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| #!/usr/bin/env python3 | |
| # -*- coding: utf-8 -*- | |
| """ | |
| trirep.py | |
| Created on Thu May 9 16:23:30 2019 | |
| @author: nolte | |
| Introduction to Modern Dynamics, 2nd edition (Oxford University Press, 2019) | |
| Population dynamics | |
| """ | |
| import numpy as np | |
| from scipy import integrate | |
| from matplotlib import pyplot as plt | |
| plt.close('all') | |
| def tripartite(x,y,z): | |
| sm = x + y + z | |
| xp = x/sm | |
| yp = y/sm | |
| f = np.sqrt(3)/2 | |
| y0 = f*xp | |
| x0 = -0.5*xp - yp + 1; | |
| plt.figure(2) | |
| lines = plt.plot(x0,y0) | |
| plt.setp(lines, linewidth=0.5) | |
| plt.plot([0, 1],[0, 0],'k',linewidth=1) | |
| plt.plot([0, 0.5],[0, f],'k',linewidth=1) | |
| plt.plot([1, 0.5],[0, f],'k',linewidth=1) | |
| plt.show() | |
| def solve_flow(y,tspan): | |
| def flow_deriv(y, t0): | |
| #"""Compute the time-derivative .""" | |
| f = np.zeros(shape=(N,)) | |
| for iloop in range(N): | |
| ftemp = 0 | |
| for jloop in range(N): | |
| ftemp = ftemp + A[iloop,jloop]*y[jloop] | |
| f[iloop] = ftemp | |
| phitemp = phi0 # Can adjust this from 0 to 1 to stabilize (but Nth population is no longer independent) | |
| for loop in range(N): | |
| phitemp = phitemp + f[loop]*y[loop] | |
| phi = phitemp | |
| yd = np.zeros(shape=(N,)) | |
| for loop in range(N-1): | |
| yd[loop] = y[loop]*(f[loop] - phi); | |
| if np.abs(phi0) < 0.01: # average fitness maintained at zero | |
| yd[N-1] = y[N-1]*(f[N-1]-phi); | |
| else: # non-zero average fitness | |
| ydtemp = 0 | |
| for loop in range(N-1): | |
| ydtemp = ydtemp - yd[loop] | |
| yd[N-1] = ydtemp | |
| return yd | |
| # Solve for the trajectories | |
| t = np.linspace(0, tspan, 701) | |
| x_t = integrate.odeint(flow_deriv,y,t) | |
| return t, x_t | |
| # model_case 1 = zero diagonal | |
| # model_case 2 = zero trace | |
| # model_case 3 = asymmetric (zero trace) | |
| print(' ') | |
| print('trirep.py') | |
| print('Case: 1 = antisymm zero diagonal') | |
| print('Case: 2 = antisymm zero trace') | |
| print('Case: 3 = random') | |
| model_case = int(input('Enter the Model Case (1-3)')) | |
| N = 3 | |
| asymm = 3 # 1 = zero diag (replicator eqn) 2 = zero trace (autocatylitic model) 3 = random (but zero trace) | |
| phi0 = 0.001 # average fitness (positive number) damps oscillations | |
| T = 100; | |
| if model_case == 1: | |
| Atemp = np.zeros(shape=(N,N)) | |
| for yloop in range(N): | |
| for xloop in range(yloop+1,N): | |
| Atemp[yloop,xloop] = 2*(0.5 - np.random.random(1)) | |
| Atemp[xloop,yloop] = -Atemp[yloop,xloop] | |
| if model_case == 2: | |
| Atemp = np.zeros(shape=(N,N)) | |
| for yloop in range(N): | |
| for xloop in range(yloop+1,N): | |
| Atemp[yloop,xloop] = 2*(0.5 - np.random.random(1)) | |
| Atemp[xloop,yloop] = -Atemp[yloop,xloop] | |
| Atemp[yloop,yloop] = 2*(0.5 - np.random.random(1)) | |
| tr = np.trace(Atemp) | |
| A = Atemp | |
| for yloop in range(N): | |
| A[yloop,yloop] = Atemp[yloop,yloop] - tr/N | |
| else: | |
| Atemp = np.zeros(shape=(N,N)) | |
| for yloop in range(N): | |
| for xloop in range(N): | |
| Atemp[yloop,xloop] = 2*(0.5 - np.random.random(1)) | |
| tr = np.trace(Atemp) | |
| A = Atemp | |
| for yloop in range(N): | |
| A[yloop,yloop] = Atemp[yloop,yloop] - tr/N | |
| plt.figure(3) | |
| im = plt.matshow(A,3,cmap=plt.cm.get_cmap('seismic')) # hsv, seismic, bwr | |
| cbar = im.figure.colorbar(im) | |
| M = 20 | |
| delt = 1/M | |
| ep = 0.01; | |
| tempx = np.zeros(shape = (3,)) | |
| for xloop in range(M): | |
| tempx[0] = delt*(xloop)+ep; | |
| for yloop in range(M-xloop): | |
| tempx[1] = delt*yloop+ep | |
| tempx[2] = 1 - tempx[0] - tempx[1] | |
| x0 = tempx/np.sum(tempx); # initial populations | |
| tspan = 70 | |
| t, x_t = solve_flow(x0,tspan) | |
| y1 = x_t[:,0] | |
| y2 = x_t[:,1] | |
| y3 = x_t[:,2] | |
| plt.figure(1) | |
| lines = plt.plot(t,y1,t,y2,t,y3) | |
| plt.setp(lines, linewidth=0.5) | |
| plt.show() | |
| plt.ylabel('X Position') | |
| plt.xlabel('Time') | |
| tripartite(y1,y2,y3) |