A PREY-PREDATOR MODEL WITH COVER FOR THE PREY AND AN ALTERNATIVE FOOD FOR THE PREDATOR AND CONSTANT HARVESTING OF BOTH THE SPECIES *

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1 Jordn Journl of Mthemtics nd Sttistics (JJMS) (), 009, pp A PREY-PREATOR MOEL WITH COVER FOR THE PREY A A ALTERATIVE FOO FOR THE PREATOR A COSTAT HARVESTIG OF BOTH THE SPECIES * K. LAKSHMI ARAYA.PATTABHIRAMACHARYULU ABSTRACT The present pper is devoted to n nlyticl investigtion of two species preypredtor model. Predtor is provided with limited resource of food in ddition to the prey nd cover to prey proportionte to its popultion to get protection from the predtor. Both the prey nd predtor re hrvested t constnt rte. The model is chrcterized by couple of first order non-liner ordinry differentil equtions. The lone equilibrium point of the model is identified nd its stbility criteri is discussed. The Globl stbility of linerized equtions is discussed by constructing suitble Lipunov s function. A threshold theorem is stted nd results re discussed.. ITROUCTIO In the clssicl Lotk - Volterr Prey - Predtor model, there is no protection for Prey from the Predtor nd Predtor sustins on the Prey lone. When the Prey popultion flls below certin level, the predtor would migrte to nother region in serch of food nd return only when the Prey popultion rises to the required level. Olinck,[] gve n introduction to Mthemticl modeling in life sciences. Kpur, [], Smith,[3], Colinvux, [4], Freedmn, [5] discussed some of the prey-predtor ecologicl models. My, [6] discussed stbility nd complexity of ecologicl models, Vrm, [7] discussed bout their exct solutions. ryn & Rmchryulu, [8], [9], [0], [] nd [] discussed different prey-predtor models.. Bsic Equtions The model equtions for two species prey-predtor system is given by coupled below. non-liner ordinry differentil equtions employing the terminology given nd re the popultions of the prey nd predtor with the nturl growth rtes nd respectively, is rte of decrese of the prey due to insufficient food, 000 Mthemtics Subject Clssifiction: 37C75. Keywords: Equilibrium point, orml Study Stte, Prey, Predtor, Stbility, Threshold igrms, nd Threshold Theorems. Copyright enship of Reserch nd Grdute Studies, Yrmouk University, Irbid, Jordn. Received on: ec. 4, 008 Accepted on: June 6,

2 44 K. LAKSHMI ARAYA A.PATTABHIRAMACHARYULU is rte of decrese of the prey due to inhibition by the predtor, is rte of increse of the predtor due to successful ttcks on the prey, is rte of decrese of the predtor due to insufficient food other thn the prey, h is rte of hrvest of the prey, h is rte of hrvest of the predtor, nd both h, h re ssumed to be positive constnts. (i)eqution for the growth rte of prey species ( ): d ( k) - h. (.) (ii)eqution for the growth rte of predtor species ( ): d ( k ) - h (.) 3. Equilibrium sttes The equilibrium sttes for the system under investigtion re given by d d 0 nd 0 { ( k) } h (3.) i.e. { ( k) } h (3.) (3.) ( k) (3.) ( k) we get h ( k) h ( k) ( k ) ( k ) ( k ) ( k ) (3.3) On rerrnging, the bove terms cn brought to the form ( k) ( ) ( k) ( ) ( k) h 4 ( ) ( k) ( h ) 0 4, (3.4) which connects the hrvesting rtes nd the norml stedy stte. From this eqution two cses cn be drwn. They re (i) Cse of exclusive hrvesting (i.e. the hrvesting rtes of prey nd predtor re independent of ech other) is given s h nd h (3.5) 4 4

3 A PREY-PREATOR MOEL WITH COVER FOR THE PREY A A ALTERATIVE FOO FOR THE PREATOR A COSTAT HARVESTIG OF BOTH THE SPECIES 45 (ii) Cse of mixed or gross hrvesting is chrcterized by ( ) ( ) 0 k h k h 4 4 ( ) ( ) (3.6) In either of the cses, the equilibrium vlues of nd re relted by ( k) ( ) ( k) ( ) 0 (3.7) The lone equilibrium point is (Hlf of the crrying cpcity of (Hlf of the crrying cpcity of ) nd (3.8) ) (3.9) 4. Stbility of Equilibrium Stte Let (, ) U ( u, u) (4.) where U ( u, u ) is smll perturbtion over the equilibrium stte ( ).The bsic equtions (.), (.) re qusi-linerized to obtin the, du equtions for the perturbed stte AU where ( k) ( k ) A ( k ) ( k) (4.) The chrcteristic eqution for the system is det [ A λi ] 0 (4.3) The equilibrium stte is stble only when the roots of the eqution (4.3) re negtive, in cse they re rel or hve negtive rel prts, in cse they re complex. Put u nd u in (.) & (.), where u nd u re smll perturbtions from the equilibrium stte. du du ( u ){ ( u ) ( k)( u )} h (4.) ( u ){ ( u ) ( k)( u )} h (4.) By neglecting products, higher powers of u, u, nd constnt terms we get du u ( ku ) (4.3)

4 46 K. LAKSHMI ARAYA A.PATTABHIRAMACHARYULU nd du u ( k) u (4.4) The chrcteristic eqution is λ ( ) λ [ ( k) ] 0 (4.5) The roots of which cn be noted to be negtive. The co-existent equilibrium stte is stble. The trjectories re u0 ( λ )- u0 ( k) u e λ t λ λ u0 ( λ )- u0 ( k) e λ t (4.6) λ λ u ( λ )- u ( k) u 0 0 λ λ u0 ( λ )- u0 ( k) λ λ The curves re illustrted in Figures &. t e λ t e λ (4.7) CASE : Initilly the prey domintes the predtor nd it continues through out its growth i.e. u 0 < u 0. In this cse the predtor lwys outnumbers the prey. It is evident tht both the species re symptotic to the equilibrium point. Hence this stte is stble. CASE : The prey domintes the predtor in nturl growth rte but its initil strength is less thn tht of predtor. i.e. u0 > u 0. In this cse, initilly the prey outnumber the predtor nd this continues the time instnt t t* given by eqution (4.8), fter which the predtor out number the prey. t t * λ λ where 3 ( b ) u ( b) u ln ( b 6 ) u0 ( 4 b ) u0 λ ; 4 λ ; 5 λ ; λ 6 (4.8) b ( k) ; b ( k). (4.9) As t both u& u pproches the equilibrium point. Hence the stte is stble.

5 A PREY-PREATOR MOEL WITH COVER FOR THE PREY A A ALTERATIVE FOO FOR THE PREATOR A COSTAT HARVESTIG OF BOTH THE SPECIES Trjectories of perturbed species The trjectories in the u-uplne re given by ( p - v 3 ) ( )( v v) ( u- uv) [ u ] d ( 3 ) ( ) p v u v u (4.0) where, p (4.) 3 nd v, v re roots of qudrtic eqution v bv c4 0with ; b ( k) (4.) ; c ( k) 4 nd d, n rbitrry constnt. When ( ) < 4 ( k), (4.3) the roots re complex with negtive rel prt. Hence the equilibrium stte is stble. The solution curves re illustrted in Figure.3. When ( ) > 4 ( k), (4.4) the roots re rel nd negtive. Hence the equilibrium stte is stble. The solution curves re illustrted in Figure Lipunov s Function for Globl Stbility The Linerized Bsic Equtions for the model under investigtion re du du The chrcteristic eqution is u ( ku ) (5.) ( ku ) u (5.) ( λ )( λ ) ( k) 0 λ pλ q 0 (5.3) where p > nd q { ( k) } > 0 0 Hence the conditions for Lipunov s function re stisfied. ow we define Eu (, u) ( u buu cu ) (5.4) ( ( k ) ) ( ) { ( k) } where (5.5)

6 48 K. LAKSHMI ARAYA A.PATTABHIRAMACHARYULU ( k) ( k) b (5.6) ( ) ( ( k) ) { ( k) } c nd (5.7) pq { }{ ( k) } (5.8) From equtions (5.5) & (5.8) it is cler tht > 0 nd > 0. Also ( c b ) ( ( k ) ) ( ) { ( k) } { ( ) ( ( k) ) { ( k) } ( k) ( k) ( k) Since > 0, c (5.9) b is lso greter thn zero. (5.0) } > 0 The function E(x, y) is positive definite. Further, du du ( u bu )[ u ( k) u ] ( bu cu )[ ( k) u u ] ( b( k) ) u ( b ( k) c ) u {[ b ( k)] [ b c ( k) } uu (5.) By substituting the vlues of, b nd c from equtions (5.5),( 5.6) &(5.7) nd on simplifiction, we get du du { k k k [ ( ) ( ) ( ) { ( k) ] } u k k [ ( ) ( )

7 A PREY-PREATOR MOEL WITH COVER FOR THE PREY A A ALTERATIVE FOO FOR THE PREATOR A COSTAT HARVESTIG OF BOTH THE SPECIES 49 ( k) ] } u { 3 ( k) ( k) ( k) 3 ( k) ( k) ( k) ( k) ( k) ( k) 3 3 ( ) ( ) ( ) k k k } uu u u du du ( u u ), (5.) which is clerly negtive definite. So E(x, y) is Lypunov s function for the liner system. Further we prove tht E ( u, u ) is lso Lypunov s function for the non liner system. If, F nd F re defined by F(, ) { ( k) } h F(, ) { ( k) } h (5.3), (5.4) we hve to prove tht F F is negtive definite. By substituting u nd u in (5.) & (5.) equtions, we get du ( u ) u k k u { ( ) ( ) }- h From (5.3), F( u, u ) du u ( ku ) f ( u, u ) (5.5) where f ( u, u ) u ( k) uu - h (5.6)

8 50 K. LAKSHMI ARAYA A.PATTABHIRAMACHARYULU Similrly F ( u, u ) du u ( k) u f ( u, u ) (5.7) where f ( u, u ) u ( k) uu - h (5.8) And we hve u bu nd bu cu (5.9) By considering the equtions (5.5),(5.7) nd (5.9), F F ( u u ) ( u bu ) f ( u, u ) ( bu cu ) f ( u, u ) (5.0) By introducing polr co-ordintes we get, F F [( cos sin ) ( ) ( cos sin ) ( )] (5.) r r θ b θ f u, u b θ c θ f u, u By denoting lrgest of the numbers, b, c by M, our ssumptions become (, ) r f u u < nd f( u, u) 6M for ll sufficiently smll r > 0, F F so 4Kr r < r < 0 6M 3 r < (5.) 6M Thus E ( u, u ) is positive definite function with the property tht F F is negtive definite. The equilibrium point is n symptoticlly stble. 6. Threshold Theorem In consonnce with the principle of competitive exclusion [Guss (934)], we deduce Threshold Theorems for the lone equilibrium point. The bsic equtions for the model under considertion cn be re-written s where d { k β } h k d { k β } h k k ; k ; β ( k) nd (6.) β ( k).

9 A PREY-PREATOR MOEL WITH COVER FOR THE PREY A A ALTERATIVE FOO FOR THE PREATOR A COSTAT HARVESTIG OF BOTH THE SPECIES 5 Theorem. Principle of Competitive Exclusion. k k When k k β > nd β >, then every solution of (), t () t of (6.) pproches the equilibrium solution () t ( 0 ) nd () t ( 0 ) s t pproches infinity. In other words, if prey nd predtor species re nerly identicl nd the microcosm cn support both the members of prey nd predtor species depending up on the initil conditions. Proof: The first step in our proof is to show tht () t nd () t cn never become negtive. To this end, observe tht () t nd is solution of (6.) for ny choice of (0) () t. The orbit of this solution in the plne is the point (0, 0) for (0) 0 ; the line 0 < < k, 0 for 0 < (0) < k ; the point ( k,0) for (0) k ; nd the line k < <, 0 for (0) k >. Thus the xis, for 0 is the union of four distinct orbits of (6.). Similrly the xis, for 0, is the union of four distinct orbits of (6.). This implies tht ll solutions (), t () t of (6.) which strt in the first qudrnt ( () t 0, 0) > > of the plne must remin there for ll future time. The second step in our proof is to split the first qudrnt into regions in which both d nd d hve fixed signs. This is ccomplished in the following mnner. Let l nd l be the lines k β h nd k β h respectively d nd the point of their intersection, is (, ). Observe tht is negtive if (, ) lies bove the line l nd positive if (, ) lies below l. Similrly, is negtive if (, ) lies bove l nd positive if (, ) lies below l. Thus the two lines l nd l split the first qudrnt of the plne into four regions in d which both d nd d hve fixed signs. (), t () t both increse with time in region I; () t increses nd () t decreses with time in region II;

10 5 K. LAKSHMI ARAYA A.PATTABHIRAMACHARYULU () t decreses nd () t increses with time in region III nd both () t nd () t decrese with time in region IV. In this region both the prey predtor compete with ech other but do not flourish nd t the sme time do not get extinct. Finlly we require the following four lemms. Lemm. Any solution of (), t () t of (6.) which strts in region I t time t t0 will remin in this region for ll future time t t0, nd ultimtely pproch the equilibrium solution () t, () t.(figure 5) Lemm. Any solution of (), t () t of (6.) which strts in region II t time t t0 will remin in this region for ll future time t t0, nd ultimtely pproch the equilibrium solution () t, () t.(figure 5) Lemm 3. Any solution of (), t () t of (6.) which strts in region III t time t t0 will remin in this region for ll future time t t0, nd ultimtely pproch the equilibrium solution () t, () t.(figure 5) Lemm 4. Any solution of (), t () t of (6.) which strts in region VI t time t t0 will remin in this region for ll future time t t0, nd ultimtely pproch the equilibrium solution () t, () t.(figure 5) Lemms,, 3 nd 4 stte tht every solution (), t () t of (6.) which strts in region I, II, III or VI t time t t0 nd remins there for ll future time must lso pproch equilibrium solution () t, () t s t pproches infinity. ext, observe tht ny solution (), t () t of () which strts on l or l must immeditely fterwrds enter regions I, II, III or VI. Finlly the solution pproches the equilibrium solution () t, () t. This is illustrted in Figure 6

11 A PREY-PREATOR MOEL WITH COVER FOR THE PREY A A ALTERATIVE FOO FOR THE PREATOR A COSTAT HARVESTIG OF BOTH THE SPECIES Trjectories 8. Future Work In the present pper it is investigted tht Prey-Predtor model with constnt hrvesting of both species, cover for prey nd limited lternte food is supplied to the predtor. There is scope to study the model with constnt hrvesting of the prey species, or constnt hrvesting of the predtor species. Further cover cn be removed to the Prey nd without n lternte food to the predtor.

12 54 K. LAKSHMI ARAYA A.PATTABHIRAMACHARYULU REFERECES [] Michel Olinck, An introduction to Mthemticl models in the socil nd Life Sciences, Addison Wesley, 978. [] Kpur, J.., Mthemticl models in Biology nd Medicine, Affilited Est- West,985. [3] Smith, J. M., Models in Ecology, Cmbridge University Press, Cmbridge, 974. [4] Pul Colinvux, Ecology, John Wiley nd Sons Inc., ew York, 977. [5]. Freedmn, H. I., eterministic Mthemticl Models in Popultion Ecology, Mrcel ecker, ew York, 980. [6] My, R. M., Stbility nd complexity in Model Eco-systems, Princeton University Press, Princeton, 973. [7] Vrm, V. S., A note on Exct solutions for specil prey-predtor or competing species system, Bull. Mth. Biol., 39(977), [8] ryn, K. L., A Mthemticl Study of Prey Predtor Ecologicl Models with Prtil Cover for the Prey nd Alterntive Food for the Predtor, Ph. thesis, JTU, Hyderbd, 005. [9] ryn, K.L. & Rmchryulu,.C.P, A Prey-Predtor Model with n Alterntive Food for the Predtor, hrvesting of both the Species nd with A Gesttion Period for Interction, IJOPCM,() ( 008 ), [0] ryn, K.L. & Rmchryulu,.C.P, A Prey-Predtor Model with n Alterntive Food for the Predtor, hrvesting of both the Species nd. A Time ely, Proceedings of ICM, UAE, ( 008 ), [] ryn, K.L. & Rmchryulu,.C.P, A Prey-Predtor Model with cover proportionl to size of the Prey nd n Alterntive Food for the Predtor, Int. J. of Mth. Sci. & Engg. Appls., (IV) (008), 9-4. [] ryn, K.L. & Rmchryulu,.C.P, Some Threshold Theorems for Prey- Predtor Model with Hrvesting, Int.J. of Mth. Sci. & Engg. Appls., ( II )(008), eprtment of Mthemtics & Humnities, SLC S IET, Hyderbd-50 5, Indi. eprtment of Mthemtics & Humnities, IT, Wrngl , Indi. E-mil ddress: nrynkunderu@yhoo.com