A Course in Mathematical and Statistical Ecology (Theory and by Anil Gore

By Anil Gore

A path in Mathematical and Statistical Ecology

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Also we have PNo(O) = 1. N, (t) dt OPNo , which after separating variables and integrating gives where C is a constant of integration. Since at t = O. Not 0 This can be written in a standard form as y' + P( x)y of such an equation is given by y = e- J P(x)dx = Q(x). The solution J Q(x)eJ P(x)dxdx. \No [e;t + c] ° where C is a constant of integration. \ , so that P(No+l)(t) = N oe->. \t. Consequently high reproductive rate active over a short time interval will have the same effect on the probability distribution of population size as a low reproductive rate active over a longer time.

Ct+l- r' CHAPTER 2 28 However there is one anomalous feature of this regressive process. rrt+l are i::;m-1 It is possible that the last element n m ,t becomes negative which is not meaningful. rrt+ 1 does not correspond to a meaningful vector in the past (see exercise 11). We made the assumption here that Fm is positive. In reality it is conceivable that females in one or more of the oldest age groups are not reproductive. It is convenient to ignore these last nonreproducing age groups (see exercise 12).

If k is less than 1 these oscillations are damped but not otherwise. 7. THE ALLEE EFFECT So far we have tried to relax many assumptions involved in the c1assical logistic model. Inst ead of a linear per capita growth rate we considered a nonlinear one. In place of instantaneous response, we considered time delays . As alternative to continuous time we used discret e time etc. The main interest was behavior of the model for large population size. But one im port ant assumption that when population is very low , it undergoes exponential growth, has gone unnoti ced.

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