₁). The benefited species (S₁) is called the commensal and the other (S₂) is called the host. Some real-life examples of commensalism are presented below.

(i). A squirrel in an oak - tree gets a place to live and food for its survival, while the tree remains neither benefited nor harmed.

(ii). A flatworm attached to the horse crab and eating the crab’s food, while the crab is not put to any disadvantage.

Notation Adopted

 : The population strength of  at time ,   : Time instant  : Natural death rates of  and   : Natural growth rate of   : Self inhibition coefficients of ,   : Interaction coefficients of  due to  and  due to  : Extinction coefficient of ,  : Carrying capacity of  Further the variables  are non-negative and the model parameters  are assumed to be non-negative constants.

2. Equilibrium States

The model equations for syn ecosystem are given by the following system of first order non-linear ordinary differential equations.

                            (1)

                           (2)

                   (3)

The system under investigation has eight equilibrium states given by

^{,,  ,,, , ,} 

3. Stability of Equilibrium States

Let where is a small perturbation over the equilibrium state .

The basic equations (1), (2) and (3) are quasi-linearized to obtain the equations for the perturbed state as,

                           (4)

where

with

The characteristic equation for the system is

                            (5)

The equilibrium state is stable, if all the roots of the equation (5) are negative in case they are real or have negative real parts, in case they are complex.

3.1. Stability of E₁

In this case, we have

The characteristic equation is

                   (6)

The characteristic roots of (6) are . Since one of these three roots is positive. Hence the state is unstable and the solutions of the equations (4) are

where  are the initial values of  respectively.

The trajectories in   and  planes are given by

3.2. Stability of E₂

In this case, we have

The characteristic roots are . Since one of these three roots is positive, hence the state is unstable and the solutions are

where

The trajectories in the ;planes are given by

3.3. Stability of E₃

In this case, we have

The characteristic roots are . Since one of these three roots is positive, hence the state is unstable. The equations (4) yield the solutions,

where 

The trajectories in the ;planes are

3.4. Stability of E₄

In this case, we get

The characteristic roots are .  Since all the three roots are negative, hence the state is stable. The equations (4) yield the solutions,

 

where and   with

The trajectories in the  ;  planes are given by  and

3.5. Stability of E₅

In this case, we get

The characteristic roots are . Since two of these three roots are positive, hence the state is unstable. The equations (4) yield the solutions,

The trajectories in the ;planes are given by

3.6. Stability of E₆

In this case,

The characteristic roots are . Since one of these three roots is positive, hence the state is unstable. The equations (4) yield the solutions,

 

where ;

with

The trajectories in the  ; planes are given by and

3.7. Stability of E₇

In this case, we get

The characteristic roots are . Since one of these three roots is positive, hence the state is unstable. The equations (4) yield the solutions,

 

where ;

with

The trajectories in the  ; planes are given byand

3.8. Stability of E₈

In this case, we get

The characteristic roots are.  Since two of these three roots are positive, hence the state is unstable. The equations (4) yield the solutions,

 

where  

with  and

The trajectories in the  ; planes are given by  and

4. Liapunov’s Function

In section 5 we discussed the local stability of all eight equilibrium states, from which only one state  is stable and rest of them are unstable. We now examine the global stability of dynamical system (1), (2) and (3) at this state by suitable Liapunov’s function.

Theorem. The equilibrium state   is globally asymptotically stable.

Proof.  Let us consider the following Liapunov’s function

Now, the time derivative of L, along with solution (3) can be written as

which is negative definite.

Hence, the steady state is globally asymptotically stable.

5. Numerical Examples

The numerical solutions of the growth rate equations (1), (2) and (3) computed employing the fourth order Runge-Kutta method for specific values of the various parameters that characterize the model and the initial conditions. The results are illustrated in Figures 1, 2 and 3.

Figure 1. Variation of N₁, N₂, N₃ against time (t) for d₁ = 0.14, a₁₁ = 0.24,  a₁₃ = 1.04, d₂ = 0.66, a₂₂ = 1.56, a₂₃ = 0.42, a₃ = 2.92, a₃₃ = 7.68, N₁ = 8,      N₂ = 5, N₃ = 2.

Figure 2. Variation of N₁, N₂, N₃ against time (t) for d₁ = 0.10, a₁₁ = 0.24,  a₁₃ = 5.74, d₂ = 2.22, a₂₂ = 0.34, a₂₃ = 4.9, a₃ = 2.82, a₃₃ = 4.2, N₁ = 2, N₂ = 4, N₃ = 8.

Figure 3. Variation of N₁, N₂, N₃ against time (t) for d₁ = 0.10, a₁₁ = 0.60, a₁₃ = 0.60, d₂ = 8.1, a₂₂ = 0.60, a₂₃ = 2, a₃ = 4.7, a₃₃ = 0.60, N₁ = 8, N₂ = 2.5, N₃ = 7.5.

6. Observations

Case 1: This is a situation at the self inhibition coefficient of the third species is highest. Initially the first and second species dominates over the third till the time instant t* = 5.5 and t* = 0.25 respectively and thereafter the dominance is reversed. Further, we notice that the initial values of S1, S2, S3 are in decreasing order. This is shown in  Figure 1.

Case 2: In this case the initial values of S1, S2, S3 are in increasing order. Initially the second species dominates over the first till the time instant t* = 0.2 and thereafter the dominance is reversed. Further, it is evident that all the three species asymptotically converge to the equilibrium point as shown in Figure 2.

Case 3: This is a situation at the self inhibition coefficient of all the three species are identical. The third species dominates over the other two throughout. The natural death rate of S1 is greater than of S2. In course of time we notice a steady variation with no appreciable growth rate in all the three species. (Figure 3).

7. Conclusion

The present paper deals with an investigation on the stability of a three species syn ecology consisting of two hosts and one commensal with mortality rate for the first and second species. In this paper we established all possible equilibrium states. It is conclude that, in all eight equilibrium states, only one state E₄ is stable. Further, the global stability is established with the help of suitable Liapunov’s function and the growth rates of the species are numerically estimated using Runge-Kutta fourth order method.

Acknowledgment

I thank to Professor (Retd), N.Ch.Pattabhi Ramacharyulu, Department of Mathematics, NIT, Warangal (T.S.), India for his valuable suggestions and encouragement.

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