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By N. Goel (Auth.)

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The reader might 45 EQUILIBRIUM THEORY find it interesting to make an analysis of graph 6(d). It can be shown that the determinant of any antisymmetrical matrix of an odd order vanishes. This is related to the facts that the eigenvalues of an antisymmetric matrix are purely imaginary and that they occur in pairs, one being the complex conjugate of the other. In the case of an odd order matrix, the only way this condition can be satisfied is to have one eigenvalue zero. Since the determinant of the matrix is the product 3 4 3 4 3 4 34 3 4 1 2 1 2 1 2 1 2 1 2 b c d e 3 4 Π K □ M H H a I 2 f FIG.

0 ) = const. 9b) when t=0. We define R2 by *Σ*Λ0)=*#. e. 16) 3 are given equal weight. This problem has been solved by Mazur and Montroll (1960) in the context of vibrations of crystal lattices. (t) = Σ MtfW-MiM). (t) = Σ AiiAjfAimAjm* exp [(X z +X m )/]. 7) we can replace \m by — \m if, at the same time, we replace Aim by Ai^ and Ajm* by Ajm.

We ask for a set of conditions such that each oy>0. Our arrangement of arrows imply 48 NONLINEAR MODELS OF INTERACTING POPULATIONS 2f 4 T 3 T |1 i ί i (i) FIG. 8. Some of the graphs involving four interacting species. The graphs represent some of the interactions with the nutrient reservoir. (iii) (ii ) £i>0, £ 3 <0, k2<0, Λ3ι>0, α 43 >0, and h<0 and O24>0. From these conditions it is clear that qz>0, q*>0, and #i>0. The stationary population q2 of species 2 can be positive only if &A043>&4&013.

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