Mathematical Methods in Biology and Neurobiology (Universitext)
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This book helps readers develop mathematical tools required for modern biology. It offers a survey of mathematics, from stochastic processes to pattern formation, and contains biological examples from the molecular to the evolutionary and ecological levels.
together with a map whose image contains all vertices of degrees 1 and 2. (In the rooted case, we do not require that the root be in the image of even though it may have degree .) The map need not be injective. For a phylogenetic (-) tree, however, we require that be a bijection onto the leaves of . In particular, such a phylogenetic tree has no vertices of degree 2. When every interior vertex has degree 3, we speak of a binary phylogenetic tree. This is a natural assumption in biology because,
between 0 and satisfies (3.3.8) Using (3.3.7), we obtain (3.3.9) Iteratively, we obtain (3.3.10) Recalling the assumption of stationary increments, this shows (3.3.2). As a consistency check, the reader should verify that a counting process given by (3.3.2) with independent increments conversely has stationary increments and is locally continuous in probability. We also observe that (3.3.11) and therefore, almost surely, only finitely many events take place in the finite interval .
a definite sign there. When this sign is positive, then for , the solution of (4.3.3)—which exists for all —satisfies . Similarly, when for , the solution with initial values in that interval satisfies . In particular, the fixed point is attracting when for and for . It is repelling when both signs are reversed. The fixed point 0 for (4.3.9) is neither attracting nor repelling, because does not change its sign there. When is the largest fixed point, then either for in which case the solution
Thus, when gets too large, this term takes over and keeps the population in check. Biological and other populations always satisfy (4.3.61) Thus, we only need to investigate solutions in the positive quadrant. For a single population, we consider the logistic or Fisher equation (see (4.3.7)) (4.3.62) This is about a population growing under the condition of limited or constrained resources, so that, when it gets too large, the capacity limits take over and keep it in balance. is an unstable
all quantities are expressed in units of cost). In fact, we can measure all items in cost units normalized so that each unit of each of the independent variables costs the same. This yields (5.1.8) We shall make this assumption for the rest of this section. In the case of two independent variables, the cost levels are then straight lines (of slope with the indicated normalization). 5.1.2 Reward Functions and Strategy Types With the above normalization (5.1.8) that the cost levels are flat,