Table of Contents
Group 4
A dangerous orchid
Wiki site of the practical exercise of the V Southern-Summer School on Mathematical Biology.
Here you will find the exercise assignment and the group's products.
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Introduction
Mimicry is an adaptation in which a species, the mimic, has a morphological and/or behavioural resemblance to another species, the model. There are two main types of mimicry: defensive mimicry, in which the mimic is avoided by predators by resembling an unpalatable/nasty model; and aggressive mimicry, in which the mimic resembles a harmless/attractive model in order to attract prey. O'Hanlon et al. (2014) described a unique type of aggressive mimicry: the orchid mantis (Hymenopus coronatus) mimics the flowers of some species in order to attract insect pollinators as prey. A surprising result found by the authors is that live mantises attract more pollinators than the live flowers themselves (Figure 3 of their paper).
Assignment
Propose a mathematical model that portrays the dynamics of the system composed by the orchid mantis, the flowers they mimic and the pollinators that serve as prey. The model should be simple but biologically realistic. Also, your model should allow to explore relevant questions about the consequences of this dynamic.
Proposed Questions
- O'Hanlon et al. show that the orchid mantis uses mimicry and not camouflage to capture pollinators. What are the differences between those two dynamics?
- The authors make no conclusions about which plant species is being mimicked. How would the dynamics of this system change if there were more than one plant species serving as model?
- Is it possible for orchid mantis to maintain higher population densities than their flower model? If so, what mechanisms would allow this surprising outcome?
Proposed Model
Our model is in this model.pdf pdf file.
For Testing the Model
If you are in windows download you can just download unzip and run this file:
For testing our model during the presentation you can copy and paste this python source code and run it for an interactive model simulator. If it doesn't work for you follow this instructions:
If you already have python:
- Run in a console (this will download the packages needed and automatically install them):
pip install PyQt4 pyqtgraph
Otherwise follow the instructions on the tutorials to install python. And then you should be able to run the line above. If the line above doesn't work, maybe you dont have pip.
To install pip (python package manager):
- In Ubuntu/Linux run:
sudo apt-get install python-pip
- In another operating system you should find how to install pip and then follow the instructions above.
- model.py
#!/bin/python from scipy.integrate import odeint import numpy as np from PyQt4.QtCore import * from PyQt4.QtGui import * from pyqtgraph.widgets.PlotWidget import PlotWidget from pyqtgraph.graphicsItems.InfiniteLine import InfiniteLine from pyqtgraph.graphicsItems.ScatterPlotItem import ScatterPlotItem from pyqtgraph import mkPen, mkBrush, setConfigOption from pyqtgraph.parametertree import ParameterTree, Parameter import pyqtgraph as pg from pylab import plot, show, xlabel, ylabel, quiver t_init, t_step, t_end = 0, 0.1, 10000 t = t_init t_values = np.arange(t_init, t_end, t_step) class MyWidget(QWidget): def __init__(self): super(MyWidget, self).__init__() self.parameter_tree = ParameterTree() self.parameter_tree.setMaximumWidth(300) self.k1, self.k2, self.k3, self.k4 = 0.4, 0.6, 0.1, 0.005 self.alpha, self.beta = 0.01, 0.1 self.Kf, self.Kp = 0, 0 self.d1, self.d2, self.d3 = 0.3, 0.2, 0.044 self.F, self.P, self.M = 100, 50, 2 self.parameter = Parameter(name='Parameters', children=[ { 'name':'Initial Conditions', 'type':'group', 'children': [{ 'name':'F', 'type':'float', 'value':self.F }, { 'name':'P', 'type':'float', 'value':self.P }, { 'name':'M', 'type':'float', 'value':self.M }] }, { 'name':'K', 'type':'group', 'children': [{ 'name':'k1', 'type':'float', 'value':self.k1 }, { 'name':'k2', 'type':'float', 'value':self.k2 }, { 'name':'k3', 'type':'float', 'value':self.k3 }, { 'name':'k4', 'type':'float', 'value':self.k4 }] }, { 'name':'D', 'type':'group', 'children': [{ 'name':'d1', 'type':'float', 'value':self.d1 }, { 'name':'d2', 'type':'float', 'value':self.d2 }, { 'name':'d3', 'type':'float', 'value':self.d3 }] }, { 'name':'Other paramters', 'type':'group', 'children': [{ 'name':'alpha', 'type':'float', 'value':self.alpha }, { 'name':'beta', 'type':'float', 'value':self.beta }, { 'name':'Kf', 'type':'float', 'value':self.Kf }, { 'name':'Kp', 'type':'float', 'value':self.Kp }, ] }] ) self.parameter_tree.setParameters(self.parameter) self.plot_widget = PlotWidget() self.plt = self.plot_widget.getPlotItem() l = self.plt.addLegend(size=(100, 100), offset=(500, 30)) self.plt.showGrid(True, True) res = self.solve() self.f_plt = self.plt.plot(t_values, res[0], name='Flowers', antialias=True) self.f_plt.setPen(mkPen('#EE02FF', width=3)) self.p_plt = self.plt.plot(t_values, res[1], name='Pollinators', antialias=True) self.p_plt.setPen(mkPen('#FFD800', width=3)) self.m_plt = self.plt.plot(t_values, res[2], name='Mantis', antialias=True) self.m_plt.setPen(mkPen('#FF0000', width=3)) print('Flowers {0}\nPollinators {1}\nMantis {2}'.format(res[0][-1], res[1][-1], res[2][-1])) self.fp_plt = PlotWidget() fp_plt = self.fp_plt.getPlotItem() fp_plt.showGrid(True, True) l = fp_plt.addLegend(size=(100, 100), offset=(500, 30)) self.fp_f_plt = fp_plt.plot([], [], name='Flowers', antialias=True) # , symbol='+') self.fp_f_plt.setPen(mkPen('#EE02FF', width=3)) self.fp_p_plt = fp_plt.plot([], [], name='Pollinators', antialias=True) # , symbol='s') self.fp_p_plt.setPen(mkPen('#FFD800', width=3)) self.fp_m_plt = fp_plt.plot([], [], name='Mantis', antialias=True) # , symbol='t') self.fp_m_plt.setPen(mkPen('#FF0000', width=3)) wid = QWidget() l2 = QVBoxLayout() self.min_value = QDoubleSpinBox() self.max_value = QDoubleSpinBox() self.parameter_name = QLineEdit('d3') run_button = QPushButton('Run') l3 = QHBoxLayout() l3.addWidget(QLabel('Parameter: ')) l3.addWidget(self.parameter_name) l3.addWidget(QLabel('Min value: ')) l3.addWidget(self.min_value) l3.addWidget(QLabel('Max value: ')) l3.addWidget(self.max_value) l3.addWidget(run_button) l2.addLayout(l3) l2.addWidget(self.fp_plt) wid.setLayout(l2) self.fpa_plt = PlotWidget() fpa_plt = self.fpa_plt.getPlotItem() fpa_plt.showGrid(True, True) yaxis = fpa_plt.getAxis('left') yaxis.setTicks([[(1, '0 0 0'), (2, 'CO'), (3, 'NM -'), (4, ('NM +'))]]) line1 = InfiniteLine(QPointF(0, 1), angle=0, pen=mkPen('#f00', width=2)) line2 = InfiniteLine(QPointF(0, 2), angle=0, pen=mkPen('#f00', width=2)) line3 = InfiniteLine(QPointF(0, 3), angle=0, pen=mkPen('#f00', width=2)) line4 = InfiniteLine(QPointF(0, 4), angle=0, pen=mkPen('#f00', width=2)) fpa_plt.addItem(line1) fpa_plt.addItem(line2) fpa_plt.addItem(line3) fpa_plt.addItem(line4) self.sc1 = ScatterPlotItem(antialias=True, brush=mkBrush('#00f'), size=20) self.sc2 = ScatterPlotItem(antialias=True, brush=mkBrush('#ff0'), size=20) self.index = 0 fpa_plt.addItem(self.sc1) fpa_plt.addItem(self.sc2) wid2 = QWidget() l4 = QVBoxLayout() l5 = QHBoxLayout() self.min_fxa = QDoubleSpinBox() self.max_fxa = QDoubleSpinBox() self.param_fxa = QLineEdit('d3') run_button_2 = QPushButton('Run') l5.addWidget(QLabel('Parameter: ')) l5.addWidget(self.param_fxa) l5.addWidget(QLabel('Min value: ')) l5.addWidget(self.min_fxa) l5.addWidget(QLabel('Max value: ')) l5.addWidget(self.max_fxa) l5.addWidget(run_button_2) l4.addLayout(l5) l4.addWidget(self.fpa_plt) wid2.setLayout(l4) lay = QHBoxLayout() self.tab = QTabWidget() l6 = QVBoxLayout() l6.addWidget(self.parameter_tree) phase_button = QPushButton('Plot phase diagram') l6.addWidget(phase_button) lay.addLayout(l6) lay.addWidget(self.tab) self.tab.addTab(self.plot_widget, 'Simulation') self.tab.addTab(wid, 'Fixed Points') self.tab.addTab(wid2, 'Fixed Points Analysis') self.setLayout(lay) self.parameter.sigTreeStateChanged.connect(self.upd) self.update_fp() run_button.clicked.connect(self.run_a_lot) run_button_2.clicked.connect(self.run_a_lot_of_points) phase_button.clicked.connect(self.plot_phase) def run_a_lot(self): interval = self.min_value.value(), self.max_value.value() values = np.linspace(interval[0], interval[1], 200) parameter_name = self.parameter_name.text() old_val = eval(str('self.' + parameter_name)) f, p, m = [], [], [] self.fp_plt.getPlotItem().setLabels(bottom=str(parameter_name), left='Fixed Points') for i in values: exec (str('self.' + parameter_name + '=' + str(i))) res = self.solve() f.append(res[0][-1]) p.append(res[1][-1]) m.append(res[2][-1]) l = len(f) print('Finished simulation for {0} = {1}'.format(parameter_name, eval(str('self.' + parameter_name)))) self.fp_f_plt.setData(values, f) self.fp_p_plt.setData(values, p) self.fp_m_plt.setData(values, m) self.fp_plt.update() exec (str('self.' + parameter_name + '=' + str(old_val))) def run_a_lot_of_points(self): interval = self.min_fxa.value(), self.max_fxa.value() values = np.linspace(interval[0], interval[1], 100) parameter_name = self.param_fxa.text() old_val = eval(str('self.' + parameter_name)) s1x, s1y, s2x, s2y = [], [], [], [] self.sc1.clear() self.sc2.clear() self.index = 0 for i in values: exec (str('self.' + parameter_name + '=' + str(i))) self.update_fp() res = self.solve() print('Finished simulation for {0} = {1}'.format(parameter_name, eval(str('self.' + parameter_name)))) f, p, m = res[0][-1], res[1][-1], res[2][-1] eps = 5 if np.abs(f - self.fp0[0]) < eps and np.abs(p - self.fp0[1]) < eps and np.abs(m - self.fp0[2]) < eps: val = 1 elif np.abs(f - self.fp1[0]) < eps and np.abs(p - self.fp1[1]) < eps and np.abs(m - self.fp1[2]) < eps: val = 2 elif np.abs(f - self.fp2[0]) < eps and np.abs(p - self.fp2[1]) < eps and np.abs(m - self.fp2[2]) < eps: val = 3 elif np.abs(f - self.fp3[0]) < eps and np.abs(p - self.fp3[1]) < eps and np.abs(m - self.fp3[2]) < eps: val = 4 else: val = 5 if ((self.k4 / self.d3) - (self.k1 / self.d1)) < self.beta \ and (self.k4 / self.d3) > self.beta \ and (self.alpha * self.k1 * self.d2) / (self.d1 * self.k2) \ < ((self.k4 / self.d3) - self.beta) * \ (self.beta + (self.k1 / self.d1) - (self.k4 / self.d3)): s2x.append(self.index) s2y.append(val) else: s1x.append(self.index) s1y.append(val) show = (4 * self.alpha * self.d1 * self.d2) / (self.k1 * self.k2) self.fpa_plt.setXRange(0, self.index, 0.5, True) self.index += 1 self.sc1.setData(s1x, s1y) self.sc2.setData(s2x, s2y) def upd(self, *args): exec ('self.' + args[1][0][0].name() + '=' + str(args[1][0][2])) self.update_fp() res = self.solve() print('Flowers {0}\nPollinators {1}\nMantis {2}'.format(res[0][-1], res[1][-1], res[2][-1])) if self.tab.currentIndex() == 0: self.f_plt.setData(t_values, res[0]) self.p_plt.setData(t_values, res[1]) self.m_plt.setData(t_values, res[2]) self.plot_widget.update() def update_fp(self): P0 = F0 = M0 = 0 P1 = self.d3 / (self.k4 - self.d3 * self.beta) F1 = (self.k1 * P1 - self.d1) / (self.alpha * P1 * self.d1) M1 = ((self.k2 * F1) / (1 + self.alpha * P1 * F1) - self.d2) * ((1 + self.beta * P1) / self.k3) M2 = 0 P2 = (self.k1 * self.k2 - np.sqrt(self.k1 * self.k2 * (-4 * self.alpha * self.d1 * self.d2 + self.k1 * self.k2))) \ / (2 * self.alpha * self.d2 * self.k1) F2 = (self.k1 * self.k2 - np.sqrt(self.k1 * self.k2 * (-4 * self.alpha * self.d1 * self.d2 + self.k1 * self.k2))) \ / (2 * self.alpha * self.d1 * self.k2) M3 = 0 P3 = (self.k1 * self.k2 + np.sqrt(self.k1 * self.k2 * (-4 * self.alpha * self.d1 * self.d2 + self.k1 * self.k2))) \ / (2 * self.alpha * self.d2 * self.k1) F3 = (self.k1 * self.k2 + np.sqrt(self.k1 * self.k2 * (-4 * self.alpha * self.d1 * self.d2 + self.k1 * self.k2))) \ / (2 * self.alpha * self.d1 * self.k2) self.fp0, self.fp1, self.fp2, self.fp3 = (F0, P0, M0), (F1, P1, M1), (F2, P2, M2), (F3, P3, M3) print('F0 = {0}\nP0 = {1}\nM0 = {2}\n' .format(F0, P0, M0)) print('F1 = {0}\nP1 = {1}\nM1 = {2}\n' .format(F1, P1, M1)) print('F2 = {0}\nP2 = {1}\nM2 = {2}\n' .format(F2, P2, M2)) print('F3 = {0}\nP3 = {1}\nM3 = {2}\n' .format(F3, P3, M3)) print() def solve(self): res = odeint(self.func(), (self.F, self.P, self.M), t_values).T return res def func(self): flower = lambda t, f, p, m:(self.k1 * p * f) / (1 + self.alpha * f * p) \ - self.d1 * f \ - self.Kf * f * f polli = lambda t, f, p, m:(self.k2 * p * f) / (1 + self.alpha * f * p) \ - (self.k3 * m * p) / (1 + self.beta * p) \ - self.d2 * p \ - self.Kp * p * p mantis = lambda t, f, p, m:((self.k4 * m * p)) / (1 + self.beta * p) \ - self.d3 * m return lambda y, t:(flower(t, *y), polli(t, *y), mantis(t, *y)) def plot_phase(self): # Equations without the mantis (simpler model for the phase space) flower = lambda t, f, p:(self.k1 * p * f) / (1 + self.alpha * f * p) \ - self.d1 * f polli = lambda t, f, p:(self.k2 * p * f) / (1 + self.alpha * f * p) \ - self.d2 * p f = lambda y, t:np.array([flower(t, *y), polli(t, *y)]) # res = odeint(f, (self.F, self.P), t_values).T # plot(res[0], res[1]) res = odeint(f, (20, 20), t_values).T plot(res[0], res[1]) res = odeint(f, (150, 20), t_values).T plot(res[0], res[1]) res = odeint(f, (150, 350), t_values).T plot(res[0], res[1]) res = odeint(f, (200, 20), t_values).T plot(res[0], res[1]) res = odeint(f, (150, 20), t_values).T plot(res[0], res[1]) res = odeint(f, (150, 0), t_values).T plot(res[0], res[1]) xlabel('Flowers') ylabel('Pollinators') res = odeint(f, (0, 50), t_values).T plot(res[0], res[1]) R, C = np.meshgrid(np.arange(-30, 200, 30), np.arange(-30, 330, 10)) dy = f(np.array([R, C]), 0) plot([self.fp0[0]], [self.fp0[1]], 'o') plot([self.fp2[0]], [self.fp2[1]], 'o') plot([self.fp3[0]], [self.fp3[1]], 'o') quiver(R, C, dy[0, :], dy[1, :], scale_units='xy', angles='xy') show() xlabel('Flowers') ylabel('Pollinators') res = odeint(f, (0.5, 0.3), t_values).T plot(res[0], res[1]) res = odeint(f, (0.3, 0.8), t_values).T plot(res[0], res[1]) res = odeint(f, (.1, 1), t_values).T plot(res[0], res[1]) res = odeint(f, (0.3, 0.7), t_values).T plot(res[0], res[1]) res = odeint(f, (0.2, 0.75), t_values).T plot(res[0], res[1]) res = odeint(f, (0.2, 0.7), t_values).T plot(res[0], res[1]) res = odeint(f, (0.334170, 0.75188), t_values).T plot(res[0], res[1]) plot([self.fp0[0]], [self.fp0[1]], 'o') plot([self.fp2[0]], [self.fp2[1]], 'o') R, C = np.meshgrid(np.arange(-1, 2, .2), np.arange(-1, 2, .2)) dy = f(np.array([R, C]), 0) quiver(R, C, dy[0, :], dy[1, :], scale_units='xy', angles='xy') show() app = QApplication([]) wid = MyWidget() wid.show() app.exec_()
References
1)
Ce n'est pas une fleur