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

KEY BENEFIT: This reference introduces a number of mathematical types for organic structures, and offers the mathematical concept and methods necessary in examining these types. fabric is geared up in accordance with the mathematical conception instead of the organic program. comprises purposes of mathematical thought to organic examples in each one bankruptcy. specializes in deterministic mathematical versions with an emphasis on predicting the qualitative answer habit over 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, comparable to a version for the Human Immunodeficiency Virus - HIV and a version for flour beetles. KEY MARKET: Readers seeking a superior history within the arithmetic at the back of modeling in biology and publicity to a wide selection of mathematical versions in biology.

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14 bria are x = Oand-~ = 1 Suppose X1+1 = (l(2)(x1 + xt) = f(x1). Then the equili . First f'(x) - 1/2 + We compute the Liapunov exponents at the equilibria so x tha~ ~ x so that f'(O) = 1/2 and f'(l) = 3/2. If Xo = x, then X1c 1 t-1 2:ln \f'(O)/ A(O) = lim -t k=O = -ln(2 ) < o t--'>oo and . 1 t-1 A(l) = 1 ~~ r,~ln \f'(l) / = ln(3/2) > O. t' L' equilibrium x -~ oh The stable . as a nega ive iapunov exponent and the unsta.. e Liapunov · is prope rty of negative and positive Liapunov . o 11111 (Exercise 20).

A change in the solution beha vior 11 derivative off or f1 equals one or m-cycle changes stability, that is, when the can occur when f' (x(r) ) = ± 1 have negative one. 17). 10). unstable, disapone and stable one bifurcation point is passed, two equilibria, sky bifurcablue a as to ed referr times pear. 10 e Figur In one. ble appear, two stable ones separ ated by an unsta with two system A ation. bifurc fork cation is referr ed to as a supercritical pitch of the ity stabil the If ility. 10 re (Figu ation tion.

Local There is an easy check for local stability in the two-d nt of the Jacobian mina deter the and stability depends on the values of the trace ). (1988 matrix. The proof follows Edelstein-Keshet figur e 2. 13 The characteristic polynomial p(A) when the eigenvalues are real and satisfy [A;[ < 1 . 5 conti nuous first-order partia l Assu me the funct ions f(x, y) and g(x, y) 2have contains the point (x, y). Then derivatives in x and yon some open set in R that system near the equil ibrium point (x, y) of the nonli 65 '- 0 '' - '' '' ' / ' ' ..