  We uncover the gradient structure to investigate the convergence of solutions
in nonlocal nonlinear dynamical systems. Mainly but not exclusively, we use the
Lojasiewicz inequality to prove convergence results in various spaces with
continuous, or discrete temporal domain, and finite, or infinite dimensional
spatial domain. To be more specific, we analyze Lotka-Volterra type dynamics
and concentration-dispersion dynamics. Lotka-Volterra equations describe the
population dynamics of a group of species. Under the assumption that the
interaction between species is symmetric, we present two different methods to
derive the convergence result. One, the entropy trapping method, is to adapt
the idea of Akin and Hofbauer (Math. Biosci. 61 (1982) 51-62) of using
monotonicity of the energy to bound the entropy, which provides the proximal
distance of the solution from the desired equilibrium. Another method, inspired
by Jabin and Liu's observation in (Nonlinearity 30 (2017) 4220) is to change
variables to resolve the singular nature of gradient structure. We apply this
idea to show the convergence result in generalized Lokta-Volterra systems, such
as regularized Lotka-Volterra systems and infinite dimensional Lotka-Volterra
equations with mutation. Concentration-dispersion dynamics is a new type of
equation that is inspired by fixed point formulations for solitary wave shapes.
As a continuous time analogue of Petviashvili iteration, we aim to dynamically
calculate the nontrivial solitary wave profile. Using the gradient structure we
deduce the existence of nontrivial solitary and periodic wave profiles.
Unfortunately, the convergence result remains open. Instead, we regularize the
concentration dispersion equations by adding a nonlinear diffusive term, and
prove the convergence of the solutions by using the Lojasiewicz convergence
framework.
