The stability of Gauss model; having harvested factor

Selçuk J. Appl. Math. Selçuk Journal of Vol. 13. No. 2. pp. 3-10, 2012 Applied Mathematics

The Stability of Gauss Model; Having Harvested Factor Mansour Saraj1_{, M.H. Rahmani Doust}2_{, F. Haghighifar}3

1_{Department of Mathematics , Faculty of Mathematical Sciences and Computer,}

Shahid Chamran University, Ahvaz, Iran e-mail:m sara j@ scu.ac.ir

2_{Department of Mathematics, Faculty of Sciences, Neyshabur University, Neyshabur,}

Iran

e-mail:m h.rahm anidoust@ neyshabur.ac.ir

3_{Department of Mathematics, Qom University, Qom, Iran}

e-mail:f.haghighifar@ yaho o.com

Received Date: September 30, 2011 Accepted Date: May 11, 2012

Abstract. Scientists are interesting to …nd out "how to use living resources without damaging the ecosystem at the same time". Since the living resources are limited therefore above question is one of important problems that math-ematician scientists try to investigate and in appropriate ways to solv this problem. Regarding to the harvested rate is used as control parameters and moreover, the study of harvested population dynamics is more realistic. In the present paper, some predator-prey Gauss models in which two ecologically in-teracting species are harvested independently with constant or variable rates has been considered and the behavior of locally and globally stability of their solutions have been investigated. The main aim is to present a mathematical analysis for the above model. Finally we investigate some examples..

Key words: Gauss Model; Growth Rate Model; Harvested Factor; Lineariza-tion; Lotka-Volterra.

2000 Mathematics Subject Classi…cation: 93A30, 37B35, 93B18, 65H1. 1. Introduction

Gauss is one of scientist that was studied predator-prey problem which is well-known and an old problem in mathematical biology. He was obtained mainly results to interpret and analyze this problem. In (1934) Gauss and in (1936) Gauss and Smaragdov was studied generalization of the following model as a model for predator-prey interactions:

(1.1) 8 < : dx dt = ax yp(x) dy dt = y( + cp(x))

Above model states that the prey growth is enhanced by its own presence and its increase growth is limited at predators present, but the predator growth is decreased by its own presence and its growth rate is enhanced at preys present. This model is analyzed in [3] by Burton.

(1.2) 8 < : dx dt = xg(x) yp(x) dy dt = y( + q(x))

Here the function g(x) is the speci…c growth rate of the prey in the absence of any predators and represents the relative increase of preys in unit of time. The function p(x) is e¢ ciency of predator on particular prey and expresses the number of prey consumed by a predator in a unit of time. The function q(x) is the predator response function with respect to that particular prey. The second statement of (1.2) describes the growth rate of the predator population and the function q(x) gives the total increase of the predator population. It is clear that in the absence of prey the predator population declines.

Let p(x) = bx, so system (1.1) convert to Lotka-Volterra system, which is one of famous models in analyzing interaction between predator-prey. This model is modularize about 1925 by Lotka and Volterra independently at …rst. Lotka-Volterra system is as follows:

(1.3) 8 < : dx dt = ax bxy dy dt = y( c + dx)

Note that in the present paper x and y are density of prey and predator popu-lations respectively and all of coe¢ cient are positive constant.

2. The Predator-Prey Gauss Model with Having Harvested Factor 2.1. Constant harvested factor

Let two species of prey and predator live in an ecosystem. For example consider two species …sh and shrink that live in a lake and …sh bait with shrink. Let Gauss model modularize interaction between …sh and shrink. Moreover let …sherman spreads dragnet on surface of lake one time at month and baits constant number of two species prey and predator. The above problem is describe with following

system, which is known as predator-prey system with having constant harvested factor: (2.1) 8 < : dx dt = ax yp(x) h dy dt = y( + cp(x)) k

Let (x; y) is arbitrary equilibrium point of above system. We analyze stability of above system by using suitable Lyapunov function.

Theorem 2.1. Let x < x and y > y. System (2.1) is globally asymptotically stable.

Proof. It is clear that v(x; y) = cx + y is a Lyapunov function for given system. Now by di¤erentiability of v with respect to time variable we have:

dv dt = c dx dt + dy dt

By substituting dx_dt and dy_dt from system (2.1) and some simplify dv_dt is given by: dv

dt = c[ax yp(x) h ax + yp(x) + h] + [y( + cp(x)) k y( + cp(x)) + k] = ca(x x) s(y y)

So if x < x and y > y, then dv

dt < 0 and system (2.1) is globally asymptotically stable.

2.2. E¤ort rate harvested factor

Consider the predator-prey Gauss model with having e¤ort rate harvested factor for prey and predator. Following system represent this model:

(2.2) 8 < : dx dt = ax yp(x) Ex dy dt = y( + cp(x)) Ey

If a > E by substituting a E by a above system convert to system (1.1), else this system is not satisfy in assumptions of Gauss model.

2.3. Harvested factor type 1

Consider following system with having harvested factor. In this system har-vested factors are x2 _{and y}2_{for prey and predator species respectively. Also we} can say following system is Gauss model with having interaction interaspecies between prey and predator species. This system is as follows:

(2.3) 8 < : dx dt = ax yp(x) x 2 dy dt = y( + cp(x)) y 2

Let (x; y) is arbitrary equilibrium point of above system. We analyze stability of above system by using suitable Lyapunov function.

Theorem 2.2. Let (x x)(a x x) < 0 and y > y. System (2.3) is globally asymptotically stable.

Proof. It is clear that v(x; y) = cx + y is a Lyapunov function for given system. Now by deviating of v with respect to time variable we have:

dv dt = c dx dt + dy dt

By substituting dx_dt and dy_dt from system (2.1) and some simplify dv_dt is given by: dv

dt = c[ax yp(x) x

2 _ax+yp(x)+x2_{]+[y( +cp(x)) y}2 _{y( +cp(x))+y}2_{] =} c(x x)(a x x) (s + y + y)(y y)

Let (x x)(a x x) < 0 and y > y, then dv_dt < 0 and system (2.3) is globally asymptotically stable.

2.4. Harvested factor type 2

Consider Gauss model with having coexistence interaspecies between prey species and interaction interaspecies between predator species. Following system rep-resent model of above problem:

(2.4) 8 < : dx dt = ax yp(x) + x2 dy dt = y( + cp(x)) y 2

Let (x; y) is arbitrary equilibrium point of above system we analyze stability of above system by using Lyapunov function.

Theorem 2.3. Let x < x and y > y. System (2.4) is globally asymptotically stable at equilibrium point (x; y).

Proof. It is clear that used Lyapunov function in theorem (2.2) is a suitable Lyapunov function for given system. Now by derivation of this function and some simplifying dv_dt is given by:

dv

dt = c(a + x + x)(x x) (s + y + y)(y y) So if x < x and y > y, then dv

dt < 0 and system (2.4) is globally asymptotically stable.

3. Examples of the Predator-Prey Gauss Model 3.1. Example 1

Consider following system of extended Gauss model. In this model density of the prey increases by logistic growth rate rx(1 x_k) and this enhance limited by term _a+xpxy which p and a are positive constants. In this system v is number of su¢ cient preys to support one predator species.

(3.1) 8 < : dx dt = rx(1 x k) pxy a+x dy dt = sy(1 y vx)

It is clear that in above system x 6= 0. Now let y = 0, Thus (k; 0) is a equilibrium point of this system. Let x; y 6= 0, so from equation two we have y = vx and by substituting y in equation one and a little simplify, x is given by:

x1= k(ra k r+pv) 2r k q (ra k r+pv)2+4ar2k 2r x2= k( ra k r+pv) 2r + k q (ra k r+pv)2+ 4ar2 k 2r

Since x1 is negative and in other hand orbit of solutions of Gauss system is intR2_{+= fx}ijxi 0; i = 1; 2g, thus x2 that is positive is studied. Then system (3.1) has equilibrium points:

(k; 0) and (x2; vx2)

We using the linearization method to study the stability of above equilibrium point for system (3.1). For this means, we …nd out the jacobian matrix, which may be found as follows:

J j(x;y)= 0 @ r 2rx_{k (a+x)}apy2 px a+x sy vx2 s 2sy vx 1 A

For equilibrium point (k; 0) jacobian matrix is given by:

J j(k;0)= 0 @ r pk a+k 0 s 1 A

So r and s are eigenvalues of above matrix, thus equilibrium point (k; 0) is a saddle point for system (3.1).

At equilibrium point (x2; vx2) jacobian matrix is …nd out as follows:

A = J j(x2;vx2)= 0 @ r 2rx2 k apvx2 (a+x2)2 px2 a+x2 s x2 s 1 A So if r < 2rx2 k + apvx2 (a+x2)2, then

detA > 0; trA < 0

therefor system (3.1) is locally asymptotically stable in said point. So following proposition is proved.

Proposition 3.1. Following statements about system (3.1) are holds: i: Equilibrium point (k; 0) is a saddle point for system (3.1). ii: Let r < 2rx2

k + apvx2

(a+x2)2, then system (3.1) is locally asymptotically stable

in equilibrium point (x2; vx2). 3.2. Example 2

Consider following system of Gauss model with having harvested factor. In this system term _a+xpxy represent e¢ ciency of the predator on preys. The term qEx is e¤ort rate harvested factor on preys. This system may be considered as extension of system(3.1). (3.2) 8 < : dx dt = rx(1 x k) pxy a+x qEx dy dt = sy(1 y vx)

Let r > qE, above system has two equilibrium points in orbit of solutions of Gauss model, as follows:

(k qkE_r ; 0)and(x; vx) where

x = k(rak r+pv+qE)

2r +

kp(ra

k r+pv+qE)2 4rk(aqE ra)

2r

We analyze equilibrium points of system (3.2) by using linearization method. First jacobian matrix is given by:

J j(x;y)= 0 @ r 2rx k apy (a+x)2 qE px a+x sy vx2 s 2sy vx 1 A

For this means …rst by using jacobian matrix and little simplify eigenvalues in equilibrium point (k qkE_r ; 0) are found as 1; 2 = qE r; s. As regarding to assumption r > qE, equilibrium point (k qkE_r ; 0) is a saddle point for system (3.2). Now A = J j(x;vx)= 0 @ r 2rx k apvx (a+x)2 qE px a+x s x s 1 A

So if r < 2rx_k +_(a+x)apvx2 + qE, then trA < 0 and detA > 0, therefor system is

locally asymptotically stable.

Following system is extended of system (3.2) and it is studied as above system.

(3.3) 8 < : dx dt = rx(1 x k) pxy a+x qEx dy dt = sy(1 y vx qE) 3.3. Example 3

Consider following system:

(3.4) 8 < : dx dt = rx(1 x k) pxy dy dt = sy(1 y vx)

If k > 1 above system has two equilibrium points (k; 0) and ( rk kpv+r; v

rk kpv+r). We study stability of above system by using linearization method. For this means jacobian matrix is found out as follows:

J j(x;y)= 0 @ r 2 rx k py px a+x sy vx2 s 2sy vx 1 A

By substituting equilibrium points of system (3.4) and simplifying we found that following proposition is true.

Proposition 3.2. Following statements about system (3.4) are holds: i: Equilibrium point (k; 0) is a saddle point for system (3.4).

ii: System (3.4) is locally asymptotically stable in equilibrium point (_kpv+rrk ; v_kpv+rrk ).

Consider following extension of system (3.4):

(3.5) 8 < : dx dt = rx(1 x k) pxy x y dy dt = sy(1 y vx)

above system has two equilibrium points (x1;x1_v1) and (x2;x2_v1), which x1=(r+ r k+2vb) 2(r k+ b v ) p (r+r k+2 b v)2 4( r k+ b v)(r+v+ b v) 2(r k+ b v) x2= (r+r k+2 b v) 2(r k+bv ) + p (r+r k+2vb)2 4(rk+bv)(r+v+bv) 2(r k+vb)

We analyze above system by using Lyapunov function. Let (x; y) is arbitrary equilibrium point of system (3.5), following theorem express conditions of sta-bility of above system.

Theorem 3.1. Global stability of the system (3.5) at equilibrium point (x; y) is related to sign of (x x)(y y).

Proof. Consider Lyapunov function

v(x; y) =R_xxs x_s ds +R_yy t y_t dt

Now by di¤erentiate of above Lyapunov function with respect to variable t dv_dt is found out as follows:

dv dt = x x x dx dt + y y y dy dt Now by instituteddx dt and dy

dt from system (3.5) and some simplifying dv dt is found out as follows: dv dt = r k(x x) 2 _{(x x)(y y) +} (x x)(y y) yy + s vy(x x)(y y) s vx)(y y)2+_yys(y y)2 Therefor sign of dv

dt is related to sign of (x x)(y y). References

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2. T. Kumar Kar, Modelling and Analysis of a Aarvested Prey–Predator System Incorporating a Prey Refuge, Journal of Computational and Applied Mathematics ,Vol. 185 , PP. 19–33, 2006.

3. T. A. Burton, Volterra Integral and Di¤erential Equations, Academic press, New York, 1983.

4. K. Hasik, On a Predator-Prey System of Gauss Type, Mathematical Biology, Vol. 60, PP. 59-74, 2010.

5. T. K. Kar, K.S. Chaudhuri, Harvesting in Two Predator One Preyb Fisheri: a Bioconomic Model , ANZIAM, Vol. 45, PP. 443–456, 2004.