To gain an understanding of how cowpea (Vigna unguiculata (L.) Walp.)-maize (Zea mays L.) mixture yield advantages arise, and how they can be improved, a series of experiments were conducted at the Tropical Agricultural Research and Training Center (C.A.T.I.E.), at Turrialba, Costa Rica during November 1979 through September 1980. A semierect cowpea cultivar and a double-cross maize hybrid were sown simultaneously in "replacement series" with N and P fertilization and population density variables;The importance of light when intercrops differ greatly in size and height was evident during the growing season. In two instances where P limited maize growth more than cowpea, cowpea was more competitive than maize, but their competitive relationship was reversed by P addition, and when maize grew taller later in the season. Therefore, greater mixture advantages may be expected where cowpea is capable of accumulating substantial dry matter before maize becomes too large;Cowpea competitiveness and LERs were often improved when cowpea was not a majority in the mixture. The reason for cowpea being more competitive at some proportions than at others is unknown, although it is conceivable that at low densities more leaf area per plant may have allowed greater N(,2)-fixation for cowpea;Competition for P was not apparent. A reduction in cowpea N(,2)-fixation caused by shading, and its weak ability to compete with maize for mineral N, are suggested to be the cause of cowpea seed yield reductions in mixture under high fertilization and high population densities. Under low P, where maize yields were somewhat limited, the largest seed LERs were obtained. Also, only under low P did an increase in population density above the levels commonly used in monoculture increase seed LER. Therefore, it is suggested that low fertilization levels should be considered in intercropping research because it is where species complementarity and mixture advantages seem to be greater;The intercropped maize population density generally imposed a limit on cowpea seed productivity that could not be overcome by increasing cowpea population density. Conversely, the yield of intercropped maize generally was a function of its own population density and fertilization.