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Most converged heterogeneous networks have complex wired mesh backbones with gateway nodes having limited resources. As the number of sessions increases, competition among the sessions at some of the gateway nodes may also increase. Even with the presence of admission control techniques, once the resources at these gateways and their surrounding nodes are exhausted, they can turn into bottlenecks that disrupt the entire heterogeneous network. Therefore, in this paper, we analyze a system maintains stability over a period of time in a resource diminishing environment, provided the resources are fairly allocated at the time of admission. The methodology we propose for the analysis is ecologically inspired. First, we define the converged heterogeneous network as an ecosystem of diverse networks with limited resource constraints. Next, resource competition theory from the field of ecology is applied for identifying and studying both winners and losers of such competition. Results indicate that the phenomenon of competitive exclusion takes place between competing sessions. Consequently, we propose a novel way to enable coexistence between the competitors to achieve a win-win situation. For this purpose a control policy is derived to be incorporated in admission control techniques in a competitive ecosystem of heterogeneous networks. Results suggest that a fair coexistence among sessions can be achieved by this approach.