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

By Linda J.S. Allen

KEY BENEFIT: This reference introduces numerous mathematical types for organic platforms, and offers the mathematical conception and methods priceless in reading these types. fabric is geared up in response to the mathematical thought instead of the organic program. includes functions of mathematical concept to organic examples in every one bankruptcy. makes a speciality of 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, similar to a version for the Human Immunodeficiency Virus - HIV and a version for flour beetles. KEY MARKET: Readers seeking an effective historical past 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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4. Recal l that, in the cobwe bbing plane. 3, the repro ductio n curve f(x) = ax/(b b. 3. Two cases are consid x conve rges then b, > a if and b < a if 1 ates x 1 conve rge mono tonica lly to zero a < b. when brium equili ve positi a exist mono tonica lly to x. There does not 2 is sketch ed for x > 0. 4, the reprod uction curve l that if 0 < r < 3/4, Two cases are considered, 0 < r < 3/4 and r > 3/4. ;)/2 is locally t in the cobwebbing and if r > 3/4 it is unstable. 3. This latter example [Figur d.

A1 .... 0 1 the Jacobian matri x J evaluis locally asym ptotic ally stable if the eigenvalues of ated at the equil ibrium satisfy )A;I < 1 iff ITr(J)I < 1 + det(J ) < 2. 19) se /A;/ < 1 and the eigenvalues are complex conjugates. Then 2 Case 2 Sup~o r < 48. 20) det(J ), Tr(J) < -1 - det(J ), or det(J ) > 1. 2 _ r ± . 19) holds and magn itude less than one. tic equat ion of the Deno te r = Tr(J) and 8 = det(J ). Then the characteris Jacobian matrix J is 2 p(A) = A - rA + 8 = 0.

O a) 0 < a :::; oo. Then x is said to be globally att:acti_ve if where f · , a . , ' . = x The equilibrium x 1s smd to for all initial conditions Xo E (0, a), hm1 ..... oo Xi · . d if - · 1 cally stable be globally asymptotically stable if xis globally attractive an xis o . 12) satisfies (i), (ii), and 0 < f(x) < x for all x E (0, a), then the origin is globally asymptotically stable. Proof The result follows by noticing that 0 < f' (x 0 ) < · · · < /2(x 0 ) < f(x 0 ) < x 0 for x 0 E (0, a). The sequence {f(x0 )}~ 0 is monotone decreasing and bounded below by zero.

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