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--- 2016:groups:g1:start [2016/01/07 16:59] – group1
+++ 2016:groups:g1:start [2024/01/09 18:45] (current) – external edit 127.0.0.1
@@ Line 2: / Line 2: @@
 ====== Less is more: when positives don't sum ======
+==== Plot 1 ====
+%matplotlib inline
+from numpy import *
+from matplotlib.pyplot import *
+from scipy.integrate import odeint
+ion()
+t = arange(0,2., 0.01)
+# parameters
+fr = 8.75
+fa = 6.25
+p = 0.7
+c = 0.01
+g = 0.001
+# initial condition: this is an array now!
+x0 = array([10., 10.]) # Pr, Pa
+# the function still receives only `x`, but it will be an array, not a number
+def LV(x, t, fr, fa, p, c, g):
+    # in python, arrays are numbered from 0, so the first element
+    # is x[0], the second is x[1]. The square brackets `[ ]` define a
+    # list, that is converted to an array using the function `array()`.
+    # Notice that the first entry corresponds to dV/dt and the second to dP/dt
+    return array([ (fr*p*x[0]) + fa*p*x[1] - ((c-g)*x[0]*(x[0]+x[1])),
+                   (fa*(1-p)*x[1]) + fr*(1-p)*x[0] - ((c-3*g)*x[1]*(x[0]+x[1])) ])
+# call the function that performs the integration
+# the order of the arguments is as below: the derivative function,
+# the initial condition, the points where we want the solution, and
+# a list of parameters
+x = odeint(LV, x0, t, (fr, fa, p, c, g))
+# plot the solution
+plot(t, x)
+xlabel('$t [years^{-1}]$', fontsize=25) # define label of x-axis
+ylabel('$Plants Population$', fontsize=25) # and of y-axis
+legend(['$P_{r}$', '$P_{a}$'], loc='lower right', fontsize=20)
+savefig('foo1.pdf')
+==== Portuguesa ====
+t = arange(0,3., 0.01)
+rFec = t = arange(0,10.,1)
+rGam = arange(0,100.,10)
+for i in range(len(rFec)):
+    for j in range(len(rGam)):
+        x = odeint(LV, x0, t, (fr, fa, p, c, g, i, j))
+        A[i,j] = x[5,1]/(x[5,1] + x[5,0])
 Wiki site of the practical exercise of the [[http://www.ictp-saifr.org/mathbio5|V Southern-Summer School on Mathematical Biology]].
@@ Line 32: / Line 82: @@
   *Godschalx //et al.// (2015) //Ants are less attracted to the extrafloral nectar of plants with symbiotic, nitrogen-fixing rhizobia// Ecology 96(2), pp.348-354 [[http://www.esajournals.org/doi/abs/10.1890/14-1178.1|link]]
+=====Maria Bonita=====
+<code>
+%matplotlib inline
+from numpy import *
+from matplotlib.pyplot import *
+from scipy.integrate import odeint
+ion()
+t = arange(0,50., 0.01)
+# parameters
+rv = 0.8
+rr = 1.5
+ra = 0.6
+kv = 5.
+kr = 5.
+ka = 5.
+a = 0.15
+b = 0.03
+c = 0.15
+d = 0.02
+n = 0.1
+p = 1
+alfa = 0.25
+beta = 0.25
+# initial condition
+x0 = array([1., 1., 1.]) # V, R, A
+def LV(x, t, rv, rr, ra, kv, kr, ka, a, b, c, d, p, n, alfa, beta):
+    return array([ rv*x[0]*(1-(x[0]/kv)) + a*x[0]*x[1] - p*(1+sin(t))*x[0]/(n+alfa*x[1]+beta*x[2]),
+                   rr*x[1]*(1-(x[1]/kr)) + b*x[0]*x[1] - c*x[1]*p*(1+sin(t))*x[0]/(n+alfa*x[1]+beta*x[2]) ,
+                   ra*x[2]*(1-(x[2]/ka)) + d*x[2]*p*(1+sin(t))*x[0]/(n+alfa*x[1]+beta*x[2])])
+# call the function that performs the integration
+# the order of the arguments is as below: the derivative function,
+# the initial condition, the points where we want the solution, and
+# a list of parameters
+x = odeint(LV, x0, t, (rv, rr, ra, kv, kr, ka, a, b, c, d, p, n, alfa, beta))
+# plot the solution
+plot(t, x)
+xlabel('t') # define label of x-axis
+ylabel('populations') # and of y-axis
+legend(['V', 'R', 'A'], loc='lower right')
+</code>
+=====Right First Model=====
+<code>
+%matplotlib inline
+from numpy import *
+from matplotlib.pyplot import *
+from scipy.integrate import odeint
+ion()
+t = arange(0,3., 0.01)
+# parameters
+fr = 8.75
+fa = 6.25
+c = 0.01
+p = 0.5
+g = 0.001
+# initial condition
+x0 = array([50., 50.]) # Pr, Pa
+def LV(x, t, fr, fa, p, c, g):
+    # in python, arrays are numbered from 0, so the first element
+    # is x[0], the second is x[1]. The square brackets `[ ]` define a
+    # list, that is converted to an array using the function `array()`.
+    # Notice that the first entry corresponds to dV/dt and the second to dP/dt
+    if t > 25.:
+        g = 0.
+    p = 0.25 * sin(6*t) + 0.5
+    return array([ (fr*p*x[0]) + fa*p*x[1] - ((c-g)*x[0]*(x[0]+x[1])),
+                   (fa*(1-p)*x[1]) + fr*(1-p)*x[0] - ((c-3*g)*x[1]*(x[0]+x[1])) ])
+# call the function that performs the integration
+# the order of the arguments is as below: the derivative function,
+# the initial condition, the points where we want the solution, and
+# a list of parameters
+x = odeint(LV, x0, t, (fr, fa, p, c, g))
+# plot the solution
+plot(t, x)
+xlabel('t') # define label of x-axis
+ylabel('populations') # and of y-axis
+legend(['Pr', 'Pa'], loc='lower right')
+</code>
+=====Killing the Ants at t = 25=====
+<code>
+%matplotlib inline
+from numpy import *
+from matplotlib.pyplot import *
+from scipy.integrate import odeint
+ion()
+t = arange(0,50., 0.01)
+# parameters
+fr = 8.75
+fa = 6.25
+c = 0.01
+p = 0.5
+g = 0.001
+# initial condition
+x0 = array([50., 50.]) # Pr, Pa
+def LV(x, t, fr, fa, p, c, g):
+    # in python, arrays are numbered from 0, so the first element
+    # is x[0], the second is x[1]. The square brackets `[ ]` define a
+    # list, that is converted to an array using the function `array()`.
+    # Notice that the first entry corresponds to dV/dt and the second to dP/dt
+    if t > 25.:
+        g = 0.
+    p = 0.25 * sin(t/5) + 0.5
+    return array([ (fr*p*x[0]) + fa*p*x[1] - ((c-g)*x[0]*(x[0]+x[1])),
+                   (fa*(1-p)*x[1]) + fr*(1-p)*x[0] - ((c-3*g)*x[1]*(x[0]+x[1])) ])
+# call the function that performs the integration
+# the order of the arguments is as below: the derivative function,
+# the initial condition, the points where we want the solution, and
+# a list of parameters
+x = odeint(LV, x0, t, (fr, fa, p, c, g))
+# plot the solution
+plot(t, x)
+xlabel('t') # define label of x-axis
+ylabel('populations') # and of y-axis
+legend(['Pr', 'Pa'], loc='lower right')
+</code>
+=====El Pato  =====
+<code>
+%matplotlib inline
+from numpy import *
+from matplotlib.pyplot import *
+from scipy.integrate import odeint
+ion()
+t = arange(0,3., 0.01)
+# parameters
+fr = 8.75
+fa = 6.25
+p = 0.7
+c = 0.01
+g = 0.001
+# initial condition: this is an array now!
+x0 = array([10., 10.]) # Pr, Pa
+# the function still receives only `x`, but it will be an array, not a number
+def LV(x, t, fr, fa, p, c, g):
+    # in python, arrays are numbered from 0, so the first element
+    # is x[0], the second is x[1]. The square brackets `[ ]` define a
+    # list, that is converted to an array using the function `array()`.
+    # Notice that the first entry corresponds to dV/dt and the second to dP/dt
+    return array([ (fr*p*x[0]) + fa*p*x[1] - ((c-g)*x[0]*(x[0]+x[1])),
+                   (fa*(1-p)*x[1]) + fr*(1-p)*x[0] - ((c-3*g)*x[1]*(x[0]+x[1])) ])
+# call the function that performs the integration
+# the order of the arguments is as below: the derivative function,
+# the initial condition, the points where we want the solution, and
+# a list of parameters
+x = odeint(LV, x0, t, (fr, fa, p, c, g))
+# plot the solution
+plot(t, x)
+xlabel('t') # define label of x-axis
+ylabel('populations') # and of y-axis
+legend(['Pr', 'Pa'], loc='lower right')
+</code>
+)
+===== Patolihno =====
+%matplotlib inline
+from numpy import *
+from matplotlib.pyplot import *
+from scipy.integrate import odeint
+ion()
+t = arange(0,50., 0.01)
+# parameters
+rv = 0.8
+rr = 1.5
+ra = 0.6
+rp = 0.2
+kv = 5.
+kr = 5.
+ka = 5.
+kp = 4.
+a = 1.5
+b = 3.
+c = 1.5
+d = 2.
+e = 2.
+n = 0.6
+alfa = 2.5
+beta = 2.5
+# initial condition
+x0 = array([10., 40., 8.,8.]) # V, R, A, P
+def LV(x, t, rv, rr, ra, rp, kv, kr, ka, kp, a, b, c, d, e, n, alfa, beta):
+    return array([ rv*x[0]*(1-(x[0]/kv)) + a*x[0]*x[1] - x[3]*x[0]/(n+alfa*x[1]+beta*x[2]),
+                   rr*x[1]*(1-(x[1]/kr)) + b*x[0]*x[1] - c*x[1]*x[3]*x[0]/(n+alfa*x[1]+beta*x[2]) ,
+                   ra*x[2]*(1-(x[2]/ka)) + d*x[2]*x[3]*x[0]/(n+alfa*x[1]+beta*x[2])  ,
+                   rp*x[3]*(1-(x[3]/kp)) - e*x[3]*x[2] + x[3]*x[0]/(n+alfa*x[1]+beta*x[2])])
+# call the function that performs the integration
+# the order of the arguments is as below: the derivative function,
+# the initial condition, the points where we want the solution, and
+# a list of parameters
+x = odeint(LV, x0, t, (rv, rr, ra, rp, kv, kr, ka, kp, a, b, c, d, e, n, alfa, beta))
+# plot the solution
+plot(t, x)
+xlabel('t') # define label of x-axis
+ylabel('populations') # and of y-axis
+legend(['V', 'R', 'A', 'P'], loc='upper left')
 ===== Code =====
 <code>
+%matplotlib inline
 from numpy import *
 from matplotlib.pyplot import *
 from scipy.integrate import odeint
+ion()
-t = arange(0, 50., 0.01)
+t = arange(0,10., 0.01)
 # parameters
-r = 2.
+fr = 1.5
-c = 0.5
+fa = 2.
-e = 0.1
+p = 0.6
-d = 1.
+c = 0.2
+g = 15.
 # initial condition: this is an array now!
-x0 = array([1., 3.])
+x0 = array([50., 50.]) # Pr, Pa
 # the function still receives only `x`, but it will be an array, not a number
-def LV(x, t, r, c, e, d):
+def LV(x, t, fr, fa, p, c, g):
-    return array([ r*x[0] - c * x[0] * x[1],
+    # in python, arrays are numbered from 0, so the first element
-                   e * c * x[0] * x[1] - d * x[1] ])
+    # is x[0], the second is x[1]. The square brackets `[ ]` define a
+    # list, that is converted to an array using the function `array()`.
+    # Notice that the first entry corresponds to dV/dt and the second to dP/dt
+    return array([ (fr*p*x[0]) + fa*p*x[1] - ((c/g)*x[0]*(x[0]+x[1])),
+                   (fa*(1-p)*x[1]) + fr*(1-p)*x[0] - ((c/(3*g))*x[1]*(x[0]+x[1])) ])
-x = odeint(LV, x0, t, (r, c, e, d))
+# call the function that performs the integration
+# the order of the arguments is as below: the derivative function,
+# the initial condition, the points where we want the solution, and
+# a list of parameters
+x = odeint(LV, x0, t, (fr, fa, p, c, g))
 # plot the solution
@@ Line 62: / Line 351: @@
 xlabel('t') # define label of x-axis
 ylabel('populations') # and of y-axis
-legend(['V', 'P'], loc='upper right')
+legend(['Pr', 'Pa'], loc='lower right')
 </code>