Community models are designed for projecting species composition over time through species interactions and evaluating whether the species may coexist.
The structure of species interaction is often characterized by a matrix A = (ay), where ay characterizes the effect exerted by species y on species i. If species i is preyed upon by species y (that includes parasitism and predation), then ayy <0 and ay > 0. If species i and j are rivals, then both ay and ayi are negative. If species y is prone to self-regulation (through intraspecific competition or other mechanisms), then ayy is negative.
The strength of species interaction may vary with time, but sign remains the same. The qualitative structure of interactions sometimes may tell us about the community behavior at large. For example, the 'predator-prey' system where either predator or prey (or both) are prone to self-regulation is always stable.
The qualitative structure of interaction is visualized by a qualitative community matrix
Thus, Q denotes the class of community matrices corresponding to a given qualitative structure of interactions. If eigenvalues of all the matrices from the class have negative real parts, then the community is said to be qualitatively stable. The necessary conditions of qualitative stability include
Taking into account these conditions we may easily detect the communities which are not stable at large. For example, the system of two competitors
is not qualitatively stable for a 2 i >0.
The sufficient conditions of qualitative stability are more complicated and therefore it is not an easy task to prove a qualitative stability of a community which satisfies the necessary conditions mentioned above.
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