When the
species abundances of an ecological system are treated with a set of differential equations, it is possible to test for stability by
linearizing the system at the equilibrium point.
Robert May used this stability analysis in the 1970s which uses the
Jacobian matrix or
community matrix to investigate the relation between the diversity and stability of an ecosystem. === May stability analysis and
random matrix theory === To analyze the stability of large ecosystems, May drew on ideas from
statistical mechanics, including
Eugene Wigner's work successfully predicting the properties of
Uranium by assuming that its
Hamiltonian could be approximated as a
random matrix, leading to properties that were independent of the system's exact interactions. May considered an ecosystem with S species with abundances N_1,\ldots,N_S whose dynamics are governed by the couples system of ordinary differential equations,\frac{\mathrm dN_i}{\mathrm d t} = g_i(N_1,\ldots,N_S),\qquad i = 1,\ldots,S. Assuming the system had a fixed point, N_1^\star,\ldots, N_S^\star, May linearized dynamics as,\frac{\mathrm d N_i}{\mathrm d t} = \sum_{j=1}^S J_{ij}(N_j - N_j^\star) ,\qquad i =1 ,\ldots,S.The fixed point will be
linearly stable if all the
eigenvalues of the
Jacobian, J_{ij}, are positive. The matrix J is also known as the
community matrix. May supposed that the Jacobian was a random matrix whose off-diagonal entries J_{ij}\;(i\neq j) are all drawn as random variates from a
probability distribution and whose diagonal elements J_{ii} are all -1 so that each species inhibits its own growth and stability is guaranteed in the absence of inter-species interactions. According to
Girko's circular law, when S\gg 1, the eigenvalues of J are distributed in the complex plane uniformly in a circle whose radius is \sqrt{S}\sigma and whose center is -1, where \sigma is the standard deviation of the distribution for the off-diagonal elements of the Jacobian. Using this result, the eigenvalue with the largest real part contained in the support of the spectrum of J is -1+\sqrt{S}\sigma. Therefore, the system will lose stability when,\sqrt{S} > \frac{1}{\sigma}. This result is known as the May stability criterion. It implies that dynamical stability is limited by
diversity, and the strictness of this tradeoff is related to the magnitude of fluctuations in interactions. Recent work has extended the approaches of May to construct
phase diagrams for ecological models, like the
generalized Lotka–Volterra model or
consumer-resource models, with large complex communities with
disordered interactions. This work has relied on uses and extensions of
random matrix theory, the
cavity method, the
replica formalism, and other methods inspired by
spin-glass physics. == Types ==