An equation describing the dynamics of single species populations is derived. The model allows for variations in the physiological characteristics of animals of different ages and sizes. An analytical solution which holds under certain specific conditions is found. It is shown that Von Foerster's equation, the logistic equation and other prior models are special cases of the new model.

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The use of rays to construct fields is illustrated by finding the field in the region 3>0 when the field is given on the plane z = 0. This construction is valid for complex rays as well as real ones. The method is applied to a gaussian field in the plane z=0, in which case a gaussian beam results. The calculation involves only complex rays. Exactly the same results are also obtained by applying the method of stationary phase to an integral representation of the field. However, the ray method is simpler than the stationary-phase method, and it is also applicable to problems for which the stationary-phase method cannot be used because no integral representation of the field is known. INDEX HEADINGS: Geometrical optics; Lasers.

We reporr the first observation of the simultaneous creation and competition between a free-carrier light-induced diffraction grating and an accompanying thermal grating in gallium phosphide. The lifetimes of the two gratings are widely different and the nonlinear index n, = I x lo-" cm'lW for the free-carrier grating is almost an order of magnitude larger rhan the thermal effect of a 100-psec mode-locked pulse. 0 1992 John Wilqv & Sons, Inc. ABSTRACT I n this article the norion of transition function is reviewed under a mathemarical form: Two methods are explored which respectively lead to two different transition functions. Thes are used to make usynprotic evaluations of an eruct integral representation. Original anulyric formulations are established. Their main properties are presented and compared with orher diffracrion coe,fjScients which alreadv exist. Several numerical cotnpu~ations allow us to give an idea of the accuracy of' each formulation. 0 1992 Johrt W i l q & Sorts. ltic.