Unrelated Species
Some species are not directly related to each other. For example, fungi actively decompose dead organic matter in an ecosystem. Carnivorous predators might not notice or ever come into contact with many of the fungi found in a particular ecosystem.
Abundant Resources
Under circumstances of abundant resources, organisms may consume the same resources but might not find it necessary to compete with each other for these resources. Over time, these species may evolve to ignore each other. For example, different species of birds might feed on the same seeds and worms alongside each other. However, if the resources get scarce, these species might begin competing with one another and become territorial.
Non-Competitive
For two different species to not coexist, they must have some way of competing with each other. Some species have not evolved this capability, but if they share the same ecosystem over a long period of time, they may develop this ability. For example, two species might develop aggressive behavior intended to scare off competitive species. One reason why biologists sometimes fear foreign species entering into an ecosystem is that native species sometimes do not have adaptations that allow them to compete with the foreign species.
Symbiotic Relationships
Some species form symbiotic relationships, meaning that the different species benefit each other in certain ways. For example, most plants rely on certain species of bacteria to fix nitrogen so that the plants are able to use the nitrogen in their biological systems.
Niche Complementarity
Instead of competing with each other for resources, some species find it easier to simply hunt for resources in different areas. This is called "niche complementarity." For example, many species of plants have different root lengths, so that the plants with the shallow roots can collect nutrients close to the surface and the plants with deeper roots can collect nutrients far below.
Competitive Coexistence
The line between coexistence and competition can blur sometimes. Species can pursue food that they are the best at catching, while leaving other prey to different species of animals. But from time to time, these species can find themselves competing for the same water, food, nutrients and sunlight.