The Dynamics Of Modified Leslie-Gower The Pest- Predator System With .

Advances in Engineering Research, volume 209𝐻2 𝑎(2𝛼 𝑟)𝐻3 𝑎𝑘𝑟 𝑎𝑚𝑟 𝑛𝐴𝑟 𝑛𝐴𝛼𝑓𝑟𝜌 𝛼 2 𝛽 𝛼𝛽𝑟𝐻4 𝑛𝐴𝛼𝑟𝜌 𝛼𝛽𝑘𝑟 𝑛𝐴𝛼𝛽𝑟 𝛼𝛽𝑚𝑟𝐻5 𝑛2 𝐴2 𝑟 2 𝜌 𝑛𝐴𝑘𝜌𝑟 2 𝑛𝐴𝑚𝜌𝑟 2A fourth-degree equation has four roots in the complexdomain (4). Omitting to write explicitly its coefficients,in view of their complexity (4). Now we find sufficientconditions for it to have at least one positive root [19].Depending on the parameters of the system (3), there istwo equilibrium point 𝐸3 (𝑥3 , 𝑦3 ) and 𝐸4 (𝑥4 , 𝑦4 ).Theorema3. LOCAL STABILITY ANALYSISKolmogorov conditions [20] at (0,The stability properties of the equilibrium point ofsystem (3) is determined by the roots of the characteristicequation. Here, the Jacobian matrix at an equilibriumpoint is𝐽 [𝑟𝛼𝑦𝛼𝑥𝑦𝑟𝑓𝑥 21 𝑓𝑦 𝑛𝐴 𝑚 𝑥 (𝑛𝐴 𝑚 𝑥) 1 𝑓𝑦𝛼𝑥𝛽𝑦 2 𝑛𝐴 𝑚 𝑥 (𝑘 𝑥)2𝜌𝑛𝐴𝑦𝑦 𝛽 (1 )(𝑛𝐴 𝑚 𝑥)2𝑘 𝑥𝛽𝑦𝜌𝑛𝐴 ]𝑘 𝑥 𝑛𝐴 𝑚 𝑥The direct evaluation of the Jacobian matrix at 𝐸1 gives𝐽(𝐸1 ) [𝑟 0 0 𝛽 𝜌𝑛𝐴𝑛𝐴 𝑚2.The(0,equilibrium𝑘(𝑛𝐴𝛽 𝑛𝐴𝜌 𝛽𝑚)𝛽(𝑛𝐴 𝑚))pointis𝐸2 locallyasymptotically stableProof.The equilibrium point 𝐸2 is locally asymptotically stableif 𝑇 0 and 𝐷 0.The trace (𝑇) and the determinant (𝐷) of the Jacobianmatrix at 𝐸2 are determined by 𝜆2 𝑇𝜆 𝐷 0, where𝑇 𝑎1 𝑎3 and 𝐷 𝑎1 𝑎3 . The system (3) satisfies the𝑘(𝑛𝐴𝛽 𝑛𝐴𝜌 𝛽𝑚)𝛽(𝑛𝐴 𝑚))making 𝐷 𝑎1 𝑎3 positive. Therefore 𝐸2 is locallyasymptotically stable if 𝑇 0 and unstable if 𝑇 0. Wehave𝑇 𝑟𝑓𝑘(𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚) 1𝛽(𝑛𝐴 𝑚)𝛼𝑘(𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚) 𝛽(𝑛𝐴 𝑚)2𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚 𝛽 (1 )𝛽(𝑛𝐴 𝑚)𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚𝜌𝑛𝐴 𝑛𝐴 𝑚𝑛𝐴 𝑚Here it is important that the term 𝑇 0 and 𝐷 is alwayspositive and this completes the proof.Similarly, the Jacobian matrix at 𝐸3 is],Theorema 1. The equilibrium point 𝐸1 (0,0) isalways unstable.𝐽(𝐸3 ) [𝑎4 𝑎5 𝑎6 𝑎7 ]Theorem 3. The equilibrium point 𝐸3 (𝑥 , 𝑦 ) islocally asymptotically stableProof.The characteristic equation of the matrix J( 𝐸1 ) are(𝜆 𝑟)(𝑛𝐴𝜆 𝑚𝜆 𝐴𝑛𝛽 𝑛𝐴𝜌 𝑚𝛽) 0 . It isObvious that one of eigenvalue 𝜆1 𝑟 is positive, theother eigenvalues is always positive 𝜆2 𝐴𝑛𝛽 𝑛𝐴𝜌 𝑚𝛽𝑛𝐴 𝑚 0. Hence, 𝐸1 is an unstable.Proof.Clearly that two eigenvalues are determined by 𝜆2 𝑇𝜆 𝐷 0, where 𝑇 𝑎4 𝑎6 and 𝐷 𝑎4 𝑎7 𝑎5 𝑎6 .Therefore 𝐸3 is stable if 𝑇 0 and 𝐷 0. The proof iscomplete.The evaluation of the Jacobian matrix at 𝐸2 gives4. NUMERICAL SIMULATIONS𝐽(𝐸2 ) [𝑎1 0 𝑎2 𝑎3 ], withWe would like to convey a numerical simulation toillustrate bifurcation phenomena. The study numeric arevery important to solving of solution the system (3)which is undertaken by Python 3.7. Numericalsimulations are carried out to support the analysis anddisplay the change in the equilibrium point solutionthrough a bifurcation diagram.𝑟𝑎1 𝑓𝑘(𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚)𝛽(𝑛𝐴 𝑚)𝑎2 1 𝛼𝑘(𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚)𝛽(𝑛𝐴 𝑚)2(𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚) 𝑛𝐴𝜌𝑘(𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚) 𝛽(𝑛𝐴 𝑚)2𝛽(𝑛𝐴 𝑚)3𝑎3 𝛽 (1 𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚𝛽𝑛𝐴 𝜌𝑛𝐴 𝛽𝑚) 𝛽(𝑛𝐴 𝑚)𝑛𝐴 𝑚𝜌𝑛𝐴 𝑛𝐴 𝑚521

Advances in Engineering Research, volume 209Table 1. The set of parameter value used for tionMaximum growth rateof pestThe fear effect of pestto predatorMaximum rate ofpredatorHalf saturation valueof the predatorThe relative ability ofthe predator to detectadditional foodAmount of additionalfood to the g capacity ofpredator2.1𝛽Maximum growth rateof the l condition 𝑁 [𝑥(0) 5, 𝑦(0) 7.2]. Thisoccurs quickly since the initial value is taken chose to 𝑁.The limit cycle encourages an almost periodic solutionwhen 𝛼 increases. The equilibrium 𝐸1 (0,0) unstableremains itself. The interior equilibrium 𝐸3 (5.763726, 8.479635) evolves into the positive periodicsolution. We find the Hopf bifurcation when 𝛼 1 0.72993019 and a stable limit cycle appears around theinteriorequilibriumpointfor𝐸3 (5.763726, 8.479635) in Figure 1b. Now, we discussfour different parameters value (𝛼) to illustrate thesolution tends to the equilibrium point. The dynamics ofthe two populations of the present system (3) are obviousat the eight frames shown in the Figure 2a to 5b.4.1 Phase PortraitFor simulation, we take the same parameter values inTable 1. Using these parameter values, it can be shownthat all equilibrium points 𝐸1 , 𝐸2 , 𝐸3 , and 𝐸4 of thesystem (3) exist. However, we perform the numericalsimulation showing that system (3) the limit cycleundergoes periodic perturbation.Figure 2a The phase portraits of system (3) which shows𝐸3 stable.Figure 2b The solution tends to 𝐸3 stable at 𝛼 0.658321.Figure 1a The phase portraits of system (3) which showsHopf Bifurcation at 𝛼 0.72993019.Figure 3a The phase portraits of system (3) whichshows 𝐸3 stable.Figure 1b Hopf Bifurcation Diagram with specify of thelimit cycle at 𝛼 0.72993019.In Figure 1a, we illustrate that the system (3) gets alimit cycle when parameter value 𝛼 0.72993019 with522

Advances in Engineering Research, volume 209Figure 3b The dynamics tends to 𝐸3 stable at 𝛼 0.7.Figure 4a The phase portraits of system (3) whichshows 𝐸3 stable.Figure 5b The solution tends to 𝐸2 (0, 3.047)at 𝛼 0.866893.The phase portraits of the system (3) which the solutiontends to 𝐸3 (5.5284, 8.24679) stable when parameter𝛼 0.658321 shown in Figure 2a and 2b. Figure 3a and3b shows that all the equilibrium points 𝐸1 , 𝐸2 , 𝐸3 , and 𝐸4exists. This simulation uses the parameters as shown inTable 1, that the stability conditions at the equilibriumpoint 𝐸3 (6.4863, 9.1956) fulfill the results of thestability conditions. The red line corresponds to the pest(𝑥) population and the black line correspond to thepredator (𝑦) population in Figure (2b – 5b). Numerically,the system (3) indicates a kind dynamic fullness.4.2 Bifurcations around the interior equilibriumpointNumerical continuity is performed on an equilibriumsolution with a variation of one parameter to indicate anindication of a change in the type of stability usesMatCont. Changes in stability indicate the emergence ofbifurcations, namely Hopf bifurcation, Saddle Nodebifurcation, and Transcritical bifurcation.Figure 4b The dynamics of the system (3) at 𝛼 0.73994233.Figure 5a The phase plane diagram of the system (3)stable.Figure 6 Bifurcation Diagram of the predator population(𝑦) with continuation parameter 𝛼.In numerical continuity, the parameter (𝛼) is to indicatechanges in the stability of several the equilibrium point.Figure 6 presents the bifurcation diagram of the solutionof the system (3) which is varying the parameter of themaximum rate of the predator predation to the pestpopulation (𝛼).523

Advances in Engineering Research, volume 209a. Saddle node BifurcationIn detail the changes that occur at the interiorequilibrium point are shown in Figure 6 diagram ofbifurcation for the predator population (𝒚) with respect toparameter 𝜶. The results of the numerical continuation ofthe parameter 𝛼 show the changes that occur in thestability at the interior equilibrium point ( 𝑬𝟑 ). TheBifurcation diagram in Figure 6 shows for𝟎. 𝟕𝟐𝟗𝟗𝟑𝟎𝟏𝟗 𝜶 𝟎. 𝟖𝟔𝟔𝟖𝟗𝟒. There are two interiorequilibrium point where of them is (𝑬𝟑 ) unstable while(𝑬𝟒 ) is stable. Both conditions of the interior equilibriumpoint crash at 𝜶 𝟎. 𝟖𝟔𝟔𝟖𝟗𝟒, namely Limit Point (LP).The phenomenon of the emergence of LP shows thatthere is a Saddle-node (Fold) bifurcation which is drivenby the parameter of the maximum value of the predatorpredation rate on the pest. if 𝜶 𝟎. 𝟖𝟔𝟔𝟖𝟗𝟑𝟓 , ( 𝑬𝟐 )stable.b. Double stability phenomenonAnother interesting dynamics behavior to observe asshown in Figure 6 is the appearance of two differentstability, known as the bi-stability phenomenon. Thisphenomenon occurs during 0.866893 𝛼 0.866894that the system (3) has a double stability. There are twostability in the solution of system (3) namely the interiorequilibrium point (𝐸4 ) and the pest extinction equilibriumpoint (𝐸2 ) which are locally asymptotically stable, whilethe other interior equilibrium point are unstable. Thephenomenonalsooccurs0.658321 𝛼 0.72993019. The solution of the interior equilibriumpoint (𝐸3 ) and the pest extinction equilibrium point (𝐸2 )which are locally asymptotically stable.c. Transcritical BifurcationThe bifurcation diagram in Figure 6 illustrates theexchange between the equilibrium interior point (𝐸3 ) andthe pest population extinction points ( 𝐸2 ). Thisphenomenon shows a Transcritical bifurcation. For𝛼 0.658321, the pest population extinction points (𝐸2 )are unstable. As it increases of the parameter 𝛼, a BranchPoint (BP) apprears at 𝛼 0.658321. This BP indicatesa Transcritical bifurcation, while a change in theequilibrium point of the pest population extinction points(𝐸2 ), which was initially unstable for 𝛼 0.658321,became stable for 𝛼 0.658321. On the other hand, thereis a change in the coexist equilibrium point 𝐸3 , which wasoriginally stable 𝛼 0.658321 , became unstable for𝛼 0.658321.four equilibrium points. There is one equilibriumcondition that is always unstable, namely the equilibriumpoint of extinction for all populations. The otherequilibrium points are stable with certain conditions thathave been proven. The results of the dynamic analysishave been confirmed through numerical simulations bycontinuing the parameter of the maximum rate of thepredator predation, namely 𝛼. The numerical simulationresults also show that the pest-predator interaction modeldescribes rich dynamics with the occurrence of Hopfbifurcation, Transcritical bifurcations, Saddle-nodebifurcations and double stability phenomena driven bythe continuation of the maximum value parameter ofMaximum rate of predator predation on pests.AUTHORS’ CONTRIBUTIONSDS: Conceptualization, Data curation, Visualization,software (simulation and continuation numeric) anddrafting manuscript: M and A; Formal analysis, Fundingacquisition, and review. M.J; Methodology, datavisualization and editing. All authors have read andapproved the final manuscript.REFERENCES[1]N.A. Herawati and Sudarmaji, Helminths of theRice-field Rat, Rattus argentiventer, In G.RSingleton (Ed.), Rat, Mice and People: RodentBiology and Management, ACIAR Canberra,2003.[2]R. Kaliky, K. Yolanda, and Sudarmaji, PeranKeanekaragaman Hayati untuk MendukungIndonesia sebagai Lumbung Pangan Dunia,Seminar Nasional dalam Rangka Dies NatalisUNS ke 42 Tahun 2018, 2(1)( 2018) c.id/6858/1/Makalah%20Seminar%20UNS Tempe Samsul%20Rizal%20UNILA REVISI.pdf[3]N.T. Haryadi, M. W. Jadmiko and T. Agustina.,Pemanfaatan Burung Hantu untuk MengendalikanTikus di Kecamatan Semboro Kabupaten ac.id/handle/123456789/73295[4]P. D. N. Srinivasu, B. S. R. V. Prasad and M.Venkatesulu, Biological Control ThroughProvision of Additional Food to Predators,Theoretical Population Biology, University ofMissouri, Kansas City, USA,2007, pp 111-120.DOI: http://dx.doi.org/10.1016/j.tpb.2007.03.011[5]Sen, et. al, Global Dynamics of an Additional FoodProvided Predator-Prey System with ConstantHarvest in Predators, Applied Mathematics andComputation 250 (2015) 193-211. DOI:http://dx.doi.org/10.1016/j.amc.2014.10.0855. CONCLUSIONWe have discussed a model describing pest-predatorinteractions taking into account the effects of fear on pestpopulations and food additives on predator populations.The predator growth model uses a modified LeslieGower model. We find that the pest-predator model has524

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model involves seasonal parameters in determining the quality and quantity of food additives. Inspired by the model [18] which discussed the Leslie-Gower model and modified by assumpting the additional food to predator in terms of the handling time. In this article, we have studied a prey-predator system with additional food for