Nonlocal competition and logistic growth: patterns, defects and fronts
  Logistic growth of diffusing reactants on spatial domains with long range competition is studied. The bifurcations cascade involved in the transition from the homogenous state to a spatially modulated stable solution is presented, and a distinction is made between a modulated phase, dominated by single or few wavenumbers, and the spiky phase, where localized colonies are separated by depleted region. The characteristic defects in the periodic structure are presented for each phase, together with the invasion dynamics in case of local initiation. It is shown that the basic length scale that controls the bifurcation is the width of the Fisher front, and that the total population grows as this width decreases. A mix of analytic results and extensive numerical simulations yields a comprehensive examination of the possible phases for logistic growth in the presence of nonlocal competition. 
Yosef E. Maruvka and Nadav M. Shnerb