By Johan A. J. Metz, O. Diekmann
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Extra resources for The Dynamics of Physiologically Structured Populations
Example text
16) the age at which an individual becomes reproductive. In words: the birth rate at t equals the cumulation of the births from all individuals born at different times between time zero and now, which are still among the living, plus the births produced by the remains of the initial population. 15) is a so-called renewal equation (or linear Volterra integral equation of the convolution type). 15) will grow exponentially for large t. 18) Which brings us full circle. 7) to rederive the stationary length distribution 1/J.
However, in practice a numerical solution of (2. 6) is about as fast as using a complicated explicit formula. 6). For example it is easily shown that 41 always increases with ~ and that if g is concave 41 is concave as well. 1: Prove the last statements. 6) r. 3 right). 3: Rashevsky's explanation of Ivlev's findings. ) Ivlev's experiments i) were of short duration, ii) were done at high Daphnia densities, and iii) were done with (initially) starved fish. 5), can be neglected. Therefore the total amount of prey eaten equals the difference between the satiation at the start and at the end of the experiment.
6: Assume that there is a lower limit a to the size at division, and that a is larger than the size of any newborn cell. What is the expected number of daughters reaching size a, of a cell that itself at this moment passes size a? 2. Formulation of the p equations z Just as for Daphnia the size density will fulfill the role of p-state. e. J n(t,x)dx is the y number of cells with sizes between y and z at time t. Drawing up the balance of growth, death and division during an arbitrarily small time interval, like we did in the previous sections, we find a n(t,x) at = -a;a (V(x)n(t,x))-l'(x)n(t,x)-b(x)n(t,x) + 4b(2x)n(t,2x).