The importance of transient dynamics in ecological systems and in the models that describe them has become increasingly recognized. However, previous work has typically treated each instance of these dynamics separately. We review both empirical examples and model systems, and outline a classification of transient dynamics based on ideas and concepts from dynamical systems theory. This classification provides ways to understand the likelihood of transients for particular systems, and to guide investigations to determine the timing of sudden switches in dynamics and other characteristics of transients. Implications for both management and underlying ecological theories emerge.

The second derivatives of prepotential with respect to Whitham time-variables in the Seiberg-Witten theory are expressed in terms of Riemann theta-functions. These formulas give a direct transcendental generalization of algebraic ones for the Kontsevich matrix model. In particular case they provide an explicit derivation of the renormalization group (RG) equation proposed recently in papers on the Donaldson theory.

A 1-matrix model is proposed, which nicely interpolates between doublescaling continuum limits of all multimatrix models. The interpolating partition function is always a KP τ -function and always obeys L −1 -constraint and string equation. Therefore this model can be considered as a natural unification of all models of 2d-gravity (string models) with c ≤ 1.

We introduce a new 1-matrix model with arbitrary potential and the matrix-valued background field. Its partition function is a τ -function of KPhierarchy, subjected to a kind of L −1 -constraint. Moreover, partition function behaves smoothly in the limit of infinitely large matrices. If the potential is equal to X K+1 , this partition function becomes a τ -function of K-reduced KP-hierarchy, obeying a set of W K -algebra constraints identical to those conjectured in [1] for double-scaling continuum limit of (K − 1)-matrix model. In the case of K = 2 the statement reduces to the early established [2] relation between Kontsevich model and the ordinary 2d quantum gravity . Kontsevich model with generic potential may be considered as interpolation between all the models of 2d quantum gravity with c < 1 preserving the property of integrability and the analogue of string equation.

We represent the partition function of the Generalized Kontsevich Model (GKM) in the form of a Toda lattice τ -function and discuss various implications of non-vanishing "negative"-and "zero"-time variables: the appear to modify the original GKM action by negative-power and logarithmic contributions respectively. It is shown that so deformed τ -function satisfies the same string equation as the original one. In the case of quadratic potential GKM turns out to describe forced Toda chain hierarchy and, thus, corresponds to a discrete matrix model, with the role of the matrix size played by the zero-time (at integer positive points). This relation allows one to discuss the doublescaling continuum limit entirely in terms of GKM, i.e. essentially in terms of finite-fold integrals.

The prepotential F (ai), defining the low-energy effective action of the SU (N ) N = 2 SUSY gluodynamics, satisfies an enlarged set of the WDVV-like equations FiF −1 k Fj = Fj F −1 k Fi for any triple i, j, k = 1, . . . , N −1, where matrix Fi is equal to (Fi)mn = ∂ 3 F ∂a i ∂am∂an . The same equations are actually true for generic topological theories. In contrast to the conventional formulation, when k is restricted to k = 0, in the proposed system there is no distinguished "first" time-variable, and the indices can be raised with the help of any "metric" η (k) mn = (F k )mn, not obligatory flat. All the equations (for all i, j, k) are true simultaneously. This result provides a new parallel between the Seiberg-Witten theory of low-energy gauge models in 4d and topological theories.

The Seiberg-Witten prepotentials for N=2 SUSY gauge theories with N_f<2N_c fundamental multiplets are obtained from conformal N_f=2N_c theory by decoupling 2N_c-N_f multiplets of heavy matter. This procedure can be lifted to the level of Nekrasov functions with arbitrary background parameters epsilon_1 and epsilon_2. The AGT relations imply that similar limit exists for conformal blocks (or, for generic N_c>2, for the blocks in conformal theories with W_{N_c} chiral algebra). We consider the limit of the four-point function explicitly in the Virasoro case of N_c=2, by bringing the dimensions of external states to infinity. The calculation is performed entirely in terms of representation theory for the Virasoro algebra and reproduces the answers conjectured in arXiv:0908.0307 with the help of the brane-compactification analysis and computer simulations. In this limit, the conformal block involving four external primaries, corresponding to the theory with vanishing beta-function, turns either into a 2-point or 3-point function, with certain coherent rather than primary external states.Comment: 7 page

We suggested in 2009 that the Nekrasov function with one non-vanishing deformation parameter is obtained by the standard Seiberg-Witten (SW) contour-integral construction. The only difference is that the SW differential pdx is substituted by its quantized version for the corresponding integrable system, and contour integrals become exact monodromies of the wavefunction. This provides an explicit formulation of the earlier guess by Nekrasov and Shatashvili in 2009. In this paper, we successfully check this suggestion in the first order in 2 and the first order in instanton expansion for the SU (N) model, where the consistency of the so-deformed SW equations is already non-trivial.