A community is said to be stable if every eigenvalue of its community matrix has negative real part. This stems from the fact that the community matrix reveals the immediate response of the community to a perturbation:
where x is the vector of deviations caused by the perturbation. The solution of the linear system of ordinary differential equations eventually converges to zero if every eigenvalue of A has negative real part.
Succession models are designed for projecting ecosystem coverage of a region over time through successions and disturbances.
The state of ecosystem coverage is described by ecosystem-distribution vector x, in which elements x(t) denote the fraction of the region occupied with ith ecosystem at time t and n
The ecosystem distribution vector at time t + 1 is projected by using transition matrix x(t + 1) = Px(t)
where pj denotes the probability that ecosystem i will replace ecosystem j within the projection period:
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