What will happen when two or more species interact, like predators and their prey?
How can oscillatory patterns emerge?
How can external stimuli trigger collective behavior within a population of independent individuals?
Understanding pattern formation requires tools from analysis. We introduce dynamical systems to model changes in time and partial differential equations to model distributions in physical or feature spaces. The combination of the two in reaction-diffusion systems leads to mathematical models like the Turing mechanism that can generate surprisingly rich patterns. Another example we treat is chemotaxis where organisms can be induced to collective behavior by following gradients of chemical substances.
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