Heteroclinic Bifurcation and Multistability in a Ratio-dependent Predator-Prey System with Michaelis-Menten Type Harvesting Rate

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1 Proceedings of the World Congress on ngineering Vol I WC Jul 4-6 London U.K. Heteroclinic Bifurcation and Multistabilit in a Ratio-dependent Predator-Pre Sstem with Michaelis-Menten Tpe Harvesting Rate N. Bairagi S. Chakrabort and S. Pal Abstract In this article we stu a ratiodependent predator-pre sstem where predator population is subjected to harvesting with Michaelis- Menten tpe harvesting rate. We stu the eistence of heteroclinic bifurcations in an eploited predatorpre sstem b using Melnikov s method. Our simulation results also show that the sstem ma ehibit monostabilit bistabilit and tristabilit depending on the initial values of the sstem populations and the harvesting effort. Kewords predator-pre model ratio-dependence harvesting Melnikov functions heteroclinic bifurcation. Introduction The standard Lotka-voltera tpe models assume that the percapita rate of predation depends on the pre numbers onl. This means that the predator s functional response is a function of pre densit onl. In general predator s functional response should certainl be a function of both pre and predator densities [ ] as there is often competition among the predators for their food. A simple alternative assumption is that the per capita rate of predation depends on the ratio of pre to predator densities. It is well known that classical pre-dependent predator pre model ehibits the parado of enrichment [3 4] which states that enriching a predator pre sstem (b increasing the carring capacit) will cause an increase in the equilibrium densit of the predator but not in that of pre and will destabilize the positive equilibrium and thus increases the possibilities of the stochastic etinction of the predator. However numerus field observations provide contrar to this parado of enrichment. It is often observed in nature that fertilization increases the pre densit but does not destabilize a stable stea state and fails to increase the amplitude of the os- The author sincerel acknowledge All India Council for Technical duction for their financial support to present the research work in WC U.K. Centre for Mathematical Biolog and colog Department of Mathematics Jadavpur Universit Kolkata 73 West Bengal India. Corresponding Author -mail: nbairagi@math.jdvu.ac.in Fa No Ph.No Baruipara High School Halisahar 4 Parganas (N) West Bengal India. Department of Mathematics Universit of Kalani West Bengal India. cillations in sstem that alrea ccle [5]. Another parado that the predator-pre model with pre-dependent Michaelis- Menten functional response ehibits is the so-called biological control parado [6] which states that we cannot have both a low and stable pre equilibrium densit. However there are man eamples of successful biological control where the pre is maintained at ver low densities compared with its carring capacit [7]. A ratio dependent predator-pre model with Michaelis-Menten functional response does not show these paradoes ([]-[5]) and assumed therefore to be superior than their pre-dependent counter part. Kuang and Beretta [] observed that ratio-dependent predator-pre models are richer in boundar namics and showed that if the positive stea state of the sstem is locall asmptoticall stable then the sstem has no nontrivial positive periodic solutions. Jost et. al [] demonstrated that the equilibrium for a ratio dependent predator pre model can either be a saddle point or an attractor. Xiao and Ruan [] and Berezovskaa et. al. [4] observed that there eist different kinds of topological structures in the vicinit of the origin of a ratio dependent predator pre model. Hsu et. al. [6] considered a ratio dependent food chain model and studied the etinction namics as well as the sensitivit of the sstem to initial population densities. Berezovskoa et. al. [5] presented an algorithmic approach to analze the behavior of ratio dependent predator pre sstem. Tang and Zhang [7] gave an analtical condition on parameters for the eistence of the hetroclinic loop. Most recentl Li and Kuang [8] appling the same ideas and techniques to a different Hamiltonian sstem obtained a new eplicit relation in higher order epansion for the bifurcation curve of a heteroclinic loop. The heteroclinic bifurcation plas an important role in understanding the namics of the sstem [3 4] because heteroclinic bifurcation ma trigger a catastrophic shift from the state of large oscillations of predator and pre populations to the state of etinction of both populations [8]. Harvesting in a predator pre sstem ma be two fold. The primar objective is optimal eploitation of the harvested stock to maimize the profit ([]-[]). In contrast some researchers ([3]-[6]) considered harvesting from ecological point of view. Xiao and Jennings [7] observed numerous kinds of bifurcation in a ratio-dependent predator-pre model where pre is being harvested at a constant rate. Xiao et al. [8] considered a constant harvesting term in predator equation and observed subcritical supercritical and the cusp bifurcation of codimension. We also studied a ratio-dependent predator-pre model in presence of parasite where the pre population was subjected to harvesting [9]. The namics of ISBN: ISSN: (Print); ISSN: (Online) WC

2 Proceedings of the World Congress on ngineering Vol I WC Jul 4-6 London U.K. zero equilibria was thoroughl investigated to find out conditions on the sstem parameters such that trajectories starting from the domain of interest can reach the zero equilibrium following an fied direction. Meza et al. [3] studied a ratiodependent predator-pre model where predator is subject to an on-off control known as threshold polic. The showed that an equilibrium on the boundar ma slide to an stable interior equilibrium point due to on-off control and thus avoid the etinction of species. One important question in the bioeconomic modeling of productive resources is the rate of harvesting. It is shown that Michaelis-Menten tpe functional form of catch rate h(t) given b h(t) = q b + l where b and l are positive constants q is the catchabilit coefficient and is the eternal effort devoted to harvesting is better than the constant rate of harvesting and catch-perunit-effort harvesting [3 3]. None of these studies ([7]- [3]) however considered the Michaelis-Menten tpe functional form of catch rate in their model sstems. The objective of this paper is to rigorousl stu the eistence of heteroclinic bifurcation in a ratio-dependent predator-pre model where the predator population is subjected to harvesting with Michaelis-Menten tpe harvesting rate. The organization of the paper is as follows: Section deals with the model development. Section 3 is devoted to the stu of heteroclinic bifurcations. Numerical studies and discussion are presented in Section 4. The model The following assumptions are made in formulating the mathematical model: Let (t) and (t) be respectivel the pre and predator densities at time t. Assume that the pre population grows logisticall to its carring capacit K with intrinsic growth rate r. Let d be the food independent death rate and b be the conversion efficienc of the predator. Assume that predator follows ratio-dependent Tpe II functional response α where α is the maimum pre consumption rate and a is the half-saturation a+ constant. Let predator is harvested following the Michaelis-Menten tpe catch rate. Based on these assumptions we formulate the following ratiodependent predator-pre model with predator harvesting: = r( ) α K = αb a+ q. (.) a+ b+l All parameters are assumed to be positive. Taking = K = K a (.) as and t = at we can write down the sstem α = α ( ) = β γ (.) where α = ar α β = ab = a q αkl γ = ad α = ab and Kl = aq. bα For convenience we replace t b t respectivel and rewrite the above sstem as = α( ) + = β γ. (.3) + + Changing the independent variable t to (+)t and replacing t b t for convenience the sstem (.3) becomes = α( )( + ) = β γ( + ) (+). + (.4) It is eas to show that the sstem (.3) in the first quadrant is equivalent to the polnomial sstem (.4) [4 7 ]. Using Briot Boughet s transformations t t (.5) the sstem (.4) can be written as = [α α ( α) α] = [(β α γ) + α + ( α γ) + α (+) ]. + (.6) For convenience we have written t in place of t respectivel in (.6). Transformation (.5) is a homomorphism in the first quadrant and its inverse maps the -ais to the point (). Now changing the variables α t t (.7) the equation (.6) can be transformed into the following simpler sstem: = [α ( α) ] = [(β α γ) + + ( α γ) + (.8) α (+) ]. α + Sstem (.8) has to be analzed with the following initial conditions: () > () >. 3 Heteroclinic Bifurcation In the sstem (.8) we simpl use α and γ = β α γ (or equivalentl α and β ) as our unfolding parameters while fiing γ and get the following transformed equations as = (α ) + (α ] ) [γ = + + ( γ ) + ( α ) α (+) α +. (3.) This sstem can then be viewed as a perturbation of the sstem = (α ) = [γ + + ( γ)] (3.) as α γ and all are ver small. Here γ = γ = β α γ. Note that the coefficients of second order terms in (3.) do not depend on α and γ and we shall assume that γ <. ISBN: ISSN: (Print); ISSN: (Online) WC

3 Proceedings of the World Congress on ngineering Vol I WC Jul 4-6 London U.K. The sstem (3.) is integrable if and in this case the function F α () = b a b ( where a = ( γ) γ = α < (3.3) γ ( γ) γ α γ ) (3.4) and b = γ γ (3.5) are constant along the solution curves. In fact when (3.3) holds along an solution curve ((t)(t)) of (3.) we have This gives [b b ( (α ) [ d Fα ( ) = b b (α ) ( [a a α γ ) ] a + b a [γ + + ( γ )] α γ ) aα a b (a + ) a b b ( γ ) ] =. ] a( γ) a b+ +{γ + + ( γ )} [ bα a b (b + ) ] γ a b b a+ b =. Using the transformations ǫ ǫ α = ǫγ and γ = ( γ ) γ ǫγ + γ ǫ and rescaling time t equation ǫ (3.) transforms into = (γ ) + ǫ(γ ) = [ ( γ ) γ γ + + ( γ ] ) + ǫ(γ γ + ) ǫ( + ǫ)( + ǫ ). γ (3.6) Multipling (3.6) b the integrating factor a b we obtain the equivalent perturbed Hamiltonian sstem = a b [(γ ) + ǫ(γ )] = a b [ b ( γ ) γ γ + + ( γ ) + ǫ(γ γ + ) ] [ ] a b b ǫ( + ǫ)( + ǫ ). γ (3.7) One can check that F γ () = b a b ( γ γ ) is the Hamiltonian for (3.7) when ǫ = where a and b are given in (3.5). We use the Melnikov theor [ ] to locate the parameter values that produce a heteroclinic ccle for (3.7) in case ǫ. In the following we have emploed the technique used in [33 34]. We can set γ = without an loss of generalit. The heteroclinic ccle for ǫ = lies on the level curve F γ ( ) = denoted b Γ which corresponds to a triangle formed b the three line segments determined b = = and + γ =. Let G( ) = ( a b ( ) a b (γ + ) ). The Melnikov function is where M(γ ) = int Γ tracedg( ) tracedg() = (a b ) a b + (b a) a b +γ b a b and int Γ denotes the region bounded b Γ. M(γ ) = has a unique solution [ ] (a b )I(a b) + (b a)i(ab) γ = bi(a b ) where I(u v) = It is eas to see that I(uv) = = Γ u v u > v >. v (v+)s v+ u ( ) v+ s u where s = γ. We also have I(u + v) = v + si(uv + ) + I(u v). v + Using integration b parts we obtain Thus we get and I(u v + ) = I(u + v) = I(uv + ) = (v + ) I(u + v). (u + )s (u + ) I(u v) (u + v + 3) (v + ) I(u v). s(u + v + 3) Therefore [ ] a(b a) γ = a b +. s(a + b + ) a + b + Now we have a + b = 4 3γ γ Putting these values we get γ = a b = s = γ. 6γ ( γ ) (4 γ ). The Melnikov theor [34] shows that if ( γ) 6γ γ = α + γ ( γ ) (4 γ ) α + O(α 3 ) (3.8) then the sstem (3.) has a heteroclinic ccle and it is stable [4]. Condition (3.8) is equivalent to ISBN: ISSN: (Print); ISSN: (Online) WC

4 Proceedings of the World Congress on ngineering Vol I WC Jul 4-6 London U.K. β = γ + α + γ + γ = γ + + α γ.3 6γ α + O(α3 ). ( γ ) (4 γ ) We thus state the following lemma. + (3.9).5..5 Lemma We assume that γ are fied and γ < < γ + <. For small α if condition (3.9) holds the sstem (3.) has a stable heteroclinic ccle connecting saddles at ( ) (α ) and (β α γ ) γ γ β γ γ In this case foloowing Tang and Zhang [7] there eists a stable heteroclinic ccle. The positive coeistence equilibrium α γ ++γ β +..5 lies inside the heteroclinic c- cle and it is a spiral source. Figure : The phase portrait of the sstem (.3) for α =.5 β =.6 γ =. =.5 and =.. Here ( ) is an attractor ( ) is a saddle and the interior equilibrium is an unstable focus. The figure shows eistence of heteroclinic loop in the sstem (.3). Conditions of Lemma 5.3. shows that for small α γ + < β < α γ + γ + γ <. (3.).5. The sstem (.8) in [3] shows that in this case a limit ccle in the sstem is alwas stable and unique once it eists. We therefore conclude that there is no limit ccle inside the heteroclinic ccle since it is attracting..5. Lemma 5.3. and the transformation used to convert (.) to (.9) give the following theorem..5 Theorem Assume that γ is fied and γ <. If for small α condition (3.) holds sstem (.3) has a stable heteroclinic ccle connecting saddles at ( ) and ( ). Simulations and discussion To illustrate the analtical results we consider the following fied parameter values: α =.5 β =.6 γ =. =.5 and =.. This parameter set satisfies conditions of the Theorem 5.3. and the sstem (.3) ehibit heteroclinic bifurcations (Fig. ). Here ( ) is an attractor ( ) is a saddle and the interior equilibrium is an unstable focus. There is a heteroclinic loop consisting of the origin ( ) the saddle equilibrium ( ) the heteroclinic orbit connecting ( ) & ( ) and the seperatrices between ( ) & ( ). The solid line denotes the seperatrices. An trajector started below the seperatrices (denoted b dash-dot line) converges to ( ) spirall and an trajector above the seperatrices (denoted b dotted line) converges to ( ) monotonicall (Fig. ). If we slightl increase the parameter value of from. to. keeping other parameter values unaltered some trajectories (denoted b dotted lines) go to ( ) and some produces limit ccle (denoted b dash-dot lines) which is unique depending on the initial values of the sstem populations (Fig. ). If we again increase the parameter value of from. to.4 keeping other parameter values unaltered the sstem ehibits bistabilit (Fig. 3). In this case some trajectories (denoted b dotted lines) go to ( ) and some converge to the interior equilibrium (denoted b dash-dot lines). ISBN: ISSN: (Print); ISSN: (Online) Figure : The phase portrait of the sstem (.3) for =.. Other parameters are as in the Fig.. Here some trajectories go to ( ) and some converge to the unique limit ccle surrounding the interior equilibrium Figure 3: The phase portrait of the sstem (.3) for =.4. Other parameters are as in the Fig.. The sstem in this case ehibits bistabilit. Here some trajectories go to ( ) and some converge to the interior equilibrium depending on the initial value. WC

5 Proceedings of the World Congress on ngineering Vol I WC Jul 4-6 London U.K Figure 4: The phase portrait of the sstem (.3) for =.5. Other parameters are as in the Fig.. The sstem in this case ehibits tristabilit. Figure 7: The phase portrait of the sstem (.3) for α =.. Other parameters are as in the Fig.. Here all trajectories independent of the initial values go to ( ) If we further increase the parameter value of from.4 to.5 the sstem ehibits tristabilit (Fig. 4). Here some trajectories (denoted b dotted lines) converge to ( ) some converge to ( ) (denoted b dashed lines) and some converge to the interior equilibrium (denoted b dash-dot lines) depending on the initial values of the sstem populations. If the parameter value of is increased again from.5 to. keeping other parameter values intact the sstem ehibits bistabilit (Fig. 5) Figure 5: The phase portrait of the sstem (.3) for =.. Other parameters are as in the Fig.. The sstem in this case ehibits bistabilit..3 It is observed that some trajectories (denoted b dotted lines) go to () and some converge to () (denoted b dashed lines). A further increment in from. to.5 leads the sstem to monostabilit (Fig. 6). All trajectories in this case converges to () (denoted b dashed lines). If we reduce the value of the parameter α from.5 to. keeping other parameter values as in the Fig. then all trajectories converge to ( ) (Fig. 7). It is to be observed that we have obtained different namics (see Fig. to Fig 6) of the sstem (.3) onl b changing the parameter value of which is directl related to the harvesting effort () of the predator population. Thus an eploited ratio-dependent predatorpre sstem where the predator population is subjected to harvesting with Michaelis-Menten tpe harvesting rate ma ehibit ver rich namics including heteroclinic bifurcation and multistabilities..5. References Figure 6: The phase portrait of the sstem (.3) for =.5. Other parameters are as in the Fig.. The sstem in this case ehibits monostabilit. [] R. Arditi and L. R. Ginzburg Coupling in predator-pre namics: ratio-dependence Jr. Theor. Biol. L.R. vol. 39 pp [] R. Arditi L. R. Ginzburg and H. R. Akcakaa Variation in plankton densities among lakes: a case for ratiodependent models Am. Nat. vol. 38 pp [3] N. G. Hairston F.. Smith and L. B. Slobodkin Communit structure population control and competition American Naturalist vol. 94 pp [4] M. L. Rosenzweig Parado of nrichment: destabilization of eploitation sstems in ecological time Science ISBN: ISSN: (Print); ISSN: (Online) WC

6 Proceedings of the World Congress on ngineering Vol I WC Jul 4-6 London U.K. vol. 7 pp [5] P. A. Abrams and C. J. Walters Invulnerable pre and the parado of enrichment colog vol. 77 pp [6] R. F. Luck valuation of natural enemies for biological control: a behavior approach Trends in colog and vol. vol. 5 pp [7] R. Arditi and A. A. Berrman The biological parado Tr. col. vol. vol. 6 pp [8] H. R. Akcakaa Population ccles of mamals: evidence for a ratio-dependent predator-pre hpothesis col. Monogr.. vol. 6 pp [9] A. P. Gutierrez The phsiological basis of ratiodependent predator-pre theor: a metabolic pool model of Nicholson s blowflies as an eample colog vol. 73 pp [] Y. Kuang and. Beretta Global qualitative analsis of a ratio-dependent predator-pre sstem J. Math. Biol. vol. 36 pp [] C. Jost O. Arino R. Arditi About deterministic etinction in ratio-dependent predator-pre models Bull. Math. Biol. vol [] D. Xiao S. Ruan Global namics of a ratio-dependent predator-pre sstem J. Math. Biol. vol. 43 pp [3] S. B. Hsu T. W. Hwang and Y. Kuang Global analsis of the Michaelis-Menten tpe ratio-dependent predatorpre sstem J. Math. Biolo. -(). [4] F. S. Berezovskaa G. P. Karev and R. Arditi Parametric analsis of the ratio-dependent predator-pre sstem J. Math. Biol. vol. 43 pp. -. [5] F. S. Berezovskaa A. S. Novozhilov and G. P. Karev Population models with singular equilibrium Mathematical Bios. vol. 8 pp [6] S. B. Hsu T. W. Hwang and Y. Kuang A ratiodependent food chain model and its applications to biological control Math. Biosci. vol. 8 pp [7] Y. Tang and W. Zhang Heteroclinic bifurcation in a ratio-dependent predator-pre sstem J. Math. Bio. no. 5 pp [8] B. Li and Y. Kuang Heteroclinic bifurcation in the MichaelisMenten-tpe ratio-dependent predator-pre sstem Siam J. Appl. Math. vol. 67 no [9] C. W. Clark Mathematical Bioeconomics: The optimal Management of Renewable Resources John Wile & Sons New York 976. [] F. Brauer and A. C. Soudack Coeistence properties of some predator-pre sstems under constant rate harvesting and Stocking J. Math. Biol. vol. pp [] M. A. Mesterton-Gibbons Technique for finding optimal two-species harvesting policies Nat. Resource Model vol. 9 pp [] P. D. N. Srinivasu Bioeconomics of a renewable resource in presence of a predator Nonlinear Analsis Real World Appl vol. pp [3] S. Chakrabort S. Pal and N. Bairagi Predator-pre interaction with harvesting: mathematical stu with biological ramifications Appl. Math. Analsis. [4] C. Azar J. Holmberg and K. Lindgren Stabilit analsis of harvesting in a predator-pre model J. Theo. Biol. vol. 74 pp [5] G. Dai and M. Tang Coeistence region and global namics of a harvested predator-pre sstem Siam J. Appl. Math. vol. 58 no pp [6] N. Bairagi S. Chaudhur and J. Chattopadha Harvesting as a disease control measure in an ecoepidemiological sstem -a theoretical stu Mathematical Biosciences pp [7] D. Xiao and L. S. Jennings Bifurcations of a ratiodependent predator-pre sstem with constant rate harvesting SIAM J. Appl. Math. Biol. vol. 659 no [8] D. Xiao W. Li and M. Han Dnamics of a ratiodependent predator-pre model with predator harvesting J. Math. Anal. Appl. vol. 34 pp [9] S. Chakrabort S. Pal and N. Bairagi Dnamics of a ratio-dependent eco-epidemiological sstem with pre harvesting Nonlinear Analsis R.W.A. vol. pp [3] M.. M. Meza A. Bhaa and. Kaszkurewicz Stabilizing control of a ratio-dependent predator-pre models Nonlinear Anal RWA vol. 7 pp [3] T. K. Kar and K. S. Chaudhuri Regulation of a prepredator fisher b taation: a namic reaction model J. Bio. Sstems vol. no. pp b. [3] S. V. Krishna P. D. N. Srinivasu Conservation of an ecosstem through optimal taation. Bulletin of Math Bio. vol. 6 pp [33] S. N. Chow C. Li. and D. Wang Normal Forms and Bifurcation of Planar Vector Fields Cambridge Universit Press New York 994. [34] J. Guckenheimer and P. Holmes Nonlinear Oscillations Dnamical Sstems and Bifurcations of Vector fields Springer-Verlag New York 983. [35] S. Wiggins Introduction to Applied Nonlinear Dnamical Sstems and Chaos Springer-Verlag new ork (3). ISBN: ISSN: (Print); ISSN: (Online) WC