Competition ) Bulmer 99 3 Erickson 97, Case 999 Park 9, Case 999 Brown and Rothery 99 Argentine ants > Harvester ants T. castaneum < > T. confusum
n, n = r # " n dn n ' = r # " n " µ n n K ' K = r # " n " µ n n K ' # " µ n " n n ' Lotka Volterra K! ij j i = r " n + # n ' ) n K " # n + n ' ) n K Lotka Volterra! /K µ Lotka, Alfred James 99), USA Alfred Lotka, chemist, demographer, ecologist and mathematician, was born in Lviv Lemberg), at that time situated in Austria, now in Ukraine. He came to the United States in 9 and wrote a number of theoretical articles on chemical oscillations during the early decades of the twentieth century, and authored a book on theoretical biology 9). He is best known for the predator-prey model he proposed, at the same time but independent from Volterra the Lotka-Volterra model, still the basis of many models used in the analysis of population dynamics). He then left academic) science and spent the majority of his working life at an insurance company Metropolitan Life). In that capacity he became president of the PAA the Population Association of America). Lotka Volterra = r " n + # n ' dn ) n " # n + n ' ) n K K http://users.pandora.be/ronald.rousseau/html/lotka.html Volterra, Vito 9), Italy Vito Volterra's interest in mathematics started at the age of when he began to study Legendre's Geometry. At the age of 3 he began to study the Three Body Problem and made some progress by partitioning the time into small intervals over which he could consider the force constant. His family were extremely poor his father had died when Vito was two years old) but after attending lectures at Florence he was able to proceed to Pisa in 7. At Pisa he studied under Betti, graduating Doctor of Physics in. His thesis on hydrodynamics included some results of Stokes, discovered later but independently by Volterra. http://www-groups.dcs.st-and.ac.uk/~history/mathematicians/volterra.html phase plane analysis n n = n = n
= r " n + # n ' n ) n = n =, n + " n = K K " # n + n ' n ) n = n =, " n + n = K n K n K /! = dn = n n K /! K K K K /! n K K /! n n =, n + " n K = n =, " n + n K = ) n >, n >, " n + # n K >, " # n + n K > n =, n + " n K = n =, " n + n K = ) n >, n >, " n + # n K <, " # n + n K > n n K /! K 3) ) = r " n + # n ' ) n > n K " # n + n ' ) n > K n K /! K 3) ) = r " n + # n ' ) n < n K " # n + n ' ) n > K n ) ) ) ) K K /! n K K /! n
3 n =, n + " n K = n =, " n + n K = 3) n >, n >, " n + # n K >, " # n + n K < n =, n + " n K = n =, " n + n K = ) n >, n >, " n + # n K <, " # n + n K < n n K /! K 3) ) = r " n + # n ' ) n > n K " # n + n ' ) n < K n K /! K 3) ) = r " n + # n ' ) n < n K " # n + n ' ) n < K n ) ) ) ) K K /! n K K /! n /, /) K /! > K, K /! > K n + " n K = " n + n K =! n *,n * ) = K " #! K, K " # K ' ) " # # " # # 3 3 3 3! n n n n n t) n t)
3 n K /! > K, K /! > K K /! > K, K /! < K n K /! > K, K /! < K K /! K /! K K K K /! n K /! K n 7.. 7.. 7.. 7.. 7.. n K n K K /! < K, K /! > K K /! < K, K /! < K 7. 7.. 7.. K /!. K /! K K /! n K /! K n n t) n t) K /! < K, K /! > K K /! < K, K /! < K 3 7.. 7.. 3 3 3 n t) n t) n t) n t)
Lotka Volterra K, K!,! r, r K /! > K K /! < K Gause K /! > K Lotka Volterra K /! < K!!!,!