Dynamics of a Holling-Tanner Model

Transcription

1 American Jornal of Engineering Research (AJER) 07 American Jornal of Engineering Research (AJER) e-issn: p-issn : Volme-6 Isse-4 pp-3-40 wwwajerorg Research Paper Open Access Dynamics of a olling-tanner Model Rashi Gpta Department of Mathematics NIT Raipr (CG) ABSTRACT: In this paper we have made an attempt to nderstand the dynamics of a olling-tanner model with olling type III fnctional responses We have carried ot stability analysis of both the non-spatial model withot diffsive spreading and of the spatial model Tring and opf stability bondaries are fond With the help of nmerical simlations we have observed the spatial and spatiotemporal patterns for the model system Non-Tring patterns are examined for some fixed parametric vales for the predator poplations Keywords: Reaction diffsion system Globally asymptotically stable Diffsion olling Tanner model I INTRODUCTION The stdy of poplation dynamics is nearly as old as poplation ecology starting with the pioneering work of Lotka and Volterra a simple model of interacting species that still bears their joint names This was a nearly linear model bt the predator-prey version displayed netrally stable cycles From then on the dynamic relationship between predators and their prey has long been and will contine to be one of dominant themes in both ecology and mathematical ecology de to its niversal existence and importance 3 In recent years pattern formation in non-linear complex systems have been one of the central problems of the natral social technological sciences and ecological systems 4-0 Particlarly many researchers have stdied the predator-prey system with reaction-diffsion The olling Tanner model describes the dynamics of a generalist predator which feeds on its favorite food item as long as it is in abndant spply and grows logistically with an intrinsic growth rate and a carrying capacity proportional to the size of the prey Starting with the pioneering work of Segel and Jackson spatial patterns and aggregated poplation distribtion are common in natre and in a variety of spatio-temporal models with local ecological interactions 3 Wolpert 4 gave a clear and non-technical description of mechanisms of pattern formation in animals Chen and Shi 5 stdied a spatial olling-tanner model and proved the global stability of a niqe constant eqilibrim nder a simple parameter condition In this paper we have considered two interacting species prey and predator for modelling the dynamics in spatially distribted poplation with local diffsion We have considered a spatial olling-tanner model with olling type III fnctional responses The characteristic featre of olling type III fnctional responses is that at low densities of the prey the predator consmes it less proportionally than is available in the environment relative to the predators' other prey 6 We have obtained the conditions for local and global stability of the model system in the absence and the presence of diffsion We also obtained the criteria for Tring and opf instability The main objective of this paper is to see the spatial interaction and the selection of spatiotemporal patterns This paper is organized as follows In section the model system and parameters are discssed The model system is analyzed in the absence as well as in the presence of diffsion in section 3 and 4 We have discssed the conditions for Tring and opf instability in section 5 In section 6 we have discssed the nmerical simlation reslts in both one and two dimensional spatial domain At the end reslts are discssed in section 7 II TE MODEL SYSTEM We consider a reaction-diffsion model for prey-predator where at any point ( x y) and time t the prey N ( x y t ) and predator P ( x y t ) poplations The prey poplation N ( x y t ) is predated by the predator poplation P ( x y t ) The per capita predation rate is described by olling type III fnctional response The model system satisfy the following: w w w a j e r o r g Page 3

2 American Jornal of Engineering Research (AJER) 07 N N a N P r N D N N T K D N P h P e P s P D P P T N () The parameters r a D e h s K D D in model () are positive constants ere r is the prey s N P intrinsic growth rate in the absence of predation a is the rate at which prey is feeded by predator and it follows olling type III fnctional response D is the half-satration constants for prey density e is the predator s intrinsic growth rate h is the nmber of prey reqired to spport one predator at eqilibrim when P eqals to N / h s is the intra-specific competition rate for the predators K is the carrying capacity of the phytoplankton parameters D and D represent the diffsion coefficients of prey and predator respectively N P The nits of the parameters are as follows Time T and length X Y [0 R ] are measred in days ( d ) and metres ( m ) respectively Frther r N P e D are sally measred in mg of dry weight per litre [ m g d w / l ] the dimension of a and h is d The dimension of s is d The diffsion coefficients D and N D are measred in [ m d ] P We introdce the following notation in order to bring the system of eqations to a non-dimensional form: N v a P t rt D x X y Y rh e sk d D d D K rk K L L a r a rl rl We obtain the system v d t () v v v v d v t with the initial condition ( x y 0 ) 0 v ( x y 0 ) ( x y ) (3) and with the no-flx bondary conditions v 0 ( x y) (4) n n III STABILITY ANALYSIS OF TE NON-SPATIAL MODEL SYSTEM In this section we restrict orselves to the stability analysis of the model system in the absence of diffsion in which only the interaction part of the model system are taken into accont We find the nonnegative eqilibrim states of the model system and discss their stability properties with respect to variation of several parameters 78 3 Local Stability Analysis We analyze model system () withot diffsion In sch case the model system redces to with ( x 0 ) 0 v ( x 0 ) 0 d v d t d v v v v d t The stationary dynamics of system (5) can be analyzed from d / dt dv / dt 0 Then we have (5) w w w a j e r o r g Page 33

3 American Jornal of Engineering Research (AJER) 07 v 0 v v v 0 It can be seen that model system has two non-negative eqilibria namely E ( 0 ) and E ( v ) It may be noted that and v satisfies the following eqations: ( )( ) v v 0 v 0 From Eqs(6) it is easy to show that the eqilibrim point E is a saddle point with stable manifold in - direction and nstable manifold in v-direction 9 In the following theorem we are able to find necessary and sfficient conditions for the positive eqilibrim matrix of model (5) at E ( v ) is v ( ) ( ) J v v v Following the Roth-rwitz criteria we get Theorem The positive eqilibrim v ( ) v A v ( ) E ( v ) to be locally asymptotically stable The Jacobian v ( ) v v B v ( ) ( ) only if the following ineqalities hold: A 0 and B 0 Remark Let the following ineqalities hold E ( v ) is locally asymptotically stable in the v plane if and Then A 0 and B 0 It shows that if ineqalities in (8) hold then stable in the v plane (8) (6) (7) E ( v ) is locally asymptotically 3 Global Stability Analysis In order to stdy the global behavior of the positive eqilibrim which establishes a region for attraction for model system (5) Lemma The set R ( v ) : 0 0 v the interior of the positive qadrant R Theorem 3 If w 0 v ( ) 4 w 0 v E ( v ) we need the following lemma is a region of attraction for all soltions initiating in hold then E is globally asymptotically stable with respect to all soltions in the interior of the positive qadrant R w w w a j e r o r g Page 34 (9) (0)

4 American Jornal of Engineering Research (AJER) 07 IV STABILITY ANALYSIS OF TE SPATIAL MODEL SYSTEM In this section we stdy the effect of diffsion on the model system abot the interior eqilibrim point Instability will occr de to diffsion when a parameter varies slowly in sch a way that a stability condition is sddenly violated and it can bring abot a sitation wherein pertrbation of a non-zero (finite) wavelength starts growing (pertrbations of zero wave nmber are stable when diffsive instability sets in 0 ) Tring instability can occr for the model system becase the eqation for predator is nonlinear with respect to predator poplation To stdy the effect of diffsion on the model system (5) we derive conditions for stability analysis in one- and two- dimensions cases 4 One-Dimension Case After non-dimensionalization of the model system () in one dimensional space takes the following form with non-zero initial condition and no-flx bondary conditions v d t x v v v v v d t x ( x 0 ) 0 v ( x 0 ) 0 for x [0 R] To stdy the effect of diffsion on the model system we pertrb the steady state The linearized form of the Eqs () obtained as: U v v V Where U t V t b U b V d b U b V d U x V x v ( ) b b ( ) v v b b v Ths the soltion of the system () of the form U a n ex p ( t ikx ) V b n 0 n ( v ) by setting where and k are the freqency and wave nmber respectively The characteristic eqation of the linearized system is given by where 0 () (4) A ( d d ) k (5) 4 v ( ) v B d d k d v d k (6) ( ) where A and B are defined in (7) From Eqs(4-6) and sing the Roth-rwitz criteria the following theorem follows immediately Theorem 4 (i)the positive eqilibrim if and only if 0 and 0 () (3) E ( v ) is locally asymptotically stable in the presence of diffsion (ii)if the ineqalities in Eq(8) are satisfied then the positive eqilibrim point stable in the presence as well as absence of diffsion E is locally asymptotically w w w a j e r o r g Page 35

5 American Jornal of Engineering Research (AJER) 07 (iii)sppose that ineqalities in Eq(8) are not satisfied ie either A or B is negative or both A and B are negative Then for strictly positive wave-nmber k 0 ie spatially inhomogeneos pertrbations by increasing d and d to sfficiently large vales and can be made positive and hence E can be made locally asymptotically stable In the following theorem we are able to show the global stability behavior of the positive eqilibrim in the presence of diffsion Theorem 5 (i) If the eqilibrim E of model system (5) is globally asymptotically stable the corresponding niform steady state of model system () is also globally asymptotically stable (ii) If the eqilibrim E of system (5) is nstable even then the corresponding niform steady state of system () can be made globally asymptotically stable by increasing the diffsion coefficients d and d to a sfficiently large vale for strictly positive wave-nmber k 0 4 Two-Dimension Case In the two-dimensional case the model system can be written as v d t x y (7) v v v v v v d t x y We analyze the above model nder the following initial and bondary conditions: ( x y 0 ) 0 v ( x y 0 ) 0 ( x y ) (8) and v 0 ( x y ) t 0 n n where n is the otward normal to We state the main reslt of this section in the following theorem Theorem 6 (i) If the eqilibrim E of model system (5) is globally asymptotically stable the corresponding niform steady state of model system (7) is also globally asymptotically stable (ii) If the eqilibrim E of system (5) is nstable even then the corresponding niform steady state of system (7) can be made globally asymptotically stable by increasing the diffsion coefficients d and d to a sfficiently large vale for strictly positive wave-nmber k 0 V TURING AND OPF INSTABILITY The Tring instability occrs if at least one of the roots of the above (4) has a positive root or positive real part or in other words R e( ) 0 for some k 0 Irrespective of the sign of and the diffsion-driven instability occrs when is a qadratic in k k k cr where k ence the condition for diffsive instability is given by ( ) 0 4 v ( ) v ( k ) d d k d v d k B 0 ( ) and the graph of ( ) is a parabola The minimm of y k (9) (0) ( ) occrs at y k v ( ) k d v d 0 cr d d Conseqently the condition for diffsive instability is ( k ) Therefore cr () w w w a j e r o r g Page 36

6 American Jornal of Engineering Research (AJER) 07 v ( ) d v d B 4 d d Now consider the set of parameter vales We obtain and v For the set of parameter vales d 0 0 d 3 the corresponding critical vale is k k The graph of c r c r ( ( )) ( ) Fig For all vales of crve for which ( k ) vs () k has been plotted for different vales of d in k lying in the range ( ) the system () is nstable The region nder the ( k ) 0 is known as Tring instability region Figre The graph of the fnction ( k ) vs k for and d 0 0 for different vales of d The Tring bifrcation occrs when Im ( ( k )) 0 and R e( ( k )) 0 at k k 0 k is the critical wave nmber If is considered as a bifrcation parameter then its critical vale eqals to where R v d ( d d ) T B B 4 R C R T T (3) B v v v b d d b d v v ( ) ( d d ) C v ( ) b v d b d v d b v d v ( ) ( d d ) The opf bifrcation occrs when Im ( ( k )) 0 and R e( ( k )) 0 at k 0 bifrcation parameter is v ( ) v ( ) v The critical vale of the (4) At the opf bifrcation threshold the temporal symmetry of the system is broken and gives rise to niform oscillations in space and periodic oscillations in time with the freqency v v ( ) w v v ( ) ( ) w and wavelength : w w w a j e r o r g Page 37

7 American Jornal of Engineering Research (AJER) 07 w v v v v ( ) ( ) ( ) VI NUMERICAL SIMULATIONS In this section we perform nmerical simlations to illstrate the reslts obtained in previos sections For this prpose we have plotted the time series spatiotemporal patterns and snapshots for one dimensional cases and two dimensional cases The dynamics of the model system () is stdied with the help of nmerical simlation The step lengths x and t of the nmerical grid are chosen sfficiently small so that the reslts are nmerically stable The temporal dynamics is stdied by observing the effect of time on space vs density plot of prey and predator poplations Spatiotemporal chaos is generated as a reslt of breaking the homogeneity and the formation of a non-stationary irreglar spatial pattern when the local kinetics of the system is oscillatory for a wide class of initial conditions In the absence of any spatial gradient period-dobling bifrcations serve as the generating mechanism for chaos in these model systems For the nmerical integration of the model system we have sed the Rnge-Ktta forth-order procedre on the MATLAB 70 platform To investigate the spatiotemporal dynamics of the model system () we solved it nmerically sing semi implicit (in time) finite difference method Application of the finite difference method gives rise to a sparse banded linear system of algebraic eqations The reslting linear system is solved by sing the GMRES algorithm for the twodimensional case Consider the set of parameter vales (5) for the model system (5) With the above set of parameters we note that the positive eqilibrim E exists and it is given by ( v ) ( ) We choose the same set of parameters as in (5) for the model system () with initial condition where ( x 0 ) ( x x )( x x ) v ( x 0 ) v ( v ) is the nontrivial state for the coexistence of prey and predator poplation and 8 0 x 0 0 x is the parameter affecting the system dynamics In order to avoid nmerical artifacts we checked the sensitivity of the reslts to the choice of the time and space steps and their vales were chosen sfficiently small In Figre we have presented the time series of the model system () for the fixed set of parameters vales (3) with initial condition (4) By increasing the time t the system shows from stable to limit cycle behavior Figre 3 shows the spatiotemporal evoltion of predator density for the system Finally Figre 4 shows the snapshots of predator density for increasing vales of diffsion coefficient d to observe the irreglar patchy non-tring spatial distribtion of predator poplation (6) Figre Time series for the model system () at the fixed set of parameters vales d d with t (a) 00 (b) 500 (c) 700 w w w a j e r o r g Page 38

8 American Jornal of Engineering Research (AJER) 07 Figre 3 Spatiotemporal patterns of predator density for the model system () at fixed set of parameters vales d with d = (a) 00 (b) 0 (c) (d) 0 Figre 4 Snapshots of predator poplations with parameters vales d and d = (a) (b) 0 (c) 5 (d) 0 VII DISCUSSION AND CONCLUSION In this paper we have considered a minimal model of predator-prey interaction with holling type-iii fnctional responses The dynamics of the olling-tanner predator-prey model is qite interesting for its mathematical properties and for its efficacy in describing real ecological systems sch as lynx and hare and sparrow and sparrow hawk The characteristic featre of olling type III fnctional responses is that at low densities of the prey the predator consmes it less proportionally than is available in the environment relative to the predators' other prey Collings 3 have sed the olling-tanner model to stdy the poplation interaction w w w a j e r o r g Page 39

9 American Jornal of Engineering Research (AJER) 07 between the predacios mite Metaseils occidentalis (Nesbitt) and its spider mite prey Tetranychs mcdanieli (McGregor) Recently a modification of the olling-tanner model by invoking the ratio-dependent fnctional response is sggested by aqe and Li 4 We have investigated the model both analytically and nmerically We have stdied the reactiondiffsion model in both one and two dimensions and investigated its stability The nontrivial eqilibrim state E of predator-prey coexistence is locally as well as globally asymptotically stable nder a fixed region of attraction when certain conditions are satisfied We also obtained the conditions for Tring instability in terms of parameters For a fixed set of parameter vales (5) we obtained the region of Tring instability We have plotted the time series spatiotemporal patterns and snapshots of the model system () We fond that for increasing vale of time t = the system shows from stable to limit cycle behavior For increasing the vale of d we have plotted the snapshots which shows the irreglar patchy non-tring spatial distribtion of predator evolved for two dimensional cases REFERENCES [] I Berenstein M Dolnik L Yang AM Zhabotinsky IR Epstein Tring pattern formation in a two-layer system: sperposition and sperlattice patterns Physical Review E 70(4) (004) [] M Banerjee Self-replication of spatial patterns in a ratio-dependent predator-prey model Mathematical and compter Modelling (-) (00) 44-5 [3] M Barmann T Gross U Fedel Instabilities in spatially extended predator-prey systems:spatio-temporal patterns in the neighborhood of Tring-opf bifrcations Jornal of Theoretical Biology 45 (007) 0-9 [4] S Chen J Shi Global stability in a diffsive olling-tanner predator-prey model Applied Mathematics Letters 5(0) 64-8 [5] JB Collings The effects of the fnctional response on the bifrcation behavior of a mite predator-prey interaction model Bll Math Biol 36 (997) [6] B Dbey N Kmari RK Upadhyay Spatio-temporal pattern formation in a diffsive predator-prey system: an analytical approach J Appl Math Compt 3(009) 43-3 [7] B Dbey J ssain Modelling the interaction of two biological species in pollted environment J Math Anal Appl 46(000) [8] B Dbey B Das J ssain A prey-predator interaction model with self and cross Diffsion Ecol Model 7 (00) [9] M aqe B Li A ratio-dependent predator-prey model with logistic growth for the predator poplation Proceedings of 0th International Conference on Compter Modeling and Simlation University of Cambridge Cambridge (008) 0-5 [0] BE Kendall Nonlinear dynamics and chaos Encyclopedia of Life Sciences Vol 3 (00) 55-6 [] TK Kar Matsda Global Dynamics and Controllability of a arvested Prey-Predator System with olling type III Fnctional Response Nonlinear Anal: ybrid Systems (007) [] RM May Simple mathematical models with very complicated dynamics Natre(976) [3] AB Medvinsky SV Petrovskii IA Tikhonova Malchow BL Li Spatio-temporal complexity of plankton and fish dynamics SIAM Review 44(3)(00) 3-70 [4] JD Mrray Mathematical Biology II: Spatial Models and Biomedical Applications Vol 8 (003) [5] M Pascal Diffsion-indced chaos in a spatial predator-prey system Proceedings of the Royal Society of London B 5(993)- 7 [6] AB Rovinsky M Menzinger Self-organization indced by the differential flow of activator and inhibitor Physical Review Letters 70(6) (993) [7] JA Sherratt An analysis of vegetation stripe formation in semi-arid landscapes Jornal of Mathematical Biology 5() (005) [8] LA Segel JL Jackson Dissipative strctre: An explanation and an ecological Example J Theoret Biol 37 (97) [9] Serizawa Amemiya T Itoh K Patchiness in a minimal ntrient-phytoplankton model J Biosci [0] GQ Sn Z Jin YG Zhao QX Li L Li Spatial pattern in a predator-prey system with both self and cross diffsion International Jornal of Modern Physics C 0()(009)7-84 [] L Wolpert The development of pattern and form in animals Carolina Biology Readers (977)-6 [] L Yang M Dolnik AM Zhabotinsky IR Epstein Pattern formation arising from interactions between tring and wave instabilities Jornal of Chemical Physics 7(5) (00) [3] RK Upadhyay W Wang NK Thakr Spatiotemporal Dynamics in a Spatial Plankton System Math Model Nat Phenom 5(5) (00) 0- [4] RK Upadhyay NK Thakr B Dbey Nonlinear non-eqilibrim pattern formation in a spatial aqatic system: Effect of fish predation J Biol Sys8(00) 9-59 w w w a j e r o r g Page 40