Introduction to the Modeling and Analysis of Complex Systems

442 CHAPTER 19. AGENT-BASED MODELS3. Describe the rules for how the environment behaves on its own. The inherentdynamics of the environment in this model are the diffusion and spontaneous decay ofthe cAMP concentration. These can be modeled by using the discrete version of theLaplacian operator, as well as an exponential decay factor, applied to the environmentalstates everywhere in the space in every iteration. This is no different from what we did forthe numerical simulations of PDEs. We can implement them in the code as follows:Code 19.10:k = 1 # rate of cAMP decayDc = 0.001 # diffusion constant of cAMPDh = 0.01 # spatial resolution for cAMP simulationDt = 0.01 # time resolution for cAMP simulationdef update():global agents, env, nextenv# simulating diffusion and evaporation of cAMPfor x in xrange(w):for y in xrange(w):C, R, L, U, D = env[x,y], env[(x+1)%w,y], env[(x-1)%w,y], \env[x,(y+1)%w], env[x,(y-1)%w]lap = (R + L + U + D - 4 * C)/(Dh**2)nextenv[x,y] = env[x,y] + (- k * C + Dc * lap) * Dtenv, nextenv = nextenv, envHere we adopt periodic boundary conditions for simplicity.4. Describe the rules for how agents interact with the environment. In this model,agents interact with the environment in two different ways. One way is the secretion ofcAMP by the agents, which can be implemented by letting each agent increase the cAMPconcentration in a discrete cell where it is located. To do this, we can add the following tothe update function:Code 19.11:f = 1 # rate of cAMP secretion by an agent# simulating secretion of cAMP by agentsfor ag in agents:env[ag.x, ag.y] += f * Dt