By Stavros N. Busenberg

ISBN-10: 0121483606

ISBN-13: 9780121483609

ISBN-10: 0323153429

ISBN-13: 9780323153423

Differential Equations and functions in Ecology, Epidemics, and inhabitants Problems.

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**Extra info for Differential Equations and Applications in Ecology, Epidemics, and Population Problems**

**Example text**

5). 6) Since (r/K) the Sturm Comparison Theorem [1] implies each y"(x) solution is a non-negative solu- has at least one zero on 0 < x < π. provides a contradiction. 5) <1. must have all maxima The existence of a positive solu- tion now follows from the global bifurcation results in [8]. 5) and (r/K) [l - (Ίί(χ) Since + rN(x))1[y(x) + (τ/Κ)\_1~(Τί(χ) + ϊί(χ))~\ = 0. 6) (in particular, N(x)) must vanish somewhere on J. This contradiction proves the uniqueness assertion. III. 3) as £-+«>. 2) is seen by the following theorem to be globally asymptotically stable.

FIGURE 1 48 J. P. E. Hodgson The sectors of the (u,v) plane in which £ must lie are illustrated in Figure 2. l I <* 2 FIGURE A,BfC, = u(l+e) = u(ek+e and D 2 + vie k-e) k)-v(ek-l)=0 2 It is a straightforward argument in plane geometry to ABCD show that a rectangle I structed only when of the required type can be con- is "steeper" than I . This requires e k < 1 . 2 e(Q) Thus if e(Q) tion in k < 1 for all 0 < Θ < 2ττ and the varia- is sufficiently small we will be able to find a box π = {a__
__

5) solution on I. , N = 0. (ii) N(x) = N(x;r/K) 0 < N(x) < 1 for r/K ■+ 1 . 5) xel and there positive M(r/K) - max χε I 22 David Green, Jr. and Harlan W. 5). 5). 6) Since (r/K) the Sturm Comparison Theorem [1] implies each y"(x) solution is a non-negative solu- has at least one zero on 0 < x < π. provides a contradiction. 5) <1. must have all maxima The existence of a positive solu- tion now follows from the global bifurcation results in [8]. 5) and (r/K) [l - (Ίί(χ) Since + rN(x))1[y(x) + (τ/Κ)\_1~(Τί(χ) + ϊί(χ))~\ = 0.

### Differential Equations and Applications in Ecology, Epidemics, and Population Problems by Stavros N. Busenberg

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