An Introduction to Mathematical Biology by Linda J.S. Allen

By Linda J.S. Allen

KEY BENEFIT: This reference introduces various mathematical types for organic platforms, and offers the mathematical thought and methods valuable in examining these versions. fabric is equipped in line with the mathematical conception instead of the organic software. includes functions of mathematical idea to organic examples in each one bankruptcy. specializes in deterministic mathematical types with an emphasis on predicting the qualitative answer habit through the years. Discusses classical mathematical types from inhabitants , together with the Leslie matrix version, the Nicholson-Bailey version, and the Lotka-Volterra predator-prey version. additionally discusses newer types, akin to a version for the Human Immunodeficiency Virus - HIV and a version for flour beetles. KEY MARKET: Readers seeking a high-quality historical past within the arithmetic at the back of modeling in biology and publicity to a wide selection of mathematical types in biology.

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E s10nal maps, mcludmg the equat ion x t+1 -- X1 , s ou one- 1men t d Th me t' an), Li and Yorke (19~51)o~:d M:se paper s are by Sharkov~~ii (1964, in Russi ematician who math an Russi a was y (1975). AN. Sharkovskn . , ing lies at the order i's ovski Shark rs. 7 Bifurcation Theory ples dividing by t, and lettin f : large t. ~en taking the logarithm of both sides, g s: follow as ed defin is Xo at off oo' the Liapunov expo nent t Yien Li. , Sharkovskii's ordering can be found in many textbooks of the term "chaos" in con1996; Elaydi, 2000; Gulick, 1992).

D cally . mati . auto . (O , x) ' then cond. itwn (vi) . nm . as no maximm Note that iffh m, oma its m ts pom two fixedthat f has at most one . has only (') ( ') imp . -: vi satisfied. X. The theorem is due to Cull (1981). EI, then x1 We show this cann ot happen. bv~u:;o ~~~~; ~~~ Global asymptotic stability implies f has no is given m t e PP and t reverse implication is a lengthy argumen Cha pter 2. -:/- 0 for all x EI, 0 1 0-:/- Lx (l + f'(x) ) dx = (x1 + f(x1 )) - (xo + f(xo )) = f 2 (xo) - Xo.

3 to the functi 1 a fixed point of f 2 , (Elayd2i, 1999). Since i is a fixed point of fit is also and x = f (x) = g(x). In additi on, g'(x) = f'(f(x ))f'(x ) g"(x) = f"(f(x ))[f'( x)]2 + f'(f(x ))f"(x ) . the solur > 3/4 so that These two soluti ons repres ent a 2-cycle. We assum e ~e of ~h e tu magm th~ by mined d~ter tions are real. Tue stability of the 2-cycle is ary · · Coro. asym locally is e 2-cycl the We leave it as an exerci se to show that ates illustr _2·2 e Figur 6). cise (Exer 5/4' > ble if 3/4 < r < 5/4 and unstab le if r verto easy ~is 5/~.

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