Analysis of optimal harvesting of a prey-predator fishery model with the limited sources of prey and presence of toxicity

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1 ES Web of Confeences 7, 68 (8) hps://doiog/5/esconf/8768 ICEIS 8 nalsis of opimal havesing of a pe-pedao fishe model wih he limied souces of pe and pesence of oici Suimin,, Sii Khabibah, and Dia nies Munawwaoh Depamen of Mahemaics, Facul of Sciences and Mahemaics, Diponegoo Univesi, Semaang - Indonesia bsac model of pe and pedao species is discussed o sud he effecs of he limied pe densi and pesence of oici The model is sudied fo susainable opimal havesing The eisence of equilibium poins is analzed o find he sabili of coeisence equilibium, and use Ponagin s maimal mehod o obain he susainable opimal havesing The esuls show ha he opimal havesing is obained fom he soluion of opimal equilibium The oici faco deceases he susainable havesing Kewods: Pe-pedao; havesing; sabili; equilibium Inoducion ilapia, goldfish Mahemaical model d α qe β K d α q E β K (),() Coesponding auho: suimin@undipacid The uhos, published b EDP Sciences This is an open access aicle disibued unde he ems of he Ceaive Commons ibuion License 4 (hp://ceaivecommonsog/licenses/b/4/)

2 ES Web of Confeences 7, 68 (8) hps://doiog/5/esconf/8768 ICEIS 8 K K α β, β α q i, i E Model nalsis Eisence of sead sae P, P,, K q Eβ q E β P, K qeβ, fo qe β P, K EKq Kβ whee and K α is he soluion of polnomial a a aa () whee, a, a EKq Kβ K, a K qeβ Kα qeα β K a Kα qeβ K + K qeβ The polnomial has eacl one posiive oo of, q E β, K K qe β α K q E β P (, ) Kα V, ln h ln V due o is, d d d d h α q E β K α q E β K α α K K α α h K K We have, α K h K K hα α α α h α d, α q E β K d,

3 ES Web of Confeences 7, 68 (8) hps://doiog/5/esconf/8768 ICEIS 8 α q E K E α q β K q K q α q β β T, C π TR TC TR: TC :, Kq p K K β α α, Kq p q qqp β β qqp α C C pq p Kq Fom he equaion (), we can see ha sum of oos is and is poduc is I can be seen ha The equaion () has picel one posiive oo if C We have, so C The bionomic soluion,, wih C is unique if 4 nalsis of Opimal Havesing The cos of evenue in ime is defined b, δ π,, E, e (7) whee π is he eal income a ime fomulaed b π(,, E,) EE CE, δ is he ae of discoun annual To maimize he objecive funcion subjec o ssem (), we use Ponagin maimal pinciple and assume E ( ) E The fomulaion fo ma Hamilonian is given as follows, H p q C Ee δ α λ qe β K α λ qe β K λλ, dλ H δ α α e λ α α β qe K K λ dλ dh d α λ β qe () K K α E β q K λ λ e δ λα d λ () d λ ( ) α P Q λ Me δ () whee, α α P, α Q, and K K M P p q Eδ p q E The quadaic equaion in η of () is, (9)

4 ES Web of Confeences 7, 68 (8) hps://doiog/5/esconf/8768 ICEIS 8 η Qη () Q η η M δ λ () C e C e e CC, δ Qδ () C C λ () M e δ M δ λ e α p q M Q δ q p λ i () e δ i, H H E H δ Ce λ () q λ q E δ π λ () q λ () q e E λ (), λ () M M p q p q C E M, δ δ δ C,, E π π CE M, M oδ π MqMq oδ o δ π δ δ δ π 4 umeical esuls, K, K 5, 9, 7, α, α, q, q, p 8, p 5, C 5, δ, 5 P, P 98, R, R5, 8 δ δ E Fig The evoluion of pe and pedao populaions fo biological equilibium Fig P 98,fo havesing effo E equals o unis In figue, i shows ha he soluions wih he diffeen iniial condiions, close o he opimal equilibium R (5, 8) I is shows ha fo pe populaion, he opimal equilibium is lage compeed o equilibium sae On he cona, fo he pedao populaion, he opimal equilibium is less compeed o equilibium sae 5 Conclusion he simulaion esuls, i was shown ha he susainable havesing fo wo species 4

5 ES Web of Confeences 7, 68 (8) hps://doiog/5/esconf/8768 ICEIS 8 (pe and pedao) can be obained The oici faco cause he deceasing of havesing So, o peven coasal ecossem is needed he effos fo educing he emission of cabon dioide in susainable economic developmen cknowledgemens Refeences BR Redd, In J Mah Tends and Tech 4(), () T Das, R Mukhejee, KS Chaudhui, ppl Mah Model (5):8-9, (9) T Das, R Mukhejee, KS Chaudhui, J Biol Dn (5): (9) Suimin, S Khabibah, D Munawwaoh, ES Web Conf, 88 (8) 5