The use of duality concepts has now become widespread in production and consumption theory. In both of these areas, one of the central and most useful concepts is the cost or expenditure function which represents tastes or technology through the minimum cost to consumer or producer of reaching a particular utility or output level at given prices. Under utility maximizing or cost minimizing assu...

This paper deals with a fully parabolic inter-species chemotaxis-competition system indirect signal production \begin{document}$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_{t} = \text{div}(d_{u}\nabla u+\chi u\nabla w)+\mu_{1}u(1-u-a_{1}v), &(x,t)\in \Omega\times (0,\infty), \\ &v_{t} d_{v}\Delta v+\mu_{2}v(1-v-a_{2}u), & w_{t} d_{w}\Delta w-\lambda w+\alpha v, \end...