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.
spaces. The combination of the two in reaction-diffusion systems leads to mathematical models like the Turing mechanism that can generate surprisingly rich patterns. Another example we treat is chemotaxis where organisms can be induced to collective behavior by following gradients of chemical substances. We consider spatiotemporal structure formation from interactions between states at points at times . This means that the state is a function of states at some or all other points at previous
term, it will still be coupled to the other components of , however, which satisfy partial differential equations when their diffusion coefficients are positive. For modelling structure formation by systems of reaction-diffusion equations, it is important to allow for different diffusion coefficients (one coefficient 1, another 1): one variable can build spatial concentrations and adapts slowly whereas the other, fast adapting one gets enslaved. We now state precise existence theorems for the
 and the references listed there. The next result addresses the issue of long-time existence. Theorem 4.4.3 We assume that the preceding assumptions hold for all . We assume furthermore that the solution of (4.4.59) satisfies the a-priori bound (4.4.61) for all times for which it exists, with some fixed constant . Then the solution exists for all times . This result again follows from the previous, because the boundedness assumption on the solution makes it possible to apply local
(4.4.94), this can only happen if If we assume (4.4.102) then this is only compatible with (4.4.93) if (4.4.103) We are now ready to show Lemma 4.4.2 Let (4.4.102) and (4.4.103) hold and suppose (4.4.104) Then there exist such that is spatially unstable w. r. t. the mode whenever . Proof We need to find a solution of (4.4.98) with We have reduced this condition to (4.4.101). Thus, we need to find a nontrivial range of for which this holds. The value of for which the left hand side
subscript 1 or 2 denotes a partial derivative w.r.t. or , resp., then from the previous section, we obtain the equilibrium condition (5.1.15) at least if that equilibrium is attained at an interior point of the positive quadrant, i.e. at a point where and are both positive. Let us understand the meaning of such an equilibrium in our biological example of sexual reproduction. In the gamete allocation case, an interior equilibrium point corresponds to hermaphroditic behavior. This means that the