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

By Linda J.S. Allen

KEY BENEFIT: This reference introduces various mathematical types for organic platforms, and provides the mathematical conception and strategies important in reading these types. fabric is geared up in line with the mathematical idea instead of the organic software. includes purposes of mathematical concept to organic examples in each one bankruptcy. makes a speciality of deterministic mathematical types with an emphasis on predicting the qualitative resolution habit through the years. Discusses classical mathematical versions from inhabitants , together with the Leslie matrix version, the Nicholson-Bailey version, and the Lotka-Volterra predator-prey version. additionally discusses more moderen versions, comparable to a version for the Human Immunodeficiency Virus - HIV and a version for flour beetles. KEY MARKET: Readers seeking a great history within the arithmetic in the back of modeling in biology and publicity to a wide selection of mathematical versions in biology.

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F(x, y) yields expansion abou t the equilibrium for the function 1 Figure 2. 12 Liapunov 0 i:i <1) ;::l 0 ~ y) ( y - y_) f( x, y ) -_ f(-x, y_) + af(x, y) (x - x_) + af(x, ay ax 2 2 a2f(x, y) (y - y)2 a f(x, Y) (x - x) + + 2 + 2! ay 2 2! 4 r to calculate the Liap~nov We used the meth od given in Elaydi (2000) , we calc~late ~(x401). The iterex onents when x 0 = 1/2. P s for t = o, ... , 400 are calculated. 99]. · w as ms syste order firstfor Liapunov exponents can be calculated unov exponents for systems 1s A brief description of how to calculate Liap prese nted in the Appe ndix to this chapter.

Ens10n ess iscrete . h A 15) (2 tion equa logistic tion is much . s we s a11 see, the behav10r of this equa (214 ) Th d1'ff tion equa al renti diffe tic logis nal origi the b her~nt from that of itude of the param~ter. 15) depend~ on the magn refer to e uatio n aAt. ~5~icasequa twn. e param eters and initial conditI'ons m me so lut10ns are nonnegative. Assu 1 0 < r < 4, O s xo < 1. and at x = 1/2 and f(l/ ) The maxi mum off (x) = rx(l - x) occurs Th 2 - r/4. Whe n . < )) l/ f(f( and 1 > ) f(l/2 2 0 · us, 1'f 0 < ' < 4 and 0 s x 0 < 1 r > 4, then then O < f(x) < 1 f(O) = f(l) _ 0 ' : .

IfR 0 < 1, then there exists only one equilibrium, the disease-free equilibrium, and it is locally asymptotically stable. Estimates of R 0 have been obtained for diseases such as measles, chickenpox, and smallpox (May, 1983). 2 Basic reproduction number as reported by May (1983). S. S. S. S. , 1955 S) f3 - + NI S. 11 Delay Difference Equations Chapter 2 Nonlinear Difference Equations, Theory, and Examples these infectious diseases and for others which resulted in large-scale epidemics. Note that the estimates depend on the location of the epidemic and the time at which the epidemic occurred.

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