Convergence problems in nonlocal dynamics with nonlinearity

We study nonlocal nonlinear dynamical systems and uncover the gradient structure to investigate the convergence of solutions. 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, in which individuals interact either competitively or cooperatively with each other. It is well-known that Lotka-Volterra equations form a gradient system with respect to the Shahshahani metric. The Shahshahani metric, unfortunately, becomes singular in the scenario where some species become extinct. This singular nature of the Shahshahani metric is an obstacle to the usual convergence analysis. 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 nonlocal semi-linear heat equations, which can be seen as an infinite dimensional Lotka-Volterra equations with mutation.

Concentration-dispersion dynamics is a new type of equation that is inspired by fifixed point formulations for solitary wave shapes. The equations are designed in a way that the solution evolves to match the shape of a concentrated and dispersed version of the solution, which is an outcome of power nonlinearity and convolution. As a continuous time analogue of Petviashvili iteration, we aim to dynamically calculate the nontrivial solitary wave profile. We show the well-posedness and compactness of the solutions. Moreover, we suggest specific initial data which stay away from the trivial equilibrium. 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. In this way, we can dynamically approximate nontrivial wave profiles with arbitrarily small error.