Wiki site of the practical exercise of the VI Southern-Summer School on Mathematical Biology.

Here you will find the exercise assignment and the group's products.

If you are a group member login to edit this page, create new pages from it, and upload files.

Predators and parasitoids negatively affect their prey and host by eliminating individuals from the population. Such antagonistic interactions (predation and parasitism) are often considered in isolation when studying species interactions. Nevertheless, different types of interactions occur concomitantly, so understanding their combined effects is crucial to describe the dynamics of a biological system.

Natural enemies threaten the persistence of the ant colonies, which often engage in a variety of interactions. A recent study by Kaitlyn Mathis and Neil Tsutsui (2016) shows two natural enemies of an ant species (Azteca sericeasur): a parasitoid phorid fly (genus Pseudacteon), and a predator, the rove beetle Myrmedonata xipe. The salient aspect of this three-species system is that predators preferentially attack parasitized ants, since these are less aggressive and hence are easy targets to the beetle. This context-dependent predation can have implications for all three species, with beetles negatively influencing the fly population, potentially with little effects on the ant species.

A parasitic fly (Apocephalus sp) approaching a Camponotus ant (Photo credit: Alex Wild www.alexanderwild.com checkout his website for more amazing pictures of ants)

Develop a mathematical model that describes this system and use it to understand the effect of beetle predation on the ant and fly species. Given that beetles prey preferentially upon parasitized ants, investigate how density of flies influences both beetle and ant species.

• Considering this system, can predation result in a positive effect for the prey species?

Here take into account that predation by beetles could actually have a negative effect for the ants if they are not completely selective on the parasitized ones, and more so in the absence of flies. That is, our model should account for this with a parameter which is as our model stands right now 0.

• Beetles preferentially prey upon parasitized ants that become less aggressive. Investigate how specialization of beetles can influence the final density of the three species.
• Along the same lines, can beetles persist preying on ants in the absence of the parasitoid flies?
• How does the behavior of parasitized ants alter the costs and benefits of predation and parasitism, from the point of view of the ant colony? For instance, if parasitized ants are still productive workers, the cost of being parasitized can be smaller than the benefit provided by predation.

Do parasitized ants go back to the colony?

• Mathis, K. A., and N. D. Tsutsui 2016 Dead ant walking: a myrmecophilous beetle predator uses parasitoid host location cues to selectively prey on parasitized ants. Proceedings of the Royal Society of London B 283:2016 1281
• Video of phorid flies in action: https://www.youtube.com/watch?v=pFVvOo1Qd_8

J =
 
[ (r*((m*(b + h))/(k*(a*h - a*g*m*(b + h))) - 1))/((a*g*m*(b + h))/(a*h - a*g*m*(b + h)) + 1) - r*((2*m*(b + h))/(k*(a*h - a*g*m*(b + h))) - 1),       0, -(m*(b + h))/h]
[                                                                                                -r*((m*(b + h))/(k*(a*h - a*g*m*(b + h))) - 1), - b - h,  (m*(b + h))/h]
[                                                                                                                                             0,       h,             -m]

We study an ant colony in the presence of parasitoid flies and beetles as predators. The flies inject their eggs into the ants. As soon as the larva hatches from the egg, it feeds on the ant and thereby slowly kills it. Since ants are highly aggressive, beetles can only consume them in the parasitised state, when the ant is weakened by the larva. We investigate two scenarios: one with a specialist beetle feeding exclusively on the parasitised ants and a second scenario with a generalist beetle, which feeds on both types of ants (normal and parasitised).

The stability criterion gives a minimum predation rate performed by beetles. If we are over that minimum level, bettles are eating so many infected ants, so that affects flies and normal ants have no problem in converging to the capacity (asymptotic stability).

If \mu(fly death rate) is greater, then the minimum predation rate is smaller and it is easier to normal ants to converge to capacity. Similar result is true, considering the parameter \gamma (incubation rate).

If we are below the minimum level of predation rate, then there exists a coexistence equilibrium, so flies have a chance to survive by attacking ants.

Formally, If \mu (fly death rate),\b (predation rate), are greater then the equilibrium number of ants N^* is bigger. If \alpha (attack rate of flies) is greater, then N^* is smaller.

1. In the absence of flies (F) and beetles (B), the ant population (N) follows logistic growth .

It is a big assumption that we use a Holling type II functional response. Why assumptions are we making by doing it?

2. Flies turn ants into parasitoid state (P) with a Holling type II functional response.

3. An ant in parasitoid state will transform into a fly at rate h.

4. A fly is subject to mortality at rate m.

5. Beetles consume ants in parasitoid state at a constant rate b, independent of their population size. This is a consequence, not an assumption: “and thereby prevent the birth of a fly.” :)

It is a huge assumption that we are not modelling the beetles population.

6. Beetles are generalists. Their size population is at its carrying capacity.

Please write down in here where did you get each of the parameters we are using. Thank you! Saturday is sensitivity day, hopefully also miracle day.

What is the criterion and meausure that we shall use here?

Take this with a grain of salt, please. Notice that this model is an extension of the previous one to include the specialization of beetles. We consider the beetles population dynamic and assume it to be very specialist, so that beetles may only grow in the presence of ants. There is a predation term of normal ants by beetles which is mediated by the proportion of normal and parasitized ants. We include two parameters /lambda_{N} and /lambda_{P} to weight the preference of beetles for parasitized ants due to their reduced aggressiveness.

Assumptions

1. Many many many…

Model

Preference

Consider the functional response between beetles and parasitized ants.

Let me tell you why this might not be such a crazy idea. The paper reports that there is a constant removal of parasitized ants by the beetles, independent of the beetles population. This functional response suggests this behaviour when beetles are close to their carrying capacity, and the preference towards parasitized ants is very high.

Here we will assume that we are dealing with a sufficiently small beetles population for interference with each other to be negligible.

As a start we may assume that the beetles search randomly and encounter the two type of ants in proportion to their densities. But as it happens, the two type of ants are not equally preyed upon per capita: the parasitized ants are easier to kill because of their reduced aggressiveness.

Our model has three equilibrium points. The trivial equilibrium (0,0,0), the semi-trivial equilibirum (K,0,0) and a coexistence equilibrium with the following form:

As it happens, there is an inequality which shows the transition of stability of the semi-trivial equilibrium by the appeareance of the coexistence equilibrium in the positive octant.

We investigated the dynamics of the model with the estimated parameters.

You may download our code (commented in portuguese) from the following link.

code.py

from numpy import *
import matplotlib.pyplot as p
from scipy.integrate import ode
import scipy as sp
 
#Determinacao de condicoes iniciais e parametros
an_0 = 100
ap_0 = 0
f_0 = 10
r = 0.1 #reproductive rate of normal ants
k = 8000 #carrying capacity
alpha = 0.01 #searching efficiency
g = 3.5 #handling time
h = 1.0/25 #hatching rate
o = 1.5 #number of eggs/host
m = 0.12 #mortality rate of flies
psi = 0.5 #less working hours for Ap
b = 0 #beetles
 
#Seja F=(x, y, z)
#An = F_0[0]
#Ap = F_0[1]
#F = F_0[2]
 
def dF_dt(t, F_0):
    #Retorna o sistema de edo's
    if (F_0[0]+F_0[1]<1):
        F_0[0]=0.0
        F_0[1]=0.0
    if(F_0[0]<1):
        F_0[0]=0.0
    eqs = zeros((3))
    eqs[0] = r*(F_0[0])*(1 - (F_0[0])/k) -   (F_0[2]*alpha*F_0[0])/(1 + alpha*g*F_0[0])  
    eqs[1] = (F_0[2]*alpha*F_0[0])/(1 + alpha*g*F_0[0])  - (h + b)*F_0[1]
    eqs[2] = o*h*F_0[1] - m*F_0[2]
    return eqs
 
t_0 = 0 #inicio e parada de tempo
t_f = 2000
t = sp.arange(t_0, t_f, 0.5)
 
def resolucao():
    #Retorna y e x resolvidos para um determinado conjunto de parâmetros definidos anteriormente.
    solver = ode(dF_dt) #mostra qual sistema de equacoes diferenciais será resolvido
    solver.set_integrator('dopri5', method = 'bdf') #escolha do metodo 'dopri5', o metodo de Runge-Kutta da biblioteca SciPy
    F_0 = [an_0, ap_0, f_0] 
    solver.set_initial_value(F_0, t_0) #mostra quais os valores iniciais do problema
 
    sol = empty((t_f/(0.5),3)) #criacao de matriz vazia para guardar a solucao
    i=0
    #resolve numericamente ate onde for pedido (no caso, t_f)
    while solver.successful() and solver.t < t_f:
        solver.integrate(solver.t + 0.5)
 sol[i] = solver.y #guarda a solucao no vetor
        i+=1
    an, ap, f = sol.T
    return an, ap, f #devolve an, ap e f
 
def showme():#funcao que mostra o grafico de acordo com um conjunto de parametros
    an,ap,f = resolucao()
    print "an_0 = ",an_0, "ap_0 = ", ap_0, "F_0 = ", f_0, "\nr = ", r, "k = ", k, "alpha = ", alpha, "g = ", g, "h = ", h, "o = ", o, "m = ", m, "psi = ", psi, "b = ", b
    p.plot(t, an, 'r-', label='$A_n$')
    p.plot(t, ap, 'b-', label='$A_p$')
    p.plot(t, f, 'k-', label='$F$')
    p.grid()
    p.legend(loc='best')
    p.xlabel('Time')
    p.ylabel('Population Size')
    p.title('Populations evolution')
    p.show()
    if ((m*(h+b)) > ((o*h*alpha*k)/(1+alpha*g*k))):
        print "actually we already knew normal ants win because"
        print    m*(h+b),">",(o*h*alpha*k)/(1+alpha*g*k)

Why do we use a functional response for the interaction between normal ants and flies? Functional Response: the function relating the number of prey eaten by a single “average” predator to the size of the prey population.

Definitely worth reading: Learning predator promotes coexistence of prey species in host–parasitoid systems. http://www.pnas.org/content/109/13/5116.full (The model is in the supplementary information.)

Switching, Functional Response, and Stability in Predator-Prey Systems.

• presentation