By James D. Murray
Mathematical Biology is a richly illustrated textbook in a thrilling and speedy becoming box. offering an in-depth examine the sensible use of math modeling, it good points workouts all through which are drawn from a number of bioscientific disciplines - inhabitants biology, developmental biology, body structure, epidemiology, and evolution, between others. It keeps a constant point all through in order that graduate scholars can use it to achieve a foothold into this dynamic study quarter.
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Additional info for Mathematical Biology. An introduction
51). It is clearly advantageous to stay on the L + branch and potentially disastrous to get onto the L − one. Let us now see what determines the branch. 19(b), the equilibrium population Nh (E) is close to K and Nh (E) > K /2, the equilibrium population for the maximum yield Y M . 50) is then approximately 1. So increasing E, and hence the yield, we are on branch L + . 19(a) when Nh (E) = K /2. As E is increased further, Nh (E) < K /2 and the recovery time is further increased but with a decreasing yield; we are now on the L − branch.
The data determine the parameters only over a small part of the growth curve. 4) where typically f (N ) is a nonlinear function of N then the equilibrium solutions N ∗ are solutions of f (N ) = 0 and are linearly stable to small perturbations if f (N ∗ ) < 0, and unstable if f (N ∗ ) > 0. This is clear from linearising about N ∗ by writing n(t) ≈ N (t) − N ∗ , | n(t) | 1 6 1. 4) becomes dN = f (N ∗ + n) ≈ f (N ∗ ) + n f (N ∗ ) + · · · , dt which to first order in n(t) gives dN ≈ n f (N ∗ ) dt ⇒ n(t) ∝ exp f N∗ t .
If we now consider small perturbations about the steady state x 0 we write u = x − x 0 and consider | u | small. 32) where V0 = d V (x 0 )/d x 0 is positive. As in the last section we look for solutions in the form u(t) ∝ eλt λ = −αV0 − αx0 V0 e−λT . 33) If the solution λ with the largest real part is negative, then the steady state is stable. Since here we are concerned with the oscillatory nature of the disease we are interested in parameter ranges where the steady state is unstable and, in particular, unstable by growing oscillations in anticipation of limit cycle behaviour.