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Mutualisms are ubiquitous in nature and mutualistic interactions are expected to increase the fitness of interacting species. However, species engage in multiple interactions at the same time. For instance, many plants engage in multiple mutualistic interactions with a diverse range of organisms. Pollinators assist in plant reproduction, frugivores disperse their seeds and microorganisms provide them nutrients ¹⁾. When multiple mutualistic interactions occur at the same time, it is important to understand the contribution of each interaction on species' fitness. The interplay between different interactions can be difficult to predict and might have synergistic effects.

A recent paper by Godschalx and colleagues (2015) found that nitrogen-fixing bacteria can influence mutualistic interactions that lima bean plants (Phaseolus lunatus) engage with protective ants. The authors show that the effects of different mutualistic interactions cannot be described by considering simply their individual effect.

Ants from the genus Crematogaster protecting lima bean plant

Propose a simple but realist mathematical model that helps to:

• Disentangle the role (and effect) of different mutualistic interactions for the persistence of species interactions.
• Understand the contribution of different types of mutualistic interactions and the effect they have on interacting species when considered both in isolation and combined.

• How different types of mutualistic interactions influence the dynamics of interacting species when considered pairwise interactions?
• How does the predictions, regarding population dynamics and species persistence, change when multiple interactions are considered at the same time?

• 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 link

%matplotlib inline
from numpy import *
from matplotlib.pyplot import *
from scipy.integrate import odeint
ion()

t = arange(0,10., 0.01)

# parameters
fr = 1.5 
fa = 2.
p = 0.6
c = 0.2
g = 15.

# initial condition: this is an array now!
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, 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')