Lotka-Voltera Impulsive Model Abstract Control or How not to fish Human intervention in nature often destroys the equilibrium that has been established in it. Vladyslav Bivziuk (Ukraine) It is very important to find controls that would not destroy the stability of the natural Fulbright – English for Graduate Studies Program 2019 equilibrium in the ecosystem. The impulsive systems theory allows us to investigate such effects on a biological system with a simpler example that do not destroy its equilibrium. Lotka–Volterra impulsive Lotka–Volterra equations model = , , = , 1 + 0 𝑑𝑑=𝑁𝑁 ( ) + ( ) + , = , 𝑑𝑑𝑑𝑑 𝛼𝛼𝑁𝑁1 − 𝛽𝛽𝑁𝑁1𝑁𝑁2 𝑡𝑡 ≠ 𝑘𝑘푘 = 𝐴𝐴𝐴𝐴 −+𝐵𝐵𝐵𝐵𝐵𝐵 ; 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑁𝑁1 𝑡𝑡 𝜌𝜌=11𝑁𝑁1 𝑡𝑡 + 𝜌𝜌12𝑁𝑁2 ,𝑡𝑡 𝜌𝜌10 , 𝑡𝑡 𝑘𝑘𝑘𝑘 • Discovered 𝑑𝑑independently𝑑𝑑 by Alfred Lotka 2 −𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷𝐷𝐷 + 0𝑑𝑑𝑁𝑁= ( ) + ( ) + , = ; and Vito Volterra𝑑𝑑𝑑𝑑 . −𝛾𝛾𝑁𝑁2 𝛿𝛿𝑁𝑁1𝑁𝑁2 𝑡𝑡 ≠ 𝑘𝑘𝑘𝑘 𝑑𝑑𝑑𝑑 • Describe interaction between two species = , = — equilibrium state. 𝑁𝑁2 𝑡𝑡 𝜌𝜌21𝑁𝑁1 𝑡𝑡 𝜌𝜌22𝑁𝑁2 𝑡𝑡 𝜌𝜌20 𝑡𝑡 𝑘𝑘𝑘𝑘 Conditions for asymptotic stability of equilibrium (prey and predator) in simplified biological ∗ 𝛾𝛾= ∗ 𝛼𝛼 , = ; 𝑁𝑁1 𝛿𝛿 𝑁𝑁2 𝛽𝛽 state are system. ∗ ∗ 1 + |tr | < det < 1, 1 1 2 2 • x and y are the numbers of prey and Since impulses𝑥𝑥 𝑁𝑁 do− not𝑁𝑁 ruin𝑦𝑦 the𝑁𝑁 equilibrium,− 𝑁𝑁 the det = det = , predator respectively; , , , are the following condition has to be satisfied − 𝛷𝛷 𝛷𝛷 constants that represent the interaction. tr = + 𝛷𝛷 cos 𝜌𝜌 + 𝜌𝜌11𝜌𝜌22 − 𝜌𝜌12𝜌𝜌21 sin , 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐷𝐷 𝜌𝜌12𝛽𝛽 𝛾𝛾 𝜌𝜌21𝛿𝛿 𝛼𝛼 + 1 + = 0, ΦConditions𝜌𝜌11 for𝜌𝜌22 stability𝛺𝛺𝛺𝛺 (nonasymtotic𝛿𝛿 𝛼𝛼 − )𝛽𝛽 of 𝛾𝛾 𝛺𝛺𝛺𝛺 Conclusion Alfred James Lotka + 1 ∗ + ∗ = 0. equilibrium state 𝜌𝜌10 𝜌𝜌11 − 𝑁𝑁1 𝜌𝜌12𝑁𝑁2 This research reveals stability and (March 2, 1880 – December 5, 1949) � ∗ ∗ det = 1, |tr | < 2. • a US mathematician, 𝜌𝜌20 𝜌𝜌22 − 𝑁𝑁2 𝜌𝜌21 𝑁𝑁1 asymptotic stability conditions of an impulsive Lotka–Volterra model. The simple example founder of mathematical = , , Φ Φ demography; 𝛽𝛽𝛽𝛽 shows that mathematical modeling of such a 𝑥𝑥̇ − 𝑦𝑦 𝑡𝑡 ≠ 𝑘𝑘𝑘𝑘 human intervention is necessary before the • born in Lviv, Ukraine; = 𝛿𝛿 , , Example • statistician at the If = = 0, = 1, = , then the actual intrusion into the ecosystem. Modern + 0 = 𝛼𝛼𝛼𝛼 + , = , Metropolitan Life Insurance 𝑦𝑦̇ 𝑥𝑥 𝑡𝑡 ≠ 𝑘𝑘𝜃𝜃 1 methods of applied mathematics and 𝛽𝛽 stability condition is Co.. + 0 = + , = ; 𝜌𝜌12 𝜌𝜌21 𝜌𝜌11𝜌𝜌22 𝜌𝜌11 𝜌𝜌22 mathematical modeling can be useful in the 𝑥𝑥 𝑡𝑡 𝜌𝜌11𝑥𝑥 𝑡𝑡 𝜌𝜌12𝑦𝑦 𝑡𝑡 𝑡𝑡 𝑘𝑘𝑘𝑘 1 Vito Volterra 21 22 + | cos | < 2. study of artificial ecological systems and the (May 3, 1860 – October 11, 1940) 𝑦𝑦 𝑡𝑡 𝜌𝜌 𝑥𝑥 𝑡𝑡= 𝜌𝜌 ;𝑦𝑦 𝑡𝑡 𝑡𝑡 𝑘𝑘𝑘𝑘 influence of anthropogenic factors on their • Italian mathematician, 11 𝜌𝜌 11 Ω𝜃𝜃 evolution. founder of functional Ω 𝛼𝛼𝛾𝛾 𝜌𝜌References Wikipedia contributors. (2018, October 1). Alfred J. Lotka. In Wikipedia, The Free Archimedes: New studies in the history of science and technology. (1996). Dordrecht: Encyclopedia. Retrieved 15:49, August 7, 2019, from analysis; Springer. https://en.wikipedia.org/w/index.php?title=Alfred_J._Lotka&oldid=862020463 Goodstein, J. R. (2007). The Volterra chronicles: The life and times of an extraordinary Wikipedia contributors. (2019, June 6). Vito Volterra. In Wikipedia, The Free • born in Ancona, Italy; cos sin mathematician, 1860-1940. London;Providence, R.I;: American Mathematical Encyclopedia. Retrieved 15:51, August 7, 2019, from = 𝛿𝛿 𝛼𝛼 . Society. https://en.wikipedia.org/w/index.php?title=Vito_Volterra&oldid=900664145 11 12 sin cos Hou, Z., Ahmad, S., Stamova, I., Zanolin, F., Lisena, B., & Pireddu, M. (2013). Lotka- • refuse to sign an oath of 𝜌𝜌 𝜌𝜌 Volterra and Related Systems : Recent Developments in Population Dynamics. Image References 𝛽𝛽 𝛾𝛾 Ω𝜃𝜃 Ω𝜃𝜃 Berlin: De Gruyter. Voltera, V. (1910). Wikipedia, The Free Encyclopedia. Retrieved from loyalty to the Fascist regime; Φ Samojlenko, A. M., & Perestjuk, N. A. (1995). Impulsive differential equations. https://upload.wikimedia.org/wikipedia/commons/6/67/Vito_Volterra.jpg − Ω𝜃𝜃 Ω𝜃𝜃 Singapore: World Scientific. Lotka, A. J. (). Wikipedia, The Free Encyclopedia. Retrieved from • professor at the University of 𝛽𝛽 𝛾𝛾 Slyn’ko, V. I., Tunç, O., & Bivziuk, V. O. (2019). Application of commutator calculus to https://upload.wikimedia.org/wikipedia/en/0/0b/Alfred_J._Lotka.jpg 21 22 the study of linear impulsive systems. Systems & Control Letters, 123, 160-165. Graphicgeoff, (n.d.). VectorStock. Retrived from Turin and the University of 𝜌𝜌 𝜌𝜌 doi:10.1016/j.sysconle.2018.10.015 https://cdn5.vectorstock.com/i/1000x1000/31/74/food-chain-vector-2013174.jpg 𝛿𝛿 𝛼𝛼 Véron, J. (2008). Alfred J. Lotka and the Mathematics of Population. Journal Rome La Sapienza. Electronique d'Histoire des Probabilités et de la Statistique, 4(1)