KU Mathematical Modeling Laboratory   

Fish Schooling

Credit: Atsushi Yagi, Osaka University, Japan

Swarming is one of commonly observed phenomena. This remarkable phenomenon has attracted interest of researchers from diverse fields including biology, physics, computer engineering.
  1. We have constructed a system of stochastic differential equations for the process of fish schooling on the basis of four local rules: attraction, repulsion, alignment, and reaction to environment.

  2. Four obstacle avoiding patterns

  3. We have studied a geometrical structure for the model including school diameter, connectedness and graph. It is shown that when the effect of noise on the system exceeds a threshold, fish can no longer from a school.

  4. Geometrical structure of fish school

  5. We have constructed a model for describing the process of fish school's obstacle avoidance. Four clear obstacle avoiding patterns have been found. Particularly, we presented a new scientific definition for fish school's cohesiveness.

  6. Four  obstacle avoiding patterns

  7. We have constructed a model for collective animal foraging in noisy environment with obstacles. It is observed that when swarm size increases, so does the probability of foraging success. On the other hand when the size surpasses an optimal value, the probability decrease. The observation then may be explained by the cohesiveness of swarms.

  8. Obstacles and odour

    Collective animal foraging

Forest Ecosystem

Conservation of forest resources is one of the most challenging problems in ecology and environmental science. For maintaining the ecological integrity of forest ecosystem and for preserving biodiversity, knowing forest dynamics is of very importance. The fundamental issue is therefore to predict the variation of tree density caused by random factors. We have constructed a stochastic forest model of young and old age class trees. The model is performed by stochastic differential equations. The following problems are then discussed.
  1. Existence, uniqueness and boundedness of global nonnegative solutions.
  2. Conditions for sustainability of the forest as well as existence of a Borel invariant measure.
  3. Decline of the forest. When the intensity of noise on the forest is large enough, then both young and old age trees decay.
  4. Numerical examples.

Animal Coat Patterns

Animal Coat Patterns
We know that every species of animals has its proper type of coat patterns but on the other hand in details each individual of a species has its own coat pattern. This uniformity and diversity of different biological level is mysterious and attracts interest of many researchers.

We have constructed a dynamical system for a reaction diffusion system due to Murray, which relies on the use of the Thomas system nonlinearities and describes the formation of animal coat patterns. The following problems are then discussed by using semigroup methods.
  1. Existence and uniqueness of global positive strong solutions to the system.
  2. Continuous dependence on initial values.
  3. Exponential attractors whose fractal dimensions can be estimated.

Stochastic Evolution Equations

Stochastic evolution equations (SEEs) can be used to describe many phenomena in the real world.
  1. We have studied mild solutions to linear and semilinear SEEs. Existence of unique solutions, maximal regularity of solution and regular dependence on initial data are discussed.
  2. We have presented our definition for strict solutions and studied existence and maximal regularity of solutions to both autonomous and non-autonomous linear SEEs.

We are now interested in rough evolution equations. The rough path theory for differential equations is first introduced by Lyons in 1998. It has then attracted the interests of many researchers all over the world. Among others, Gubinelli-Tindel generalized the theory in order to give a pathwise meaning to some nonlinear infinite-dimensional EEs driven by an irregular noise}, for example, a Wiener process. Hairer developed the theory of Lyons to construct a robust solution theory for the KPZ equation whose universality class includes many models in the field of interacting particle systems. He then introduced a generalization which is known as the theory of regularity structures.

We aim to build a theory of regular structures for non-autonomous and quasi-linear rough evolution equations.